Fuzzy Geometric Programming Techniques and Applications 9789811358227

Table of contents :
Front Matter ....Pages i-xxi
Preliminary Concepts of Geometric Programming (GP) Model (Sahidul Islam, Wasim Akram Mandal)....Pages 1-25
Signomial Geometric Programming (GP) Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 27-45
Introduction to Fuzzy Set Theory (Sahidul Islam, Wasim Akram Mandal)....Pages 47-73
Fuzzy Numbers and Fuzzy Optimization (Sahidul Islam, Wasim Akram Mandal)....Pages 75-131
Fuzzy Unconstrained Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 133-153
Fuzzy Unconstrained Modified Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 155-174
Fuzzy Constrained Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 175-189
Constrained Fuzzy Modified Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 191-207
Fuzzy Signomial Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 209-231
Goal Geometric Programming (Sahidul Islam, Wasim Akram Mandal)....Pages 233-258
Fuzzy Multi-objective Geometric Programming (FMOGP) Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 259-286
Geometric Programming Problem Under Uncertainty (Sahidul Islam, Wasim Akram Mandal)....Pages 287-330
Intuitionistic and Neutrosophic Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 331-355
Back Matter ....Pages 357-359

Citation preview

Forum for Interdisciplinary Mathematics

Sahidul Islam Wasim Akram Mandal

Fuzzy Geometric Programming Techniques and Applications

Forum for Interdisciplinary Mathematics Editor-in-chief P. V. Subrahmanyam, Indian Institute of Technology Madras, Chennai, India

Editorial Board Members Yogendra Prasad Chaubey, Concordia University, Montréal (Québec), Canada Jorge Cuellar, Principal Researcher, Siemens, Germany Janusz Matkowski, University of Zielona Góra, Poland Thiruvenkatachari Parthasarathy, Chennai Mathematical Institute, Kelambakkam, India Bhu Dev Sharma, Jaypee Institute of Information Technology, Noida, India Mathieu Dutour Sikirić, Rudjer Bosković Institute, Zagreb, Croatia

The Forum for Interdisciplinary Mathematics series publishes high-quality monographs and lecture notes in mathematics and interdisciplinary areas where mathematics has a fundamental role, such as statistics, operations research, computer science, financial mathematics, industrial mathematics, and bio-mathematics. It reflects the increasing demand of researchers working at the interface between mathematics and other scientific disciplines.

More information about this series at http://www.springer.com/series/13386

Sahidul Islam Wasim Akram Mandal •

Fuzzy Geometric Programming Techniques and Applications

123

Sahidul Islam Department of Mathematics University of Kalyani Kalyani, Nadia, West Bengal, India

Wasim Akram Mandal Beldanga D.H. Senior Madrasah Beldanga, Murshidabad, West Bengal, India

ISSN 2364-6748 ISSN 2364-6756 (electronic) Forum for Interdisciplinary Mathematics ISBN 978-981-13-5822-7 ISBN 978-981-13-5823-4 (eBook) https://doi.org/10.1007/978-981-13-5823-4 Library of Congress Control Number: 2018965460 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Do not worry about your problems with mathematics, I assure you mine are far greater. Albert Einstein

Dedicated to our parents and families

Preface

Nonlinear analysis plays an ever-increasing role in theoretical and applied mathematics, as well as in many other areas of science such as engineering, statistics, computer science, economics, finance, and medicine. Most of the problems in our real life are nonlinear. There are many techniques used for solving nonlinear problems. Geometric programming is one of the best techniques to solve this nonlinear optimization problem. Geometric programming was introduced in 1967 by Duffin, Peterson, and Zener. It is very useful in the applications of a variety of optimization problems and falls under the general class of signomial problems. Geometric programming is a special method used to solve a class of nonlinear programming problems, mainly to solve optimal design problems where we minimize cost and/or weight, maximize volume and/or efficiency, etc. It is an important technique to solve the special type of nonlinear optimization problems. The global optimum of a convex problem can be achieved more quickly than the result of a nonlinear problem. Since its inception, geometric programming has been closely associated with applications in engineering analysis and design problem. Uncertainty in the problem data often cannot be avoided when dealing with practical problems. Errors occur in real-world data for a host of reasons. However, over the last 30 years, the fuzzy set approach has been proved to be useful in such situations. Thousands of research papers on geometric programming problem are published under the fuzzy environment, but only one book, Fuzzy Geometric Programming, by B. Y. Cao of geometric programming under fuzzy environment is published, which mainly focuses on theoretical aspects. Organized into 13 chapters, this book discusses an overall concept of geometric programming, goal geometric programming, and multi-objective geometric programming problem under a crisp and fuzzy environment. The main aim of this book is to develop the concepts of some optimization techniques: geometric programming, modified geometric programming, fuzzy geometric programming, fuzzy modified geometric programming, signomial geometric programming, goal programming, and fuzzy multi-objective geometric programming.

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In Chap. 1, we have presented geometric programming, modified geometric programming problem, constrained geometric programming problem, and modified geometric programming problem. In Chap. 2, we have presented convexity of signomial functions, unconstrained nonlinear programming problem, primal modified signomial geometric programming problem, unconstrained modified signomial function, constrained nonlinear programming problem and application of problem, and signomial geometric programming technique. In Chap. 3, we have described uncertainty and imprecision, basic fuzzy set theory, operations on fuzzy sets, geometrical interpretation of fuzzy sets, fuzzification, extension principle and its application, definition (extension principle), Cartesian product, and definition (extension principle on the n-dimensional universe). In Chap. 4, we have discussed fuzzy number, generalized fuzzy number, generalized trapezoidal fuzzy number, integral value, fuzzy number and its nearest interval approximation, fuzzy equation, and fuzzy optimization. In Chap. 5, we have presented unconstrained geometric programming problem with fuzzy parametric interval-valued function, geometric programming problem with simple fuzzy parametric coefficients, and geometric programming problem with Zimmermann max-min operators. In Chap. 6, we have studied the unconstrained modified geometric programming problem with fuzzy parametric interval-valued function, unconstrained MGP problem with simple fuzzy parametric coefficients, and unconstrained MGP problem with Zimmermann max-min operator. In Chap. 7, we have discussed constrained geometric programming problem with a fuzzy coefficient, fuzzy parametric geometric programming, and constrained geometric programming under max-min operator. In Chap. 8, we have analyzed constrained modified geometric programming problem with a fuzzy coefficient, fuzzy parametric modified geometric programming, and fuzzy modified geometric programming. In Chap. 9, we have developed an introduction to fuzzy signomial geometric programming problem, unconstrained problem, and constrained problem. In Chap. 10, we have discussed an introduction to goal programming, strength and weakness of goal programming, the importance of weighted goal programming, Chebyshev goal programming model, multi-objective problem, goal programming with logarithmic deviational variable, and fuzzy goal programming problem. In Chap. 11, we have presented an introduction to nonlinear programming function, fuzzy nonlinear programming technique, multi-objective optimization, and inventory models through fuzzy multi-objective geometric programming approach. In Chap. 12, we have presented an uncertain chance-constrained geometric programming model, geometric programming approach under zigzag uncertainty distribution, and multi-objective geometric programming problem under uncertainty. Lastly, in Chap. 13, we have studied intuitionistic fuzzy posynomial geometric programming problem, intuitionistic fuzzy goal programming model, and neutrosophic goal geometric programming problem.

Preface

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Geometric programming has applications across a variety of fields from engineering to economics and will continue to be useful in the future. Geometric programming is an essential course at the postgraduate level in mathematics, applied mathematics, engineering, as well as at the research level in many universities. The topic also forms a course to postgraduate students in engineering in reliability, operations research, circuit design, mathematics optimizations, and engineering problems. Kalyani, India Murshidabad, India

Sahidul Islam Wasim Akram Mandal

Acknowledgements

We would like to give special thanks to our mentor Dr. Tapan Kumar Roy, Professor and Head of the Department of Mathematics, Indian Institute of Engineering Science and Technology (IIEST), Shibpur, India, for his continuous encouragement, help, and appreciation. We thank Dr. Samares Pal, Professor and Head of the Department of Mathematics, University of Kalyani, India, for his continuous encouragement during the preparation of this book. We are also thankful to all faculty members, staff and scholars of the department for their continuous support and encouragement. We appreciate the work of all those researchers who have contributed to the geometric programming technique in a crisp and fuzzy environment in several of their books and papers I referred to. We would like to express our sincere thanks to Shamim Ahmad, Shubham Dixit, and all staff of Springer Nature for their patience and cooperation in all respects, during the preparation of the final version of the manuscripts. We would like to thank anonymous referees for their valuable comments and suggestions for the improvement of this book. Special thanks to our parents, family members, and relatives for their support and encouragement.

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Contents

Preliminary Concepts of Geometric Programming (GP) Model 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Geometric Program (GP) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Modified Geometric Programming (MGP) Problem . . . . . . . 1.4 Constrained Geometric Programming (CGP) Problem . . . . . 1.5 Constrained Modified Geometric Programming (CMGP) Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Signomial Geometric Programming (GP) Problem 2.1 Introduction and History . . . . . . . . . . . . . . . . 2.2 Definition and Example . . . . . . . . . . . . . . . . . 2.3 Convexity of Signomial Functions . . . . . . . . . 2.4 Unconstrained NLP Problem . . . . . . . . . . . . . 2.5 Unconstrained Modified Signomial Function . 2.6 Constrained NLP Problem . . . . . . . . . . . . . . . 2.7 Primal Modified Signomial GP Problem . . . . . 2.8 Application . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction to Fuzzy Set Theory . . . . . . . . . 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 3.2 Uncertainty and Imprecision . . . . . . . . . 3.3 Basic Fuzzy Set Theory . . . . . . . . . . . . 3.4 Operations on Fuzzy Sets . . . . . . . . . . . 3.5 Geometrical Interpretation of Fuzzy Sets 3.6 Fuzzification . . . . . . . . . . . . . . . . . . . . . 3.7 Extension Principle and Its Application .

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3.8 Definition (Extension Principle) . . 3.9 Cartesian Product . . . . . . . . . . . . 3.10 Definition (Extension Principle on 3.11 Conclusions . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . 4

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Fuzzy Numbers and Fuzzy Optimization . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fuzzy Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Generalized Fuzzy Number (GFN) . . . . . . . . . . . . . . . . . 4.4 Generalized Trapezoidal Fuzzy Number (GTrFN) . . . . . . 4.5 Integral Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Fuzzy Number and Its Nearest Interval Approximation . . 4.6.1 a-Level Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Interval Number . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Nearest Interval Approximation . . . . . . . . . . . . . 4.6.4 Parametric Interval-Valued Function . . . . . . . . . 4.7 Fuzzy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Single-Objective Mathematical Programming . . . . . . . . . 4.8.1 Calculus-Based Method: (Newton–Raphson Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Decision-Making in Fuzzy Environment of Mathematical Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Max-Min Operator . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Max Additive Operator . . . . . . . . . . . . . . . . . . . 4.9.3 Max Product Operator . . . . . . . . . . . . . . . . . . . 4.9.4 Fuzzy Optimization Approach . . . . . . . . . . . . . . 4.10 Fuzzy Optimization (Zimmerman Approach) . . . . . . . . . 4.11 Some Basic Concepts of Uncertainty Theory . . . . . . . . . 4.12 Intuitionistic Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . 4.13 Neutrosophic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Hesitant Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Pythagorean Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Type-2 Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Unconstrained Geometric Programming Problem . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 GP Problem with Fuzzy Parametric Interval-Valued Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 GP Problem with Simple Fuzzy Parametric Coefficients

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5.4 GP Problem with Zimmermann Max-Min Operators . . . . . . . . . 145 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6

Fuzzy Unconstrained Modified Geometric Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Unconstrained MGP Problem with Fuzzy Parametric Interval-Valued Function . . . . . . . . . . . . . . . . . . . . . 6.3 Unconstrained MGP Problem with Simple Fuzzy Parametric Coefficients . . . . . . . . . . . . . . . . . . . . . . 6.4 Unconstrained MGP Problem with Zimmermann Max-Min Operator . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fuzzy Constrained Geometric Programming Problem . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Geometric Programming Problem with Fuzzy Coefficient 7.3 Fuzzy Parametric Geometric Programming . . . . . . . . . . . 7.4 Constrained GP Under Max–Min Operator . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Constrained Fuzzy Modified Geometric Programming Problem . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Modified Geometric Programming Problem with Fuzzy Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Fuzzy Parametric Modified Geometric Programming . . . . . . . 8.4 Constrained MGP Under Max-Min Operator . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fuzzy Signomial Geometric Programming Problem . . . . . 9.1 Introduction and History . . . . . . . . . . . . . . . . . . . . . . 9.2 Unconstrained Proble . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Unconstrained Fuzzy Signomial GP Problem . 9.2.2 Unconstrained Modified Fuzzy Signomial GP Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Constrained GP Problem . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Constrained Fuzzy Signomial GP Problem . . . 9.3.2 Constrained Modified Fuzzy Signomial GP Problem . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Goal 10.1 10.2 10.3 10.4 10.5

Geometric Programming . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strength and Weakness of Goal Programming . . . . . . . . . . . Importance of Weighted Goal Programming (WGP) . . . . . . . Chebyshev Goal Programming Model . . . . . . . . . . . . . . . . . Multi-objective Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Multi-objective Goal Geometric Programming (MOGP) Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Multi-objective Weighted Goal Programming (MOWGP) Formula . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Formulation of Goal GP Programming Using Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Dual Form of Goal GP Problem . . . . . . . . . . . . . . . 10.6 Goal Programming with Logarithmic Deviational Variables . . 10.6.1 Weighted Goal Programming Problem with Logarithmic Deviational Variables . . . . . . . . . . . . . . 10.7 Fuzzy Goal Programming Problem . . . . . . . . . . . . . . . . . . . 10.7.1 Fuzzy Multi-objective Goal Programming (FMOGP) in Parametric Form . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.2 Fuzzy Goal Programming Problem with Logarithmic Deviational Variables . . . . . . . . . . . . . . . . . . . . . . . 10.7.3 Fuzzy Weighted Goal Programming Problem with Logarithmic Deviational Variables . . . . . . . . . . 10.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fuzzy Nonlinear Programming (FNLP) . . . . . . . . . . . . . . . . . 11.3 Multi-objective Mathematical Programming (MOMP) Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Multi-objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Fuzzy Multi-objective Mathematical Programming (FMOMP) Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Inventory Models Through Geometric Programming Approach 11.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Geometric Programming Problem Under Uncertainty . . . . . . . . . . 287 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 12.2 Uncertain Chance-Constrained Geometric Programming (UCCGP) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Contents

xix

12.2.1 UCCGP Model with Linear Uncertainty Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 UCCGP Model with Normal Uncertainty Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 UCCGP Model with Zigzag Uncertainty Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Geometric Programming Approach Under Expected, Variance, 2-ND Moment, and Entropy-Based Zigzag Uncertainty Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Solution Procedure Under Expected-Based UVs . . . . 12.3.2 Solution Procedure Under Variance-Based UVs . . . . 12.3.3 Solution Procedure Under 2-ND Moment-Based UVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Solution Procedure Under Entropy-Based UVs . . . . . 12.3.5 Applying the Proposed UCCGP to a Two-Bar Truss Structural Model . . . . . . . . . . . . . . . . . . . . . . 12.3.6 Numerical Example and Solution . . . . . . . . . . . . . . 12.4 Multi-objective Geometric Programming Problem Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Uncertain Multi-objective Programming . . . . . . . . . . 12.4.2 UCCMOGP Model with Linear Uncertainty Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 UCCMOGP Model with Normal Uncertainty Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 UCCMOGP Model with Zigzag Uncertainty Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Intuitionistic and Neutrosophic Geometric Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Intuitionistic Fuzzy Posynomial Geometric Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Posynomial Geometric Programming with Intuitionistic Fuzzy Coefficient . . . . . . . . . . . . . . 13.2.2 Numerical Example (Design of a Two-Bar Truss) (Kheiri and Cao 2016) . . . . . . . . . . . . . . . . . . . . 13.3 Intuitionistic Fuzzy Goal Programming Model . . . . . . . . . 13.3.1 Goal Geometric Programming Model . . . . . . . . .

. . 289 . . 290 . . 291 . . 292

. . 295 . . 298 . . 299 . . 301 . . 302 . . 303 . . 310 . . 312 . . 312 . . 315 . . 317 . . . .

. . . .

319 321 329 330

. . . . 331 . . . . 331 . . . . 331 . . . . 333 . . . . 336 . . . . 338 . . . . 340

xx

Contents

13.3.2 Illustrative Numerical Example (Ghosh and Roy 2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Neutrosophic Goal Geometric Programming Problem . . . 13.4.1 Neutrosophic Multi-objective Goal Geometric Programming Problem . . . . . . . . . . . . . . . . . . . 13.4.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . 13.4.3 Neutrosophic Goal Geometric Programming Technique on Reliability Optimization Model . . 13.4.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . 13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 341 . . . . . 344 . . . . . 344 . . . . . 350 . . . .

. . . .

. . . .

. . . .

. . . .

351 353 355 355

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

About the Authors

Sahidul Islam is an assistant professor at the Department of Mathematics, University of Kalyani, India. He earned his Ph.D. from the Indian Institute of Engineering Science and Technology, Shibpur, India. Dr. Islam received the Junior Research Fellowship and Senior Research Fellowship from the Council of Scientific and Industrial Research (CSIR), India, in 2003 and 2005, respectively. He has published two books and over 52 papers in various reputed journals. His main research interests are information theory, fuzzy optimizations, operations research and optimization techniques. Wasim Akram Mandal is an assistant teacher of mathematics at the Beldanga D.H. Senior Madrasah, Murshidabad, India. He obtained his Ph.D. on the topic “Some inventory model approaches to decision making in fuzzy environment” and M.Sc. degree in applied mathematics from the University of Kalyani, India. He has published three books and over 15 research articles in several respected journals. His research interests are fuzzy optimizations and operations research.

xxi

Chapter 1

Preliminary Concepts of Geometric Programming (GP) Model

1.1 Introduction Geometric programming (GP) was introduced by Duffin, Peterson, and Zener in their famous book “Geometric programming” Theory and Application in 1967. Numerical methods for (computer) solution of GPs were developed by Duffin (1970). Zener (1971), have discussed the basic concepts and theories of GP with application in engineering. Beightler and Philips (1976) have also published a famous book on GP and its application. Feigin and Passy (1981) developed the geometric programming dual to the extinction probability problem in simple branching processes. Sinha et al. (1987) developed GP problems with negative degrees of difficulty. Klafszky et al. (1992) presented a GP approach to the channel capacity problem. Bazaraa et al. (1993) given some basic ideas of geometric programming in their book “Nonlinear Programming: Theory and Algorithms”. Floudas (1999) discussed covers global optimization methods for generalized geometric programming problems. Chu and Wong (2001) developed VLSI circuit performance optimization by GP. Jung and Klein (2001) worked optimal inventory policies under decreasing cost functions via GP. A brief history of the development of GP can be found in Peterson’s survey (2001). It is natural to guess that the name “GP” comes from the many geometrical problems that can be formulated as GPs. But in fact, this comes from the arithmetic— geometric mean inequality (A.M.-G.M. inequality). This inequality plays a central role in the analysis of GPs. It is important to distinguish between geometric programming (GP) and geometric optimization (GOP). GP ⇒ family of optimization problems of the form Min g0 (t) subject to g j (t) ≤ 1 j  1, 2, . . . , m h j (t)  1 j  1, 2, . . . , p t > 0, © Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_1

(1.1) 1

2

1 Preliminary Concepts of Geometric Programming (GP) Model

where gj (t) (j  1, 2, …, m) are posynomial functions, hj (t) (j  1, 2,…, p) are monomials, and t is the decision variable vector of n components t i (i  1, 2, …, n). GOP ⇒ Optimization problems involving geometry. Note 1.1 Unfortunately, a few authors use “GP” to mean optimization problems involving geometry and vice versa. The problem (1.1) may be written as Min g0 (t) subject to g j (t) ≤ 1 j  1, 2, .., m t >0 [since g j (t) ≤ 1, h j (t)  1 ⇒ g j (t) ≤ 1 where g j (t) (=gj (t)/hk (t)) be a posynomial. (j  1, 2, …, m; k  1, 2, …, p)] • Monomial: The word “monomial” comes from the Latin word, mono meaning only one and mial meaning term. So, a monomial is an “expression in algebra that contains only one term.” The term monomial as used in the context of geometric programming is similar to, but differs from the standard definition of monomial used in algebra. In algebra, a monomial can be a constant, a variable, or the product of one or more constant and variables but the exponent of variables cannot have a negative or fractional. Throughout, the exponent can be any real number, including fractional and negative. So if x1 , x2 , . . . , xn denote n real positive variable, then a real-valued function f of x, in the form F(x)  cx1a1 x2a2 . . . xnan , where c > 0 and ai ∈ R is called monomial function. √ Example 1.1 If x, y, and z are positive variables then 5, 0.25, 5x 2 y −3 , 10 z/x are monomial but 5 + x, 2x − 3z, and 2(x + 5x 2 y −3 ) are not monomials. Properties: 1. Addition: addition of two or more monomials is not monomial. Examples: 3, x are monomial but 3 + x is not monomial. 2. Subtraction: subtraction of two or more monomials is not monomial. Examples: 3, x are monomial but 3 − x is not monomial. 3. Multiplication: multiplication of two or more monomials is a monomial. Examples: 3, x are monomial and 3x is also a monomial. 4. Division: division of a monomial by one or more other monomials is a monomial. Examples: 3, x are monomial and 3/x is also a monomial.

1.1 Introduction

3

• Polynomial: The word “polynomial” comes from the Latin word, polyno meaning many and mial meaning term. So, a polynomial is an “expression in algebra that contains many terms, i.e., many monomial.” So sum of one or more monomials, i.e., any function of the form F(x) 

m 

ci x1a1i x2a2i . . . xnani ,

i1

is called polynomial function or simply polynomial. Example 1.2 If x, y and z are positive variables then 5, 0.25, 5x 2 y −3 , − 6zx , 2x − 3z, and 2(x + 5x 2 y −3 ) are polynomials. Properties: 1. Addition: addition of two or more polynomials is also a polynomial. Examples: 3, x are polynomial and 3 + x is also a polynomial. 2. Subtraction: subtraction of two or more polynomials is also a polynomial. Examples: 3, x are polynomial and 3 − x is also a polynomial. 3. Multiplication: multiplication of two or more polynomials is a polynomial. Examples: (3x + 2), 2x are polynomial and (6x 2 + 4x) is also a polynomial. 4. Division: division of a polynomial by one or more other monomials is a polynomial. is also a polynomial. Examples: (x − 2), x are polynomial and (x−2) x • Posynomial: If the coefficients ci > 0, then a polynomial is called a posynomial. So, sum of one or more monomials, i.e., any function of the form F(x) 

m 

ci x1a1i x2a2i . . . xnani ,

i1

where ci > 0, is called posynomial function or simply posynomial. √ Example 1.3 If x, y and z are positive variables then 5, 0.25 + x, 5x 2 y −3 , 10 z/x + x 3 + y 5 are posynomial but 5 − x, 2x − 3z, and 2(x − 5x 2 y −3 ) are not posynomial. Properties: 1. Addition: addition of two or more posynomials is a posynomial. Examples: 3, x are posynomials and 3 + x is also a posynomial. 2. Subtraction: subtraction of two or more posynomials may not be a posynomial. Examples: 3x 2 +5, 2x +7 are posynomial but (x 2 +5)−(2x +7)  (x 2 −2x −2) is not posynomial. 3. Multiplication: multiplication of two or more posynomials is a posynomial. Examples: 3x 2 + 5, 2x are posynomial and (3x 2 + 5) × 2x  6x 3 + 10x is also a posynomial.

4

1 Preliminary Concepts of Geometric Programming (GP) Model

4. Division: division of a posynomial by one or more other posynomial is a posynomial. 2 Examples: x 2 + x, x are posynomials and x x+x is also a posynomial. Note 1.2 The term posynomial is mean to suggest a combination of positive and polynomial POSITIVE + POLYNOMIAL  POSYNOMIAL. Note 1.3 Any monomial is also a posynomial. Posynomials can be divided by monomials and the result becomes a posynomial. Monomials (also posynomials) → 2x/y, 0.27, 3xy2 z Posynomials (but not monomials) → 0.24 + 2x 3 /yz, 3(2 + x/z)3 , 2x + y + 5z Not posynomials (therefore, not monomials)→ −2.3, x − 2y, x 2 + ez − tan y Posynomials (also polynomial) → 3x 2 + 4y3 + 7z, 2.7xy + 5y2 z Posynomials (but not polynomial) → 3x 0.5 + 2y−2 z Polynomials (but not posynomial) → x 3 − 3yz + 7z2 Comparison between monomial, polynomial, and posynomial

Monomial

Polynomial

Posynomial

(1) Contains only one term

(1) Contains one or more term

(1) Contains one or more term

(2) Addition of two or more monomials is not monomial

(2) Addition of two or more polynomials is polynomial

(2) Addition of two or more posynomials is posynomial

(3) Subtraction of two or more monomials is not monomial

(3) Subtraction of two or more polynomials is a polynomial

(3) Subtraction of two or more posynomials is not posynomial

(4) Multiplication of two or more monomials is a monomial

(4) Multiplication of two or more polynomials is a polynomial

(4) Multiplication of two or more posynomials is a posynomial

(5) Division of a monomial by one or more other monomials is a monomial

(5) Division of a polynomial by one or more other monomials is a polynomial

(5) Division of a posynomial by one or more other monomials is a posynomial

(6) A monomial is of the form, F(x)  cx1a1 x2a2 . . . xnan , c > 0

(6) A polynomial is of the form, F(x)   a1i a2i ani m i1 ci x 1 x 2 . . . x n

(6) A posynomial is of the form,  m F(x) a1i a2i ani i1 ci x 1 x 2 . . . x n , ci > 0

(7) Examples: 5, 2x, x 2 y4

(7) Examples: 5, 2x + y, x 2 − y 4

(7) Examples: 5, 2x + y, x 2 + y 4

1.2 Geometric Program (GP)

5

1.2 Geometric Program (GP) Primal Problem: Primal geometric programming (PGP) problem is Minimize g(t) 

T0  k1

ck

n 

α

t j kj

j1

subject to t j > 0, ( j  1, 2, . . . , n).

(1.2)

Here, ck (>0) and let α kj (j  1, 2, …, n; k  1, 2, …, T 0 ) be any real number. Here, degrees of difficulty (DD) of a GP  No. of terms in PGP − (1 + No. of variables in PGP). The problem (1.2) is an unconstrained posynomial GP problem with DD  T 0 − (n + 1). Dual Problem (DP): Dual programming (DP) problem of (1.2) is Maximize v(δ) 

 T0   ck δk k1

δk

subject to T0  k1 T0 

δk  1

(Normality condition)

αk j δk  0, ( j  1, 2, . . . , n) (Orthogonality conditions)

(1.3)

k1

δk > 0, (k  1, 2, . . . , T0 )

(Positivity conditions),

where δ  (δ1 , δ2 , . . . , δT0 )T . Note 1.4 The DP is linearly constrained and concave objective problem. To each posynomial term of PGP, there corresponds a dual variable (i.e., weight) δk . Note 1.5 In general, the difference between the number of variables and the number of independent linear equation is called the number of degrees of freedom. In DP (1.3), there are n orthogonality conditions (one for each primal variable t j of PGP (1.2)), a single normality condition, and T 0 dual variables (one for each term of PGP (1.2)). Hence, these equations have T 0 − (n + 1) degrees of freedom. Duffin, Peterson, and Zener, in their famous book (1967), called it as degree of difficulty (DD) as the larger this number, the harder the problem is to solve. Case I T 0  n + 1, (i.e., DD  0), so DP presents a system of linear equations for the dual variables. A unique solution vector of dual variable exists. In this case, no optimization is necessary. There is a single feasible solution of DP.

6

1 Preliminary Concepts of Geometric Programming (GP) Model

Case II T 0 > n + 1, (i.e., DD > 0), so the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is less than the number of dual variables. More solutions of dual-variable vector exist. In order to find an optimal solution of DP, we need to use some algorithmic methods. Case III T 0 < n + 1, (i.e., DD < 0), so the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is greater than the number of dual variables. In this case, generally no solution vector exists for the dual variables. However, using least square (LS) or min-max (MM) method, one can get an approximate solution for this system. Once optimal dual-variable vector δ * is known, the corresponding values of the primal variable vector t are found from the following relations: ck

n 

α

t j k j  δk∗ v ∗ (δ ∗ ), (k  1, 2, . . . , T0 ).

(1.4)

j1

Taking logarithms in (1.4), T 0 log-linear simultaneous equations are transformed as  δk∗ v ∗ (δ ∗ ) , (k  1, 2, . . . , T0 ). αk j (log t j )  log ck j1 

n 

(1.5)

It is a system of T 0 linear equations in x j (=log t j ) for j  1, 2, …, n. Note 1.6 If there are more primal variables t j than the number of terms T 0 (>1), many solutions t j (j  1, 2, …, n) may exist. So, to find the optimal primal variables t j (j  1, 2, …, n), it remains to minimize the primal objective function with respect to reduced n − T0 ( 0) variables. When n − T 0  0, i.e., number of primal variables  number of log-linear equations, primal variables can be determined uniquely from log-linear equations. Theorem 1.1 If x is a feasible vector for the constraints PGP and δ is a feasible vector for the corresponding DP, then g(t) ≥ d(δ) (Primal–dual inequality). Proof The expression g(t) can be written as g(t) 

T0  k1

 δk

ck

n

αk j j1 t j



δk

here weights are δ1 , δ2 , . . . , δT0 and corresponding positive terms are c2

n

α2 j

j1 t j δ2

, . . .,

cT0

n

αT j 0

j1 t j δT0

.

c1

n

α1 j j1 t j

δ1

,

1.2 Geometric Program (GP)

7

Now applying A.M-G.M inequality 

  αT j (δ1 +δ2 +···+δT0 ) α + c2 nj1 t j 2 j + · · · + cT0 nj1 t j 0

 δ1 + δ2 + · · · + δT0  n  n  αT j δT0 α δ1  α δ2 cT0 j1 t j 0 c1 j1 t j 1 j c2 nj1 t j 2 j ... ≥ δ1 δ2 δT0

c1

n

α1 j j1 t j

Or 

g(t) T0

T0 k1 δi

k1 δi

 n α δi T0 T0   ck j1 x j k j ≥ δk  1 as δi k1 k1

Or  g(t) ≥

ck δk

T0 k1 n δk 

T0

xj

k1

αk j δk

j1

Or g(t) ≥

 T0   ck δk n k1



δk

 T0   ck δk k1

δk

j1

T0

xj

k1

 v(δ)

as

αk j δk

T0 

αk j δk  0 ,

k1

i.e., g(t) ≥ v(δ). Example 1.4 Min f (x1 , x2 )  4x1 +

x1 4x2 + x1 x22

Subject to x1 , x2 > 0. This problem is easily solved through GP Here DP is      4 w1 1 w2 4 w3 Max d(w1 , w2 )  w1 w2 w3 subject to w1 + w2 + w3  1 w1 + w 2 − w3  0 

8

1 Preliminary Concepts of Geometric Programming (GP) Model

− 2w2 + w3  0 w1 , w2 , w3 > 0. Here w1∗  41 , w2∗  14 , and w3∗  21 .



 1 1 1 f ∗ x1∗ , x2∗  d ∗ w1∗ , w2∗  (4.4) 4 (4) 4 (4.2) 2  8 1 1 4x1∗  w1∗ f ∗  × 8 ⇒ x1∗  , 4 2 x1∗ 1 1 ∗ ∗ ∗  w2 f  × 8 ⇒ x 2  . x2∗ 4 2 But this problem is not easily attacked by ordinary NLP method. Here, critical points are solutions of the system of nonlinear equations δ f (x1 , x2 ) 1 4x2  0 ⇒ 4 + 2 − 2  0, δx1 x2 x1 2x1 4 δ f (x1 , x2 ) 0⇒− 3 +  0. δx2 x1 x2 Solving this system of nonlinear equations is by no means an easy task. Also, Hessian matrix  H ( f (x1 , x2 )) 

δ2 f δ2 f δx1 δx2 δx12 δ2 f δ2 f δx2 δx1 δx22



 

8x2 x13

− x23 − 2

− x23 − 4 x12

2

6x1 x24

4 x12



is a frightening prospect. • LS and MM Method: A system of linear equations (say m equations and n unknown variables with m > n) takes the form α11 δ1 + α12 δ2 + · · · + α1n δn  b1 α21 δ1 + α22 δ2 + · · · + α2n δn  b2 . ... αm1 δ1 + αm2 δ2 + · · · + αmn δn  bm Let r1  α11 δ1 + α12 δ2 + · · · + α1n δn − b1 r2  α21 δ1 + α22 δ2 + · · · + α2n δn − b2 . ... rm  αm1 δ1 + αm2 δ2 + · · · + αmn δn − bm

(1.6)

1.2 Geometric Program (GP)

9

• LS Method: Approximate solution for the system of linear equation (1.6) can be determined by the following LS method. Find δ j (j  1, 2, …, n), in which minimize r (δ) 

m  j1

ri2 

m 

(αi1 δ1 + αi2 δ2 + · · · + αin δn − bi )2 .

i1

The following n normal equations are obtained from ∂r∂δ(δ)j  0 for j  1, 2, . . . , n m  i1 m 

2 (αi1 )δ1 +

m  i1

(αi2 αi1 )δ1 +

i1 m 

(αi1 αi2 )δ2 + · · · + m  i1

(αin αi1 )δ1 +

i1

m  i1

2 (αi2 )δ2 + · · · +

...,

m  i1 m 

(αi1 αin )δn  (αi2 αin )δn 

i1

(αin αi2 )δ2 + · · · +

m  i1 m 

(αi1 bi ), (αi2 bi ),

i1 m  i1

2 (αin )δn 

m 

(1.7)

(αin bi ).

i1

It is a symmetric positive definite system of n equations with n unknowns. So, one can determine n dual variables δ j (j  1, 2, …, n). • MM Method: Approximate solution for the system of linear equations (1.6) can also be determined by MM method. By this method, one can obtain the dual-variable vector δ for which the absolutely largest of r 1 , r 2 , …, r m becomes minimum. That is, Min(Max(|r1 |, |r2 |, . . . , |rm |)) subject to α11 δ1 + α12 δ2 + · · · + α1n δn − b1  r1 α21 δ1 + α22 δ2 + · · · + α2n δn − b2  r2 . ... αm1 δ1 + αm2 δ2 + · · · + αmn δn − bm  rm δ j ≥ 0 ( j  1, 2, . . . , n)

(1.8)

Considering an auxiliary variable r, the above problem (1.8) is equivalent to Minimize r subject to αi1 δ1 + αi2 δ2 + · · · + αin δn − bi ≤ r, −αi1 δ1 − αi2 δ2 − · · · − αin δn + bi ≤ r, (i  1, 2, . . . .., m) δ j ≥ 0, ( j  1, 2, . . . , n).

(1.9)

10

1 Preliminary Concepts of Geometric Programming (GP) Model

  [since Min r is subject to r j  ≤ r j  1, 2, . . . , m, i.e., Min r is subject to −r ≤ r j ≤ r ]. δ Denoting rj  δ j , (j  1, 2, …, m) and r1  δ0 , (1.9) becomes a linear programming problem (LPP) as follows: Maximize δ0 subject to αi1 δ1 + αi2 δ2 + · · · + αin δn − bi δ0 ≤ 1, −αi1 δ1 − αi2 δ2 − · · · − αin δn + bi δ0 ≤ 1, (i  1, 2, . . . .., m) . δ j ≥ 0, ( j  0, 1, 2, . . . , n).

(1.10)

After calculating δ0 , δ1 , δ2 , . . . , δn from this LPP (1.10), approximate solutions   of dual variables δ j ( δ0 /δ j ) (j  1, 2, …, n), can be obtained. So, applying either LS or MM method dual variables can be determined. Then, corresponding optimal dual objective function of (1.3) is found. Example 1.5 (Unconstrained GP problem) Min Z (t)  t1 t2 t3−2 + 2t3 t1−1 t2−1 + 5t3 subject to t1 , t2 , t3 > 0. Here, DD  3 − (3 + 1)  −1( 0. It is a system of four linear equations with three unknown variables. Approximate solution of this system of linear equations (by LS method) is δ1∗  0.333, δ2∗  0.333, δ3∗  0.333, and corresponding optimal dual objective value, i.e., v* (δ * )  6.463. So for primal decision variables, the following system of nonlinear equations is found: t1 t2 t3−2  2.1522, 2t3 t1−1 t2−1  2.1522,

1.2 Geometric Program (GP) Table 1.1 Optimal solutions of Example 1.5

Table 1.2 Crisp input values

11

Methods

t1∗

t2∗

t3∗

Z * (t * )

GP

0.3525

1.135

0.431

6.463016

NLP

0.6266

0.6383

0.431

6.463306

a($/m2 )

b($/m2 )

c($/m2 )

d(m3 )

80

10

20

80

5t3  2.1522, and corresponding system of log-linear equations are x1 + x2 − 2x3  0.7664, x3 − x1 − x2  0.0733, x3  −0.8431, where xi  Log ti (i  1, 2, 3). ∗ Solving this system of linear equations, the optimal primal variables ti∗ ( e xi ) (i  1, 2, 3) are obtained. These are given in Table 1.1. Example 1.6 (Grain-box problem) “It has been decided to shift grain from a warehouse to a factory in an open rectangular box of length x1 meters, width x2 meters, and height x3 meters. The bottom, side, and ends are of the box cost $a, $b, and $c/m2 , respectively. It costs $1 for each round trip of the box. Assuming that the box will have no salvage value, find the minimum cost of transporting d m3 of grains.” This problem can be formulated as d + ax1 x2 + 2bx1 x3 + 2cx2 x3 x1 x2 x3 Sub to x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

Min g0 (x, s) 

Let the input values be (Table 1.2). Solution Here the primal problem is 80 + 80x1 x2 + 2.10x1 x3 + 2.20x2 x3 x1 x2 x3 Sub to x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. Min g0 (x) 

(1.11)

12

1 Preliminary Concepts of Geometric Programming (GP) Model

Table 1.3 Optimal solution of the problem (crisp method)

x1∗

x2∗

x3∗

d ∗ (δ)

g0 (x)∗

1

0.5

2

200

200

Corresponding dual form is  Max d(δ) 

80 δ1

δ1 

80 δ2

δ2 

2.10 δ3

δ3 

2.20 δ4

δ4

subject to δ1 + δ2 + δ3 + δ4  1 −δ1 + δ2 + δ3  0 −δ1 + δ2 + δ4  0 −δ1 + δ3 + δ4  0 δ1 , δ2 , δ3 , δ4 ≥ 0.

(1.12)

From (1.12), we get δ1  25 , δ2  15 , δ3  15 , and δ4  15 . The optimal solution of the model through the parametric approach is given by d ∗ (δ) 



5.80 2

 25 

5.80 1

 15 

2.5.10 1

 15 

2.5.20 1

 15

.

From primal–dual relation, we get 80  δ1 × d ∗ (δ), x1 x2 x3 80x1 x2  δ2 × d ∗ (δ), 2.10x1 x3  δ3 × d ∗ (δ), 2.20x2 x3  δ4 × d ∗ (δ). The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 1.3.

1.3 Modified Geometric Programming (MGP) Problem Primal Program: Special type of unconstrained PGP problem is

1.3 Modified Geometric Programming (MGP) Problem

Minimize g(t) 

n 

gi (ti ) 

i1

T0 n   i1 k1

Cik

13 m 

α

ti jik j

j1

subject to ti j > 0, (i  1, 2, . . . . . . , n; j  1, 2, . . . . . . , m) t  (t11 , t12 , . . . ., tnm )T , ti  (ti1 , ti2 , . . . ., tim )T (i  1, 2, . . . , n),

(1.13)

where C ik (>0) and α ikj (i  1, 2, …, n; j  1, 2, …, m; k  1, 2, …, T 0 ) are real numbers. Here, g(t) is a sum of n separable posynomial functions gi (t i ) (i  1, 2, …, n) of distinct variables. Dual Program: Applying the GP techniques to (1.13), the enlarged pre-dual function would be written in the following form:  T0  n   cik

δik

Maximize v(δ) 

i1 k1

δik

subject to T0  k1 T0 

δik  1

(Normality conditions)

αik j δik  0 ( j  1, 2, . . . , m) (Orthogonality conditions)

(1.14)

k1

δik > 0, (k  1, 2, . . . ., T0 )

for i  1, 2, …, n. Here, δ is a decision (δ11 , δ12 , . . . , δik , . . . , δnT0 )T .

variable

(Positivity conditions)

vector

of

nT 0

components,

i.e.,

Case I nT 0  nm + n, so DP presents a system of (mn + n) linear equations with nT 0 (=mn + n) variables. So, a unique solution set of dual variables exists. Case II nT 0 > nm + n, so the DP presents a system of (n + mn) linear equations for the nT 0 dual variables. A solution vector for the dual variables exists. Here, the number of linear equations is less than the number of dual variables. So more solution of dual variable exists. Case III nT 0 < nm + n, in this case, generally no solution of dual variables exists. However, using either the LS or MM method, one can get an approximate solution vector for this system. Theorem 1.2 If t is a feasible vector for the unconstrained PGP (1.13) and if δ is a feasible vector for the corresponding DP (1.14), then g(t) ≥ n

 n

v(δ) (Primal-Dual Inequality).

14

1 Preliminary Concepts of Geometric Programming (GP) Model

Proof Islam and Roy (2005). ∗ ∗ Theorem 1.3 If t ∗  (ti1 , . . . , tim ) for i  1, 2, …, n is a solution to the PGP (1.13), then the corresponding DP (1.14) is consistent. Moreover, the vector δ ∗  ∗ , . . . , δi∗T0 ) for i  1, 2, . . . , n defined by (δi1 ∗ δik 

u ik (t ∗ ) , (i  1, 2, . . . , n; k  1, 2, . . . , T0 ), gi (t ∗ )

 ∗α where u ik (t ∗ )  cik mj1 ti j ik j is the kth term of g(t) for ith item is a solution for √ DP and equality holds in the primal-dual inequality, i.e., g(t ∗ )  n n v(δ ∗ ). Proof Islam and Roy (2005). We can get dual variables which optimize v* (δ * ) such that cik

m 

 α ti jik j  δik n v(δ), (i  1, 2, . . . , n; k  1, 2, . . . , T0 )

(1.15)

j1

Taking logarithms in (1.15), the linear simultaneous equations are transformed as √  δik n v(δ) , (i  1, 2, . . . , n; k  1, 2, . . . , T0 ). αik j (Log ti j )  Log cik j1

m 



(1.16) It is a system of linear equations in x ij (=Log t ij ) (i  1, 2, …, n; j  1, 2, … m). Since there are more primal variables t ij than the number of terms nT 0, many solutions t ij (i  1, 2, …, n; j  1, 2, …, m) may exist. So, to find the optimal primal variables t ij (i  1, 2, …, n; j  1, 2, …, m), it remains to minimize the primal objective function with respect to reduced nm − nT 0 ( 0) variables. When nm − nT 0  0, primal variables can be determined uniquely from log-linear equations. Example 1.7 (Unconstrained MGP problem) −2 −1 −1 −3 −2 −2 Minimize Z  t11 t21 t31 + 2t31 t11 t21 + 5t31 + t12 t22 t32 + 3t32 t22 t12 + 4t32

subject to ti j > 0, (i  1, 2; j  1, 2, 3). This problem can be written as Minimize Z (t) 

2  3  i1 k1

Cik

3 

α

ti jik j

j1

subject to ti j > 0, (i  1, 2; j  1, 2, 3),

1.3 Modified Geometric Programming (MGP) Problem

15

where C 11  1, C 12  1, C 13  2, C 21  3, C 22  5, C 23  4, α 111  1, α 211  1, α 121  −2, α 221  1, α 131  1, α 231  −3, α 112  −1, α 212  −1, α 122  1, α 222  −2, α 132  −2, α 232  1, α 113  0, α 213  0, α 123  1, α 223  0, α 133  0, α 233  1. Dual program: Dual programming (DP) of above MGP is as follows:  2  3   cik

δik

Maximize v(δ) 

i1 k1

δik

subject to δ11 + δ12 + δ13  1 δ21 + δ22 + δ23  1 δ11 − δ12  0 −2δ11 + δ12 + δ13  0 , δ11 − 2δ12  0 δ21 − δ22  0 δ21 − 2δ22  0 −3δ21 + δ22 + δ23  0 δ11 , δ12 , δ13 , δ21 , δ22 , δ23 > 0 where δ  (δ 11 , δ 12 , δ 13 , δ 21 , δ 22 , δ 23 )T . There is a system of eight linear equations with six unknown variables. Applying LS method, the above system of linear equations reduces to δ11 − δ12  0 − 2δ11 + δ12 + δ13  0 δ11 + δ12 + δ13  1 δ21 − 2δ22  0 − 3δ21 + δ22 + δ23  0 δ21 + δ22 + δ23  1 δ11 , δ12 , δ13 , δ21 , δ22 , δ23 > 0. ∗ ∗  0.333, δ12  Approximate solutions of this system of linear equations are δ11 ∗ ∗ ∗ ∗ 0.333, δ13  0.333, δ21  0.25, δ22  0.125, δ23  0.625, and optimal dual objective value, i.e., v* (δ * )  13.17. √ The value of the objective function g(t ∗ )  2 13 × 17  7.23. So Theorem 1.1 is verified. And the following system of nonlinear equations gives optimal primal variables

16

1 Preliminary Concepts of Geometric Programming (GP) Model

Table 1.4 Optimal solutions of example 1.7 ∗ t21

∗ t31

∗ t12

∗ t22

∗ t32

Z * (t * )

Methods

∗ t11

MGP

0.349

1.145

0.431

2.506

0.773

1.0489

7.23

NLP

0.6416

0.6235

0.431

1.39

1.393

1.0489

7.31

Fig. 1.1 Multi-grain-box problem

−2 t11 t21 t31  4.3856, −2  3.2925, t12 t22 t32 −1 −1 t21  4.3856, 2t31 t11 −1 −1 t22  1.6463, 2t32 t12

5t31  8.2313, 5t32  4.3856. Solving this system of nonlinear equations, optimal solutions are determined. The optimal solutions (by MGP) of this problem are given in Table 1.4. Here, optimal solutions of Example 1.7 by NLP are also given in Table 1.4. It is seen that MGP method gives better than NLP method, which is expected. Example 1.8 MGP problem (Multi-grain-box problem) Suppose that to shift grains from a warehouse to a factory in a finite number (say n) of open rectangular boxes of lengths x1i meters, widths x2i meters, and heights x3i meters (i  1,2, …, n). The bottom, side, and ends of the box cost $ai , $bi , and $ci /m2 , respectively. It costs $1 for each round trip of the box. Assuming that the box will have no salvage value, find the minimum cost of transporting di m3 of grains (Fig. 1.1). This problem can be formulated as an unconstrained MGP problem Min g0 (x, s) 

n   i1

di + ai x1i x2i + 2bi x1i x3i + 2ci x2i x3i x1i x2i x3i

Sub to x1i ≥ 0, x2i ≥ 0, x3i ≥ 0. (i  1, 2 . . . , n).



(1.17)

1.3 Modified Geometric Programming (MGP) Problem Table 1.5 Crisp input values

17

ith box

ai ($/m2 )

bi ($/m2 )

ci ($/m2 )

d(m3 )

i1

80

10

20

80

i2

60

20

30

50

In particular, here, we assume the transporting di m3 of grains by the two different open rectangular boxes whose bottom, sides, and the ends of each box costs are given in Table 1.5. Solution Here the primal problem is Min g0 (x) 

2  i1

di + ai xi1 xi2 + 2bi x1 x3 + 2ci xi2 xi3 xi1 xi2 xi3

Sub to xi1 ≥ 0, xi2 ≥ 0, xi3 ≥ 0.

(1.18)

Corresponding dual form is Max d(δ) 

       2   di δi1 ai δi2 2bi δi3 2ci δi4 i1

δi1

δi2

δi3

δi4

subject to δ11 + δ12 + δ13 + δ14  1 δ21 + δ22 + δ23 + δ24  1 −δ11 + δ12 + δ13  0 −δ11 + δ12 + δ13  0 −δ11 + δ12 + δ14  0 −δ11 + δ12 + δ14  0 −δ11 + δ13 + δ14  0 −δ11 + δ13 + δ14  0 δ11 , δ12 , δ13 , δ14 δ21 , δ22 , δ23 , δ24 ≥ 0.

1 5

(1.19)

From (1.19), we get δ11  25 , δ12  15 , δ13  15 , δ14  15 , δ21  25 , δ22  15 , δ23  and δ24  15 . The optimal solution of the model through the parametric approach is given by 1  1   2  1 5.80 5 2.5.10 5 2.5.20 5.50 5 5.50 5 1 1 1 2 1   15   15 51 2.5.20 2.5.30 . 1 1

d ∗ (δ) 



5.80 2

 25 

18

1 Preliminary Concepts of Geometric Programming (GP) Model

Table 1.6 Optimal solution of the model (crisp method) ∗ x11

∗ x12

∗ x13

∗ x21

∗ x22

∗ x23

d ∗ (δ)

g0 (x)∗

1.026

0.513

2.053

0.962

0.639

0.961

38,980

393.618

From primal–dual relation, we get  80   δ11 × 2 d ∗ (δ), x11 x12 x13   × 2 d ∗ (δ), 80x11 x12  δ12   2.10x11 x13  δ13 × 2 d ∗ (δ),   2.20x12 x13  δ14 × 2 d ∗ (δ),  50   δ21 × 2 d ∗ (δ), x21 x22 x23   × 2 d ∗ (δ), 60x21 x22  δ22   2.20x21 x23  δ23 × 2 d ∗ (δ),   2.30x22 x23  δ24 × 2 d ∗ (δ). The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 1.6.

1.4 Constrained Geometric Programming (CGP) Problem Primal geometric programming (PGP): Minimize g0 (t) 

T0 

c0k

k1

m 

α

t j 0k j

j1

subject to gr (t) 

Tr  k1+Tr −1

cr k

m 

α

t j rkj ≤ 1

j1

t j > 0, j  1, 2, . . . , m,

(1.20)

where crk (>0) and α rkj (k  1, 2, …, 1 + T r −1 , …, T r ; r  0, 1, 2, …, l; j  1, 2, …, m) are real numbers.

1.4 Constrained Geometric Programming (CGP) Problem

19

It is a constrained posynomial PGP problem. The number of terms in each posynomial constraint function varies and it is denoted by T r for each r  0, 1, 2, …, l. Let T  T 0 + T 1 + T 2 + ··· + T l be the total number of terms in the primal program. The degree of difficulty (DD)  T − (m + 1). Dual Program: The dual programming of (1.20) is as follows: Maximize v(δ) 

 Tr  l   cr k δr k r 0 k1

δr k

⎛ ⎝

Tr 

⎞δr k δr s ⎠

s1+Tr −1

subject to T0 

δ0k k1 Tr l  

 1,

Normality condition

Orthogonality conditions (one per primal variable) r 0 k1 δr k > 0, (r  0, 1, 2, . . . , l; k  1, 2, .., Tr ). Positivity conditions (1.21) αr k j δr k  0, ( j  1, 2, . . . , m)

Case I T 0  m + 1, so DP presents a system of linear equations for the dual variables. A unique solution vector of dual variable exists. Case II T 0 > m + 1, so the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is less than the number of dual variables. More solutions of dual-variable vector exist. Case III T 0 < m + 1, so the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is greater than the number of dual variables. In this case, generally no solution vector exists for the dual variables. However, using least square (LS) or min-max (MM) method, one can get an approximate solution for this system. The solution procedure of this GP problem is the same as the unconstrained GP problem. Example 1.9 Min g(x, y, z)  2y 2 z 4 + 5x −2 Subject to 2x 2 y −2 + yz −1 ≤ 1 x, y, z > 0.

(1.22)

20

1 Preliminary Concepts of Geometric Programming (GP) Model

Solution Corresponding dual of (1.22)  Maximize v(δ) 

2 δ1

δ1 

5 δ2

δ2 

2 δ01

δ01 

1 δ02

δ02

λλ1 1

Subject to δ1 + δ2  1, −2δ2 + 2δ01  0, 2δ1 − 2δ01 + δ02  0, 4δ1 − δ02  0, λ1  δ01 + δ02 δ1 , δ2 , δ01 , δ02 > 0. From (1.22), δ1  14 , δ2  43 , δ01  43 , δ02  1, and λ1  47 . Putting the value of δ1 , δ2 and δ01 in (1.22), the corresponding optimal dual value, i.e.,          4.2 1/4 4.5 3/4 4.2 3/4 1 1 7 7/4 v (δ)  1 3 3 1 4  38.77. ∗



From primal–dual relation, 2y 2 z 4  δ1 v ∗  0.25 × 38.77, 5x −2  δ2 v ∗  0.75 × 38.77, δ01 3 2x 2 y −2   , λ1 7 δ 4 02 yz −1   . λ1 7 Taking logarithm of each side yields equations which are linear in the logarithms of the primal variables ln 2 + 2 ln y + 4 ln z  ln 9.69, ln 5 − 2 log x  ln 29.08, ln 2 + 2 log x − 2 log y  ln 0.43, ln y − ln z  ln 0.57. From above, optimal solutions are x  0.415,

y  0.897, z  1.570, g(x, y, z)  38.77.

1.4 Constrained Geometric Programming (CGP) Problem

21

Table 1.7 Optimal solutions of Example 1.10 Methods

t1∗

GP NLP

Z * (t * )

t2∗

t3∗

t4∗

1.31

1.52

1.28

1.21

14.26

1.25

1.52

1.18

1.23

14.85

Example 1.10 (Constrained GP problem) Minimize Z (t)  20t3−1 t4−1 + 30t2−6 t42 subject to t13 t42 + t1−1 t24 t32 t42 ≤ 12 t1 , t2 , t3 , t4 > 0. It is a constrained posynomial geometric programming problem with degree of difficulty −1. This problem is also solved by GP and NLP as before and optimal solutions are given in Table 1.7.

1.5 Constrained Modified Geometric Programming (CMGP) Problem Primal problem: Minimize g0 (t) 

n 

gi0 (ti ) 

i1

T0 n  

C0ik

i1 k1

m 

α

ti j0ik j

j1

subject to n 

gir (ti ) 

i1

n 

Tr 

crik

i1 kTr −1 +1

m  j1

α

ti jrik j ≤ 1,

(1.23)

ti j > 0 (i  1, 2, . . . , n; j  1, 2, . . . , m), where C ik (>0) and α rikj (i  1, 2, …, n; j  1, 2, …, m; r  0, 1, 2, …, l; k  1, 2, …, T r −1 + 1, …, T r ) are real number. Dual Program: Applying the GP techniques to Eq. (1.23), the enlarged pre-dual function could be written in the following form: Maximize v(δ) 

δik  n 

 Tr  n   cik i1 k1

subject to

δik

s1

δik δsk

22

1 Preliminary Concepts of Geometric Programming (GP) Model T0  k1 T0 

δik  1

(Normality conditions)

α0ik j δik +

k1

l 

Tr 

r 1 kTr −1 +1

αrik j δik  0 ( j  1, 2, . . . , m) (Orthogonality conditions)

δik > 0, for i  1, 2, .., n

(Positivity conditions) (1.24)

where δ  (δ11 , δ12 , . . . , δik , . . . , δnT0 )T . Case I nT 0  nm + n, so DP presents a system of (mn + n) linear equations with nT 0 (=mn + n) variables. So, a unique solution set of dual variables exists. Case II nT 0 > nm + n, so the DP presents a system of (n + mn) linear equations for the nT 0 dual variables. A solution vector for the dual variables exists. Here, the number of linear equations is less than the number of dual variables. So more solution of dual variable exists. Case III nT 0 < nm + n, in this case, generally no solution of dual variables exists. However, using either the LS or MM method, one can get an approximate solution vector for this system. The solution procedure of this MGP problem is the same as the unconstrained MGP problem. Example 1.11 −3 −1 −3 −1 Min Z (x) 10x11 + 10x11 x12 + 6x21 + 5x21 x22 Subject to x11 x12 + x21 x22 ≤ 1 x11 , x12 , x21 , x22 > 0.

(1.25)

Solution Here primal problem is −3 −1 −3 −1 Min Z (x) 10x11 + 10x11 x12 + 6x21 + 5x21 x22

Subject to x11 x12 + x21 x22 ≤ 1 x11 , x12 , x21 , x22 > 0.

(1.26)

And corresponding dual problem is 

10 δ01 Subject to d(δ) 

δ01 

10 δ02

δ02 

6 δ11

δ11 

5 δ12

δ12 

1 δ21

δ21 

1 δ22

δ22

(δ21 + δ22 )(δ21 +δ22 )

1.5 Constrained Modified Geometric Programming (CMGP) Problem

δ01 + δ02  1, δ11 + δ12  1, δ01 − 3δ02 + δ21  0, −δ02 + δ21  0, δ11 − 3δ12 + δ22  0, −δ12 + δ22  0.

23

(1.27)

Approximate solutions of this system of linear equations are δ01 

2 1 1 , δ02  , δ21  , 3 3 3

δ11 

2 1 1 , δ12  , δ22  . 3 3 3

and

Then the objective function is              3.10 2/3 3.10 1/3 3.6 2/3 3.5 1/3 3 1/3 3 1/3 2 2/3 d(δ)  2 1 2 1 1 1 3  320.12. 

And from primal–dual relation, the following system of equations gives optimal primal variables 2 √ × 320.12  11.928, 3 1 √ −3 −1 10x11 x12  × 320.12  5.964, 3 2 √ 6x21  × 320.12  11.928, 3 1 √ −3 −1 5x21 x22  × 320.12  5.964, 3 1 x11 x12  , 2 1 x21 x22  . 2 10x11 

Solving the system of equations, optimal solutions are x11  1.193, x12  0.419, x21  1.988, x22  0.252. and the value of objective function is Z (x)  40.439.

24

1 Preliminary Concepts of Geometric Programming (GP) Model

Table 1.8 Optimal solutions of Example 1.12 Methods

∗ t11

∗ t21

∗ t31

∗ t41

∗ t12

∗ t22

∗ t32

∗ t42

Z * (t * )

MGP

1.62

1.43

1.56

1.06

1.01

1.41

1.52

0.81

21.97

NLP

1.78

1.34

2.16

0.88

1.11

1.32

2.09

0.68

22.83

Example 1.12 (Constrained MGP problem) −1 −1 −6 2 −1 −1 −6 2 Minimize Z (t)  20t31 t41 + 30t21 t41 + 5t32 t42 + 20t22 t42 −1 4 2 2 −1 4 2 2 3 2 3 2 subject to t11 t41 + t11 t21 t31 t41 + t12 t42 + t12 t22 t32 t42 ≤ 25

t11 , t21 , t31 , t41 , t12 , t22 , t32 , t42 > 0. It is solved by MGP and also NLP method, and the optimal results are given in Table 1.8. Theorem 1.4 If t is a feasible vector for the constrained primal geometric program (PGP) and if δ is a feasible vector for the corresponding dual program (DP), then g0 (t) ≥ n

 n

v(δ) (Primal-Dual Inequality).

Proof Islam and Roy (2005) Theorem 1.5 Suppose that the constrained geometric program (GP) is superconsistent and that t * is a solution for (GP), then the corresponding dual program (DP) is consistent and has a solution δ ∗ which satisfies  go (t ∗ )  n n v(δ ∗ ) and  δi∗k



u i k (t ∗ ) , (i  1, 2, . . . , n; k  1, 2, . . . , T0 ) g0 (t ∗ ) λi r (δ ∗ ) u i k (t ∗ ), (i  1, 2, . . . , n; k  Tr −1

+ 1, . . . Tr ; r  1, 2, . . . , l).

1.6 Conclusion Geometric programming was introduced in 1967 by Duffin, Peterson, and Zener. It is very useful in the applications of a variety of optimization problems. It can be used to solve large-scale, practical problems by quantifying them into a mathematical optimization model. Geometric programs (GPs) are useful in the context of geometric design and models well approximated by power laws. Applications of GP include electrical circuit design and other topics such as finance and statistics. GP is a very

1.6 Conclusion

25

powerful type of application that can be used to solve a variety of different optimization problems. There are many different methods to solve GPs, and it depends on the different conditions for the specific GP. Although it may be difficult to quantify a problem into a GP, doing so can be very useful to get an approximate answer, if not an exact answer, which still can be valuable. This kind of programming has applications across a variety of fields from engineering to economics and will continue to be useful in the future as more problems are formatted into GPs.

References M. Bazaraa, C. Shetty, H. Sherali, Non-Linear Programming: Theory and Algorithms (Wiley, New York, 1993) C.S. Beightler, D.T. Philips, Applied Geometric Programming (Wiley, New York, 1976) C. Chu, D. Wong, VLSI circuit performance optimization by geometric programming. Ann. Oper. Res. 105(1–4), 37–60 (2001) R. Duffin, Linearizing geometric programs. SIAM Rev. 12(2), 668–675 (1970) R. Duffin, E. Peterson, C. Zener, Geometric Programming—Theory and Application (Wiley, New York, 1967) P. Feigin, U. Passy, The geometric programming dual to the extinction probability problem in simple branching processes. Ann. Probab. 9(3), 498–503 (1981) C. Floudas, Deterministic Global Optimization: Theory, Algorithms and Applications (Kluwer Academic, Dordrecht, 1999) S. Islam, T.K. Roy, Modified Geometric programming problem and its applications, J. Appt. Math and comput, 17(1–2), 121–144 (2005). https://doi.org/10.1007/BF02936045 H. Jung, C. Klein, Optimal inventory policies under decreasing cost functions via geometric programming. Eur. J. Oper. Res. 132(3), 628–642 (2001) E. Klafszky, J. Mayer, T. Terlaky, A geometric programming approach to the channel capacity problem. Eng. Optim. 19, 115–130 (1992) E. Peterson, The origins of geometric programming. Ann. Oper. Res. 105(1–4), 15–19 (2001) S.B. Sinha, A. Biswas, M.P. Biswal, Geometric programming problems with negative degrees of difficulty. Eur. J. Oper. Res. 28, 101–103 (1987) C. Zener, Engineering Design by Geometric Programming (Wiley, New York, 1971)

Chapter 2

Signomial Geometric Programming (GP) Problem

2.1 Introduction and History Richard J. Duffin and Elmor L. Peterson introduced the term “signomial” in their original joint work general algebraic optimization, published in the late 1960s and early 1970s. In the other hand Passy and Wilde (1967) developed the generalized polynomial optimization. Jefferson and Scott (1978) applied generalized geometric programming applied to problems of optimal control. Rijckaert and Martens (1978) developed the comparison of generalized geometric programming algorithms. Sherali and Tuncbilek (1992) developed a global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. Horst and Tuy (1993) briefly discussed global optimization problems in their book. Sherali (1998) defined global optimization of non-convex polynomial programming problems having rational exponents. Horst et al. (2000) gave brief discussion on deterministic global optimization. Qu et al. (2007) demonstrated a new global optimization algorithm for signomial geometric programming via Lagrangian relaxation. An ongoing introductory exposition is optimization problems. In spite of the fact that nonlinear optimization problems with constraints and/or objectives defined by signomials are normally harder to unravel than those defined by posynomials (in light of the fact that not at all like posynomials, signomials are not ensured to be globally convex) but a signomial optimization problem, often give a substantially more accurate mathematical representation of present real-world, nonlinear optimization problems.

2.2 Definition and Example A “signomial” is an algebraic function of at least one independent function. It is one of most effortlessly thought of as a mathematical expansion of multidimensional polynomials an augmentation that permits exponents to be arbitrary real numbers (as © Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_2

27

28

2 Signomial Geometric Programming (GP) Problem

opposed to simply non negative integers) while requiring the independent variables to be strictly positive. That is, a signomial function differs from a posynomial function in that the coefficient need not be positive. Let X be a vector of real, positive numbers. X  (x1 , x2 , . . . , xm )T . Then, signomial function has the form as follows: g0 (x)  f (x1 , x2 , . . . , xm ) 

k  i1

σi ci

m 

a

x jij ,

(2.1)

j1

where ci  absolute value of coefficient, σi  sign of coefficient (+1 or − 1). Signomial functions are closed under addition, subtraction, multiplication, and scaling. Example 2.1 g0 (x)  2x12 x2−1 − 5x1 x2 . For this signomial σ1  +1, σ2  −1, c1  2, c2  5.

2.3 Convexity of Signomial Functions A signomial function is, in general, not convex. However, the necessary conditions for when a signomial function is convex can be derived. These conditions rely on the fact that a sum of convex terms is convex by determining whether the individual terms in the signomials are convex, the convexity of the whole function can be attained. As will be shown in the following two theorems, the conditions for when a signomial term is convex depend on the sign of the term. The convexity requirements presented here have previously been given in Maranas and Floudas (1995). However, the proof here is different. The first theorem gives the convexity requirements for positive signomial terms. In this chapter, we have discussed different types of signomial geometric programming as follows.

2.3 Convexity of Signomial Functions

29

Signomial geometric programming (SGP)

Unconstrained programming

Constrained programming

Unconstrained SGP Unconstrained MSGP

Constrained SGP p

p

Constrained MSGP

p

Theorem 2.1 The signomial term f (x)  c.x1 1 x2 2 . . . x NN , where c > 0 is convex if one of the following two conditions is fulfilled: (i) All powers Pi are negative, or (ii)  One power Pk is positive, the rest of the power pi , i  k are negative and N i1 pi > 1. Proof A functions f (x) is convex if the Hessian matrix H(x) of the function is positive semi-definite. The second-order partial derivatives of f (x) are  p p c. f (x). xii x jj if i  j, ∂ f (x)  pi ( pi −1) c. f (x). x 2 if i  j. ∂ xi ∂ x j i

So H(x) will consist of the element more, it can be shown that det H (x)  (−c)

N

∂ f (x) ∂ xi ∂ x j

 N 

at position (i, j) in the matrix. Further

N p −2 pi xi i

1−

N 

i1

 pi .

i1

For the matrix H(x) to be positive semi-definite, all principal minors, det Hl , l  1, 2, . . . , N , of H(x) must be positive, i.e., for all l  1, 2, . . . , N : det Hl > 0. The determinant of the l-th principal minor is det(x)Hl (x)  (−c)

l

 l  i1

 l p −2 pi xi i

1−

l 

 pi .

i1

(i) If c > 0, x i > 0 and pi ≤ 0 are fulfilled, then det H l (x) > 0, ∀l  1, . . . , N . Since all principal minors of H(x) are positive, the function  f (x) is convex. (ii) If c < 0, xi > 0, pi ≤ 0, i  k, pk ≥ 0, and NI1 pi ≥ 1 are fulfilled, then ∀l  1, 2, . . . , N : det Hl (x) > 0. Since all principal minors of H(x) are positive, the function f (x) is convex.

30

2 Signomial Geometric Programming (GP) Problem

Theorem 2.2 (Convexity of a negative signomial term) p p p The signomial term f (x)  c.x1 1 x2 2 . . . x NN , where c < 0 is convex if all powers Pi are positive and 0
0, j  1, 2, . . . , n. Example 2.2 Min 2x12 − 3x2−3 Subject to x1 , x2 > 0. Primal Unconstrained Signomial GP Problem Primal program: A primal unconstrained signomial GP programming problem is of the form Minimize g0 (x1 , x2 , . . . , xm ) Subject to x j > 0, j  1, 2, . . . , m. where g0 (x) 

k i1

σi ci

m

a

j1

x jij .

Dual signomial GP problem: Dual GP problem of the given primal GP problem is Maximize ζ0

n  ζ0  ci σi δi i1

Subject to

k  i1

δi

σi δi  ζ0 ,

(2.2)

2.4 Unconstrained NLP Problem

31

k 

σi ai j δi  0,

j  1, 2, . . . , m

i1

δi > 0.

(2.3)

Case I n > m + 1, (i.e., DD > 0), so the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is less than the number of dual variables. More solutions of dual-variable vector exist. In order to find an optimal solution of DP, we need to use some algorithmic methods. Case II n < m + 1, (i.e., DD < 0), so the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is greater than the number of dual variables. In this case, generally no solution vector exists for the dual variables. However, using least square (LS) or min-max (MM) method one can get an approximate solution for this system. Furthermore, the primal–dual relation is ci

m 

∗ai j

xj

 ζ0 δi∗ v(δ ∗ , λ∗ ).

(2.4)

j1

Taking logarithms in (2.4), T 0 log-linear simultaneous equations are transformed as n  j1

αi j (log x j )  log

 ζ0 δi∗ v ∗ (δ ∗ , λ∗ ) , (i  1, 2, . . . , n). ci

(2.5)

It is a system of n linear equations in t j (=log x j ) for j  1, 2, …, m. Note 2.1 A Weak Duality theorem would say that g0 (x) ≥ v(δ) for any primalfeasible x and dual-feasible δ but this is not true for the pseudo-dual signomial GP problem. Corollary 2.1 When the value of σi is 1, then a signomial geometric programming problem transform to ordinary geometric programming problem. Theorem 2.3 When σi is 1, then go (x) ≥ v(δ) (Primal–Dual Inequality). Proof The expression for g0 (x) can be written as  m α  n  ci j1 x j k j go (x)  δk . δk i1

32

2 Signomial Geometric Programming (GP) Problem

Here, the weights are δ1 , δ2 , . . . , δn and positive terms are c2

m

α

j1

x j 2j

δ2

,...,

cn

m

c1

m

α1 j

j1

xj

,

δ1

α

j1

x j nj

.

δn

Now applying A.M.-G.M. inequality, we get 

c1

m

α

j1

⎛ ≥⎝

x j 1 j + c2

m

α

x j 2 j + · · · + cn

j1

m

(δ1 + δ2 + · · · + δn )   α δ1  α δ2 c1 mj1 x j 1 j c2 mj1 x j 2 j δ1

α

j1

δ2

x j nj 

...

(δ01 +δ02 +···+δn )

cn

m j1

α

x j nj

δn ⎞

δn

or

g0 (x) n i1 δi

n i1 δi

 m α δi n n   ci j1 x j n j ≥ δk  1 , as δi i1 i1

or

g0 (x) ≥

ci δ0k

n i1 m δi 

n

xj

i1

αi j δi

,

j1

or g0 (x) ≥

n δi  m  ci i1



n

 i1

δi

n

xj

i1

αi j δi

j1

  T0 , ci δi α0k j δok  0 as δi k1

 v(δ) i.e., g0 (x) ≥ v(δ).. This completes the proof. Example 2.3 Min Z (x)  10x + 5x y − 10x 3 y Subject to x, y > 0.

⎠,

2.4 Unconstrained NLP Problem

33

Solution Assume σ00  1, Corresponding dual of the given problem is

Maximize v(δ) 

10 δ1

δ1

5 δ2

δ2

10 δ3

−δ3

,

(2.6)

Subject to δ1 + δ2 − δ3  1, δ1 + δ2 − 3δ3  0, δ2 − δ3  0, δ1 , δ2 , δ3 > 0.

(2.7)

From (2.7), δ1  1, δ2  21 and δ3  21 . Putting the value of δ1 , δ1 , and δ3 in (2.6), the corresponding optimal dual value, i.e., v(δ)  7.071. So, for primal decision variables, the following equations found: 10x  δ1 v(δ)  1 × 7.071, 1 5x y  δ2 v(δ)  × 7.071, 2 1 3 10x y  δ3 v(δ)  × 7.071. 2 Taking logarithm of each side yields equations, which are linear in the logarithms of the primal variables ln 10 + ln x  ln 7.071, ln 5 + log x + log x  ln 3.535, ln 10 + 3 log x + log y  ln 3.535. From the set of above equations, optimal solutions are x  0.707, y  1. and corresponding optimum value, Z  7.071.

2.5 Unconstrained Modified Signomial Function An unconstrained modified signomial function differs from a posynomial function in that the coefficient need not be positive.

34

2 Signomial Geometric Programming (GP) Problem

gk (x) 

n  k 

σli cli

l1 i1

m 

a

xl jli j ,

(2.8)

j1

where cli  absolute value of coefficient, σli  sign of coefficient (+1 or −1). 2 −1 2 −1 Example 2.4 g0 (x)  2x11 x12 − 5x11 x12 + 4x21 x22 − 3x21 x22 . For this signomial

σ11  +1, σ12  −1, σ21  +1, σ22  −1, c11  2, c12  5, c21  2, c22  5, Primal Unconstrained Modified Signomial GP Problem Primal program: A primal unconstrained modified signomial GP programming problem is of the form Minimize g0 (xl j ) Subject to xl j > 0, j  1, 2, . . . , m, where g0 (x) 

n k l1

i1

σli cli

m j1

(2.9)

a

xl jli j .

Dual signomial GP problem: Dual GP problem of the given primal GP problem is Maximize ζ0

 k

n   cli σli δli l1 i1

Subject to

k 

ζ0

δli

σli δli  ζ0 , (l  1, 2, . . . , n)

i1 k 

σli ali j δli  0, (l  1, 2, . . . , n; j  1, 2, . . . , m).

i1

δli > 0; ζ0  ±1.

(2.10)

Case I nk ≥ nm + n, (i.e., DD > 0), the dual signomial program presents a system of linear equations for the dual variables. A solution vector exists for the dual signomial variables. Case II nk < nm + n, (i.e., DD < 0), in this case, generally no solution vector exists for the dual signomial variables. But using least square (LS) or min-max (MM) method one can get an approximate solution for this system.

2.5 Unconstrained Modified Signomial Function

35

Furthermore the primal–dual relation is n  k 

σli cli

l1 i1

m 

 a xl jli j  ζ0 δli∗ n v(δ ∗ , λ∗ ).

j1

√ Note 2.2 A Weak Duality theorem would say that g0 (xl j ) ≥ n n v(δ) for any primalfeasible x and dual-feasible δ but this is not true for the pseudo-dual signomial GP problem. Corollary 2.2 When the values of σli is 1, then a modified signomial geometric programming problem transform to ordinary modified geometric programming problem. √ Theorem 2.4 When σi is 1, then g0 (xl j ) ≥ n n v(δ) (Primal–Dual Inequality). Proof The expression for g0 (xl j ) can be written as g0 (xli ) 

n  k 

 δik

cli

m j1

α

xl jli j

 .

δik

l1 i1

Here, the weights are δl1 , δl2 , . . . , δlk and positive terms are cl2

m j1

α

xl jl2 j

δl2

,...,

clk

m j1

δlk

c1

m j1

α1 j

xj

δ1

,

α

xl jln j

.

Now applying A.M.-G.M. inequality, we get n  ⎞ ⎛ n   m (δl1 +δl2 +···+δlk ) m m αl1 j αl2 j αlk j i1 l1 cl1 j1 xl j + cl2 j1 xl j + · · · + cln j1 xl j ⎝ ⎠ n l1 (δl1 + δl2 + · · · + δlk ) ⎛  ⎞  m  α δl1  α δl2 α δlk n  cl1 mj1 xl ji1 j clk j1 xl jlk j cl2 mj1 xl jl2 j ⎝ ⎠, ≥ ... δ δ δ l1 l2 lk l1

or 

  g0 xl j n k l1

or

i1 δli

k  n l1 δli i1

≥

 m α δik n  k  cli j1 xl jli j l1 i1

δik

,

36

2 Signomial Geometric Programming (GP) Problem



g0 (xl j ) n

k

n ≥

 n

 cli δlk

l1



m 

l1 δlk

i1

xl j

αli j δli

as

j1

 m n  k

 cli δli  l1 i1



k

δli

k 

δli  1

i1 k

xl j

i1

αli j δli

,

j1

or

g0 (xl j ) n

n ≥

 k

n   cli δli l1 i1

as

δli

k 

αli j δli  0

i1

 v(δ), i.e.,  g0 (xl j ) ≥ n n v(δ).

(2.11)

This completes the proof. Example 2.5 3 3 x12 + 6x21 + 10x21 x22 − 5x21 x22 Min Z (x)  10x11 + 5x11 x12 − 10x11

Subject to x11 , x12 , x21 , x22 > 0. Solution Assume ζ0  1, Corresponding dual of the above problem is

Maximize v(δ) 

10 δ1

δ1

5 δ2

δ2

10 δ3

−δ3

6 δ01

δ01

10 δ02

δ02

5 δ03

−δ03

. (2.12)

Subject to δ1 + δ2 − δ3  1, δ1 + δ2 − 3δ3  0, δ2 − δ3  0, δ01 + δ02 − δ03  1, δ01 + δ02 − 3δ03  0, δ02 − δ03  0, δ1 , δ2 , δ3 , δ01 , δ02 , δ03 > 0.

(2.13)

From (2.13), δ1  1, δ2  21 , δ3  21 , δ01  1, δ02  21 , and δ03  21 . Putting the value of δ1 , δ2 , δ3 , δ01 , δ02 and δ03 in (2.12), the corresponding optimal dual value, i.e.,

2.5 Unconstrained Modified Signomial Function



10 v(δ)  1  60.

1

5 1/2

1/2

10 1/2

37

−1/2 1



 6 10 1/2 5 −1/2 1 1/2 1/2

So for primal decision variables, the following equations found:  10x11  δ1 v(δ)  1 × 7.75,  1 5x11 x12  δ2 v(δ)  × 7.75, 2  1 3 10x11 x12  δ3 v(δ)  × 7.75, 2  6x21  δ1 v(δ)  1 × 7.75,  1 10x21 x22  δ2 v(δ)  × 7.75, 2  1 3 5x22 x22  δ3 v(δ)  × 7.75. 2 From the above set of nonlinear equation, optimal solutions are x11  0.775, x12  1, x21  1.292, x12  0.300. Corresponding optimal value is Z (x)  7.071.

2.6 Constrained NLP Problem A constrained NLP problem is of the following form: Min g0 (x1 , x2 , . . . xn ) Subject to gi (x1 , x2 , . . . xn ) ≤ 1 x j > 0, j  1, 2, . . . , n. Example 2.6 Min 2x12 + 3x2−3 Subject to 5x12 − 2x1 x2 x1 , x2 > 0.

(2.14)

38

2 Signomial Geometric Programming (GP) Problem

Primal Signomial GP Problem Primal program: A primal signomial GP programming problem is of the form Minimize g0 (x1 , x2 , . . . , xm ) Subject to gk (x1 , x2 , . . . , xm ) ≤ ζk , k  1, 2, . . . , p x j > 0, j  1, 2, . . . , m. where gk (x) 

k i1

σi ci

m

(2.15)

a

j1

x j i j , and ζk  ±1.

Dual signomial GP problem: Dual GP problem of the given primal GP problem is Maximize ζ0

n  p  ci σi δi  i1

Subject to

k 

δi

ζ0 ζ λ λkk k

k1

σi δi  ζk λk , k  0, 1, . . . , p,

i1 k 

σi ai j δi  0,

j  1, 2, . . . , m,

i1

δi > 0, λ0  1.

(2.16)

Case I n ≥ m + 1, (i.e., DD > 0), the dual signomial program presents a system of linear equations for the dual variables. A solution vector exists for the dual signomial variables. Case II n < m + 1, (i.e., DD < 0), in this case, generally no solution vector exists for the dual signomial variables. But using least square (LS) or min-max (MM) method one can get an approximate solution for this system. Furthermore the primal–dual relation is ci

m 

∗ai j

xj

 ζ0 δi∗ v(δ ∗ , λ∗ ) for i  0,

j1

and ci

m  j1

∗ ai j

xj

m  j1

 δi∗ /λ∗k for i ∈ [k], i ≥ 1.

(2.17)

2.6 Constrained NLP Problem

39

Example 2.7 Min g(x, y, z)  2y 2 z 4 − 5x 2 Subject to 2x 2 y −2 − y −1 z ≤ −1 x, y, z > 0. Solution Assume ζ0  1, Corresponding dual of the above signomial problem is

Maximize v(δ) 

2 δ1

δ1

5 δ2

−δ2

2 δ01

δ01

1 δ02

−δ02

1 λ−λ 1 .

(2.18)

Subject to δ1 − δ2  1, − 2δ2 + 2δ01  0, 2δ1 − 2δ01 + δ02  0, 4δ1 − δ02  0 λ1  −δ02 + δ02 δ1 , δ2 , δ01 , δ02 > 0. From (2.19), δ1  − 21 , δ2  − 23 , δ01  ζ0  −1. Corresponding dual of the problem is

−3 2

(2.19)

and δ02  −2, but δi > 0, i.e.,

     −1 2 δ1 5 −δ2 2 δ01 1 −δ02 −λ1 λ1 . Maximize v(δ)  − δ1 δ2 δ01 δ02

(2.20)

Subject to δ1 − δ2  −1, − 2δ2 + 2δ01  0, 2δ1 − 2δ01 + δ02  0, 4δ1 − δ02  0, λ1  −δ01 + δ02 δ1 , δ2 , δ01 , δ02 > 0.

(2.21)

From (2.21), δ1  21 , δ2  23 , δ01  23 , δ02  2 and λ1  21 . Putting the value of δ1 , δ2 and δ01 in (2.20), the corresponding optimal dual value, i.e., v ∗ (δ)  −0.349. From primal–dual relation 2y 2 z 4  ζ0 δ1 v ∗ ,

40

2 Signomial Geometric Programming (GP) Problem

5x 2  ζ0 δ2 v ∗ , δ01 2x 2 y −2  , λ1 δ02 . y −1 z  λ1 Taking logarithm of each side yields equations, which are linear in the logarithms of the primal variables ln 2 + 2 ln y + 4 ln z  ln 0.175, ln 5 + 2 log x  ln 0.522, ln 2 + 2 log x  2 log y  ln 3, − ln y + ln z  ln 4. From the set of above equation, optimal solutions are x  0.323,

y  0.364, z  1.456.

2.7 Primal Modified Signomial GP Problem Primal program: A primal-modified signomial GP programming problem is of the form Minimize g0 (xl j ) Subject to gli (x) ≤ ζli , k  1, 2, . . . , p xl j > 0, j  1, 2, . . . , m. where gli (x) 

n k l1

i1

σli cli

m j1

a

xl jli j , and ζli  ±1.

Dual signomial GP problem: Dual GP problem of the given primal GP problem is Maximize ζ00

p n

p   cli σli δli  l1 i1

Subject to

k 

δli

ζ00 ζ λ λkk k

k1

σli δli  ζli λli , (l  0, 1, . . . , n).

i1 k  i1

σli ali j δli  0, (l  0, 1, . . . , n; j  1, 2, . . . , m).

(2.22)

2.7 Primal Modified Signomial GP Problem

δli > 0, λl0  1.

41

(2.23)

Case I nk ≥ nm + n, (i.e., DD > 0), the dual signomial program presents a system of linear equations for the dual variables. A solution vector exists for the dual signomial variables. Case II nk < nm + n, (i.e., DD < 0), in this case generally no solution vector exists for the dual signomial variables. But using least square (LS) or min-max (MM) method one can get an approximate solution for this system. Furthermore, the primal–dual relation cli

m 

∗ai j

xl j

  ζ0 δi∗ n v(δ ∗ , λ∗ ), for i  0,

j1

and cli

m 

∗ai j

xl j

 δli∗ /λli∗ for i ∈ [k], i ≥ 1.

j1

Example 2.8 3 3 x12 + 6x21 − 5x21 x22 Min Z(x)  10x11 − 10x11

Subject to x11 x12 + x21 x22 ≤ 1 x11 , x12 , x21 , x22 > 0. Solution Taking ζ00  1, Here, primal problem is 3 3 Min Z (x)  10x11 − 10x11 x12 + 6x21 − 5x21 x22 Subject to x11 x12 + x21 x22 ≤ 1

x11 , x12 , x21 , x22 > 0. and corresponding dual problem is      10 δ01 10 −δ02 6 δ11 5 −δ12 1 δ21 δ01 δ02 δ11 δ12 δ21

δ22 1 (δ21 + δ22 )(δ21 +δ22 ) δ22 Subject to δ01 − δ02  1,

v(δ) 

δ11 − δ12  1,

(2.24)

42

2 Signomial Geometric Programming (GP) Problem

δ01 − 3δ02 + δ21  0, − δ02 + δ21  0, δ11 − 3δ12 + δ22  0, − δ12 + δ22  0.

(2.25)

Approximate solutions of this system of linear equations are δ01  2, δ02  1, δ21  1, and δ11  2, δ12  1, δ22  1. Then, the objective function is

10 d(δ)  2  18.

2

10 1

−1 2 −1 1 1 6 5 1 1 (2)2 2 1 1 1 (2.26)

And from primal–dual relation following system of equations gives optimal primal variables √ 10x11  2. 18  8.49, √ 3 10x11 x12  1. 18  4.25, √ 6x21  2. 18  8.49, √ 3 5x21 x22  1. 18  4.25, 1 x11 x12  , 2 1 x21 x22  . 2 Solving the system of equations, optimal solutions are x11  0.849, x12  0.589, x21  1.415, x22  0.353, and the value of objective function is Z (x)  8.38.

2.8 Application

43

2.8 Application Problem 2.1 (Unconstrained SGP) The demand (d) of an item is uniform at the rate of 10 units per month. The setup cost (c0 ) of a production run is Rs. 20, and the inventory holding cost (ch ) is Rs. 50 per item per month. If the shortages cost (cs ) is Rs. 50 per item per month, determine economic lot size (q) for one run also determine what is inventory level (s) at the beginning of each month. As stated, this problem can be formulated as the unconstrained SGP problem  2 2 minimize 10.20 + 50s + 50(q−s) q 2q 2q where q > 0, s > 0. It is unconstrained signomial geometric programming problem. The optimal solution is q  4, s  2 and minimum average total cost is Rs. 100/month. Problem 2.2 (Unconstrained MSGP) In a multi-item (n items) inventory model, demand of an item is uniform at the rate of d i units per month. The setup cost of a production run is Rs. c0i , and the inventory holding cost is Rs. chi per item per month. If the shortages cost is Rs. csi per item per month, determine economic lot size (qi ) for one run also determine what is inventory level (si ) at the beginning of each month. As stated, this problem can be formulated as the unconstrained MSGP problem ⎧ n ⎨ minimize g(q, s)   di c0i + chi s 2 + csi (q−s)2 qi 2qi 2qi i1 ⎩ where qi > 0, si > 0. In particular, here we assume n  2 and input data of this problem is given below

i

di

c0i

1

10

20

50

50

2

15

10

125

25

chi

csi

and the output value is

i

qi

si

g(q, s)

1

4.00

2.00

177.45

2

3.87

1.94

44

2 Signomial Geometric Programming (GP) Problem

Problem 2.3 (Constrained SGP) A constrained signomial geometric programming (SGP) problem is of the following form: ⎧ −1 −1 −1 0.5 ⎨ minimize −2x1 x2 x34 x4 + x2 x3 + 5x1 x4 1/3 0.5 subject to −x4 + x2 x3 ≤ −1 ⎩ where x1 , x2 , x3 , x4 > 0. It is constrained signomial geometric programming problem. The optimal solution is x1  0.408, x2  0.004, x3  18.570, x4  16, and minimum value is 38.322. Problem 2.4 (Constrained SMGP) A constrained modified signomial geometric programming (MSGP) problem is of the following form: ⎧ n  ⎪ ⎪ minimize g(x)  ai x1i x2i x3i4 x4i−1 + bi x2i−1 x3i−1 + ci x1i0.5 x4i ⎪ ⎪ ⎨ i1 n  1/3 subject to −x4i0.5 + x2i x3i ≤ w ⎪ ⎪ ⎪ i1 ⎪ ⎩ where x1 , x2 , x3 , x4 > 0. It is constrained MSGP problem. In particular, here we assume n  2 and input data of this problem is given below

i

ai

bi

ci

w

1

−2

1

5

−2

2

−1

1

3

and the output value is

i

x1i

x2i

x3i

x4i

g(x)

1

0.408

0.004

18.57

16.00

69.09

2

0.716

0.006

7.663

16.00

2.9 Conclusion In the previous work, in geometric programming (GP) was, for the most part, concerned with minimizing posynomial functions subject to inequality constraints on such functions which was called posynomial geometric programming. In the present

2.9 Conclusion

45

time, because a number of models abstracted from application fields were not posynomial geometric programming, the theory had to be generalized to a much broader class of optimization problems called generalized geometric programming or signomial geometric programming (SGP) problem. A SGP has wide variety of application since its initial development in real life problem. It has great impact in the area of (1) Operations Research (2) Engineering design (3) Manufacturing (4) Chemical Equilibrium (5) Economic and Statistics, etc.

References M. Avriel, A.C. Williams, An extension of geometric programming with applications in engineering optimization. J. Eng. Math. 5(2), 187–194 (1971) R.J. Duffin, E.L. Peterson, Duality theory for geometric programming. SIAM J. Appl. Math. 14, 1307–1349 (1966) R.J. Duffin, E.L. Peterson, Geometric programming with signomials. J. Optim. Theory Appl. 11, 3–35 (1973) R. Horst, H. Tuy, Global Optimization: Deterministic Approaches, 2nd edn. (Springer, Berlin, 1993) R. Horst, P.M. Pardalos, N.V. Thoai, Introduction to Global Optimization (Kluwer Academic Publishers, Dordrecht, 2000) T.R. Jefferson, C.H. Scott, Generalized geometric programming applied to problems of optimal control. I. Theory. J. Optim. Theory Appl. 26(1), 117–129 (1978) C.D. Maranas, C.A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations, J. Global Optim. 7, 143–182 (1995) U. Passy, D.J. Wilde, Generalized polynomial optimization. SIAM J. Appl. Math. 15, 1344–1356 (1967) S.-J. Qu, K.-C. Zhang, Y. Ji, A new global optimization algorithm for signomial geometric programming via Lagrangian relaxation. Appl. Math. Comput. 184(2), 886–894 (2007) M.J. Rijckaert, X.M. Martens, Comparison of generalized geometric programming algorithms. J. Optim. Theory Appl. 26, 205–241 (1978) H.D. Sherali, Global optimization of nonconvex polynomial programming problems having rational exponents. J. Global Optim. 12(3), 267–283 (1998) H.D. Sherali, C.H. Tuncbilek, A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Global Optim. 2(1), 101–112 (1992)

Chapter 3

Introduction to Fuzzy Set Theory

3.1 Overview This present reality is mind boggling; complexity in the planet by and large emerges from vulnerability as vagueness. Because of the element of multifaceted nature and equivocalness, the people have been looked by social, technical, and economic problems. Why at that point are PCs, which have been outlined by people all things considered, not equipped for tending to and vague issues? By what method would humans be able to reason about genuine systems, when the total depiction of a real system regularly requires more point-by-point information than a human would ever want to perceive at the same time and acclimatize with comprehension? The answer is that people have the ability to reason roughly, a capacity that PCs as of now do not have. In thinking about a complex system, people reason approximately about its behavior, in this manner keeping up just a generic understanding about the problem. The fuzzy set theory is produced to enhance the oversimplified model, in this way, building up a robust and flexible model keeping in mind the end goal to tackle certifiable complex systems including human angles. Moreover, it helps the decisionmaker not exclusively to think about the current options under given constraints (optimizing a given system) yet in addition to grow new alternatives (design a system). Fuzzy set theory has picked up its significance as a tool for uncertain modeling. The fuzzy set theory has been connected in numerous fields, for example, Economics, Operation Research, Management Science, Control Theory, Game Theory, Expert System, Reliability Analysis, Artificial Intelligence, Medical Diagnosis, Human Behavior, and so on. In this study, we will focus on fuzzy mathematical programming and fuzzy multi-objective decision-making problem. To do as such, we will initially present the required learning of the fuzzy set theory and fuzzy optimization.

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_3

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3 Introduction to Fuzzy Set Theory

3.2 Uncertainty and Imprecision Because of the incomplete learning and data, precise mathematics is not adequate to display a complex system. Historically, the probability theory is the way to avoid dealing with handle this inadequacy/ uncertainty in numerical models. Along these lines, all uncertainty was accepted to take after the attributes of irregular uncertainty. An arbitrary procedure is one where the results of a specific acknowledgment of the procedure are entirely a matter of possibility; an expectation of a sequence of occasions is not possible. What is possible for arbitrary process is an exact depiction of the statistic of the long-run averages of the procedure. One of the essentials in the probability theory is the law of excluded middle ((P(A∪Ac )  1) and contradiction (P(A∩Ac )  0). For example, a fruit is either an apple or not an apple, a toss is either head or tail. For this case, the probability is surely a decent way to deal with or speak to some learning or data whose boundaries can be obviously defined. Tossing coins into the air, you can figure that either heads or tails will be up. Rotating a dice, the outcome will be 1, 2, 3, 4, 5, or 6 however never be 2.5 or 4.2. Obviously, there are considerable measures of problems fulfilling the laws of the excluded middle and contradiction but, intuitively and commonsensical, this is not valid in different problems. A confirmation supporting a specific hypothesis to some degree does not simultaneously disconfirm it to any degree, since it may not give any help despite what might be expected. For example, a student may very good, not good, or good. A rose may be red color, not red, or reddish. Therefore, it is hard to characterize the sets of good students and red roses with sharp/crisp environment. The majority of our traditional tools for forma demonstrating, thinking, and processing are crisp, deterministic, and exact in character. By crisp, we mean dichotomous, i.e., truly or no type instead of more or less type. In customary dual logic, for example, a statement can be true or false and nothing in the middle. In set theory, a component can either have a place with the set or not and in optimization an answer is either feasible or not. To clearly distinguish the fuzzy sets from traditional (crisp) sets, let us initially think about the following examples: Example 3.1 In the traditional set theory, a component may either “belong to” set A or “not belong to” the set A in the given planet. For instance, assume there is a fairly extensive target and shooters dependably hit inside the target (see Fig. 3.1). A circle is situated in the center of the target. In the event that a shooter hit inside the circle, region A, he is given the title “great shooter”. Else, he is known as a “poor shooter.” In Fig. 3.1, a1 and a3 are absolutely great shooters. On the other hand, a4 and a5 are not totally great shooters or totally poor shooters. By giving a numerical measure which is assumed linearly proportional to the distance, d, of every shooter from the area A, one can state that the shooter a4 has 0.8 degree of membership in the set of good shooters however a5 has 0.2 degree of membership in the set of poor shooter, μ(a4 )  0.8 and μ(a5 )  0.2. Obviously, the numerical measure μ, can be any number. A normalized measure which is in [0, 1] is always adjusted.

3.2 Uncertainty and Imprecision

Fig. 3.1 Graphical representation for the fuzzy set of GOOD SHOOTER

49

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3 Introduction to Fuzzy Set Theory

Fig. 3.2 The membership (possibility) function of the interest rate

Example 3.2 Consider the announcement of “interest rate will be around 8–8.2%.” The most considerable interest rate is between 8 and 8.2%. The least considerable interest rate is 6.5% (optimistic value) and 10% (pessimistic value). It is not likely that the interest rate will be under than 6.5% or bigger than 10% in the opinion of the decision-maker. In this way, we give 8–8.2% interest rate a 1 possibility, 6.5% or less and 10% or more a 0 possibility. Between them, let us accept the grades shown in Fig. 3.2. Here, we did not know performance concept to grade different interest rates, yet utilized the possibility concept: most possible, minimum possible, or in between. Along these lines, we call this membership function as a possibility function or as a possibility distribution.

3.3 Basic Fuzzy Set Theory Most of the time the investigation and demonstrating of a real-world system take an intrinsic uncertainty which is not because of randomness and wherein some imprecision exist whose treatment cannot be performed utilizing probability theory. This imprecision might be vagueness (i.e., absence of limits of the set of objects), ambiguity (i.e., the relationship of a given object with various elective implications), or generality (i.e., the use of the symbols meaning to a variety of the objects). As per dictionaries and furthermore use in ordinary dialect, the words fuzzy, uncertain, vague, and imprecise are firmly related as far as their implications. The idea of “fuzziness” was introduced in 1965 as “fuzzy set theory” by Prof. L. A. Zadeh, a Professor of electrical engineering and computer science, University of California, USA. Seising (1970) presented some basic idea of fuzzy set theory and its initial applications—developments up to the 1970s. Brown (1971) presented a

3.3 Basic Fuzzy Set Theory

51

note on fuzzy sets. Zadeh (1975) demonstrated the idea of a linguistic variable and its application to approximate reasoning. Bezdek (1978) defined fuzzy partitions and relations an axiomatic basis for clustering. Klir and Yuan (1995) have given the fuzzy sets and fuzzy logic in their book. Yao (1998) presented a comparative study of fuzzy sets theory and rough sets theory. Zimmermann (2001) discussed on fuzzy sets theory broadly in his book “Fuzzy set theory—and its applications”. Deschrijver and Kerre (2003) worked on the relationship between some extensions of fuzzy set theory. A fuzzy set is a class of objects in which there are no sharp limits between those objects that belong with the class and those that do not. Here, some valuable definitions are: • Crisp Set: Crisp set X is a set of all objects with same characteristic(s). This X is known as universal set. Here, elements of the universe can be discrete or continuous and limited or unlimited. If a set “A” makes with collections of some elements of universal set X; and is a subset of the universal set X, it might be described as far as its elements x in X as μ A (x)  1 if x ∈ A  0 if x ∈ / A, where μA (x) is known as the valuation function (or characteristic function) that accept values from a valuation space comprising of just two points 0 and 1. The limitation of the characteristic function to 0 and 1 restrains the relevance of this crisp set theory. In classical or crisp sets, the transition between membership and nonmembership in a given set for an element in the universe is abrupt and well defined. But there are in many practical situations, say “set of tall men,” “class of all intelligent students” in which this transition is not abrupt and well defined. Here, the membership value of an object in a class cannot be represented satisfying by 0 and 1 only. These situations generate fuzzy sets. In classical or crisp set theory, the progress among membership and nonmembership in a given set for an object in the universe is abrupt and well defined. In any case, there are in numerous reasonable circumstances, say “set of fat men,” “class of all poor students” in which this progress is not abrupt and all around characterized. Here, the membership value of an element in a class cannot be represented to fulfilling by 0 and 1 only. These circumstances produce fuzzy sets theory. • Fuzzy Set: If the transmission between membership and nonmembership for a component in a given set is gradual, at that point the said set is called a fuzzy set. This transmission among different degrees of membership can be thought of as adjusting to the way that the limits of the fuzzy sets are ambiguous or vague. Let X be a collection of objects or elements called the universe of discourse. A  is a subset of X is a set of ordered pairs fuzzy set denoted by A

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3 Introduction to Fuzzy Set Theory

 A



  x, μ A˜ (x) :x ∈ X ,

 and the function μ A˜ : where μ A˜ (x) is termed as the grade of membership of x in A, X → M is called the membership function from X to the membership space M. It is assumed that M is the closed interval [0, 1]. A notation for representing a fuzzy set  when the universe of discourse, X, is discrete and finite, is as follows: A   μ A˜ (x1 )/x1 + μ A˜ (x2 )/x2 + · · · + μ A˜ (xn )/xn A 

n 

μ A˜ (xi )/xi

(3.1)

i1

 is denoted by When the universe, X, is continuous and infinite, the fuzzy set A    μ A˜ (x)/x A (3.2) Here, the symbol “/” is not a divisional sign but indicates delimiter. In both (3.1) and (3.2), the top number μ A˜ (x) is the membership value of the element x in the bottom. In 3.1, the “+” symbol is not for algebraic summation. It represents the fuzzy union and in 3.2, the sign “∫” is not an integral. It is a set union notation for continuous variable.  is also represented by a vector (called fuzzy vector or fit) with components of A membership functions as

  μ A˜ (x1 ), μ A˜ (x2 ), . . . , μ A˜ (xn ) . A “fit” contracts fuzzy units as a fit value which is a number between 0 and 1. Note 3.1 Classical sets (i.e., Cantor sets) can be completely described by characteristic functions, and fuzzy sets can also be described by membership functions. If the range of μ A˜ admits only two values 0 and 1, then μ A˜ degenerates to a classical sets characteristic function. • Universal fuzzy set: The fuzzy set where all the elements of the universe of discourse X have the grade of membership equal to 1 is called the universal set and denoted by U, i.e., U  {x ∈ X :μU (x)  1, ∀x ∈ X }. See Fig. 3.3. Note 3.2 Classical set may be taken as a particular case of fuzzy set. Example 3.3 Consider the set S  {Karpov, Kasparov, Anand, Kamaski, Fisher, Korchonoy} of famous chess players. Among these players, the set of “intelligent players” defined below is a fuzzy set.

3.3 Basic Fuzzy Set Theory

53

  {(Karpov, 0.85), (Kasparov, 0.91), (Anand, 0.78), (Kamaski, 0.69), (Fisher, A 0.74), (Korchonoy, 0.65)}.   0.85/Karpov + 0.91/Kasparov + 0.78/Anand + 0.69/Kamaski + 0.74/Fisher Or, A + 0.65/Korchonoy.   “real numbers close to 7” Example 3.4 A   0.1/1 + 0.4/3 + 0.6/5 + 1/7 + 0.8/8 + 0.5/10. A   “70 year age is old” Example 3.5 A  A  0.1/20 + 0.6/50 + 0.9/65 + 1/70 + 0.8/80 + 0.5/100   “real numbers larger than 5” Example 3.6 A  A  {(x, μA (x)): x  X}, where  μ A (x) 

0 1

1+| x−5 a |

2b

x ≤5 x >5

 on X is empty, denoted by ϕ, if and only if • Empty fuzzy set: A fuzzy set A μ A˜ (x)  0 for all x ∈ X. So   φ  x:μ A˜ (x)  0, ∀x ∈ X See Fig. 3.4.  on X, denoted by S( A)  or supp( A),  • Support: The support of a fuzzy set A   is   the set of points in  X at which μ A˜ (x) is positive, i.e., S( A) or supp A  x ∈ X :μ A˜ (x)> 0 .  on X, denoted by crossover( A),  is the set • Crossover: The crossover of a fuzzy set A       x ∈ X :μ A˜ (x)  0.5 . of points in X at which μ A˜ (x)  0.5, i.e., crossover A  on X, denoted by hgt( A),  is the least upper • Height: The height of a fuzzy set A bound of

Fig. 3.3 Universal fuzzy set

54

3 Introduction to Fuzzy Set Theory

  sup μ A˜ (x). μ A˜ (x), i.e., hgt( A) x∈X

 is said to be normal if there exists at least one x ∈ X attaining • Normal: A fuzzy set A   1) otherwise it is subnormal. the maximum membership grade 1 (i.e., hgt( A)  max μ A˜ (x)  1. For normalized fuzzy set A, x∈X

 be a non-normalized (subnormal) fuzzy set. So Note 3.3 Let a fuzzy set A  means to normalize its memmax μ A˜ (x) < 1. To make non-normalized fuzzy set A bership function μ A˜ (x), i.e., to divide it by max μ A˜ (x) μ (x) i.e., max A˜μ ˜ (x) . A

 be a fuzzy set in X. Then A  is convex if and only if, for Convex fuzzy sets: Let A  satisfies the inequality. any x 1 , x 2 ∈ X, the membership function of A See Figs. 3.5 and 3.6.   μ A˜ (λx1 + (1 − λ)x2 ) ≥ Min μ A˜ (x1 ), μ A˜ (x2 ) for 0 ≤ λ ≤ 1.  is a non-convex Otherwise, for some membership function μ A˜ (x) of fuzzy set, A set.

Fig. 3.4 Empty fuzzy set

Fig. 3.5 Convex fuzzy set

3.3 Basic Fuzzy Set Theory

55

Fig. 3.6 Non-convex fuzzy set

Fig. 3.7 Normalised fuzzy set

Fig. 3.8 Non-normalised fuzzy set

See Figs. 3.7 and 3.8.  on X, denoted by core( A),  Core: The core of a fuzzy set A   is the set of points in ˜ X at which μ A˜ (x)  1, i.e., core ( A)  x ∈ X :μ A˜ (x)  1 .  on X, denoted by boundary ( A),  • Boundary: The boundary of a fuzzy set A ˜ is the set of points in X at which 0 < μ A˜ (x) < 1, i.e., boundary A    x ∈ X : 0 < μ A˜ (x) < 1 . • α—level set or α-cut of a fuzzy set: The α—level set (or interval of confidence  of X is a crisp set Aα that contains all the at level α or α-cut) of the fuzzy set A  greater than or equal to α, i.e., elements of X that have membership values in A

56

3 Introduction to Fuzzy Set Theory

Fig. 3.9 Standard fuzzy set

See Fig. 3.9.   Aα  x:μ A˜ (x) ≥ α, x ∈ X, α ∈ [0, 1] . Example 3.7 From the example-2, we have A1  {70} A0.8  {65, 70} A0.5  {50, 65, 70, 80, 100} A0.1  {20, 50, 65, 70, 80, 100} And A∗0.9 A∗0.8 A∗0.5 A∗0.1

 {70}  {65, 70, 80}  {50, 65, 70, 80}  {50, 65, 70, 80, 100} Example 3.8

x

μA (x)

a

0.3

b

0.6

c

1.0

Then the α- cut of A at α  0.6 is A0.6  {b, c}. And strong α- cut of A at α  0.6 is A∗0.6  {c}. Note 3.4 α—level set Aα can be defined by the characteristic function χ Aα (x)  1 if μ A˜ (x) ≥ α  0 if μ A˜ (x) < α.

3.3 Basic Fuzzy Set Theory

57

Fig. 3.10 α-cut

  If Aα¯  x:μ A˜ (x) > α, x ∈ X, α ∈ [0, 1] then Aα¯ is called a strong α-level set. • Proofs by using α-cuts From the definition of the α-level and strong α-level sets, the following basic properties hold: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

( A ∪ B)α  Aα ∪ Bα ( A ∩ B)α  Aα ∩ Bα ( A ∪ B)α¯  Aα¯ ∪ Bα¯ ( A ∩ B)α¯  Aα¯ ∩ Bα¯ Aα¯ ⊂ Aα If α 1 ≤ α 2 then Aα1 ⊇ Aα2 and Aα¯ 1 ⊃ Aα2 ⊃ Aα¯ 2 ( Ac )α  Ac1−α ( Ac )α¯  Ac1−α A0  X, A1  φ Amax(α1 ,α2 )  Aα1 ∩ Aα2 Amin(α ,α )  Aα1 ∪ Aα2 . 1

2

See Fig. 3.10. ... ˜ A... ˜ ˜ Note 3.5 A1  core( A), 0  supp( A), A 0 \A1  boundary( A). Proof of (i)     ( A ∪ B)α  x ∈ X :max μ A˜ (x), μ B˜ (x) ≥ α    x ∈ X :μ A˜ (x) ≥ α or μ B˜ (x) ≥ α      x ∈ X :μ A˜ (x) ≥ α ∪ x ∈ X :μ B˜ (x) ≥ α  Aα ∪ Bα  Cardinality:     Cardinality (or scalar cardinality) of a discrete fuzzy set A, denoted  is the summation of the membership grades of all elements of x in  or c A by  A  i.e., A,

58

3 Introduction to Fuzzy Set Theory

     ˜ μ A˜ (x) i.e. μ A˜ (x).  A  ˜ x∈S( A)

x∈X

 is calculated by using the number The cardinal number of a continuous fuzzy set A i.e.,      ˜ μ A˜ (x) i.e. μ A˜ (x).  A  ˜ x∈S( A)

x∈X

as

   denoted by  A  , is defined The relative cardinality of a discrete finite fuzzy set A, rel   A 

rel



   ˜  A |x|

   is a continuous fuzzy set  A   When A rel

.

 x∈X

μ A˜ (x)dx  . dx

x∈X

3.4 Operations on Fuzzy Sets In the following are given some important definitions on fuzzy set properties. Con and B˜ in the universe of discourse X. sider two fuzzy sets A and B˜ are said to be equal if and only if μ A˜ (x)  μ B˜ (x) Equality: Two fuzzy sets A   B. ˜ for each x ∈ X. It is denoted by A Degree of Equality  and  Previously, we define two fuzzy sets A B are equal if and only if μ A˜ (x)  μ B˜ (x)    B. This definition for each x ∈ X. Hence if μ A˜ (x)  μ B˜ (x) for some x ∈ X, then A is true for crisp problem. To check the degree of equality of two fuzzy sets, we can use the similarity measure (or degree of equality)    ˜  A ∩ B   . E( A, B)   A ∪  B A| 1 ˜     A)   | A∩ Note 3.6 (i) E( A, B)  1 if and only if A B, E( A,  A |  1  1. | A∪

  ˜   and  ∩  B do not overlap at all, E A, B  0, i.e., (ii)  A B   0 if and only if A  and  A B have disjoint support. ˜  (iii) In general case 0 ≤ E( A, B) ≤ 1.    (iv) E( A, B)  E(  B, A).  

3.4 Operations on Fuzzy Sets

59

Fig. 3.11 Complement of fuzzy set A

 is said to be containment of  Containment: A fuzzy set A B if and only if μ A˜ (x) ≤ μ B˜ (x) for each x ∈ X. Degree of containment (or inclusion) ⊆  ⊆  Previously, we defined A B if and only if μ A˜ (x) ≤ μ B˜ (x) for each x ∈ X. If A B     and A  B then A is a proper subset of B. Again, this definition of “containment”  is the subset of  is true for crisp problem. To check the degree that A B, we can use ˜   the subsethood measure I( A, B) (or degree of A ⊆  B or degree of inclusion).   A ∩  B ˜  I ( A, B)    . A  Note 3.7 ˜  I ( A, B)  0 if and only if μ A˜ (x)μ B˜ (x)  0 ⊆  B  1 if and only if μ A˜ (x) ≤ μ B˜ (x), i.e. A ˜ A) ˜ ≤ I (C, ˜ C) ⊆     ≤ I (  for any fuzzy set C˜ and If A B then I (C, B), I ( A, B, C) ˜  0 ≤ I ( A, B) ≤ 1.  c of a fuzzy set A Complement: The membership function of the complement A of X is defined as μ A˜ c (x)  1 − μ A˜ (x) for each x ∈ X. c  A



  x,μ A˜ c (x) : x ∈ X,μ A˜ c (x)  1 − μ A˜ (x)

See Fig.  3.11.  c (1 − μ A˜ (x))/x for any continuous fuzzy set A. or A x∈X

Note 3.8 The membership function μ A˜ c (x) is symmetrical to μ A˜ (x) with respect to the line μ  0.5.

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3 Introduction to Fuzzy Set Theory

 and  Fig. 3.12 Union of A B

 and B˜ is a fuzzy set in X, denoted by A ∪  Union: The union of A  B, whose membership function is μ A∪ ˜ B˜ (x)  μ A˜ (x) ∨ μ B˜ (x)  max μ A˜ (x), μ B˜ (x) for each x ∈ X.      ∪  So A B  x,μ A∪ ˜ B˜ (x) :μ A∪ ˜ B˜ (x)  max μ A˜ (x), μ B˜ (x) , ∀x ∈ X See Fig. 3.12.  ∪  max(μ A˜ (x),μ B˜ (x))/x for any continuous fuzzy set A. or A B x∈X

μ A˜ (x)

μ B˜ (x),

μC˜ (x),

0

0

0

1

0

1

0

1

1

1

1

1

 and  ∩  Intersection: The intersection of A B is a fuzzy set in X, denoted by A B,  whose membership function is μ A∩ ˜ B˜ (x)  μ A˜ (x) ∧ μ B˜ (x)  min μ A˜ (x), μ B˜ (x) for each x ∈ X.

∩  Fig. 3.13 A B

3.4 Operations on Fuzzy Sets

61

So ∪  A B 

∩  or A B



    x, μ A∩ ˜  ˜ B˜ (x)  min μ A˜ (x), μ B˜ (x) , ∀x ∈ X B (x) :μ A∩

 min(μ A˜ (x), μ B˜ (x))/x for any continuous fuzzy set A.

x∈X

See Fig. 3.13. i.e.,

μ A˜ (x)

μ B˜ (x)

μC˜ (x)

0

0

0

1

0

0

0

1

0

1

1

1

 and  ∩  Disjoint Sets: Two fuzzy sets A B are disjoint if A B  ϕ. If there are very ∩  ∩  few elements in A B, then in crisp problem A B  ϕ, but for fuzzy set we say  and  that A B are overlapped with some degree. To check this degree of overlap of two fuzzy sets, we can use the following formula:   A ∩  B ˜  Degree of overlap, i.e. O( A, B)     .  ˜  ˜   A B  ˜  ∩  Note: (i) O( A, B)  1 if and only if A B  φ. ˜   and  (ii) O( A, B)  0 if and only if A B have disjoint support. ˜  ˜ (commutative). (iii) O( A, B)  O(  B, A) ˜  Where |x| is the cardinality of the universe of discourse. Here and 0 ≤ O( A, B) ≤   1 when A overlapped B with some degree.  with respect Difference: The operation difference, i.e., relative complement of A    to B, denoted as B\ A, is defined as 

      x, μ B\ B\ A ˜ A˜ (x) : μ B\ ˜ A˜ (x)  μ B∩ ˜ A˜ c (x)  min μ B˜ (x), 1 − μ A˜ (x) , ∀x ∈ X  or  B\ A



 and  min(μ B˜ (x), 1 − μ A˜ (x))/x for continuous fuzzy sets A B.

x∈X

See Fig. 3.14. Distance between two fuzzy sets   Let X be a discrete universe and A, B are two fuzzy sets on X. Then  and  (i) (a) the linear (Hamming) distance between two fuzzy sets A B is denoted by

62

3 Introduction to Fuzzy Set Theory

 and  Fig. 3.14 Difference of fuzzy sets A B

  H ( A, B) and defined as ˜  H ( A, B) 

n    μ ˜ (xi ) − μ ˜ (xi ) A

B

i1

and  and  (b) the normalized linear (Hamming) distance between two fuzzy sets A B is defined as   Hn ( A, B) 

n  1   μ A˜ (xi ) − μ B˜ (xi ). n i1

 and  (ii) (a) The quadratic (Euclidean) distance between two fuzzy sets A B is denoted   by D( A, B) and defined as   n   2 ˜  D( A, B)   μ A˜ (xi ) − μ B˜ (xi ) i1

and  and  (b) the normalized quadratic (Euclidean) distance between two fuzzy sets A B is defined as   n 1  2 ˜  μ A˜ (xi ) − μ B˜ (xi ) . Dn ( A, B)   n i1 Note: General definition of L p distance (Minkowski distance) between two fuzzy  and  sets A B is defined as

3.4 Operations on Fuzzy Sets

63

  n 1  p p ˜  p μ A˜ (xi ) − μ B˜ (xi ) L ( A, B)   for discrete universe x, n i1 ⎡



⎣

⎤ 1p p  μ ˜ (x) − μ ˜ (x) dx ⎦ for continuous universe x, A B

x

where 1 ≤ p 1 ˜  A˜ p .. i.e., CON( A) Concentration results in reduction of the values of the membership function μ A˜ (x) as follows: For small values of μ A˜ (x) (i.e., 0 < μ A˜ (x) < 0.5), the reduction is relatively large and for large values of μ A˜ (x) (i.e., 0.5 < μ A˜ (x) < 1), the reduction is relatively small.

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3 Introduction to Fuzzy Set Theory

Dilation Dilation stretch or dilate a fuzzy set by increasing the membership of elements. For  is defined by example, the operator dilation of a fuzzy set A     ˜  A˜ .5  x, μ A˜ (x) .5 . DIL( A) As concentration, we may generalize that dilation operation by using a power r(0 < r < 1)    

˜  A˜ r  x, μ A˜ (x) r : x ∈ A, ˜ 0 0, ka, klμ , , krμ , kωa , (ka, klv , krv , kva ), ka, −klμ , −krμ , kωa , (ka, −klv , −krv , kva ), for k < 0.

(ii) If a˜ I  (a1 , a2 , γμ , τμ , ωa ), (a1 , a2 , γv , , τv , va ) be GTrIFN, then K a˜ I 

  for k > 0, ka1 , ka2 , kγμ , kτμ , kωa , (ka1 , ka2 , kγv , kτv , kva ), (ka1 , ka2 , −kγμ , −kτμ , kωa ), (ka1 , ka2 , −kγv , −kτv , kva ), for k < 0.

Definition 4.24 A (α, β)-cut set of GTIFN a˜ I  (a, lμ , rμ , ωa ), (a, lv , rv , va ) is defined as I a˜ α,β  {x: ωa˜ (x) ≥ α, va˜ (x) ≤ β},

where 0 ≤ α ≤ ωa , va ≤ β ≤ 1. A α-cut set of GTIFN a˜ I is a crisp subset of R, which is defined as 

 lμ α  rv α . aˆ α  [a L (α), a R (α)]  a − lμ + , (a + rv ) − ωa ωa

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4 Fuzzy Numbers and Fuzzy Optimization

And β-cut of GTIFN a˜ I is a crisp subset of R, which is defined as 

(1 − β)lv (1 − β)rv . aˆ β  [a L (β), a R (β)]  (a − lv ) + , (a + rv ) − 1 − va 1 − va Definition 4.25 The (α, β)-cut set of (a1 , a2 , γμ , τμ , ωa ), (a1 , a2 , γv , τv , va ) is defined as

GTrIFN

a˜ I



I a˜ α,β  {x: ωa˜ (x) ≥ α, va˜ (x) ≤ β},

where 0 ≤ α ≤ ωa, va ≤ β ≤ 1. A α-cut set of GTrIFN a˜ I is a crisp subset of R, which is defined as 

 γμ α   τμ α  . aˆ α  [a L (α), a R (α)]  a1 − γμ + , a2 + τμ − ωa ωa And β-cut of GTrIFN a˜ I is a crisp subset of R, which is defined as

 (1 − β)γv (1 − β)τv aˆ β  [a L (β), a R (β)]  (a1 − γv ) + , (a2 + τv ) − . 1 − va 1 − va Note 4.8 Let a˜ I be any GTIFN or GTrIFN. For any α ∈ [0, ωa ] and β ∈ [va , 1], where 0 ≤ α + β ≤ 1 the following equality is valid: I a˜ α,β  aˆ α ∩ aˆ β .

According Note 4.8 and definition of the intersection between aˆ α and aˆ β , we have the following result: a˜ α,β  [a L , a R ], where a L  max{a L (α), a L (β)}, and a R  min{a R (α), a R (β)}. Note 4.9 For each a > 0, the exponential function f (x)  a x is continuous. Note 4.10 Multiplication of two continuous functions is continuous. Definition 4.26 Let A  [a L , a R ] and B  [b L , b R ] be two closed intervals, and then order relation between two closed intervals be

4.12 Intuitionistic Fuzzy Numbers

121

A ≤ B, iff a L ≤ b L and a R ≤ b R . Definition 4.27 (Interval-valued function) Let a > 0, b > 0 and consider the interval [a, b]. From a mathematical point of view, any real number can be represented on a line. Similarly, we can represent an interval by a function. If the interval is of the form [a, b], the interval-valued function is taken as h(ρ)  a (1−ρ) bρ for ρ ∈ [0, 1]. The choice of the parameter ρ reflects some attitude on the part of the decisionmaker. Lemma 4.14 For given [a, b], a > 0, b > 0, then h(ρ)  a 1−ρ bρ for ρ ∈ [0, 1] is a strictly monotone increasing continuous function. Proof according to Note 4.8 and Note 4.9, h(ρ) is continuous. Since 0 ≤ ρ ≤ 1, then d(h(ρ)) 1  ρ(1 − ρ) ρ (1−ρ) ≥ 0, dρ a b then h(ρ) is monotone increasing and the proof is complete. Lemma 4.15 Let A  [a L , a R ] and B  [b L , b R ] are two closed intervals, if A ≤ B then for ρ ∈ [0, 1] h A (ρ) ≤ h B (ρ). Proof From Definition 4.8, a L ≤ b L and a R ≤ b R , since ρ ∈ [0, 1], we obtain two following inequalities: a L(1−ρ) ≤ b(1−ρ) , and L a ρR ≤ bρR . ρ

ρ

b R , hence h A (ρ) ≤ h B (ρ). Then we have a L(1−ρ) a R ≤ b(1−ρ) L

4.13 Neutrosophic Set Definition 4.28 (Neutrosophic set) Let X be a space of points and x ∈ X . A neutrosophic set (NS) A˜ N in X having the form A˜ N  {xμ A (x), ν A (x), σ A (x)|x ∈ X }, where μ A (x), ν A (x) , and σ A (x) denote the truth membership degree, falsity membership degree, and indeterminacy membership degree of x, and they are real standard or nonstandard subsets of ]0− , 1+ [, i.e.,

122

4 Fuzzy Numbers and Fuzzy Optimization

μ A (x): X →]0− , 1+ [ ν A (x): X →]0− , 1+ [ and σ A (x): X →]0− , 1+ [. There is no restriction on the sum of μ A (x), ν A (x) and σ A (x). So, 0− ≤ sup μ A (x) + sup ν A (x) + sup σ A (x) ≤ 3+ . From the philosophical point of view, the NS takes the value from the real standard or nonstandard subsets of ]0− , 1+ [. But in real-life application in scientific and engineering problems, it is difficult to use NS with value from the subsets of ]0− , 1+ [. Ye (2013) reduced NSs of nonstandards intervals into a kind of simplified neutrosophic sets of standard intervals that will preserve the operations of NSs. Definition 4.29 (Single-valued neutrosophic set) Let X be a space of points with a generic element x in X. A single-valued neutrosophic set (SVNS) A˜ N in X is characterized by μ A (x), ν A (x) and σ A (x), and having the form A˜ N  {xμ A (x), ν A (x), σ A (x)|x ∈ X } where μ A (x): X → [0, 1] ν A (x): X → [0, 1] and σ A (x): X → [0, 1] with 0 ≤ μ A (x) + ν A (x) + σ A (x) ≤ 3 for all x ∈ X.

4.14 Hesitant Fuzzy Set A HFS is defined in terms of a function that returns a set of membership values for each element in the domain. Definition 4.30 Let X be a reference set, a HFS on X is a function h that returns a subset of values in [0, 1]: h: X → ℘([0, 1]) A HFS can also be constructed from a set of fuzzy sets.

(4.22)

4.14 Hesitant Fuzzy Set

123

Definition 4.31 Let M  {μ1 , . . . , μn } be a set of n membership functions. The HFS associated with M, h M is defined as h M : X → ℘([0, 1]) h M (x) 



{μ(x)},

(4.23)

μ∈M

where x ∈ X. It is remarkable that this definition is quite suitable to decision-making, when experts have to assess a set of alternatives. In such a case, M represents the assessments of the experts for each alternative and h M the assessments of the set of experts. However, note that it only allows to recover those HFSs whose memberships are given by sets of cardinality less than or equal to n. Afterward, Xia and Xu (2011) completed the original definition of HFS by including the mathematical representation of a HFS as follows: E  {x, h E (x): x ∈ X }, where h E (x) is a set of some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set E. For convenience, Xia and Xu noted h  h E (x) and called it hesitant fuzzy element (HFE) of E and H  ∪h E (x), the set of all HFEs of E. In some papers, the concepts HFS and HFE are used indistinctively, even though both concepts are different. A HFS is a set of subsets in the interval [0, 1], one set for each element of the reference set X. An HFE is one of such sets, the one for a particular x ∈ X. Given a function ϕ on n HFEs, the following definition expresses how to build a function ϕ  on HFS from ϕ: Definition 4.32 Let {H1 , . . . , Hn } be a set of n HFSs on X and ϕ an n-ary function on HFEs, we define ϕ  (H1 , . . . , Hn )(x)  ϕ(H1 (x), . . . , Hn (x)).

(4.24)

However, we can define functions for HFSs that do not correspond to functions on HFEs. The following definition illustrates this case. Definition 4.33 Let {H1 , . . . , Hn } be a set of n HFSs on X, ϕ  (H1 , . . . , Hn )(x) 

(maxi max(Hi ) + minxi min(Hi ))  ∧ Hi (x), 2 i

where for any α ∈ [0, 1], α ∧ h corresponds to {s|s ∈ h, s ≤ α}.

(4.25)

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4 Fuzzy Numbers and Fuzzy Optimization

Definition 4.34 Let H ⊆ ℘([0, 1]) be the set of all finite non-empty subsets of the interval [0, 1], and let X be a non-empty set. A THFS A over X is given by Eq. (4.22) where h E : X → H. Each h E (x) ∈ H is called a typical hesitant fuzzy element of H (THFE). Definition 4.35 Given an HFE, h, its lower and upper bounds are h −  min{γ |γ ∈ h}

(4.26)

h +  max{γ |γ ∈ h}.

(4.27)

Definition 4.36 Let h be an HFE, its complement is defined as hc 



{1 − γ}.

(4.28)

γ ∈h

Definition 4.37 Let h 1 and h 2 be two HFEs; their union is defined as h1 ∪ h2 



max{γ1 , γ2 }.

(4.29)

γ1 ∈h 1 ,γ2 ∈h 2

Definition 4.38 Let h 1 and h 2 be two HFEs; their intersection is defined as h1 ∩ h2 



min{γ1 , γ2 }.

(4.30)

γ1 ∈h 1 ,γ2 ∈h 2

Definition 4.39 Let X be a reference set, an Atanassov’s IFS A on X, is defined by A  {x, μ A (x), v A (x)|x ∈ X },

(4.31)

where the values μ A (x) and v A (x), belonging to [0, 1], represent the membership degree and nonmembership degree of the element x to the set A, respectively, with the condition 0 ≤ μ A (x) + v A (x) ≤ 1, ∀x ∈ X . The envelope of an HFE is represented in the following definition. Definition 4.40 Let h be an HFE, and the Atanassov’s IFS Aenv (h) is defined as the envelope of h, where Aenv (h) can be represented as   Aenv (h)  { h − , 1 − h + }.

(4.32)

Definition 4.41 Let E  {H1 , . . . , Hn } be a set of n HFSs and  a function, : [0, 1]n → [0, 1], we then export  on fuzzy sets to HFSs defining E 

 γ ∈H1 (x)×H2 (x)×···×Hn (x)

{(γ )}.

(4.33)

4.14 Hesitant Fuzzy Set

125

Definition 4.42 Given two HFSs H1 and H2 on X of the same cardinality, we define that H1 ≥ H2 if H1 (x) ≥ H2 (x) for all x. Note that H1 (x) and H2 (x) are HFEs. σ σ Here, h 1 ≥ h 2 for HFEs h 1 and h 2 if h 1 j ≥ h 2 j for all j  {1, . . . , |H1 |} where h σ j is the j th element in h when they are ordered in a decreasing order. Definition 4.43 Let ϕ be a function on HFSs such that the cardinality of ϕ is the same for all HFSs. We then say that ϕ is monotonic when ϕ(E) ≥ ϕ(E  ) for all E  {H1 , . . . , Hn } and E   {H1 , . . . , Hn } such that Hi ≥ Hi for all i  {1, …, n}. Proposition 4.2 Let E  {H1 , . . . , Hn } and E   {H1 , . . . , Hn } such that Hi ≥ Hi for all i  {1, …, n}. Then, if  is a monotonic function,  E is monotonic. To establish an order between HFEs, Xia and Xu introduced a comparison law by defining a score function, which is defined under the following assumptions: • The values of all the HFEs are arranged in an increasing order. • The HFEs have the same length when they are compared. Therefore, if any two HFEs have different lengths, the shorter one will be extended by adding the maximum element until both HFEs have the same length. Definition 4.44 Let h be an HFE, and the score function of h is defined as follows: s(h) 

1 " γ, l(h) γ ∈h

(4.34)

where l(h) is the number of elements in h. Let h 1 and h 2 be two HFEs, then if s(h 1 ) > s(h 2 ), then h 1 > h 2 ; if s(h 1 )  s(h 2 ), then h 1  h 2 . Definition 4.45 Let h 

/ γ ∈h

{γ } be an HFE, and the score function S of h is defined

by #l(h)

j1 δ( j)γ j S(h)  #l(h) , j1 δ( j)

where

#l(h) j1

(4.35)

δ( j) is a positive-valued monotonic increasing sequence of index j.

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4 Fuzzy Numbers and Fuzzy Optimization

4.15 Pythagorean Fuzzy Set In the following, some basic concepts related to IFS and PFS are introduced. Definition 4.46 Let X be a universe of discourse. An IFS I in X is given by I  {x, μ1 (x), v1 (x)|x ∈ X },

(4.36)

where μ1 : X → [0, 1] denotes the degree of membership and v1 : X → [0, 1] denotes the degree of nonmembership of the element x ∈ X to the set I, respectively, with the condition that 0 ≤ μ1 (x)+v1 (x) ≤ 1. The degree of indeterminacy π1 (x)  1 − μ1 (x) − v1 (x). Yager (2014) proposed a novel concept of PFS to model the condition that the sum of the degree to which an alternative xi satisfies and dissatisfies with respect to the attribute c j is bigger than 1, while the IFS cannot deal with it. Definition 4.47 Let X be a universe of discourse. A PFS P in X is given by P  {x, μ P (x), v P (x)|x ∈ X },

(4.37)

where μ P : X → [0, 1] denotes the degree of membership and v P : X → [0, 1] denotes the degree of nonmembership of the element x ∈ X to the set P, respectively, 2 with the condition that 0 ≤ μ P (x)2 + v P (x) 0 ≤ 1. The degree of indeterminacy π P (x)  1 − μ P (x)2 − v P (x)2 . For convenience, Zhang and Xu called (μ P (x), v P (x)) a Pythagorean fuzzy number (PFN) denoted by p  (μ P , v P ). Based on above definition, Zhang and Xu defined the distance between p1 and p2 as follows: d( p1 , p2 ) 

1  2  2   2  2   2  2   μ p1 − μ p2  +  v p1 − v p2  +  π p1 − π p2  . 2 (4.38)

The main difference between PFN and IFN is their corresponding constraint conditions which can be easily shown in Fig. 4.18. Definition 4.48 For any PFN p  (μ P , v P ), the score function of p is defined as follows: s( p)  (μ P )2 − (v P )2 , where s(p) ∈ [−1, 1]. For any two PFNs p1 , p2 , (i) if s( p1 ) < s( p2 ), then p1 ≺ p2 , (ii) if s( p1 ) > s( p2 ), then p1  p2 , and

(4.39)

4.15 Pythagorean Fuzzy Set

127

Fig. 4.18 Comparison of spaces of the PFNs and IFNs

(iii) if s( p1 )  s( p2 ), then p1 ∼ p2 . It is easily known that the score function defined in above is unreasonable, if we consider two PFNs p1  (0.5, 0.5) and p2  (0.6, 0.6), based on Definition 4.48, p1 ∼ p2 . But in fact, it fails to meet people’s intuition, so Peng and Yang proposed the accuracy function and modify the comparison rules. Definition 4.49 For any PFN p  (μ P , v P ), the accuracy function of p is defined as follows: a( p)  (μ P )2 + (v P )2 ,

(4.40)

where a(p) ∈ [0, 1]. For any two PFNs p1 , p2 : (i) if s( p1 ) > s( p2 ), then p1  p2 , (ii) if s( p1 )  s( p2 ), then (ii1) if a( p1 ) > a( p2 ), then p1  p2 and (ii2) if a( p1 )  a( p2 ), then p1 ∼ p2 . Definition 4.50 Let p  (μ, v), p1  (μ1 , v1 ), and p2  (μ2 , v2 ) be three PFNs, and λ > 0, and then their operations are defined as follows: (i) p1 ∪ p2  (max{μ1 , μ2 }, min{v1 , v2 }), (ii) p1 ∩ p2  (min{μ1 , μ2 }, max{v1 , v2 }), (iii) p c  (v, μ),  1 (iv) p1 ⊕ p2  μ21 + μ22 − μ21 μ22 , v1 v2 ,   1 (v) p1 ⊗ p2  μ1 μ2 , v12 + v22 − v12 v22 ,

128

4 Fuzzy Numbers and Fuzzy Optimization

 1  λ 1 − 1 − μ2 , v λ , and   1 λ  λ 2 (vii) pλ  μ , 1 − 1 − v . (vi) λ p 

Definition 4.51 Let p  (μ, v), p1  (μ1 , v1 ), and p1  (μ2 , v2 ) be three PFNs, and λ > 0, λ1 > 0, λ2 > 0, then their operations are defined as follows: (i) (ii) (iii) (iv) (v) (vi)

p1 ⊕ p2  p2 ⊕ p1 , p1 ⊗ p2  p2 ⊗ p1 , λ( p1 ⊕ p2 )  λp1 ⊕ λp2 , λ1 ⊕ λ2 p  (λ1 ⊕ λ2 ) p, ( p1 ⊗ p2 )λ  p1λ ⊗ p2λ , and p λ1 ⊗ p λ2  p (λ1 +λ2 ) .

4.16 Type-2 Fuzzy Set Definition 4.52 A type-2 fuzzy set is a fuzzy set whose membership values are type-1 fuzzy sets on [0, 1]. A type-2 fuzzy set A˜ is characterized by the membership function: A˜ 

 !  (x, u), μ A˜ (x, u) |∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]

(4.41)

in which 0 ≤ μ A˜ (x, u) ≤ 1. In fact Jx ⊆ [0, 1] represents the primary membership of x, and μ A˜ (x, u) is a type-1 fuzzy set known as the secondary set. Hence, a type-2 membership grade can be any subset in [0, 1], the primary membership, and corresponding to each primary membership, there is a secondary membership (which can also be in [0, 1]) that defines the possibilities for the primary membership. Uncertainty is represented by a region, which is called the footprint of uncertainty (FOU). When μ A˜ (x, u)  1, ∀u ∈ Jx ⊆ [0, 1] we have an interval type-2 membership function, as shown in Fig. 2.3. The uniform shading for the FOU represents the entire interval type-2 fuzzy set and it can be described in terms of an upper membership function μ¯ A˜ (x) and a lower membership function μ A˜ (x) (Figs. 4.19 and 4.20). Definition 4.53 A type m fuzzy set is a fuzzy set in X whose membership values are type m − 1, m > I fuzzy sets on [0, 1]. Operations for Type-2 Fuzzy Sets The extension principle can be used to define set-theoretic operations for type-2 fuzzy sets. We shall consider only fuzzy sets of type-2 with discrete domains. Let two fuzzy sets of type-2 be defined by

4.16 Type-2 Fuzzy Set

129

Fig. 4.19 An example of a type-1 membership function

5_ 4_ 3_ 2_ 1_ | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 11

Fig. 4.20 Interval type-2 membership function

5 _

Upper MF function

4 _

Lower MF function

3 _ 2 _ 1 _

0

|

1

|

2

| 3

|

4

|

5

|

6

|

7

|

8

|

9

|

10

˜ ˜ A(x)  {(x, μ A˜ (x))} and B(x)  {(x, μ B˜ (x))} where μ A˜ (x)  {(u i , μui (x))/x ∈ X, u i , μui (x) ∈ [0, I ]}. μ B˜ (x)  {(v j , μv j (x))/x ∈ X, v j , μv j (x) ∈ [0, I ]} The u i , and v j are degrees of membership of type-1 fuzzy sets and the μui (x) and μv j (x), respectively, their membership functions. Using the extension principle, the set-theoretic operations can be defined as follows. Definition 4.54 Let two fuzzy sets of type-2 be defined as above. The membership function of their union is then defined by μ A∪ ˜ B˜ (x)  μ A˜ (x) ∪ μ B˜ (x)  {(w, μ A∪ ˜ B˜ (w))|w  max{u i v j , }, u i , v j ∈ [0, I ]} ! sup where μ A∪ ˜ B˜ (w)  wmax{u ,v } min μui (x), μv j (x) . i j

130

4 Fuzzy Numbers and Fuzzy Optimization

Their intersection is defined by μ A∩ ˜ B˜ (x)  μ A˜ (x) ∩ μ B˜ (x)  {(w, μ A∩ ˜ B˜ (w))|w  max{u i v j , }, u i , v j ∈ [0, I ]} sup

where μ A∪ ˜ B˜ (w)  wmin

{u i ,v j }

! min μui (x), μv j (x) .

And the complement of A˜ by μCA˜ (x)  {[(1 − u i ), μ A˜ (u i )]}.

Usefulness of Type-2 Fuzzy sets • Type-2 fuzzy sets empower us to manage linguistic uncertainties, which can be conveyed as: “words can mean different things to different people”. • A fuzzy sets association of higher type (e.g., type-2) has been seen as one way to deal with to build the fuzziness of a relation. • According to Hisdal, “expanded fuzziness in a description implies expanded capacity to deal with inexact data in a logically correct way”.

4.17 Conclusion Each procedure of decision-making in a real-life problem can be spoken to because of a last decision, and the output can be represented as an activity or as an opinion of the final decision. Distinctive types of vagueness can be found in a wide variety of optimization and basic decision-making problems identified with arranging and activity of power systems and subsystems. Considering various types of fuzzy member ship functions makes decisions more realistic and sensible. The mixture of the uncertainty or vagueness or fuzziness factor in the development of various models serves for increasing their adequacy and, subsequently, the unwavering quality and verifiable proficiency of decisions based on their analysis. The fuzzy set theory concepts have some specific parameterization features which are certain expansions of crisp set theory and fuzzy set theory relations, respectively, and have a great potential to be applied to the decision-making problems.

References K.T. Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications (Springer Physica Verlag, Heidelberg, 1999) H. Becker, H.R. Tizhoosh, Fuzzy Image Processing in Handbook of Computer Vision and Applications, vol. 2. Academic Press (1999)

References

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R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment. Manage. Sci. 17, 141–164 (1970) N.N. Karnik, J.M. Mendel, Q. Liang, Type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 7(4), 643–658 (1999) G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall (1995) Q. Liang, J.M. Mendel, Interval Type-2 Fuzzy Logic Systems: Theory and Design. IEEE Trans. Fuzzy Syst. 8(5), 535–550 (2000) B. Liu, Uncertainty Theory, 4th edn. (Springer, Berlin, 2015) J.M. Mendel, R.I.B. John, Type-2 fuzzy sets made simple. IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002) Z.S. Xu, M.M. Xia, Distance and similarity measures for hesitant fuzzy sets. Inform. Sci. 181(11), 2128–2138 (2011) R.R. Yager, Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 22(4), 958–965 (2014). https://doi.org/10.1109/TFUZZ.2013.2278989 J. Ye, Multi-criteria decision making method using the correlation coefficient under single-value neutrosophic environment. Int. J. Gen. Syst. 42(4), 386–394 (2013) H.-J. Zimmermann, Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1, 45–55 (1978) H.J. Zimmermann, Application of fuzzy set theory to mathematical programming. Inf. Sci. 36, 29–58 (1985) H.J. Zimmermann, Methods and applications of fuzzy mathematical programming, in An introduction to Fuzzy Logic Application in Intelligent Systems, ed. by R.R. Yager, L.A. Zadeh. Kluwer Publishers, Boston (1992), 97–120

Chapter 5

Fuzzy Unconstrained Geometric Programming Problem

5.1 Introduction Since 1960s, geometric programming (GP) is utilized in different fields (like operations management, engineering science, and so on). Geometric programming (GP) is one of the powerful techniques to solve a specific type of nonlinear programming problem (NLP). The theory of geometric programming (GP) is first introduced in 1961 by Duffin and Zener. The first publication on geometric programming (GP) was published by Duffin and Zener in 1967. There are numerous references to applications and strategies for GP in the survey paper by Ecker. In the standard geometric model, all coefficients are considered as fixed. In real circumstances, it will have some little fluctuations. So the fuzzy coefficients are considered to be more realistic. In 1990, R. K. Varma has contemplated fuzzy programming technique to solve geometric programming (GP) problem. Biswal (1992) presented fuzzy programming with nonlinear membership functions technique to deal with multi-objective GP problem. Cao (1993) is the initial one to change geometric programming (GP) to its comparing fuzzy state and has demonstrated that fuzzy program is a helpful method to solve a multi-objective optimization problem. An optimization model with a posynomial objective function to a fuzzy max-min relation equation is presented by Yang and Cao (2005). Cao (1987) worked on solution and theory of question for a kind of fuzzy positive geometric program Lingo (1999) demonstrated the Modeling Language and Optimizer. Liu et al. (2003) developed a fuzzy-stochastic robust programming model for regional air quality management under uncertainty. Mandal and Mendoça et al. (2004) presented optimization problems in multivariable fuzzy predictive control. Yang and Cao (2005) worked on geometric programming with fuzzy relation equation constraints. Islam and Roy (2006) developed an inventory model with L-R fuzzy number. Shivanian and Khorram (2009) presented monomial geometric programming with fuzzy relation inequality constraints with max-product composition. Yousef et al. (2009) developed geometric programming problems with fuzzy parameters and its application to crane load sway. © Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_5

133

134

5 Fuzzy Unconstrained Geometric Programming Problem

In this chapter, we have discussed three types of fuzzy unconstrained geometric programming method that are (1) GP problem with fuzzy parametric interval-valued function. (2) GP problem with simple fuzzy parametric coefficients. (3) GP problem with Zimmermann max-min operator. Definition 5.1 Let D ⊂ Rm be the (convex) subset of Rm defined by   D  (x1 , x2 , . . . , xm ) ∈ Rm |x > 0, j  1, 2, . . . m . A function g: D → Rm of the form T0 m     α g xj  ck x j kj , k1

j1

where ck > 0 for k  1, 2, . . . , T0 and αk j ∈ R for k  1, 2, . . . , T0 and j  1, 2, . . . , m, is called posynomial. A problem of the form Minimum g(x) Subject to x > 0, is called unconstrained geometric programming (GP) problem. When coefficients, exponents, or objective are fuzzy in nature, then it is called unconstrained fuzzy GP problem which is defined as follows: Minimize  g(x) Subject to x > 0. where  g(x) 

T0

ck k1

m

α

j1

x j kj .

Example 5.1 3x13 x2 + 7x12 x25 is fuzzy posynomial. And, the problem of the form Minimize 3x13 x2 + 7x12 x25 Subject to x1 , x2 > 0, is a fuzzy unconstrained GP problem.

5.2 GP Problem with Fuzzy Parametric Interval-Valued Function

135

5.2 GP Problem with Fuzzy Parametric Interval-Valued Function Primal Problem Consider a primal unconstrained fuzzy geometric programming problem, which is of the form Min g0 (x) 

T0 

cok

k1

m 

α

x j 0k j

j1

Subject to x j > 0,

(5.1)

c0k are triangular fuzzy numbers (TFN). where α0k j are real numbers and coefficients 1 2 3 , c0k , c0k ). Here, we have considered c˜0k  (c0k Using nearest interval approximation (NIA) method, we transform all trianguU L , c0k ]. The geometric programming lar fuzzy number into interval number, i.e., [c0k problem with imprecise parameters is of the following form: Min gˆ 0 (x) 

T0 

cˆ0k

k1

m 

α

x j 0k j

j1

subject to x j > 0,

(5.2)

U U L L , c0k ].c0k > 0, c0k > 0, where cˆok denotes the interval counterparts, i.e., cˆ0k ∈ [c0k for all k. Using parametric interval-valued functional form, the problem (5.2) reduces to

Min g0 (x, s) 

T0  k1

L 1−s U s (c0k ) (c0k )

m 

α

x j 0k j

j1

Subject to x j > 0 for j  1, 2, . . . , m. Dual Problem Corresponding to dual programming (DP), the problem of (5.3) is

Max d(δ, s)  Subject to T0  k1

δok  1,

T0  L 1−s U s δ0k  (c0k ) (c0k ) δ0k k0

(5.3)

136

5 Fuzzy Unconstrained Geometric Programming Problem T0 

αk j δ0k  0,

k1

δ0k > 0.

(5.4)

Case 1 For T 0 ≥ M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables. More or unique solutions exist for the dual vectors. Case 2 For T 0 < M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case generally, no solution vectors exist for the dual variables. However, one can get an approximate solution vector for the system using either the latest square (SQ) or max-min (MN) method. These are applied to solve such a system of linear equations. Once, optimal dualvariable vector δ ∗ is known, and the corresponding values of the primal variable vector x is found from the following relations: m  L 1−s  U s    α c0k x j k j  δk∗ v∗ δ ∗ , (k  1, 2, . . . , T0 ). c0k j1

Theorem 5.1 If x is a feasible vector for the constraints PGP and δ is a feasible vector for the corresponding DP, then go (x, s) ≥ d(δ, s) (Primal–Dual Inequality). Proof The expression for g0 (x, s) can be written as go (x, s) 

T0 

δ0k

L 1−s U s (c0k ) (c0k )

m

δ0k

k1

α

j1

x j 0k j

 .

Here, the weights are δ01 , δ02 , . . . , δ0T0 and the positive terms are L 1−s U s ) (c01 ) (c01

δ01

m j1

α

L 1−s U s x j 01 j (c02 ) (c02 ) , δ02

m j1

α

x j 02 j

,...,

L 1−s U s ) (c0T0 ) (c0T 0

m

α0T0 j

j1

xj

δ0TO

Now applying A.M.–G.M. inequality, we get ⎛

α α α L 1−s (cU )s m x 01 j + (c L )1−s (cU )s m x 02 j + · · · + (c L )1−s (cU )s m x 0T0 j ⎜ (c01 ) 02 02 01 0T0 0T0 j1 j j1 j j1 j ⎜ ⎝ (δ01 + δ02 + · · · + δ0T ) 0 ⎛

⎞(δ +δ +...+δ 01 02 0T0 ) ⎟ ⎟ ⎠

⎛ ⎞δ ⎛ α α01 j ⎞δ01 ⎛

01 L 1−s (cU )s m x 0T0 j U )s m x α02 j ⎜ (c L )1−s (cU )s m x (c L )1−s (c02 ⎜ (c0T0 ) 0T0 j1 j 01 j1 j j1 j ⎟ ⎟ ⎜⎜ 01 ⎜ 02 ⎜ ≥ ⎜⎝ ...⎝ ⎠ ⎠ ⎝ ⎝ δ01 δ02 δ0 T0

or

⎞ ⎞δ 0T0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠

.

5.2 GP Problem with Fuzzy Parametric Interval-Valued Function

 0 δ0k T

go (x, s) T0 k1 δik

k1

≥

137

L

α δ0k  T0 U s  (c0k )1−s (c0k ) mj1 x j 0k j δ0k

k1

as

T0 

 δ0k  1

k1

or  g0 (x, s) ≥

L 1−s U s ) (c0k ) (c0k

0 δok  m T

T0 

k1

k1

δ0k

α0k j δok

xj

j1

or

g0 (x, s) ≥

T0  L 1−s U s δik  m  (c ) (c )

δik

k1



ik

ik

T0  

α0k j δok

x k1 j

j1

L 1−s U s (cik ) (cik )

δik

k1

T0 

 δik   T0 α0k j δok  0 , as k1

 d(δ, s) i.e., g0 (x, s) ≥ d(δ, s). Theorem 5.2 If δ is a feasible vector for the dual programming (DP) problem, then d(δ, 1) ≥ d(δ, 0). U U  c0k . Proof We have c0k So U L c0k ≥ c0k , for all k, (k  1, 2, . . . , T0 ).

or L 1−1 U 1 L 1−0 U 0 ) (c0k ) ≥ (c0k ) (c0k ) (c0k

or U 0 L 1−1 U 1 ) (c0k ) ) (c0k (c L )1−0 (c0k ≥ 0k δ0k δ0k

or 

L 1−1 U 1 ) (c0k ) (c0k δ0k

δ0k

 ≥

L 1−0 U 0 ) (c0k ) (c0k δ0k

δ0k

138

5 Fuzzy Unconstrained Geometric Programming Problem

or T0  L 1−1 U 1 δ0k T0  L 1−0 U 0 δ0k   (c0k ) (c0k ) (c0k ) (c0k ) ≥ , δ δ0k 0k k0 k0

i.e., d(δ, 1) ≥ d(δ, 0). Application 5.1 ˜ y + cx ˜ y −2 Min Z (x)  ax ˜ −1 y + bx Subject to x, y > 0. where crisp input data is given as a  2, m  5, and c  2. Solution When the input data is taken as triangular fuzzy number, i.e., a˜  (1, 2, 3), b˜  (4, 5, 6) and c˜  (1, 2, 3). Using nearest interval approximation (NIA) method, we get the corresponding interval number and interval-valued function, i.e., a˜  [1.5, 2.5], ⇒ aˆ  (1.5)1−s (2.5)s ∈ [1.5, 2.5], b˜  [4.5, 5.5], ⇒ bˆ  (4.5)1−s (5.5)s ∈ [4.5, 5.5], c˜  [1.5, 2.5], ⇒ cˆ  (1.5)1−s (2.5)s ∈ [1.5, 2.5], where s ∈ [0, 1]. Here, the primal problem is Min Z (x, s)  (1.5)1−s (2.5)s x −1 y + (4.5)1−s (5.5)s x y + (1.5)1−s (2.5)s x y −2 Sub to x ≥ 0, y ≥ 0. The corresponding dual form is  Max d(δ, s) 

(1.5)1−s (2.5)s δ1

δ1 

(4.5)1−s (5.5)s δ2

δ2 

(1.5)1−s (2.5)s δ3

δ3

Subject to δ1 + δ2 + δ3  1, − δ1 + δ2 + δ3  0, δ1 + δ2 − 2δ3  0, δ1 , δ2 , δ3 > 0.

(5.5)

From (5.5), δ1  21 , δ2  16 and δ3  13 . The optimal solution of the model through the parametric approach is given by

5.2 GP Problem with Fuzzy Parametric Interval-Valued Function

139

Table 5.1 Optimal solution of the Application 5.1 (fuzzy method) S

x1∗

x2∗

x3∗

d ∗ (δ, s)

g0 (x, s)∗

0.0

0.991

0.496

2.066

184.463

184.463

0.2

0.995

0.497

2.042

190.285

190.285

0.4

0.997

0.499

2.017

196.291

196.291

0.6

1.000

0.500

1.992

202.486

202.486

0.8

1.004

0.502

1.964

208.876

208.876

1.0

1.007

0.503

1.945

215.469

215.469

 d(δ, s) 

2.(1.5)1−s (2.5)s 1

( 21 ) 

6.(4.5)1−s (5.5)s 1

( 16 ) 

3.(1.5)1−s (2.5)s 1

( 13 )

.

From the primal–dual relations, we get 2x −1 y  δ1 v(δ)  δ1 × d(δ, s), 5x y  δ2 v(δ)  δ2 × d(δ, s), 5x y −2  δ3 v(δ)  δ3 × d(δ, s). The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 5.1. Application 5.2 (Grain-box problem) “It has been decided to shift grain from a warehouse to a factory in an open rectangular box of length x1 meters, width x2 meters and height x3 meters. The bottom, side and ends of the box cost $a, $b and $c/m 2 respectively. It cost $1 for each round trip of the box. Assuming that the box will have no salvage value, find the minimum cost of transporting dm 3 of grains” (Fig. 5.1). This problem can be formulated as

m m

m Fig. 5.1 Grain-box problem

140

5 Fuzzy Unconstrained Geometric Programming Problem

Table 5.2 Crisp input values

a($/m 2 )

b($/m 2 )

c($/m 2 )

d(m 3 )

80

10

20

80

d + ax1 x2 + 2bx1 x3 + 2cx2 x3 x1 x2 x3 Sub to x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. Min g0 (x, s) 

(5.6)

The input values are given in Table 5.2. Solution When the input data is taken as triangular fuzzy number, i.e., a˜  (70, 80, 90), b˜  (8, 10, 12), c˜  (16, 20, 24), and d˜  (70, 80, 90). Using the nearest interval approximation method, we get the corresponding interval number and interval-valued function, i.e., a  [75, 85], ⇒ aˆ  (75)1−s (85)s ∈ [75, 85], b  [9, 11], ⇒ bˆ  (9)1−s (11)s ∈ [9, 11], c  [18, 22], ⇒ cˆ  (18)1−s (22)s ∈ [18, 22], d  [75, 85], ⇒ dˆ  (75)1−s (85)s ∈ [75, 85], where s ∈ [0, 1]. Here, the primal problem is (75)1−s (85)s + (75)1−s (85)s x1 x2 + 2(9)1−s (11)s x1 x3 + 2(18)1−s (22)s x2 x3 x1 x2 x3 Sub to x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. Min g0 (x, s) 

The corresponding dual form is

Max d(δ, s) 

(75)1−s (85)s δ1

δ1

(75)1−s (85)s δ2

δ2

2(9)1−s (11)s δ3

δ3

2(18)1−s (22)s δ4

δ4 ,

subject to δ1 + δ2 + δ3 + δ4  1, − δ1 + δ2 + δ3  0, − δ1 + δ2 + δ4  0, − δ1 + δ3 + δ4  0, δ1 , δ2 , δ3 , δ4 > 0.

(5.7)

From (5.7), we get δ1  25 , δ2  15 , δ3  15 , and δ4  15 . The optimal solution of the model through the parametric approach is given by

d ∗ (δ, s) 

5(75)1−s (85)s 2

2 5

5(75)1−s (85)s 1

1 5

From the primal–dual relations, we get

2 × 5(9)1−s (11)s 1

1 5

2 × 5(18)1−s (22)s 1

1 5

.

5.2 GP Problem with Fuzzy Parametric Interval-Valued Function

141

Table 5.3 Optimal solution of the Application 5.2 S

x1∗

x2∗

x3∗

d ∗ (δ, s)

g0 (x, s)∗

0.0

0.991

0.496

2.066

184.463

184.463

0.2

0.995

0.497

2.042

190.285

190.285

0.4

0.997

0.499

2.017

196.291

196.291

0.6

1.000

0.500

1.992

202.486

202.486

0.8

1.004

0.502

1.964

208.876

208.876

1.0

1.007

0.503

1.945

215.469

215.469

(75)1−s (85)s  δ1 × d ∗ (δ, s), x1 x2 x3 (75)1−s (85)s x1 x2  δ2 × d ∗ (δ, s), 2(9)1−s (11)s x1 x3  δ3 × d ∗ (δ, s), 2(18)1−s (22)s x2 x3  δ4 × d ∗ (δ, s). The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 5.3.

5.3 GP Problem with Simple Fuzzy Parametric Coefficients Consider a fuzzy geometric programming problem, which is as follows:  Min g(x) Subject to x > 0. T0

α Its objective of the form g(x)  k1 c˜k mj1 x j k j is all posynomials of x in which coefficients c˜k are fuzzy numbers. A real number c˜k described as fuzzy subset on the real line R, whose membership function μ ck (x) has the following characteristics with − ∝< ck1 ≤ ck2 ≤ ck3 0. Which is equivalent to Min

T0 

ck L (α)

k1

m 

α

x j kj

j1

.

Subject to x j > 0. Dual Problem The corresponding dual programming (DP) problem is

Max d(δ, s) 

T0   ck L δ0k k0

δ0k

Subject to T0 

δok  1,

k1 T0 

αk j δ0k  0,

k1

δ0k > 0.

(5.8)

Case 1 For T 0 ≥ M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables. More or unique solutions exist for the dual vectors. Case 2 For T 0 < M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case generally, no solution vectors exist for the dual variables. However, one can get an approximate solution vector for the system using either the latest square (SQ) or max-min (MN) method. These are applied to solve such a system of linear equations. Once optimal dualvariable vector δ ∗ is known, the corresponding values of the primal variable vector x is found from the following relations: ck L

m  j1

  α x j k j  δk∗ v∗ δ ∗ , (k  1, 2, . . . , T0 ).

5.3 GP Problem with Simple Fuzzy Parametric Coefficients

143

Application 5.3 ˜3 ˜ 3 t1−1 t2−1 + 5t Min Z (t)  t1 t2 t3−2 + 2t subject to t1 , t2 , t3 > 0. Solution Here, DD  3 − (3 + 1)  −1(< 0). It is an unconstrained posynomial PGP with negative DD  −1. Consider the fuzzy coefficients, which are as follows: 2˜  1.6 + (1 − α) × 0.8, 5˜  4 + (1 − α) × 2. DP is  Maximize v(δ) 

1 δ1

δ1 

1.6 + (1 − α) × 0.8 δ2

δ2 

4 + (1 − α) × 2 δ3

δ3

subject to δ1 + δ2 + δ3  1, δ1 − δ2  0, δ1 − δ2  0, − 2δ1 + δ2 + δ3  0, δ1 , δ2 , δ3 > 0. It is a system of four linear equations with three unknown variables. The approximate solution of this system of linear equations (by LS method) is δ *1  0.333, δ *2  0.333, δ *3  0.333 and the corresponding optimal dual objective value, i.e., v* (δ * )  6.463. So for primal decision variables, the following system of nonlinear equations is found: t1 t2 t3−2  δ1∗ v∗ (δ ∗ ), (1.6 + (1 − α) × 0.8)t3 t1−1 t2−1  δ1∗ v∗ (δ ∗ ), (4 + (1 − α) × 2)t3  δ1∗ v∗ (δ ∗ ). Solving the above nonlinear equation for different values of “α” by Lingo, the optimal primal variables are obtained. These are given in Table 5.4. Application 5.4 Consider the Application 5.1 (Grain-box problem). Here, the primal problem is

144

5 Fuzzy Unconstrained Geometric Programming Problem

Table 5.4 Optimal solution of the Application 5.3 S. No.

α

t1

t2

t3

v ∗ (δ, α)

g0 (t)

1

0.0

2.18

0.18

0.41

7.30

7.30

2

0.2

2.18

0.18

0.41

6.97

6.97

3

0.4

2.18

0.18

0.42

6.63

6.63

4

0.6

2.18

0.18

0.44

6.29

6.29

5

0.8

2.18

0.18

0.45

5.94

5.94

6

1.0

2.18

0.18

0.46

5.57

5.57

80 1 x2 + 2.10x 1 x3 + 2.20x 2 x3 + 80x x1 x2 x3 Sub to x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. Min g0 (x) 

Transforming the fuzzy coefficient in parametric form  8 + (1 − α) × 4, 10  15 + (1 − α) × 10, 20  70 + (1 − α) × 20. 80 The corresponding dual form is  70 + (1 − α) × 20 δ1 70 + (1 − α) × 20 δ2 δ1 δ2  δ3  2.(8 + (1 − α) × 4) 2.(15 + (1 − α) × 10) δ4 δ3 δ4 

Max d(δ, α) 

subject to δ1 + δ 2 + δ 3 + δ 4  1 − δ1 + δ2 + δ3  0 − δ1 + δ2 + δ4  0 − δ1 + δ3 + δ4  0 δ1 , δ2 , δ3 , δ4 ≥ 0.

(5.10)

From (5.10), we get δ1  25 , δ2  15 , δ3  15 , and δ4  15 . The optimal solution of the model through the parametric approach is given by ∗

d (δ, α) 



5(70 + (1 − α) × 20) 2

25 

5(70 + (1 − α) × 20) 1

15

5.3 GP Problem with Simple Fuzzy Parametric Coefficients

145

Table 5.5 Optimal solution of the Application 5.4 S. No.

α

x1

x2

x3

d ∗ (δ, α)

g0 (x)

1

0.0

1.453

1.163

1.869

232.777

301.776

2

0.2

1.381

1.088

1.915

219.714

284.239

3

0.4

1.297

0.961

1.969

206.590

259.049

4

0.6

1.199

0.834

2.033

193.389

228.833

5

0.8

1.084

0.684

2.110

180.092

213.926

6

1.0

0.945

0.504

2.205

166.673

200.406



5.2(8 + (1 − α) × 4) 1

15 

5.2(15 + (1 − α) × 10) 1

15

.

From primal–dual relations, we get 70 + (1 − α) × 20  δ1 × d ∗ (δ, α), x1 x2 x3 (70 + (1 − α) × 20)x1 x2  δ2 × d ∗ (δ, α), 2.(8 + (1 − α) × 4))x1 x3  δ3 × d ∗ (δ, α), 2.(15 + (1 − α) × 10))x2 x3  δ4 × d ∗ (δ, α). Solving the above nonlinear equation for different values of “α” by Lingo, the optimal primal variables are obtained. These are given in Table 5.5.

5.4 GP Problem with Zimmermann Max-Min Operators Some Definitions and Theorems Definition 5.2 For n-th parabolic flat fuzzy number (a1 , a2 , a3 , a4 )PfFN containing cik is the coefficients cik (0 ≤ i ≤ n; 1 ≤ k ≤ T i ), the membership function of  n ⎧ a2 −cik ⎪ 1 − ⎪ a2 −a1 ⎪ ⎪ ⎨ 1  n μc˜ik (c˜ik )  cik −a3 ⎪ 1 − ⎪ ⎪ a4 −a3 ⎪ ⎩ 0

for a1 ≤ cik ≤ a2 for a1 ≤ cik ≤ a2 for a3 ≤ cik ≤ a4

(5.11)

for otherwise.

Similarly, we can determine the membership function of the indexes α˜ ik j (0 ≤ i ≤ n; 1 ≤ k ≤ T i ; 1 ≤ i ≤ m).

146

5 Fuzzy Unconstrained Geometric Programming Problem

Note: (a) (b) (c) (d)

when n  1, c˜ik becomes trapezoidal fuzzy number (TrFN), when n  1, and a3  a4 , c˜ik becomes triangular fuzzy number (TFN), when n  2, c˜ik becomes parabolic flat fuzzy number (PfFN), when n  2, and a3  a4 , c˜ik becomes parabolic fuzzy number (pFN).

Definition 5.3 Here, δ-cut of c˜ik (0 ≤ i ≤ n; 1 ≤ k ≤ T i ) is given by     √ √ n n −1 −1 μ−1 (δ)  μ (δ), μ (δ)  a + 1 − δ(a −a − 1 − δ(a −a a ), ) 1 2 1 4 4 3 . c˜ik c˜ik L c˜ik R (5.12) Similarly, we can determine the δ-cut of αik j (0 ≤ i ≤ n; 1 ≤ k ≤ T i ; 1 ≤ i ≤ m). Proposition 5.1 When the coefficient and indexes of the fuzzy geometric programming problem are taken as fuzzy numbers T0 

 Min

ck

k1

m 

α˜

x j kj

j1

Subject to x j > 0,

(5.13)

Then using δ-cut of fuzzy numbers coefficients and indexes, the above problem reduces to  Min

T0  

−1 μ−1 c˜k L (δ), μc˜ R (δ)

m 

k1



xj

 μ−1 (δ),μ−1 (δ) α˜ α˜ kjL

kj R

.

j1

Subject to x j > 0. Which is equivalent to  Min

To  k1

μ−1 c˜k L (δ)

m 

xj

μ−1 α˜

kjS

(δ).

j1

where  μ −1 αk j S (δ)



˜ k j L > 0, μ−1 α˜ k j L (δ) when α −1 μα˜ k j R (δ) when α˜ k j L < 0.

Definition 5.4 For any x ∈ Rm and feasible index di ∈ R (R is the real number μ −1 T0

m αik j S set), if g0 ((x, δ)  k1 μ −1 x (δ) ≤ 1, then the linear membership (δ) cik L j1 j function are given by

5.4 GP Problem with Zimmermann Max-Min Operators

μ0 (g0 (x, δ)) 

⎧ ⎪ ⎨ 1 ⎪ ⎩

z 0 +d0 −g0 (x,δ) d0

147

 if g0 (x, δ) ≤ z 0 , if z 0 ≤ g0 (x, δ) ≤ z 0 + d0 ,

(5.13)

if g0 (x, δ) ≥ z 0 + d0 .

0

There are many nonlinear membership functions of which here we take only linear membership function. Based on Zimmerman, first finding δ-cut of the fuzzy numbers in coefficients and indexes then we built membership functions of both objective and constraints goals and using max-min operator the above problem (5.13) is reduced to a fuzzy nonlinear programming (FNLP) problem Max λ

⎞ ⎛ T0 m   μ−1 α ˜ μ−1 x j k j S (δ)⎠ ≥ λ subject to μ⎝ c˜k L (δ) k1

j1

x j > 0, λ, δ ∈ [0, 1],

(5.14)

which is equivalent to a geometric programming problem with parameters λ, δ variation Min λ−1

⎛ ⎞ T0 m   μ−1 α ˜ subject to μ⎝ μ−1 x j k j S (δ)⎠ ≥ λ c˜k L (δ) k1

j1

x j > 0, λ, δ ∈ [0, 1].

(5.15)

  Theorem 5.3 Let the membership function μ(g(x, δ)), μc˜k (ck ), μα˜ k j αk j be all continuous and strictly monotone. Then (5.15) is equivalent with Min λ−1 T0 subject to

−1 k1 μc˜k L (δ)

m

μ−1 α˜

j1 x j

kjS

μi−1 (δ)

(δ)

≤1

x > 0, λ, δ ∈ [0, 1], (1 ≤ j ≤ m). Proof Please see reference [13] Islam and Roy (2006).   Corollary 5.1 Let the membership function μ(g(x, δ)), μ ck (ck ), μα˜ k j αk j be all continuous and strictly monotone and the problem is Min λ−1

148

5 Fuzzy Unconstrained Geometric Programming Problem

T0 k1

μ−1 c˜k L (δ)

m j1

μ−1 α˜

xj

kjS

(δ)

≤ 1, μ−1 (δ) x j > 0, λ, δ ∈ [0, 1], (1 ≤ j ≤ m).

Subject to

(5.16)

Which is a classical posynomial geometric programming with parameters γ , δ. Dual Problem The dual form of the problem (5.16) is

−1 ωk  −1 ω00  T0 μc˜k (δ)/μ−1 (λ) λ Max d(ω)  ω00 ωk k1 subject to ω00  1, ω00 

T0 

ω0k

k1 T0 

α˜ k−1 j S (δ)ω0k  0, λ, δ ∈ [0, 1],

K 1

ω0k ≥ 0 where ω0k  ω0k (δ, λ). Application 5.5 ˜ y + 2x ˜ y −2 ˜ −1 y + 5x Min Z (x, y)  2x Subject to x, y > 0.

(5.17)

Solution The solution of Example 4.5 by any simple crisp method is x  0.365, and y  0.585 and minimum cost Z (x, y)  6.406. The coefficients are taken as triangular fuzzy number (TFN), i.e., 2˜  (1.5, 2, 2.5), 5˜  (4, 5, 6). The lower bounds of the tolerance interval are z 0 (t)  6. Spreads of tolerance intervals d0  4. Taking the membership function as follows: ⎧ ⎪ 1, z(x) ≤ z 0 ⎨ , z < g(x) ≤ z 0 + d0 , μ(z(x))  z0 +dod−g(x) 0 0 ⎪ ⎩ 0. g(x) > z 0 + d0 Then, the problem (5.17) by corollary 5.4.1 is

5.4 GP Problem with Zimmermann Max-Min Operators

149

Min λ−1 −(1.5 + α(2 − 1.5))x −1 y − (4 + α(5 − 4))x y − (1.5 + α(2 − 1.5))x y −2 ≤ 1, [−(6 + 4 − 1) + 4λ] x, y>0, λ, δ ∈ [0, 1].

Subject to

i.e., the problem is Min λ−1 −(1.5 + 0.5α)x −1 y − (4 + α)x y − (1.5 + 0.5α)x y −2 ≤ 1, [−9 + 4λ] x, y > 0, λ, α ∈ [0, 1]. (5.18)

Subject to

The dual of (5.18) is δ   δ02 λ−1 00 (1.5 + 0.5α) δ01 (4 + α) δ00 (9 − 4λ)δ01 (9 − 4λδ02 )  δ03 (1.5 + 0.5α) (δ01 + δ01 + δ03 )(δ01 +δ01 +δ03 ) (9 − 4λ)δ03

 Max d(δ) 

Subject to δ00  1, δ01 + δ02 + δ03  δ00  1, − δ01 + δ02 + δ03  0, δ01 + δ02 − 2δ03  0, δ01 , δ02 , δ03 > 0. Solving the above equations, we have δ01  21 , δ02  16 , and δ03  13 . Putting the value of δ01 , δ02 and δ03 , the corresponding optimal dual value, i.e.,  d(δ) 

λ−1 1

1 

2(1.5 + 0.5α) (9 − 4λ)

21 

6(4 + α) (9 − 4λ)

16 

3(1.5 + 0.5α) (9 − 4λ)

13

and it can be determined λ by the aid of d(δ)  λ−1 . Then, the above equation reduces to 

2(1.5 + 0.5α) (9 − 4λ)

21 

6(4 + α) (9 − 4λ)

16 

3(1.5 + 0.5α) (9 − 4λ)

13

 1.

150

5 Fuzzy Unconstrained Geometric Programming Problem

Table 5.6 Optimal solution of the Application 5.5 S. No.

α

λ

x

y

Z(x, y)

1

0.0

0.99

0.316

0.531

6.441

2

0.2

0.95

0.350

0.568

6.409

3

0.4

0.88

0.358

0.577

6.407

4

0.6

0.81

0.366

0.586

6.406

5

0.8

0.73

0.366

0.586

6.406

6

1.0

0.65

0.366

0.586

6.406

Solving the above nonlinear equation of λ for a given α ∈ [0, 1] by Newton–Rapshon method, we obtain the value of λ∗ . Putting the value of λ∗ , we obtain the value of dual-objective function. Again from the primal–dual relations, we get δ∗ −(1.5 + 0.5α)x −1 y 1  01 ∗  , −9 + 4λ δ00 2 −(4 + α)x y δ∗ 1  02 ∗  , −9 + 4λ δ00 6 −(1.5 + 0.5α)x y −2 δ∗ 1  03 ∗  . −9 + 4λ δ00 3 Solving the above nonlinear equation for different values of “α” by Lingo, the optimal primal variables are obtained. These are given in Table 5.6. Application 5.6 Consider the Application 5.1 (Grain-box problem). Solution The solution of Example 1.6 by any simple crisp method is x1  1, x2  0.5, x3  2 and minimum cost g0 (x)  200.  The coefficients are taken as triangular fuzzy number (TFN), i.e.,10 (6, 10, 14), 20  (12, 20, 28), 80  (70, 80, 90). The lower bounds of the tolerance interval g0 (x)  195. Spreads of tolerance intervals d0  10. Taking the membership function as follows:

5.4 GP Problem with Zimmermann Max-Min Operators

μ(z(x)) 

⎧ ⎪ ⎨ ⎪ ⎩

1,

z 0 +do −g(x) , d0

0.

151

z(x) ≤ z 0 z 0 < g(x) ≤ z 0 + d0 . g(x) > z 0 + d0

Then, the problem (1.11) by corollary 5.1 is Min λ−1 −(70 + α(80 − 70))/(x1 x2 x3 ) − (80 + α(80 − 70))x1 x2 − 2(6 + α(10 − 6))x1 x3 − 2(12 + α(20 − 12))x2 x3 ≤ 1, [−(195 + 10 − 1) + 10λ]

subject to

x, y > 0, λ, δ ∈ [0, 1].

i.e., the problem is Min λ−1 subject to

−(70 + α(80 − 70))/(x1 x2 x3 ) − (80 + α(80 − 70))x1 x2 − 2(6 + α(10 − 6))x1 x3 − 2(12 + α(20 − 12))x2 x3 ≤ 1, [−(195 + 10 − 1) + 10λ]

x, y > 0, λ, δ ∈ [0, 1].

i.e., the problem is Min λ−1 −(70 + 10α)/(x1 x2 x3 ) − (80 + 10α)x1 x2 − 2(6 + 4α)x1 x3 − 2(12 + 8α)x2 x3 ≤ 1, [−204 + 10λ] x, y > 0, λ, α ∈ [0, 1].

subject to

The dual is

δ00 

δ02 δ01  (70 + 10α) (70 + 10α) (204 − 10λ)δ01 (204 − 10λδ02 )  δ03  δ04 2(6 + 4α) 2(12 + 8α) (δ01 + δ01 + δ03 + δ04 )(δ01 +δ01 +δ03 +δ04 ) . (204 − 10λ)δ03 (204 − 10λ)δ04

Max d(δ) 

λ−1 δ00

Subject to δ00  1, δ01 + δ02 + δ03 + δ04  δ00  1, − δ01 + δ02 + δ03  0, − δ01 + δ02 + δ04  0, − δ01 + δ03 + δ04  0, δ01 , δ02 , δ03 , δ04 > 0.

Solving the above systems of the linear equation, we get δ01  25 , δ02  15 δ03  15 and δ04  15 . Putting the value of δ01 , δ02 and δ03 , the corresponding optimal dual value, i.e., d(δ)  λ−1



5(70 + 10α) 2(204 − 10λ)δ01

2  5

5(70 + 10α) (204 − 10λδ02 )

1  5

2.5(6 + 4α) (204 − 10λ)δ03

It can be determined λ by the aid of d(δ)  λ−1 . Then, the above equation reduces to

1  5

2.5(12 + 8α) (204 − 10λ)δ04

1 5

.

152

5 Fuzzy Unconstrained Geometric Programming Problem

Table 5.7 Optimal solution of the Application 5.6 S. No.

α

λ

x1

x2

x3

Z(x)

1

0.0

0.99

0.557

0.279

1.626

365.291

2

0.2

0.99

0.650

0.325

1.720

281.793

3

0.4

0.99

0.746

0.373

1.817

234.709

4

0.6

0.99

0.848

0.424

1.916

209.882

5

0.8

0.99

0.952

0.476

2.019

200.576

6

1.0

0.40

1.000

0.500

2.000

200.000



5(70 + 10α) 2(204 − 10λ)

25 

5(70 + 10α) (204 − 10λ)

15 

2.5(6 + 4α) (204 − 10λ)

15 

2.5(12 + 8α) (204 − 10λ)

15

 1.

Solving the above nonlinear equation of λ for given α ∈ [0, 1] by Newton–Rapshon method, we obtain the value of λ∗ . Putting the value of λ∗ in (), we obtain the value of dual-objective function. Again from the primal–dual relations, we get δ∗ 2 −(70 + 10α)/(x1 x2 x3 )  01 ∗  , −204 + 10λ δ00 5 ∗ −(70 + 10α)x1 x2 δ 1  02 ∗  , −204 + 10λ δ00 5 ∗ −2(6 + 4α)x1 x3 δ03 1  ∗  , −204 + 10λ δ00 5 ∗ −2(‘12 + 8α)x2 x3 δ04 1  ∗  . −204 + 10λ δ00 5 Solving the above nonlinear equation for different values of “α” by Lingo, the optimal primal variables are obtained. These are given in Table 5.7.

5.5 Conclusion In this chapter, we have discussed unconstrained fuzzy geometric programming (GP) technique with a negative or positive integral degree of difficulty. Three different types of unconstrained fuzzy GP techniques (GP problem with fuzzy parametric intervalvalued function, fuzzy parametric geometric programming, and unconstrained GP under Zimmermann max-min operators) are presented here. This technique can be applied to solve the different decision-making problems (like in optimization engi-

5.5 Conclusion

153

neering, operations research inventory, and other areas). In fuzzy, we have considered triangular fuzzy number (TFN). In future, the other type of membership functions such as piecewise linear hyperbolic, L-R fuzzy number, trapezoidal fuzzy number (TrFN), parabolic flat fuzzy number (PfFN), parabolic fuzzy number (pFN), pentagonal fuzzy number (PFN), etc., can be considered to construct the membership function and then, the model can be easily solved.

References R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment. Manage. Sci. 17, B141–B164 (1970) B.Y. Cao, in Solution and Theory of Question for a Kind of Fuzzy Positive Geometric Program. Proceedings of the 2nd IFSA Congress, Tokyo, Japan 1 (1987), pp. 205–208 B.Y. Cao, Fuzzy geometric programming (I). Fuzzy Sets Syst. 53(2), 135–154 (1993) S. Islam, T.K. Roy, A fuzzy EPQ model with flexibility and reliability consideration and demand dependent unit production cost under a space constraint: a fuzzy geometric programming approach. Appl. Math. Comput. 176(2), 531–544 (2006) N.K. Mandal, T.K. Roy, A displayed inventory model with L-R fuzzy number. Fuzzy Optim. Decis. Mak. 5(3), 227–243 (2006) L.F. Mendoça, J.M. Sousa, J.M.G. Sá da Costa, Optimization problems in multivariable fuzzy predictive control. Int. J. Approximate Reasoning 36, 199–221 (2004) LINGO: The Modeling Language and Optimizer (1999). Lindo system Inc., Chicago, IL 60622, USA L. Liu, G.H. Huang, Y. Liu, G.A. Fuller, G.M. Zeng, A fuzzy-stochastic robust programming model for regional air quality management under uncertainty. Eng. Optim. 35, 177–199 (2003) E. Shivanian, E. Khorram, Monomial geometric programming with fuzzy relation inequality constraints with max-product composition. Comput. Ind. Eng. 56(4), 1386–1392 (2009) J.-H. Yang, B.-Y. Cao, in Geometric Programming with Fuzzy Relation Equation Constraints. FUZZ-IEEE (2005), pp. 557–560 S. Yousef, N. Badra, T.G. Yazied Abu-El, Geometric programming problems with fuzzy parameters and its application to crane load sway. World Appl. Sci. J. 7(1), 94–101 (2009) L.A. Zadeh, Fuzzy sets. Inf. Control 8, 338–353 (1965) M.P. Biswal, Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy Sets Syst. 51(1), 67–71 (1992)

Chapter 6

Fuzzy Unconstrained Modified Geometric Programming Problem

6.1 Introduction The theory of modified geometric programming (MGP) was first proposed in 2005 by Islam and Roy. There are many references to applications and methods of MGP in the research paper by Roy and Islam, but they developed the problem under crisp environment. Wasim and Islam (2016) developed the modified geometric programming (MGP) under fuzzy environment. In an ordinary GP model, all coefficients are crisp. However, in real-life situation, it will have some little uncertainty. Therefore, the consideration of fuzzy coefficients is more realistic. Liu et al. (2003) defined a fuzzy-stochastic robust programming model for regional air quality management under uncertainty. Mendoça et al. (2004) developed optimization problems in multivariable fuzzy predictive control. Yang and Cao (2005) worked on geometric programming with max-product fuzzy relation equation constraints. Panda et al. (2008) presented multi-item EOQ model with hybrid cost parameters under fuzzy geometric programming approach. Panda and Maiti (2009) developed multi-item inventory models with price dependent demand under geometric programming approach. In this chapter, we have discussed three types of fuzzy unconstrained modified geometric programming (MGP) methods that are (1) Unconstrained MGP problem with fuzzy parametric interval-valued function. (2) Unconstrained MGP problem with simple fuzzy parametric coefficients. (3) Unconstrained MGP problem with Zimmermann max-min operator. Definition 6.1 Let D ⊂ Rm be the (convex) subset of Rm defined by   D  (x1 , x2 , . . . , xm ) ∈ Rm |x > 0, j  1, 2, . . . m . A function g:D → Rm of the form

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_6

155

156

6 Fuzzy Unconstrained Modified Geometric Programming Problem T0 m     α g xj  ck x j kj , k1

j1

where ck > 0 for k  1, 2, . . . , T0 and αk j ∈ R for k  1, 2, . . . , T0 and j  1, 2, …, m, is called posynomial. A problem of the form Minimum

n    g xj i1

Subject to x j > 0, is called unconstrained modified geometric programming (MGP) problem. When coefficients, exponents, or objective are fuzzy in nature, then it is called unconstrained fuzzy MGP problem and it defined as follows: Minimize

n 

 g(x) , i1

Subject to x j > 0. where g˜ i (x) 

n i1

T0

k1 c˜ik

m j1

α˜

x j ik j .

3 2 5 2 5 ˜ 3 x22 + 2x ˜ 11 ˜ 21 ˜ 11 x12 + 7x x12 + 5x x22 is fuzzy modified posynomial. Example 6.1 3x 21

And, the problem of the form 3 2 5 3 2 5 ˜ 21 ˜ 11 ˜ 11 ˜ 21 Minimize 3x x12 + 7x x12 + 5x x22 + 2x x22 Subject to x11 , x12 , x21 , x22 > 0,

is a fuzzy unconstrained MGP problem.

6.2 Unconstrained MGP Problem with Fuzzy Parametric Interval-Valued Function Primal Program A primal fuzzy modified geometric programming problem has the form Min g˜ i (x) 

T0 n   i1 k1

Subject to xi j > 0,

c˜ik

m 

α

x j ik j

j1

(6.1)

6.2 Unconstrained MGP Problem with Fuzzy Parametric Interval-Valued Function

157

Here, α0k j are real numbers and coefficients c˜ik are triangular fuzzy numbers 1 2 3 , cik , cik ). (TFN) as c˜ik  (cik Using nearest interval approximation (NIA) method, we transform all triangular U L , cik ]. The geometric programming fuzzy numbers into interval numbers, i.e., [cik problem with imprecise parameters is of the following form: Min gˆ i (x) 

T0 n   i1 k1

cˆik

m  j1

α

x j ik j

,

(6.2)

Subject to xi j > 0, U U L L where cˆik denotes the interval counterparts, i.e., cˆik ∈ [cik , cik ].cik > 0, cik > 0 for all i and k. Using parametric interval-valued functional form, the problem (6.2) reduces to

Min gi (x, s) 

T0 m n    L 1−s  U s  α x j ik j cik cik i1 k1

j1

Subject to xi j > 0 for j  1, 2, . . . m.

(6.3)

This is a parametric geometric programming form. Dual Program The corresponding dual of (6.3) is

 1−s  s δik U L cik cik Max di (δ, s)  δik i0 k0 T0 n  

Subject to T0 

δik  1,

.

(6.4)

k1 T0 

αik j δik  0,

k1

δik > 0. Case 1 For nT 0 ≥ nM + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables. In this case, more or unique solution exists for the dual vectors. Case 2 For nT 0 < nM + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case, generally, no solution vector exists for the dual variables. However, one can get an approximate solution vector for the system using either the latest square (SQ) or max-min method.

158

6 Fuzzy Unconstrained Modified Geometric Programming Problem

These are applied to solve such a system of linear equations. Once optimal dual variable vectors δ ∗ are known, the corresponding values of the primal variable vector x is found from the following relations: m  L 1−s  U s  α ∗ n ∗ cik cik x j k j  δik v (δ), (i  1, 2, . . . , n; k  1, 2, . . . , T0 ). j1

Theorem 6.1 If x is a feasible vector for the constraints PGP and δ is a feasible vector for the corresponding DP, then gi (x, s) ≥ n n di (δ, s) (Primal-Dual Inequality) Proof The expression for gi (x, s) can be written as gi (x, s) 

 L 1−s  U s m αik j  cik cik j1 x j δik . δik k1

T0 n   i1

α01 j L 1−s U s (ci1 ) (ci1 ) mj1 x j δi1

Here, the weights are δi1 , δi2 , . . . , δi T0 and positive terms are αi T j (ciLT )1−s (ciUT )s mj1 x j 0 0 0 δi TO

αi2 j L 1−s U s (ci2 ) (ci2 ) mj1 x j δi2

,

,..., . Now applying A.M.-.G.M Inequality, we get 

⎛ ⎜ ⎜ ⎝

≥

n i1

 1−s    L 1−s  U s m  L 1−s  U s m αi T0 j αi1 j αi2 j m U s + ci2 + · · · + ciLT0 ci1 ci2 c0i ci1 j1 x j j1 x j j1 x j n (δ + δ + · · · + δ ) 02 0T0 i1 01

n  ⎞ (δi1 +δi2 +···+δ0T i1

0

)

⎟ ⎟ ⎠

⎛ ⎞δi T0 ⎞ ⎛ 1−s  s ⎞δi1 ⎛     ⎞δi2 ⎛    αi T0 j m αi1 j αi2 j 1−s U s m m n L 1−s cU s ciLT0 ciUT0  ci2 cL ⎜ ci1 j1 x j j1 x j j1 x j i1 ⎟ ⎟ ⎜⎝ ⎠ ⎝ i2 ⎠ ...⎜ ⎠ ⎟ ⎝ ⎝ ⎠ δi1 δi2 δi T0 i1

or

n  0  δik T

gi (x, s) n T0 i1

i1 k1

k1 δik



L α δik  T0 T0 n  U s   (cik )1−s (cik ) mj1 x j ik j ≥ δik  1 as δik i1 k1 k1

or 

gi (x, s) n

 1−s  s  0 δik m n U L   k1 cik cik T

n ≥

i1



αik j δik

xj

j1

 1−s  s δik m T0 n  U L   cik cik i1 k1

or

δik

T0  k1

δik

j1

T0 

x k1 j

αik j δik

6.2 Unconstrained MGP Problem with Fuzzy Parametric Interval-Valued Function



gi (x, s) n

n ≥

T0  L 1−s U s δik n   (c ) (c ) ik

i1 k1



ik

δik

as

T0 

159

 αik j δik  0

k1

 di (δ, s) √ i.e., gi (x, s) ≥ n n di (δ, s). This completes the proof. Theorem 6.2 δ is a feasible vector for the dual programming (DP) problem, then di (δ, 1) ≥ di (δ, 0). U L Proof We have cik ≥ cik , for all k, (k  1,2, …, T 0 ). or

 L 1−1  U 1  L 1−0  U 0 cik cik ≥ cik cik or  L 1−1  U 1  L 1−0  U 0 cik cik cik c ≥ ik δik δik or

 1−1  1 δik  1−0  0 δik U U L L cik cik cik cik ≥ δik δik or

 1−1  1 δik

 1−0  0 δik T0 U U L L  cik cik cik cik ≥ δik δik k0 k0

Ti 

or

 1−1  1 δik

 1−0  0 δik T0 n  U U L L  cik cik cik cik ≥ δik δik i1 k0 i1 k0

Ti n  

i.e., di (δ, 1) ≥ di (δ, 0). Application 6.1 MGP Problem (Multi-Grain-Box Problem) Suppose that to shift grains from a warehouse to a factory in a finite number (say n) of open rectangular boxes of lengths x1i meters, widths x2i meters and heights x3i meters (i  1, 2, …., n). The bottom, side, and ends of the box cost $ai , $bi and

160

6 Fuzzy Unconstrained Modified Geometric Programming Problem

Fig. 6.1 Multi-grain-box problem Table 6.1 Crisp input values for Application 6.1

i-th Box

ai ($/m 2 )

bi ($/m 2 )

ci ($/m 2 )

d (m 3 )

i1

80

10

20

80

i2

60

20

30

50

$ci /m 2 , respectively. It costs $1 for each round trip of the box. Assuming that the box will have no salvage value, find the minimum cost of transporting di m 3 of grains (Fig. 6.1). This problem can be formulated as an unconstrained MGP problem

Min g0 (x, s) 

n   i1

 di + ai x1i x2i + 2bi x1i x3i + 2ci x2i x3i x1i x2i x3i .

(6.5)

Sub to x1i ≥ 0, x2i ≥ 0, x3i ≥ 0 (i  1, 2, . . . , n). In particular, here, we assume that the transporting di m 3 of grains by the two different open rectangular boxes whose bottom, sides, and the ends of each box costs are given in Table 6.1. Solution When the input data is taken as triangular fuzzy numbers, i.e., a˜ 1  (70, 80, 90), a˜ 2  (50, 60, 70), b˜1  (8, 10, 12), b˜2  (16, 20, 24), c˜1  (16, 20, 24), c˜2  26, 30, 34 and d˜1  (70, 80, 90), d˜2  (40, 50, 60). Using nearest interval approximation method, we get the corresponding interval number and interval-valued function, i.e., a˜ 1  [75, 85], ⇒ aˆ 1  (75)1−s (85)s ∈ [75, 85], a˜ 2  [75, 85], ⇒ aˆ 2  (55)1−s (65)s ∈ [55, 65], b˜1  [9, 11], ⇒ bˆ1  (9)1−s (11)s ∈ [9, 11], b˜2  [18, 22], ⇒ bˆ2  (18)1−s (22)s ∈ [18, 22], c˜1  [18, 22], ⇒ cˆ1  (18)1−s (22)s ∈ [18, 22], c˜2  [28, 32], ⇒ cˆ2  (28)1−s (32)s ∈ [28, 32], d˜1  [75, 85], ⇒ dˆ1  (75)1−s (85)s ∈ [75, 85], d˜2  [45, 55], ⇒ dˆ2  (45)1−s (55)s ∈ [45, 55],

6.2 Unconstrained MGP Problem with Fuzzy Parametric Interval-Valued Function

161

where s ∈ [0, 1]. Here, the primal problem is Min g0 (x, s) 

2  i1

dˆi + aˆ i xi1 xi2 + 2bˆi x1 x3 + 2cˆi xi2 xi3 xi1 xi2 xi3

Sub to xi1 ≥ 0, xi2 ≥ 0, xi3 ≥ 0, i  1, 2.

(6.6)

The corresponding dual form is

δi1  

δi3   2  aˆ i δi2 2bˆi 2cˆi δi4 dˆi Max d(δ, s)  δi1 δi2 δi3 δi4 i1 Subject to δ11 + δ12 + δ13 + δ14  1 δ21 + δ22 + δ23 + δ24  1 − δ11 + δ12 + δ13  0 − δ21 + δ22 + δ23  0 − δ11 + δ12 + δ14  0 − δ21 + δ22 + δ24  0 − δ11 + δ13 + δ14  0 − δ21 + δ23 + δ24  0 δ11 , δ12 , δ13 , δ14 δ21 , δ22 , δ23 , δ24 ≥ 0. From the above, we get δ11  25 , δ12  15 , δ13  15 , δ14  15 , δ21  25 , δ22  15 , δ23  15 and δ24  15 . The optimal solution of the model through the parametric approach is given by

d ∗ (δ, s) 

5(75)1−s (85)s 2

×

2

5(45)1−s (55)s 2

5

5(75)1−s (85)s 1

2

5

1

5(55)1−s (65)s 1

5

2 × 5(9)1−s (11)s 1

1

5

1

5

2 × 5(18)1−s (22)s 1

2 × 5(18)1−s (22)s 1

1

5

From primal–dual relations, we get (75)1−s (85)s   δ11 × 2 d ∗ (δ, s), x11 x12 x13 1−s  × 2 d ∗ (δ, s), (75) (85)s x11 x12  δ12  × 2(9)1−s (11)s x11 x13  δ13

2 d ∗ (δ, s),

1 5

2 × 5(28)1−s (32)s 1

1 5

162

6 Fuzzy Unconstrained Modified Geometric Programming Problem

Table 6.2 Optimal solution of the Application 6.2 S

∗ x11

∗ x12

∗ x12

∗ x21

∗ x22

∗ x23

g0 (x, s)∗

0

1.032

0.516

2.149

0.963

0.619

0.946

363.489

0.2

1.030

0.515

2.112

0.963

0.627

0.952

375.474

0.4

1.027

0.514

2.077

0.962

0.635

0.958

387.917

0.6

1.025

0.513

2.042

0.962

0.644

0.964

400.820

0.8

1.023

0.512

2.007

0.962

0.652

0.970

414.233

1.0

1.021

0.511

1.975

0.961

0.661

0.976

428.160

Table 6.3 Comparison result

S

g0 (x, s)∗

d ∗ (δ, s)

√ 2 2 d(δ, s)

0

363.489

32,713.23

361.736

0.2

375.474

34,986.31

374.093

0.4

387.917

37,417.33

386.871

0.6

400.820

40,017.26

400.086

0.8

414.233

42,797.86

413.753

1.0

428.160

45,771.67

427.886

 2(18)1−s (22)s x12 x13  δ14 ×

2

d ∗ (δ, s),

(45)1−s (55)s   δ21 × 2 d ∗ (δ, s), x21 x22 x23  × 2 d ∗ (δ, s), (55)1−s (65)s x21 x22  δ22  2(18)1−s (22)s x21 x23  δ23 ×  × 2(28)1−s (32)s x22 x23  δ24

2

d ∗ (δ, s),

2

d ∗ (δ, s).

The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 6.2. Theorem 6.1 in MGP problem (multi-grain-box Problem) is verified in Table 6.3. √ From the table, we see that g0 (x, s)∗ ≥ 2 2 d(δ, s) for all s ∈ [0, 1], so Theorem 6.1 is verified. For s  0, the lower bound of the interval value of the parameter is used to find the optimal solution. For s  1, the upper bound of interval value of the parameter is used for the optimal solution. These results yield the lower and upper bounds of the optimal solution. The main advantage of the proposed technique is that one can get the intermediate optimal result using proper value s.

6.3 Unconstrained MGP Problem with Simple Fuzzy Parametric Coefficients

163

6.3 Unconstrained MGP Problem with Simple Fuzzy Parametric Coefficients Consider a fuzzy geometric programming problem as follows:  Min g(x) Subject to x > 0

(6.7)

n T0 m αik j Its objective of the form g(x)  i1 are all posynomial of k1 c˜ik j1 x j x in which coefficients c˜ik are fuzzy numbers. A real number c ik described as fuzzy subset on the real line R, whose membership function μcik (x) has the following characteristics with − ∝< cik1 ≤ cik2 ≤ cik3 0.

(6.8)

Which is equivalent to Min

T0 n  

cik L (α)

i1 k1

m 

α

x j ik j

j1

Subject to x j > 0.

(6.9)

Dual problem The corresponding dual programming (DP) problem of (6.9) is Max d(δ, s) 

 T0   cik L (α) δ0k k0

Subject to

δ0k

164

6 Fuzzy Unconstrained Modified Geometric Programming Problem T0 

δok  1,

k1 T0 

αk j δ0k  0,

k1

δ0k > 0.

(6.10)

Case 1 For T 0 ≥ M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables. More or unique solutions exist for the dual vectors. Case 2 For T 0 < M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case, generally, no solution vectors exist for the dual variables. However, one can get an approximate solution vector for the system using either the least square (SQ) or max-in method. These are applied to solve such a system of linear equations. Once the optimal dual variable vectors δ ∗ are known, the corresponding values of the primal variable vector x is found from the following relations: cik L (α)

m 

  α x j k j δk∗ v ∗ δ ∗ , (i  1, 2, . . . , n; k  1, 2, . . . , T0 ).

j1

Application 6.2 (Unconstrained MGP Problem) −3 −1 −2 −3 −1 −2 ˜ 31 + 1t ˜ 11 t21 t31 ˜ 31 t11 ˜ 12 t22 t32 ˜ 32 t12 ˜ 32 Minimize Z  1t + 2t t21 + 5t + 3t t22 + 4t subject to ti j > 0, (i  1, 2; j  1, 2, 3).

This problem can be written as

Minimize Z (t) 

2  3  i1 k1

C˜ ik

3 

α

ti jik j

j1

subject to ti j > 0, (i  1, 2; j  1, 2, 3), ˜ c˜21  1, ˜ c˜12  2, ˜ c˜13  5, ˜ c˜22  3, ˜ c˜23  4, ˜ α111  1, α121  −2, where c˜11  1, α221  1, α131  1, α231  −3, α112  −1, α212  −1, α122  1, α222  −2, α132  −2, α232  1, α113  0, α213  0, α123  1, α223  0, α133  0, α233  1. Let us consider fuzzy coefficients as

6.3 Unconstrained MGP Problem with Simple Fuzzy Parametric Coefficients

165

1˜  0.8 + (1 − α) × 0.4, 2˜  1.6 + (1 − α) × 0.8, 3˜  2 + (1 − α) × 2, 4˜  3 + (1 − α) × 2, 5˜  4 + (1 − α) × 2, Dual Program The dual programming (DP) of the above MGP is as follows: 

     0.8 + (1 − α) × 0.4 δ11 1, 6 + (1 − α) × 0.8 δ12 4 + (1 − α) × 2 δ13 δ11 δ12 δ13 δ21  δ22    2 + (1 − α) × 2 3 + (1 − α) × 2 δ23 0.8 + (1 − α) × 0.4 × δ21 δ22 δ23

v(δ) 

subject to δ11 + δ12 + δ13  1 δ21 + δ22 + δ23  1 δ11 − δ12  0 − 3δ11 + δ12 + δ13  0 δ11 − 2δ12  0 δ21 − δ22  0 δ21 − 2δ22  0 − 3δ21 + δ22 + δ23  0 δ11 , δ12 , δ13 , δ21 , δ22 , δ23 > 0 where δ  (δ11 , δ12 , δ13 , δ21 , δ22 , δ23 )T . There is a system of eight linear equations with six unknown variables. By applying LS method, the above system of linear equations reduces to δ11 − δ12  0 − 2δ11 + δ12 + δ13  0 δ11 + δ12 + δ13  1 δ21 − 2δ22  0 − 3δ21 + δ22 + δ23  0 δ21 + δ22 + δ23  1

166

6 Fuzzy Unconstrained Modified Geometric Programming Problem

Table 6.4 Optimal solution of the Application 6.2 S. No.

α

t11

t21

t31

t12

t22

t32

v ∗ (δ* )

z ∗ (t)

1

0.0

0.34

1.15

0.45

2.50

0.78

1.00

64.94

21.940

2

0.2

0.34

1.14

0.44

2.51

0.78

1.02

55.81

20.191

3

0.4

0.35

1.14

0.44

2.51

0.77

1.02

47.35

18.921

4

0.6

0.35

1.14

0.44

2.51

0.76

1.03

39.58

17.371

5

0.8

0.35

1.15

0.43

2.52

0.76

1.05

32.49

15.971

6

1.0

0.36

1.15

0.43

2.52

0.76

1.06

26.07

14.342

δ11 , δ12 , δ13 , δ21 , δ22 , δ23 > 0. The approximate solutions of this system of linear equations are δ *11  0.333, δ *12  0.333, δ *13  0.333, δ *21  0.25, δ *22  0.125, δ *23  0.625, and optimal dual-objective value, i.e.,      0.8 + (1 − α) × 0.4 0.333 1, 6 + (1 − α) × 0.8 0.333 4 + (1 − α) × 2 0.333 0.333 0.333 0.333       0.8 + (1 − α) × 0.4 0.25 2 + (1 − α) × 2 0.125 3 + (1 − α) × 2 0.625 × 0.25 0.125 0.625

v ∗ (δ ∗ ) 



And, the following system of nonlinear equations gives optimal primal variables: −2  0.333 × (0.8 + (1 − α) × 0.4)t11 t21 t31

v ∗ (δ∗),

−1 −1 t21  0.333 × (1.6 + (1 − α) × 0.8)t31 t11

v ∗ (δ∗),

(4 + (1 − α) × 2)t31  0.333 ×

v ∗ (δ∗)

−1 −1 t22  0.125 × (0.8 + (1 − α) × 0.4)t32 t12 −1 −1 t22  0.125 × (0.8 + (1 − α) × 0.4)t32 t12

(3 + (1 − α) × 2)t32  0.625 ×



v ∗ (δ∗),



v ∗ (δ∗),

v ∗ (δ∗).

Solving the above nonlinear equation for different values of “α” by Lingo, the optimal primal variables are obtained. These are given in Table 6.4. Application 6.3 Consider the Application 6.1 (Multi-Grain-Box Problem). Here, the primal problem is Min g0 (x) 

2  i1

d˜i + a˜ i xi1 xi2 + 2b˜i x1 x3 + 2c˜i xi2 xi3 xi1 xi2 xi3

Sub to xi1 ≥ 0, xi2 ≥ 0, xi3 ≥ 0.

(6.11)

6.3 Unconstrained MGP Problem with Simple Fuzzy Parametric Coefficients

167

Taking fuzzy coefficient in parametric form   8 + (1 − α) × 4, 10   15 + (1 − α) × 10, 20   24 + (1 − α) × 12, 30   40 + (1 − α) × 20, 50   50 + (1 − α) × 20, 60   70 + (1 − α) × 20. 80 The corresponding dual form is 

Max d(δ, α) 

               d1 δ11 a1 δ12 2b1 δ13 2c1 δ14 d2 δ21 a2 δ22 2b2 δ23 2c2 δ24 δ11 δ12 δ13 δ14 δ21 δ22 δ23 δ24

Subject to δ11 + δ12 + δ13 + δ14  1 δ11 + δ12 + δ13 + δ14  1 − δ11 + δ12 + δ13  0 − δ11 + δ12 + δ13  0 − δ11 + δ12 + δ14  0 − δ11 + δ12 + δ14  0 − δ11 + δ13 + δ14  0 − δ11 + δ13 + δ14  0

(6.12)

δ11 , δ12 , δ13 , δ14 δ21 , δ22 , δ23 , δ24 ≥ 0.

1 5

From (6.12), we get δ11  25 , δ12  15 , δ13  15 , δ14  15 , δ21  25 , δ22  15 , δ23  and δ24  15 . The optimal solution of the model through the parametric approach is given by 1

d ∗ (δ, α) 



5.80 2

2  5

5.80 1

1  5

2.5.10 1

1  5

2.5.20 1



5.50 2

2  5

5.50 1

1  5

2.5.20 1

1 

From primal–dual relation, we get 70 + (1 − α) × 20   δ11 × 2 d ∗ (δ, α), x11 x12 x13  × 2 d ∗ (δ, α), (70 + (1 − α) × 20)x11 x12  δ12  2.(8 + (1 − α) × 4)x11 x13  δ13 ×

2

d ∗ (δ, α),

5

2.5.30 1

1 5 5

168

6 Fuzzy Unconstrained Modified Geometric Programming Problem

 2.(15 + (1 − α) × 10)x12 x13  δ14 ×

2 d ∗ (δ, α),

40 + (1 − α) × 20   δ21 × 2 d ∗ (δ, α), x21 x22 x23  × 2 d ∗ (δ, α), (50 + (1 − α) × 20)x21 x22  δ22  × 2.(15 + (1 − α) × 10)x21 x23  δ23  × 2.(24 + (1 − α) × 12)x22 x23  δ24

2

d ∗ (δ, α),

2

d ∗ (δ, α).

6.4 Unconstrained MGP Problem with Zimmermann Max-Min Operator Primal fuzzy modified geometric programming problem has the form Min g˜ i (x) 

T0 n  

c˜ik

i1 k1

m 

α

x j ik j

j1

Subject to xi j > 0,

(6.13)

Here, α0k j are real numbers and coefficients c˜ik are fuzzy triangular numbers. Proposition 6.1 When, the coefficient and indexes of the fuzzy geometric programming problem are taken as fuzzy numbers  Min

T0 n   i1 k1

c˜ik

m 

α˜

x j ik j

j1

Subject to x j > 0. Then using δ-cut of fuzzy numbers coefficients and indexes, the above problem reduces to T0 n  m  μα˜ ik j L −1 (δ),μα˜ ik j R −1 (δ)    μc˜ik L −1 (δ), μc˜ik R −1 (δ) xj Min i1 k1

Subject to x j > 0, which is equivalent to

j1

!

6.4 Unconstrained MGP Problem with Zimmermann Max-Min Operator

 Min

To n  

μc˜ik L −1 (δ)

i1 k1

m 

μα˜ ik j S −1

xj

169

(δ)

j1

where " μα˜ ik j S −1 (δ) 

˜ ik j L > 0, μ−1 α˜ ik j L (δ) when α μ−1 when α ˜ (δ) ik j L < 0, α˜ ik j R

Definition 6.2 For any x ∈ Rm and feasible index di ∈ R (R is the real number set), μ−1 n Ti m α˜ ik j S −1 if gi (x, δ)  i1 μ x (δ) ≤ 1, (1 ≤ i ≤ n), then the linear (δ) k1 c˜ik L j1 j membership functions are given by

μ0 (gi (x, δ)) 

⎧ ⎪ ⎨ 1 ⎪ ⎩

z 0 +d0 −gi (x,δ) d0

 if gi (x, δ) ≤ z 0 , if z 0 ≤ gi (x, δ) ≤ z 0 + d0 ,

(6.14)

if gi (x, δ) ≥ z 0 + d0

0

Based on Zimmerman max-min approach, the first finding δ-cut of the fuzzy numbers in coefficients and indexes, then we built membership functions of both objective and constraints goals and using max-min operator the above problem (6.13) reduced to a fuzzy nonlinear programming (FNLP) problem Max

λ

subject to μ

Ti n  

i1 k1

μ−1 c˜k L (δ)

m

μ−1 α˜

j1

xj

kjS

 (δ) ≥ λ

(6.15)

x j > 0, λ, δ ∈ [0, 1] which is equivalent to a geometric programming problem with parameters λ, δ variation λ−1

 Ti n  m μ−1  α˜ ik j S −1 subject to μ μc˜ik L (δ) xj (δ) ≥ λ Min

i1 k1

(6.16)

j1

x j > 0, λ, δ ∈ [0, 1]. Theorem 6.3 Let the membership function μ(g(x, δ)), μc˜k L (cik ), μα˜ k j S (αk j ) be all continuous and strictly monotone. Then (6.16) is equivalent with Min

λ−1 n

subject to

i1

Ti k1

−1

μ m α˜ ik j S μ−1 (δ) j1 x j c˜ L (δ) ik n −1 μ (δ) i1 i

≤ 1,

x > 0, λ, δ ∈ [0, 1], (1 ≤ j ≤ m).

170

6 Fuzzy Unconstrained Modified Geometric Programming Problem

Proof See Islam and Roy (2006). Corollary 6.1 Let the membership function μ(g(x, δ)), μc˜k (cik ), μα˜ k j (αk j ) be all continuous, strictly monotone and the problem is λ−1

Min

n

subject to

−1

Ti

i1

k1

μ m α˜ ik j S μ−1 (δ) j1 x j c˜ik L (δ) n −1 μ (δ) i1 i

≤ 1,

x j > 0, λ, δ ∈ [0, 1], (1 ≤ j ≤ m). which is a classical posynomial geometric programming with parameters γ , δ. Its dual form is  Max d(ω) 

λ−1 ω00

−1 ωk ω00  T0 n  μc˜ik j S (δ)/μ−1 (λ) i1 k1

ωk

subject to ω00  1, ω00 

T0 

ωik

k1 T0 

−1 α˜ ik j S (δ)ωik  0, λ, δ ∈ [0, 1],

K 1

ωik (≥ 0 where ωik  ωik (δ, λ) Application 6.4 Consider the Application 6.1 (Multi-Grain-Box Problem). Solution The solution of Application 6.1 by any simple crisp method is x11  1.026, x12  0.513, x13  2.053, x21  0.962, x22  0.639, x23  0.961 and minimum cost Z (x)  393.618.   When the coefficients are taken as triangular fuzzy number (TFN), i.e., 10   (40, 50, 60), 60   (15, 20, 25), 30   (24, 30, 36), 50   (6, 10, 14), 20   (70, 80, 90). (50, 60, 70), 80 The lower bounds of the tolerance interval z 0 (t)  393. Spreads of tolerance intervals d0  10. Taking the membership function

6.4 Unconstrained MGP Problem with Zimmermann Max-Min Operator

μ(g(x)) 

⎧ ⎪ ⎨1

g0 +do −g(x) , d0

⎪ ⎩ 0,

g(x) ≤ g0 g0 < g(x) ≤ g0 + d0 . g(x) > g0 + d0

Then, the problem transforms to Min λ−1 Subject to

$ 1 (70 + α(80 − 70)) − − (70 + α(80 − 70))x1 x2 x1 x2 x3 [−(393 + 10 − 1) + 10λ]

− 2(6 + α(10 − 6))x1 x3 − 2(15 + α(20 − 15))x2 x3 −

(40 + (50 − 40)α) x1 x2 x3

 − (50 + α(60 − 50))x1 x2 − 2(15 + α(20 − 15))x1 x3 −2(24 + α(30 − 24))x2 x3 ≤ 1,

i.e., the problem is Min λ−1 Subject to

$ 1 (70 + 10α) − (70 + 10α)x1 x2 − 2(6 + 4α)x1 x3 − −402 + 10λ x1 x2 x3

−2(15 + 5α)x2 x3 −

% (40 + 10α) − (50 + 10α)x1 x2 − 2(15 + 5α)x1 x3 − 2(24 + 6α)x2 x3 ≤ 1, x1 x2 x3

x1 , x2 .x3 > 0, λ, δ ∈ [0, 1].

171

172

6 Fuzzy Unconstrained Modified Geometric Programming Problem

The dual of the above problem is δ01  δ02 (70 + 10α) (70 + 10α) (204 − 10λ)δ01 (204 − 10λδ02 ) δ03  δ04  δ05  2(12 + 8α) 2(6 + 4α) (70 + 10α) × (204 − 10λ)δ03 (204 − 10λ)δ04 (204 − 10λ)δ05 δ07  δ08 δ06   2(6 + 4α) 2(12 + 8α) (70 + 10α) × (204 − 10λδ06 ) (204 − 10λ)δ07 (204 − 10λ)δ08 

Max d(δ) 

λ−1 δ00

δ00 

× (δ01 + δ01 + δ03 + δ04 + δ05 + δ06 + δ07 + δ08 )(δ01 +δ01 +δ03 +δ04 +δ05 +δ06 +δ07 +δ08 ) Subject to δ00  1, δ01 + δ02 + δ03 + δ04 + δ05 + δ06 + δ07 + δ08  1, − δ01 + δ02 + δ03 − δ05 + δ06 + δ07  0, − δ01 + δ02 + δ04 − δ05 + δ06 + δ08  0, − δ01 + δ03 + δ04 − δ05 + δ07 + δ08  0, δ01 , δ02 , δ03 , δ04 , δ5 , δ06 , δ07 , δ08 > 0. 2 1 1 1 Solving the above equations, we get δ01  10 , δ02  10 δ03  10 , δ04  10 , 2 1 1 1 δ05  10 , δ06  10 , δ07  10 , δ08  10 . Putting the value of δ01 , δ02 , δ03 , δ04 , δ5 , δ06 , δ07 and δ08 into the corresponding optimal dual value, i.e., 1  2  1  1 10(70 + 10α) 10(70 + 10α) 2.10(6 + 4α) 2.10(12 + 8α) 10 10 10 10 2(402 − 10λ)δ01 (402 − 10λδ02 ) (402 − 10λ)δ03 (402 − 10λ)δ04 1  2  1  1  10(50 + 10α) 2.10(24 + 6α) 10 10(40 + 10α) 10 10 2.10(15 + 5α) 10 × 2(402 − 10λ)δ01 402 − 10λ)δ03 (402 − 10λδ02 ) (402 − 10λ)δ04

d(δ)  λ−1



it can be determined λ by the aid of d(δ)  λ−1 . Then, the above equation reduces to 

1  2  1  1 10(70 + 10α) 10(70 + 10α) 2.10(6 + 4α) 2.10(12 + 8α) 10 10 10 10 2(402 − 10λ)δ01 (402 − 10λδ02 ) (402 − 10λ)δ03 (402 − 10λ)δ04 1  2  1  1  10(50 + 10α) 2.10(15 + 5α) 10 2.10(24 + 6α) 10 10(40 + 10α) 10 10 ×  1. 2(402 − 10λ)δ01 (402 − 10λδ02 ) (402 − 10λ)δ03 (402 − 10λ)δ04

Solving the above nonlinear equation of λ for given α ∈ [0, 1] by Newton–Raphson method, we obtain the value of λ∗ . Putting the value of λ∗ , we obtain the value of dual objective function. Again from the primal–dual relations, we get δ∗ −(70 + 10α)/(x1 x2 x3 ) 2  01 , ∗  −402 + 10λ δ00 10

6.4 Unconstrained MGP Problem with Zimmermann Max-Min Operator

173

Table 6.5 Optimal solution of the Application 6.4 S. No.

α

λ

x11

x12

x13

x21

x22

x23

Z(x, y)

1

1

0.99

1.040

0.520

2.081

1.171

0.650

0.976

429.292

δ∗ −(70 + 10α)x1 x2 1  02 , ∗  −402 + 10λ δ00 10 −2(6 + 4α)x1 x3 1 δ∗  03 , ∗  −402 + 10λ δ00 10 δ∗ 1 −2(‘12 + 8α)x2 x3  04 . ∗  −402 + 10λ δ00 10 δ∗ −(40 + 10α)/(x1 x2 x3 ) 2  05 , ∗  −402 + 10λ δ00 10 −(50 + 10α)x1 x2 δ∗ 1  06 , ∗  −402 + 10λ δ00 10 −2(15 + 5α)x1 x3 δ∗ 1  07 , ∗  −402 + 10λ δ00 10 −2(‘24 + 6α)x2 x3 δ∗ 1  08 . ∗  −402 + 10λ δ00 10 Solving the above nonlinear equation for “α  1” by Lingo, the optimal primal variables are obtained. These are given in Table 6.5.

6.5 Conclusion In this chapter, we have discussed about fuzzy unconstrained modified geometric programming (MGP) technique with negative or positive integral degree of difficulty (DD). Three different types of fuzzy unconstrained MGP techniques (unconstrained MGP problem with fuzzy parametric interval-valued function, unconstrained MGP problem with simple fuzzy parametric coefficients, and unconstrained MGP problem with Zimmermann max-min operator) are presented here. This technique can be applied to solve the distinctive decision-making problems (inventory, optimization engineering, and different areas). The parameters, for the most part, are assessed in view of the past trial and managerial judgment. In this way, all things considered, circumstance fuzzy set theory is more sensible than crisp set theory or probability theory.

174

6 Fuzzy Unconstrained Modified Geometric Programming Problem

References S. Islam, T.K. Roy, Modified geometric programming problem and its applications. J. Appt. Math. Comput. 17(1–2), 121–144 (2005) S. Islam, T.K. Roy, A fuzzy EPQ model with flexibility and reliability consideration and demand dependent unit production cost under a space constraint: a fuzzy geometric programming approach. Appl. Math. Comput. 176(2), 531–544 (2006) L. Liu, G.H. Huang, Y. Liu, G.A. Fuller, G.M. Zeng, A fuzzy-stochastic robust programming model for regional air quality management under uncertainty, engineering. Optimization 35, 177–199 (2003) L.F. Mendoça, J.M. Sousa, J.M.G. Sá da Costa, Optimization problems in multivariable fuzzy predictive control. Int. J. Approximate Reasoning 36, 199–221 (2004) D. Panda, M. Maiti, Multi-item inventory models with price dependent demand under flexibility and reliability consideration and imprecise space constraint: a geometric programming approach. Math. Comput. Model. 49, 1733–1749 (2009) (September 11, 2012 18:42 Atlantis Press Book 9.75in x 6.5in book_Kahraman Bibliography 449) D. Panda, S. Kar, M. Maiti, Multi-item EOQ model with hybrid cost parameters under fuzzy/fuzzystochastic resource constraints: a geometric programming approach. Comput. Math. Appl. 56, 2970–2985 (2008) A.M. Wasim, S. Islam, Fuzzy unconstrained parametric geometric programming problem and its application. J. Fuzzy Set Valued Anal. 2016(2), 125–139 (2016). http://dx.doi.org/10.5899/2016/ jfsva-00301 J.H. Yang, B.Y. Cao, Geometric programming with max-product fuzzy relation equation constraints, in Proceedings of the 24th North American Fuzzy Information Processing Society, Ann Arbor, Michigan, June 22–25, 650–653 (2005)

Chapter 7

Fuzzy Constrained Geometric Programming Problem

7.1 Introduction Duffin et al. (1967) put a foundation stone to solve extensive variety of engineering problems by creating fundamental theories of geometric programming and its application in their book. Geometric programming infers its name from its close association with geometrical ideas since the method depends on inequality of geometric and their properties that relate with products and sums of positive numbers. The use of geometric inequality has additionally been exceptionally helpful in the development of the condensation technique for posynomial problems. Following Zadeh (1965), critical commitments toward this direction have been connected in numerous fields including production and engineering-related areas. Sommer (1981) connected fuzzy dynamic programming to an inventory and production scheduling problem in which the administration cuisses to satisfy an agreement for giving an item and after that pull back from the market. Kacprzyk et al. (1982) presented the assurance of ideal of firms from a global view purpose of best administration in a fuzzy environment with fuzzy imperatives enhanced reappointments and a fuzzy goal for ideal stock levels to be accomplished. In a constrained fuzzy geometric programming, the constraints must be of the form posynomial ≤1, but this is not so restrictive an assumption as it might first appear, since many constraints may be transformed into this given form  ⇔ 20x  1−1 x2−1 ≤ 1, x1 x2 ≥ 20  8x2 ≤ x3 ⇔  4x1 x3−1 +  8x2 x3−1 ≤ 1, 4x1 +   2x −  3y ≤ z 2 ⇔  2x z −2 −  3yz −2 ≤ 1, . . . etc. where “~” denotes the fuzzy number.

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_7

175

176

7 Fuzzy Constrained Geometric Programming Problem

Definition 7.1 Let D ⊂ Rm be the (convex) subset of Rm defined by   D  (x1 , x2 , . . . , xm ) ∈ Rm |x > 0, j  1, 2, . . . m . A function g0 : D → Rm of the form T0 m     α go x j  ck x j kj , k1

j1

where ck > 0 for k  1, 2, . . . , T0 and αk j ∈ R for k  1, 2, . . . , T0 and j  1, 2, …, m, is called posynomial. A constrained geometric program in standard form looks like this Minimum g0 (x) Subject to gi (x) ≤ 1, i  i, 2, . . . , m, f i (x) ≤ 1, i  1, 2, . . . p, x > 0,

(7.1)

where gi are posynomials, f i are monomials, and x i are optimization variables. The objective must be to minimize a posynomial. Often times the geometric program must be reformulated into standard form. If presented with a maximizing problem, the inverse can be taken to convert it into a minimizing problem. When the objective and constraint goals, coefficients, and exponents become fuzzy sets and fuzzy numbers respectively, then we transform (6.1) into a fuzzy geometric programming as follows: Minimum  g0 (x) Subject to  gi (x) ≤ 1, i  i, 2, . . . , m,  f i (x) ≤ 1, i  1, 2, . . . p, x > 0,

(7.2)

f i are fuzzy monomials, and x i are optimization where  g0 are fuzzy posynomials,  variables. Example 7.1 Consider the following example problem: 

2

Maximize 2xy 2 Subject to  2 ≤ x ≤ 5,  3 ≤ y ≤ 4,  2y 2 +  5x 2 +  3x 2 y 3 ≤  6 x, y > 0. The equivalent standard form FGP is as follows:

7.1 Introduction

177

Minimize  2−1 x 2−1 y 2 Subject to  2x −1 ≤ 1 x ≤1  5  3y −1 ≤ 1 y ≤1  4 2  6≤1 5x 2 +  3x 2 y 3 / 2y +  x, y > 0.

7.2 Geometric Programming Problem with Fuzzy Coefficient Primal fuzzy geometric programming problem is of the form Min  g0 (x) Subject to  gi (x) ≤ 1, i  1, 2, . . . , n. x j > 0, j  1, 2, . . . , m.

(7.3)

where gi (x) 

Ti  k1

 cik

m 

α

x j ik j

j1

Here cik are fuzzy triangular numbers, as  numbers and coefficients   1αik j 2are 3real , cik , cik .  cik  cik Using nearest interval approximation we transform all triangular fuzzy

L Umethod, , cik . The geometric programming problem number into interval number, i.e., cik with imprecise parameters is of the following form: Min gˆ 0 (x) subject to gˆ i (x) ≤ 1 x j > 0,

(7.4)

Ti α cˆik mj1 x j ik j and cˆik denotes the interval counterparts, i.e., where gˆ i (x)  k1 L U L U cik ∈ cik , cik . cik > 0, cik > 0, for all k. Using parametric interval-valued functional form, the problem (7.4) reduces to

Min g0 (x, s) Subject to gi (x, s) ≤ 1 xj > 0

f or j  1, 2, . . . m.

(7.5)

178

7 Fuzzy Constrained Geometric Programming Problem

Ti  L 1−s  U s m αik j cik cik where gi (x, s)  k1 j1 x j . This is a parametric geometric programming (PGP) problem. Dual program Corresponding dual programming (DP) problem of (7.5) is   1−s  s δik  T δik r U L  cik cik Max d(δ, s)  δis δik s1 i0 k1 Tr n  

(7.6)

Subject to T0 

δok  1

k1 Tr n  

αik j δik  0,

j  1, 2, . . . m.

i0 k1

δik > 0,

(i  0, 1, . . . n; k  1, 2, . . . , Tr ).

Case 1 For Tr ≥ M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables. More or unique solution exist for the dual vectors. Case 2 For Tr < M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case generally, no solution vectors exist for the dual variables. However one can get an approximate solution vector for the system using either the latest square (SQ) or max-min method. These are applied to solve such a system of linear equations. Once optimal dual variable vector δ ∗ are known, the corresponding values of the primal variable vector x is found from the following relations: m  L 1−s  U s    α ∗ ∗ ∗ c0k c0k x j 0k j  δ0k v δ ,

(k  1, 2, . . . , T0 ).

j1

And m  L 1−s  U s  δ∗ α cik cik x j ik j  Tr ik j1

s1 δis

,

(i  1, 2, . . . n; k  1, 2, . . . , Tr ).

Application 7.1 (Constrained GP problem) ˜ 2−6 t42 ˜ 3−1 t4−1 + 30t Minimize Z (t)  20t subject to t13 t42 + t1−1 t24 t32 t42 ≤ 12˜

7.2 Geometric Programming Problem with Fuzzy Coefficient

179

t1 , t2 , t3 , t4 > 0.

(7.7)

It is a constrained posynomial geometric programming problem with degree of difficulty −1. Considered fuzzy coefficients in a triangular form and transformed it as fuzzy interval parametric form, as   (10, 12, 14) ≈ [11, 13] ≈ (11)1−s (13)s , 12   (16, 20, 24) ≈ [18, 22] ≈ (18)1−s (22)s , 20   (24, 30, 36) ≈ [27, 33] ≈ (27)1−s (33)s . 30 Then the problem (7.7) transforms to Minimize Z (t)  (18)1−s (22)s t3−1 t4−1 + (27)1−s (33)s t2−6 t42 subject to t13 t42 + t1−1 t24 t32 t42 ≤ (11)1−s (13)s t1 , t2 , t3 , t4 > 0. Dual of the above primal problem is  d(δ, α)  

(18)1−s (22)s δ01

δ01 

1   1−s (11) (13)s δ12

(27)1−s (33)s δ02

δ02 

δ12

1   (11)1−s (13)s δ11

δ11

(δ11 + δ12 )(δ11 +δ12 )

subject to δ01 + δ02  1, − δ01 + 2δ12  0, − δ01 + 2δ02 + 2δ11 + 2δ12  0, − 6δ02 + 4δ12  0, 3δ11 − δ12  0. ∗ ∗  0.74, δ02  Approximate solutions of this system of linear equations are δ01 ∗ ∗ 0.15, δ11  0.05, δ12  0.24 and optimal dual objective value is i.e.,

  v∗ δ ∗ 



(18)1−s (22)s 0.74

0.74 

(27)1−s (33)s 0.15

0.15 

1 0.05.(11)1−s (13)s

0.05

180

7 Fuzzy Constrained Geometric Programming Problem

Table 7.1 Optimal solution of the Application 7.1 S. No.

s

t1

t2

t3

t4

d ∗ (δ∗ , s)

z(t)

1

0.0

1.876

1.330

0.895

2.062

13.178

13.178

2

0.2

1.884

1.334

0.899

2.082

13.526

13.526

3

0.4

1.910

1.340

0.915

2.075

13.883

13.883

4

0.6

1.913

1.342

0.915

2.104

14.248

14.248

5

0.8

1.969

1.354

0.953

2.049

14.624

14.624

6

1.0

2.025

1.393

0.991

1.999

15.010

15.010



1 0.24.(11)1−s (13)s

0.24 0.290.29 .

By primal–dual relation (18)1−s (22)s t3−1 t4−1  δ01 v∗ (δ∗); (27)1−s (33)s t2−6 t4−1  δ02 v∗ (δ∗); δ01  ; δ11 + δ12 (11) (13)s t1−1 t24 t32 t42 δ02  . 1−s s δ11 + δ12 (11) (13) t13 t42 1−s

The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 7.1.

7.3 Fuzzy Parametric Geometric Programming When the objective and constraint goals, coefficients and exponents become fuzzy sets and fuzzy numbers respectively, then a fuzzy geometric programming as follows:  Min

g0 (x)

subject to x > 0.

gi (x)1 (1 ≤ i ≤ n) (7.8)

Ti α˜ Its objective and constraints of the form gi (x)  k1 c˜ik mj1 x j ik j (0 ≤ i ≤ n) are all posynomials of x in which coefficients c˜ik are fuzzy numbers. whose membership A real number c ik described as fuzzy subset on the real line function μcik (x) has the following characteristics with − ∝< cik1 ≤ cik2 ≤ cik3 0. Which is equivalent to T0 

Min Subject to

k1 Ti 

c0k L (α)

m j1

cik L (α)

k1

m

j1

α

x j 0k j (7.10)

α

x j ik j

x j > 0. Dual program The dual program of (7.10) as follows:

Max

d(δ, s) 

  Tr Tr  n   cik L δik  i0 k1

δik

δik δis

(7.11)

s1

Subject to T0 

δok  1,

k1 Tr n  

αik j δik  0,

j  1, 2, . . . m.

i0 k1

δik > 0, (i  0, 1, . . . n; k  1, 2, . . . , Tr ). Case 1 For Tr ≥ M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables. More or unique solution exist for the dual vectors.

182

7 Fuzzy Constrained Geometric Programming Problem

Case 2 For Tr < M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case generally, no solution vectors exist for the dual variables. However one can get an approximate solution vector for the system using either the latest square (SQ) or max-min (MN) method. These are applied to solve such a system of linear equations. Once optimal dual variable vector δ ∗ are known, the corresponding values of the primal variable vector x is found from the following relations: c0k L

m 

  α ∗ ∗ ∗ x j ik j  δ0k v δ ,

(k  1, 2, . . . , T0 ).

j1

And cik L

m  j1

δ∗ α x j ik j  Tr ik

s1 δis

, (i  1, 2, . . . n; k  1, 2, . . . , Tr ).

Application 7.2 (Constrained GP problem) ˜ 2−6 t42 ˜ 3−1 t4−1 + 30t Minimize Z (t)  20t subject to t13 t42 + t1−1 t24 t32 t42 ≤ 12˜ t1 , t2 , t3 , t4 > 0. It is a constrained posynomial geometric programming problem with the degree of difficulty −1. Transformed fuzzy coefficients in a parametric form, as   10 + (1 − α) + 4, 12   16 + (1 − α) + 8, 20   25 + (1 − α) + 10. 30 Dual form of the above problem is  d(δ, α)  

subject to

16 + (1 − α) + 8 δ01

δ01 

1 (10 + (1 − α) + 4)δ12

25 + (1 − α) + 10 δ02

δ12

(δ11 + δ12 )δ11 +δ12

δ02 

1 (10 + (1 − α) + 4)δ11

δ11

7.3 Fuzzy Parametric Geometric Programming

183

δ01 + δ02  1, − δ01 + 2δ12  0, − δ01 + 2δ02 + 2δ11 + 2δ12  0, − 6δ02 + 4δ04  0, 3δ11 − δ12  0. ∗ ∗  0.74, δ02  Approximate solutions of this system of linear equations are δ01 ∗ ∗ 0.15, δ11  0.05, δ12  0.24 and optimal dual objective value is i.e., 0.05     1 (16 + (1 − α) × 8) 0.74 (25 + (1 − α) × 10) 0.15 0.74 0.15 0.05(10 + (1 − α) × 4) 0.24    1 0.290.29 0.24(10 + (1 − α) × 4)

v ∗ (δ ∗ ) 



by primal–dual relation (16 + (1 − α).8)t3−1 t4−1  δ01 v∗ (δ∗) (25 + (1 − α).10)t2−6 t4−1  δ02 v∗ (δ∗) δ01 t13 t42  8 + (1 − α).4 δ11 + δ12 δ02 t1−1 t24 t32 t42  8 + (1 − α).4 δ11 + δ12 Solving above nonlinear equation for different values of “α” by Lingo, the optimal primal variables are obtained. These are given in Table 7.2.

Table 7.2 Optimal solution of the Application 7.2 S. No.

α

t1

t2

t3

t4

v ∗ (δ∗ )

z(t)

1

0.0

1.917

1.329

0.984

2.085

15.807

15.807

2

0.2

1.895

1.287

0.975

2.049

15.144

15.144

3

0.4

1.835

1.303

0.938

2.072

14.462

14.462

4

0.6

1.785

1.257

0.909

2.075

13.758

13.758

5

0.8

2.724

1.389

1.732

1.054

13.030

13.030

6

1.0

2.639

1.370

1.671

1.054

12.274

12.274

184

7 Fuzzy Constrained Geometric Programming Problem

7.4 Constrained GP Under Max–Min Operator Proposition 7.1 When the coefficient and indexes of the fuzzy geometric programming problem are taken as fuzzy numbers  Min

T0 

c˜ok

k1

m 

α˜

x j ok j

j1

subject to

Ti 

c˜ik

k1

m 

α˜

x j ik j 1

(1 ≤ i ≤ n),

j1

x j > 0,

(7.12)

using δ-cut of fuzzy numbers coefficients and indexes, the above problem reduces to 



T0  m −1   μ−1 α˜ ok j L (δ),μα˜ ok j R (δ) −1  μ−1 Min x μ (δ), (δ) j c˜ok L c˜ok R k1

subject to

j1 Ti  

−1 μ−1 c˜ik L (δ), μc˜ik R (δ)

m 

k1



xj

μ−1 α˜

ik j L

(δ),μ−1 α˜

ik j R



(δ)

1

(1 ≤ i ≤ n),

j1

x j > 0,

(7.13)

Which is equivalent to  Min

To 

μ−1 c˜ok L (δ)

k1

m 

μ−1 α˜

xj

ok j S

(δ)

j1

subject to

Ti  k1

μ−1 c˜ik L (δ)

m 

μ−1 (δ)≤1 α˜

xj

ik j S

(1 ≤ i ≤ n)

(7.14)

j1

where  μ−1 α˜ ik j S (δ)



˜ ik j L > 0, μ−1 α˜ ik j L (δ) when α −1 μα˜ ik j R (δ) when α˜ ik j L < 0,

(1 ≤ i ≤ n)

Definition 7.2 For any x ∈ Rm and feasible index di ∈ R (R is the real number μ−1 Ti m α˜ ik j S −1 set), if gi (x, δ)  (δ) ≤ 1(1 ≤ i ≤ n), then the linear k1 μc˜ik L (δ) j1 x j membership function are given by

7.4 Constrained GP Under Max-Min Operator

μ0 (g0 (x, δ)) 

⎧ ⎪ ⎨ 1 ⎪ ⎩

z 0 +d0 −g0 (x,δ) d0

0

μi (gi (x, δ)) 

⎧ ⎪ ⎨ 1 ⎪ ⎩

185

 if g0 (x, δ) ≤ z 0 , if z 0 ≤ g0 (x, δ) ≤ z 0 + d0 ,

(7.15)

if g0 (x, δ) ≥ z 0 + d0 , 1+di −i(x,δ) d0

 if gi (x, δ) ≤ z 0 , if 1 ≤ gi (x, δ) ≤ 1 + di ,

(7.16)

if gi (x, δ) ≥ 1 + di ,

0

Based on Zimmermann max-min approach (1978, 1984), first finding δ-cut of the fuzzy numbers in coefficients and indexes then we built membership functions of both objective and constraints goals and using max-min operator the above problem (7.17) reduced to a fuzzy Nonlinear Programming (FNLP) problem Max

λ 

subject to μi

Ti 

k1

μ−1 c˜ik L (δ)

m j1



μ−1 α˜

xj

ik j S

(δ) ≥ λ(1 ≤ i ≤ n),

(7.17)

x > 0, λ, δ ∈ [0, 1], which is equivalent to a geometric programming problem with parameters λ, δ variation Min

λ−1

subject to μi

Ti 

k1

μ−1 c˜ik L (δ)

m j1

μ−1 α˜

xj



ik j S

(δ) ≥ λ(1 ≤ i ≤ n),

(7.18)

x > 0, λ, δ ∈ [0, 1],   Theorem 7.1 Let the membership function μi (gi (x, δ)), μc˜ik (cik ), μα˜ ik j αik j be all continuous and strictly monotone. Then (7.18) is equivalent with Min

λ−1 Ti

subject to

k1

μ−1 (δ) c˜ ik L

m j1

μ−1 α˜ ik j S

xj

(δ)

μi−1 (δ)

≤ 1,

x > 0, λ, δ ∈ [0, 1], (0 ≤ i ≤ n, 1 ≤ j ≤ m). Proof See Islam and Roy (2006).   Corollary 7.1 Let the membership function μi (gi (x, δ)), μc˜ik (cik ), μα˜ ik j αik j be all continuous and strictly monotone and the problem is Min

λ−1 Ti

subject to

k1

μ−1 (δ) c˜ ik L

m j1

μi−1 (δ)

μ−1 α˜ ik j S

xj

(δ)

≤ 1,

x > 0, λ, δ ∈ [0, 1], (0 ≤ i ≤ n, 1 ≤ j ≤ m).

186

7 Fuzzy Constrained Geometric Programming Problem

which is a classical posynomial geometric programming with parameters γ , δ. Its dual form is Max

d(ω) 



λ−1 ω00

 −1 ωik n ω00 Ti n μc˜ (δ)/μi−1 (λ) ik

subject to ω00  1, T0  ω00  ω0k Ti n   i0 k1

i0 k1

ωik

(ωi0 )ωi0

i1

k1 −1 α˜ ik j S (δ)ωik  0 λ, δ ∈ [0, 1],

ωik ≥ 0

where ωik  ωik (δ, λ). Application 7.3 (Constrained MGP problem) 

 80  2 x3 Minimize f (x)  + 2.40x x1 x2 x3 Subject to (2x1 x3 + x1 x2 ) ≤ 5,



where x1 > 0, x2 > 0, x3 > 0. Solution The solution of Application 7.3 by any simple crisp method is x1  2.67, x2  0.94, x3  0.47 and minimum cost Z (x)  103.16. When the coefficients are taken as triangular Fuzzy Number (TFN), i.e., 5˜    (34, 40, 46), 80   (70, 80, 90). (4, 5, 6), 40 If lower bounds of the tolerance interval z 0 (t)  103. Spreads of tolerance intervals d0  28, d1  3. Taking the membership function ⎧ ⎪ 1 z 0 (x) ≤ z 0 ⎨ 0 (x) z < z μ(z(x))  z0 +dod−z 0 0 (x) ≤ z 0 + d0 , 0 ⎪ ⎩ 0 z 0 (x) > z 0 + d0 ⎧ ⎪ 1 z 1 (x) ≤ z 1 ⎨ 1+d1 −z(x) z < z μ1 (z(x))  1 1 (x) ≤ z 1 + d1 . d1 ⎪ ⎩ 0 z 1 (x) > z 1 + d1

7.4 Constrained GP Under Max-Min Operator

187

Then the problem transforms to Min

λ−1

Subject to

1 [−(103+28−1)+28λ] 2x1 x3 +x1 x2 ≤1 5+3λ



−(70+α(80−70)) x1 x2 x3

 − 2(34 + α(40 − 34))x2 x3 ≤ 1

(7.19)

x1 , x2 , x3 > 0. i.e., the problem is λ−1

Min



(70+α(80−70)) 1 x1 x2 x3 [130−28λ] 2x1 x3 +x1 x2 ≤ 1 5+3λ

Subject to

 + 2(34 + α(40 − 34))x2 x3 ≤ 1

x1 , x2 , x3 > 0. The dual of (7.20) is 

Max d(δ)  ×

λ−1 δ00 

δ

00 

    δ 11 2 (70 + α(80 − 70)) δ01 2(34 + α(40 − 34)) δ02 (130 − 28λ)δ01 (130 − 28λ)δ02 ) (5 + 3λ)δ11 ) δ 12 × (δ01 + δ02 )(δ01 +δ02 ) (δ11 + δ12 )(δ11 +δ12 )

1 (5 + 3λ)δ12

Subject to δ00  1, δ01 + δ02  δ00  1, − δ01 + δ11 + δ12  0, − δ01 + δ02 + δ12  0, − δ01 + δ02 + δ11  0, And corresponding approximate solutions are δ00  1, δ01 

2 1 1 1 , δ02  , δ11  , δ12  . 3 3 3 3

i.e., 2  1 3(70 + α(80 − 70)) 3 3.2(34 + α(40 − 34)) 3 d(δ)  2(130 − 28λ) (130 − 28λ)   ( 23 )  13   13 2.3 3 2 . (1)(1) 3 (5 + 3λ) (5 + 3λ) 

λ−1 1

1 

Again from the relation between primal–dual variables

(7.20)

188

7 Fuzzy Constrained Geometric Programming Problem

Table 7.3 Optimal solution of the Application 7.3

Method

α

λ

x1

x2

x3

z(x)

FGP

0.5

0.812

2.17

1.71

0.28

107.62

GP

–

–

2.67

0.94

0.47

103.16

δ01 (70 + α(80 − 70))  , δ00 (130 − 28λ)x1 x2 x3 δ02 2(34 + α(40 − 34))x2 x3  , δ00 (130 − 28λ) δ11 2x1 x3  , 5 + 3λ δ11 + δ11 δ12 x1 x2  . 5 + 3λ δ11 + δ11 We can determine λ by the aid of d(δ)  λ−1 . Then the above equation reduces to 

3(70 + α(80 − 70)) 2(130 − 28λ)

2  3

3.2(34 + α(40 − 34)) (130 − 28λ)

1  3

2.3 (5 + 3λ)

1  3

3 (5 + 3λ)

1 3

 

(1)(1)

  2 2 3 3

 1.

(7.21)

For α  0.5, putting the input data in Eq. (7.21) we get the nonlinear equation in λ. Solving this nonlinear equation by Newton–Raphson method we obtain λ  0.812. Solving above nonlinear equation by Lingo, the optimal primal variables are obtained. These are given in Table 7.3.

7.5 Conclusion In this chapter, we have talked about fuzzy constrained geometric programming (FCGP) technique with positive or negative integral degree of difficulty. The upside of this technique is that we can get directly optimal solution of the objective function without understanding two-level numerical programs. This technique is straightforward and takes minimum time. Three distinction kind of constrained fuzzy GP techniques (GP problem with fuzzy parametric interval-valued function, fuzzy parametric geometric programming, constrained GP under Zimmermann max-min operators) are presented here. This procedure can be applied to solve the different types of decision-making problems (like in optimization engineering, inventory, and other areas).

References

189

References R.J. Duffin, E.L. Peterson, C.M. Zener, Geometric Programming Theory and Applications, (Wiley, New York, 1967) S. Islam, T.K. Roy, A fuzzy EPQ model with flexibility and reliability consideration and demand dependent unit production cost under a space constraint: a fuzzy geometric programming approach. Appl. Math. Comput. 176(2), 531–544 (2006) J. Kacprzyk, P. Staniewski, Long-term inventory policymaking through fuzzy decision-making models. Fuzzy Sets Syst. 8, 117–132 (1982) G. Sommer, Fuzzy inventory scheduling, Applied Systems and Cybernetics. Vol. VI, ed. by G. Lasker. Academic New York (1981) L.A. Zadeh, Fuzzy Sets. Inf. Control. 8, 338–353 (1965) H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1, 45–55 (1978) H.J. Zimmermann, Fuzzy mathematical programming. Comput. Oper. Res. 10, 1–10 (1984)

Chapter 8

Constrained Fuzzy Modified Geometric Programming Problem

8.1 Introduction In this chapter, we have discussed modified Modified Geometric Programming (MGP) geometric programming approach in fuzzy environment. S. Islam and T. K. Roy introduced modified geometric programming (MGP) model under crisp environment in 2005. Rajgopal and Bricker (2002) solved the posynomial geometric programming problems via generalized linear programming. Liu (2007) worked on geometric programming with fuzzy parameters in engineering optimization. Yang and Cao (2007) developed posynomial fuzzy relation geometric programming. Roy (2008) discussed fuzzy geometric programming with numerical examples. Maiti (2008) presented fuzzy inventory model with two warehouses under possibility measure on fuzzy goal. Mandal and Islam (2016) demonstrated fuzzy unconstrained Parametric Geometric programming problem and its application. In this chapter, three types of fuzzy constrained modified geometric programming (CMGP) methods are discussed, that is (1) CMGP problem with fuzzy parametric interval-valued function, (2) CMGP problem with simple fuzzy parametric coefficients, and (3) CMGP problem with Zimmermann max-min operator. Definition 8.1 Let D ⊂ Rm be the (convex) subset of Rm defined by   D  (x1 , x2 , . . . , xm ) ∈ Rm |x > 0, j  1, 2, . . . m . A function g0 : D → Rm of the form T0 m     α g0 x j  ck x j kj , k1

j1

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_8

191

192

8 Constrained Fuzzy Modified Geometric …

where ck > 0 for k  1, 2, . . . , T0 and αk j ∈ R for k  1, 2, . . . , T0 and j  1,2,…,m, is called posynomial. n A problem of the form,    g0 x j

Minimum

Subject to

i1 n 

  gi x j ≤ 1

i1

x j > 0,

is called constrained modified geometric programming (MGP) problem. When coefficients, exponents, or objective are fuzzy in nature, then it is called unconstrained fuzzy MGP problem and it is defined as follows: Minimize Subject to

n  i1 n 

g 0 (x)   g i xj ≤ 1

i1

x j > 0.

  n T0 m α˜ ik j where g i xj  i1 j1 x j . k1 c˜ik 3 2 5 2 5 ˜ 3 x22 + 2x ˜ 11 ˜ 21 ˜ 11 x12 + 7x x12 + 5x x22 is fuzzy modified posynomial. Example 8.1 3x 21 annd the problem of the form, 3 2 5 3 2 5 ˜ 21 ˜ 11 ˜ 11 ˜ 21 Minimize 3x x12 + 7x x12 + 5x x22 + 2x x22 ˜ ˜ 11 x12 + 5x21 x21 ≤ 1. Subject to 3x

x11 , x12 , x21 , x22 > 0, is a fuzzy unconstrained MGP problem.

8.2 Modified Geometric Programming Problem with Fuzzy Coefficient Primal fuzzy modified geometric programming problem is of the form Min g˜ 0 (x) Subject to g˜ i (x) ≤ 1, i  1, 2, . . . , n. x j > 0, j  1, 2 . . . , m.

(8.1)

 p Ti α c˜ir k mj1 x j ir k j . where gi (x)  r 1 k1 Here, αir k j are real numbers and coefficients c˜ir k are triangular fuzzy numbers (TFN), as c˜ir k  cir1 k , cir2 k , cir3 k .

8.2 Modified Geometric Programming Problem with Fuzzy Coefficient

193

Using nearest interval approximation (NIA) method, we transform all triangular 

U . The geometric programming fuzzy number into interval number, i.e., cirL k , cir k problem with imprecise parameters is of the following form: Min gˆ 0 (x) Subject to gˆ i (x) ≤ 1 x j > 0,

(8.2)

where gˆ i (x) 

p Ti   r 1 k1

cˆir k

m 

α

x j ir k j ,

j1

 L U U where cˆir k denotes the interval counterparts, i.e., cˆir k ∈ cirL k , cir k ·cir k > 0, cir k > 0, for all k. Using parametric interval-valued functional form, the problem (8.2) reduces to Min g0 (x, s) Subject to gi (x, s) ≤ 1 x j > 0, for j  1, 2, . . . , m.

(8.3)

 p Ti  L 1−s  U s m αir k j where gi (x, s)  r 1 k1 cir k cir k . j1 x j This is a parametric geometric programming (PGP) problem. Dual program Corresponding dual programming (DP) problem of (8.3) is  p δr k p Tr L 1−s U s δr k   (c ) (c )  ir k ir k δsk Max d(δ, s)  δr k r 1 k1 s1

(8.4)

Subject to T0 

δok  1,

k1 Tr n  

αik j δik  0,

j  1, 2, . . . m.

i0 k1

δik > 0,

(i  0, 1, . . . n; k  1, 2, . . . , Tr ).

Case 1 For Tr ≥ M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to dual variables. More or unique solution exists for the dual vectors.

194

8 Constrained Fuzzy Modified Geometric …

Table 8.1 Input data of the application 8.1 i-th Box

c˜i

d˜i

i1

 40

 80

12

 30

 90

i3

 20

 70

w˜  15

Case 2 For Tr < M + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case, generally no solution vectors exist for the dual variables. However, one can get an approximate solution vector for the system using either the latest square (SQ) or max-min(MN) method. These are applied to solve such a system of linear equations. Once optimal dual variable vector δ ∗ are known, the corresponding values of the primal variable vector x is found from the following relations: m  L 1−s  U s    α ∗ ∗ ∗ c0k c0k x j 0k j  δ0k v δ , (k  1, 2, . . . , T0 ), j1

and m  L 1−s  U s  α cik cik x j ik j  j1

∗ δik , Tr  δis

(i  1, 2, . . . , n; k  1, 2, . . . , Tr ).

s1

Application 8.1 (Constrained MGP Problem)

Minimize f (x) 

n  i1

Subject to

n 



d˜i + 2c˜i x2i x3i x1i x2i x3i



˜ (2x1i x3i + x1i x2i ) ≤ w,

i1

where x1i > 0, x2i > 0, x3i > 0(i  1, 2, . . . , n). In particular, here we assume transporting di m3 of grains by the three different open rectangular boxes. The end of each box cost is ci $/m2 and amount of the transport in grains by three open rectangular boxes are di m3. Input data of this problem is given in Table 8.1. It is a constrained posynomial MGP problem. Optimal results of this problem are shown in Table 8.2. Consider fuzzy coefficient in triangular fuzzy number then by NIA technique, we have

8.2 Modified Geometric Programming Problem with Fuzzy Coefficient

195

Table 8.2 Optimal solution of the application 8.1 S. No.

s

x11

1

0.1

3.55 1.07 0.35 2.99 1.19 0.42 1.55 0.93 0.81 731539.6 256.44

2

0.3

3.49 1.06 0.36 2.94 0.85 0.59 1.53 0.92 0.82 750717.2 264.01

3

0.5

3.43 1.05 0.37 2.89 0.86 0.59 1.50 0.92 0.83 770397.5 273.83

4

0.7

3.37 1.04 0.37 2.84 1.17 0.44 1.47 0.91 0.85 790593.8 284.84

5

0.9

3.31 1.10 0.38 2.79 1.16 0.45 1.45 0.90 0.86 811319.6 293.62

x12

x13

x21

x22

x23

x31

x32

x33

d(δ, α)

z(x)

  (13, 15, 17) ≈ [14, 16] ≈ (14s 161−s ) ∈ [14, 16], 15   (16, 20.24) ≈ [18, 22] ≈ (18s 221−s ) ∈ [18, 22], 20   (24, 30, 36) ≈ [27, 33] ≈ (27s 331−s ) ∈ [27, 33], 30     34, 40, 46 ≈ [37, 43] ≈ (37s 431−s ∈ [37, 43], 40   (60, 70, 80) ≈ [65, 75] ≈ (65s 751−s ) ∈ [65, 75], 70   (70, 80, 90) ≈ [75, 85] ≈ (75s 851−s ) ∈ [75, 85], 80   (80, 90, 100) ≈ [85, 975] ≈ (85s 951−s ) ∈ [85, 95], s ∈ [0, 1]., 90 i.e., the problem is

(75s 851−s ) s 1−s Minimize f (x)  + 2(37 43 )x21 x31 x11 x21 x31 

  s 1−s  s 1−s 65 75 (85 95 ) + + 2(27s 331−s x22 x32 + + 2(18s 221−s x23 x33 x12 x22 x32 x13 x23 x33   Subject to (2x11 x31 + x11 x21 ) + (2x12 x32 + x12 x22 ) + (2x13 x33 + x13 x23 ) ≤ 14s 161−s ,

where x1i > 0, x2i > 0, x3i > 0(i  1, 2, 3), and the dual is   δ01   1−s s  δ02   1−s s  δ11 2 37 43 85 95 751−s 85s d(δ, α)  δ01 δ02 δ11   δ12   δ22      δ22 2 271−s 33s 651−s 75s 2 181−s 22s δ12 δ21 δ22  δ31  δ32  δ41 2 1 2     ×  1−s s  14 16 δ31 141−s 16s δ32 141−s 16s δ41

196

8 Constrained Fuzzy Modified Geometric …



1   141−s 16s δ42

δ42 

2   141−s 16s δ51

δ51 

1   141−s 16s δ52

δ52 λλk k .

With the condition δ01 + δ02  1, −δ01 + δ31 + δ32  0, −δ01 + δ02 + δ32  0, −δ01 + δ02 + δ31  0, δ11 + δ12  1, −δ11 + δ41 + δ42  0, −δ11 + δ12 + δ42  0, −δ11 + δ12 + δ41  0, δ21 + δ22  1, −δ21 + δ51 + δ52  0, −δ21 + δ22 + δ52  0, −δ21 + δ22 + δ51  0, λk  δ31 + δ32 + δ41 + δ42 + δ51 + δ52 . and corresponding approximate solutions are 2 1 2 1 2 1 1 , δ02  , δ11  , δ12  , δ21  , δ22  , δ31  , 3 3 3 3 3 3 3 1 1 1 1 1  , δ41  , δ42  , δ51  , δ52  . 3 3 3 3 3

δ01  δ32 i.e.,

  2   1   2 3 751−s 85s 3 3.2 371−s 43s 3 3 851−s 95s 3 d(δ, s)  2 1 2 1 2           1 3.2 271−s 33s 3 3 651−s 75s 3 3.2 181−s 22s 3 1 2 1 1 1  3  3   13 2.3 3 2.3     ×  1−s s  14 16 141−s 16s 141−s 16s   13   13   13 3 2.3 1       22 . 141−s 16s 141−s 16s 141−s 16s by the primal–dual relation

8.2 Modified Geometric Programming Problem with Fuzzy Coefficient

197

 (75s 851−s )  δ01 3 d(δ, s) x11 x21 x31    2(37s 431−s )x21 x31  δ02 3 d(δ, s)  (85s 951−s )  δ11 3 d(δ, s) x12 x22 x32    2(27s 331−s )x22 x32  δ12 3 d(δ, s)  (65s 751−s )  δ21 3 d(δ, s) x13 x23 x33  2(18s 221−s )x23 x33  δ22 3 d(δ, s) δ31 2x11 x31  δ31 + δ32 + δ41 + δ42 + δ51 + δ52 δ32 x11 x21  δ31 + δ32 + δ41 + δ42 + δ51 + δ52 δ41 2x12 x33  δ31 + δ32 + δ41 + δ42 + δ51 + δ52 δ42 x12 x22  δ31 + δ32 + δ41 + δ42 + δ51 + δ52 δ51 2x13 x33  δ31 + δ32 + δ41 + δ42 + δ51 + δ52 δ52 x13 x23  δ31 + δ32 + δ41 + δ42 + δ51 + δ52 The optimal solution of the fuzzy model by interval-valued parametric geometric programming is presented in Table 8.2.

8.3 Fuzzy Parametric Modified Geometric Programming When the objective and constraint goals, coefficients and exponents become fuzzy sets and fuzzy numbers, respectively, then a fuzzy geometric programming as follows:  Min g0 (x) subject to gi (x)1 (1 ≤ i ≤ n) x >0 Its objective and constraints of the form

(8.5)

198

8 Constrained Fuzzy Modified Geometric …

gi (x) 

Ti n  

c˜ik

i1 k1

m 

α

x j ik j (0 ≤ i ≤ n)

j1

are all posynomials of x in which coefficients c˜ik are fuzzy numbers. whose membership A real number c ik described as fuzzy subset on the real line function μcik (x) has the following characteristics with − ∝< cik1 ≤ cik2 ≤ cik3 0.

(8.6)

which is equivalent to Min

T0 n  

c0k L (α)

i1 k1

Subject to

Ti n   i1 k1

m 

α

x j 0k j

j1

cik L (α)

m 

α

x j ik j

j1

x j > 0. Application 8.2 Consider the application 8.1 (Constrained MGP problem). Taking fuzzy coefficient in a parametric form   9 + (1 − α) × 2, 10   26 + (1 − α) × 8, 30   35 + (1 − α) × 10, 40   70 + (1 − α) × 20, 80

(8.7)

8.3 Fuzzy Parametric Modified Geometric Programming

199

  80 + (1 − α) × 20 90 i.e., the problem is

(70 + (1 − α) × 20) Minimize f (x)  + 2(35 + (1 − α) × 10)x21 x31 x11 x21 x31

(80 + (1 − α) × 20) + + 2(26 + (1 − α) × 8)x22 x32 x12 x22 x32 Subject to (2x11 x31 + x11 x21 ) + (2x12 x32 + x12 x22 ) ≤ (9 + (1 − α) × 2) where x1i > 0, x2i > 0, x3i > 0(i  1, 2), and the dual is



(70 + (1 − α) × 20) δ01 2(35 + (1 − α) × 10) δ02 δ01 δ02



(80 + (1 − α) × 20) δ11 2(26 + (1 − α) × 8) δ12 δ11 δ12

δ

δ 22 31 1 2 λ × × λk k . (9 + (1 − α) × 2)δ22 (9 + (1 − α) × 2)δ21

d(δ, α) 

With the condition δ01 + δ02  1, − δ01 + δ21 + δ22  0, − δ01 + δ02 + δ22  0, − δ01 + δ02 + δ21  0, δ11 + δ12  1, − δ11 + δ31 + δ32  0, − δ11 + δ12 + δ32  0, − δ11 + δ12 + δ32  0, and corresponding approximate solutions are δ01 

2 1 2 1 1 1 1 1 , δ02  , δ11  , δ12  , δ21  , δ22  , δ31  , δ32  . 3 3 3 3 3 3 3 3

i.e., d(δ, α) 

3(70 + (1 − α) × 20) 2

3.2(26 + (1 − α) × 8) 1

2 3

1 3

3.2(35 + (1 − α) × 10) 1

3.2 9 + (1 − α) × 2)



1 3

1

×

3

3.(80 + (1 − α) × 20) 2

3 9 + (1 − α) × 2)

1 3

2

3  

4 4 6 × . 6

200

8 Constrained Fuzzy Modified Geometric …

Table 8.3 Optimal solution of the application 8.2 S. No.

α

x11

x12

x13

x21

x22

x23

d(δ, α)

1

0.1

3.05

0.91

0.44

2.56

1.02

0.53

11422.48 214.00

2

0.3

2.89

0.92

0.45

2.24

1.04

0.53

10969.70 215.95

3

0.5

2.67

0.94

0.47

2.31

1.05

0.54

10510.05 205.90

4

0.7

2.59

0.95

0.46

2.19

1.07

0.55

10043.14 200.51

5

0.9

2.42

0.95

0.48

2.07

1.09

0.56

9568.61 195.69

z(x)

The optimal solution of the fuzzy model parametric modified geometric programming is presented in Table 8.3.

8.4 Constrained MGP Under Max-Min Operator Consider a nonlinear programming as follows: Min g0 (x) Subject to gi (x) ≤ 1 (1 ≤ i ≤ n), x > 0.

(8.8)

Its objective and constraints of the form gi (x) 

Ti n   i1 k1

cik

m 

α

x j ik j (0 ≤ i ≤ n),

j1

x j > 0, (J  1, 2, . . . , m). Here, cik (> 0), (k  1, 2, . . . , Ti ) and αikj be any real numbers. When the objective and constraint goals, coefficients and exponents become fuzzy sets and fuzzy numbers, respectively, then we transform into a fuzzy geometric programming as follows:  Min g0 (x) subject to gi (x)1 (1 ≤ i ≤ n) x >0

(8.9)

Its objective and constraints of the form gi (x)  m n Ti α˜ ik j i1 j1 x j (0 ≤ i ≤ n) are all posynomials of x in which coefk1 c˜ik ficients c˜ik and indexes α˜ ik j are fuzzy numbers.

8.4 Constrained MGP Under Max-Min Operator

201

Some Definitions and Theorems Definition 8.2 For n-th parabolic flat fuzzy number (a1 , a2 , a3 , a4 )PfFN containing the coefficients c˜ik (0 ≤ i ≤ n; 1 ≤ k ≤ Ti ), the membership function of c˜ik is  n ⎧ a2 −cik ⎪ 1 − ⎪ a2 −a1 ⎪ ⎪ ⎨ 1  n μc˜ik (c˜ik )  cik −a3 ⎪ 1 − ⎪ ⎪ a4 −a3 ⎪ ⎩ 0

for a1 ≤ cik ≤ a2 for a1 ≤ cik ≤ a2 for a3 ≤ cik ≤ a4 for otherwise.

Similarly, we can determine the membership function of the indexes α˜ ik j (0 ≤ i ≤ n; 1 ≤ k ≤ Ti ; 1 ≤ i ≤ m). Note (a) (b) (c) (d)

when n  1, c˜ik become Trapezodial Fuzzy Number (TrFN), when n  1, and a3  a4 , c˜ik become triangular fuzzy number (TFN), when n  2, c˜ik become parabolic flat fuzzy number (PfFN), when n  2, and a3  a4 , c˜ik become parabolic fuzzy number (PrFN),

Definition 8.3 n; 1 ≤ k ≤ Ti ) is given by  Here, δ-cut of c˜ik (0 ≤ i ≤  √ √ n n −1 −1 −1 μc˜ik (δ)  μc˜ik L (δ), μc˜ik R (δ)  a1 + 1 − δ(a2 − a1 ), a4 − 1 − δ(a4 − a3 ) . (8.10) Similarly, we can determine the δ-cut of α˜ ik j (0 ≤ i ≤ n; 1 ≤ k ≤ Ti ; 1 ≤ i ≤ m). Proposition 8.1 When the coefficient and indexes of the fuzzy geometric programming problem are taken as fuzzy numbers T0 m n   α˜  Min c˜ok x j ok j subject to

i1 k1 Ti n  

j1

c˜ik

i1 k1

m

j1

α˜

x j ik j 1(1 ≤ i ≤ n),

(8.11)

x j > 0, using δ-cut of fuzzy numbers coefficients and indexes, the above problem is reduces to   T0  n  m −1   μ−1 α˜ ok j L (δ),μα˜ ok j R (δ) −1 −1  Min μc˜ok L (δ), μc˜ok R (δ) xj i1 k1

subject to

Ti  n   i1 k1

x j > 0,

j1 −1 μ−1 c˜ik L (δ), μc˜ik R (δ)

m 



xj

μ−1 α˜

ik j L

(δ),μ−1 α˜

ik j R

 (δ)

1

(1 ≤ i ≤ n),

j1

(8.12)

202

8 Constrained Fuzzy Modified Geometric …

which is equivalent to Min

To n  

μ−1 c˜ok L (δ)

i1 k1

subject to

m 

μ−1 α˜

ok j S

xj

(δ)

j1

Ti n  

μ−1 c˜ik L (δ)

m 

i1 k1

μ−1 α˜

ik j S

xj

(δ) ≤ 1

(1 ≤ i ≤ n)

(8.13)

j1

where  μ−1 α˜ ik j S (δ)



˜ ik j L > 0, μ−1 α˜ ik j L (δ) when α (1 ≤ i ≤ n) −1 μα˜ ik j R (δ) when α˜ ik j L < 0,

Definition 8.4 For any x ∈ Rm and feasible index di ∈ R (R is the real number set), if gi (x, δ) 

Ti 

μ−1 c˜ik L (δ)

k1

m 

μ−1 α˜

ik j S

xj

(δ) ≤ 1

(1 ≤ i ≤ n),

j1

then the linear membership function are given by

μ0 (g0 (x, δ)) 

⎧ ⎪ ⎨ 1 ⎪ ⎩

z 0 +d0 −g0 (x,δ) d0

0

μi (gi (x, δ)) 

⎧ ⎪ ⎨ 1 ⎪ ⎩

 if g0 (x, δ) ≤ z 0 , if z 0 ≤ g0 (x, δ) ≤ z 0 + d0 ,

(8.14)

if g0 (x, δ) ≥ z 0 + d0 , 1+di −i(x,δ) d0

 if gi (x, δ) ≤ z 0 , if 1 ≤ gi (x, δ) ≤ 1 + di ,

(8.15)

if gi (x, δ) ≥ 1 + di ,

0

Based on Zimmerman, first finding δ-cut of the fuzzy numbers in coefficients and indexes then we built membership functions of both objective and constraints goals and using max-min operator the above problem (8.13) reduced to a fuzzy nonlinear programming (FNLP) problem Max

λ 

subject to μi

Ti n  

i1 k1

μ−1 c˜ik L (δ)

m j1

μ−1 α˜

xj

ik j S

 (δ) ≥ λ(1 ≤ i ≤ n),

(8.16)

x > 0, λ, δ ∈ [0, 1], which is equivalent to a geometric programming problem with parameters λ, δ variation

8.4 Constrained MGP Under Max-Min Operator

λ−1

Min

subject to μi

Ti n  

i1 k1

μ−1 c˜ik L (δ)

m j1

203

μ−1 α˜

xj

ik j S

 (δ) ≥ λ(1 ≤ i ≤ n),

(8.17)

x > 0, λ, δ ∈ [0, 1],     Theorem 8.1 Let the membership function μi gi (x,δ) , μc˜ik (cik ), μα˜ ik j αikj be all continuous and strictly monotone. Then, (8.17) is equivalent with λ−1

Min

n

subject to

−1

Ti

i1

k1

μ m α˜ ik j S μ−1 (δ) j1 x j c˜ik L (δ) n −1 i1 μi (δ)

(8.18)

≤ 1,

x > 0, λ, δ ∈ [0, 1], (0 ≤ i ≤ n, 1 ≤ j ≤ m). Proof Please see reference Islam and Roy (2005).     Corollary 8.1 Let the membership function μi gi (x,δ) , μc˜ik (cik ), μα˜ ik j αikj be all continuous and strictly monotone and the problem is λ−1

Min

n

subject to

−1

Ti

i1

k1

μ m α˜ ik j S μ−1 (δ) j1 x j c˜ik L (δ) n −1 i1 μi (δ)

≤ 1,

x > 0, λ, δ ∈ [0, 1], (0 ≤ i ≤ n, 1 ≤ j ≤ m). which is a classical posynomial geometric programming with parameters γ , δ. Its dual form is Max

d(ω) 

λ−1 ω00



ω00  Ti n  i0 k1

−1 μ−1 c˜ik (δ)/μi (λ)

ωik

ωik

n 

(ωi0 )ωi0

i1

subject to ω00  1, ω00 

T0 

ω0k

k1 T0 

−1 α˜ 0k j S (δ)ωik +

k1

ωik ≥ 0 where ωik  ωik (δ, λ).

Ti n   i1 k1

−1 α˜ ik j S (δ)ωik  0 λ, δ ∈ [0, 1],

204

8 Constrained Fuzzy Modified Geometric …

Application 8.3 (Constrained MGP Problem)  Minimize f(x) 

   d˜1 d˜2 + 2c˜1 x21 x31 + + 2c˜2 x22 x32 x11 x21 x31 x12 x22 x32

Subject to (2x11 x31 + x11 x21 ) + (2x12 x32 + x12 x22 ) ≤ w, ˜ where x1i > 0, x2i > 0, x3i > 0(i  1, 2). Solution The solution of () by any simple crisp method is x11  2.67, x21  0.94, x31  0.47, x21  2.31, x22  1.05, x32  0.54 and minimum cost Z (x)  205.90.   When the coefficients are taken as Triangular fuzzy number (TFN) i.e., 10     (8, 10, 12), 30  (26, 30, 34), 40  (34, 40, 46), 80  (70, 80, 90) and 90  (80, 90, 100). If lower bounds of the tolerance interval; z 0 (t)  205. Spreads of tolerance intervals; d0  56, d1  5.

Taking the membership function; ⎧ ⎪ ⎨1 0 (x) μ(z(x))  z0 +dod−z 0 ⎪ ⎩0 ⎧ ⎪ ⎨1 μ1 (z(x))  1+d1d−z(x) 1 ⎪ ⎩0

z 0 (x) ≤ z 0 z 0 < z 0 (x) ≤ z 0 + d0 , z 0 (x) > z 0 + d0 z 1 (x) ≤ z 1 z 1 < z 1 (x) ≤ z 1 + d1 . z 1 (x) > z 1 + d1

Then the problem is transformed to Min λ−1

 1 −(70 + α(80 − 70)) − 2(34 + α(40 − 34))x21 x31 x11 x21 x31 [−(205 + 56 − 1) + 56λ]  (80 + α(90 − 80)) − − 2(26 + α(30 − 26))x22 x32 ≤ 1 x12 x22 x32

Subject to

x11 , x21 , x31 , x12 , x22 , x32 > 0.

i.e., the problem is

8.4 Constrained MGP Under Max-Min Operator

205

λ−1

Min

 1 (70 + α(80 − 70)) Subject to + 2(34 + α(40 − 34))x21 x31 x11 x21 x31 [260 − 56λ]  (80 + α(90 − 80)) + + 2(26 + α(30 − 26))x22 x32 ≤ 1 x12 x22 x32 (2x11 x31 + x11 x21 ) + (2x12 x32 + x12 x22 ) ≤1 10 + 5λ x11 , x21 , x31 , x12 , x22 , x32 > 0. The dual of the above primal problem is



(70 + α(80 − 70)) δ01 2(34 + α(40 − 34)) δ02 (260 − 56λ)δ01 (260 − 56λ)δ02

δ11



δ21 2(26 + α(30 − 26)) δ12 2 (80 + α(90 − 80)) (260 − 56λ)δ11 (260 − 56λ)δ12 (10 + 5λ)δ21 )

δ22

δ32

δ31 1 2 1 × (10 + 5λ)δ22 (10 + 5λ)δ13 ) (10 + 5λ)δ14

Max d(δ) 

λ−1 δ00

δ00

(δ01 + δ01 + δ11 + δ12 )(δ01 +δ01 +δ11 +δ12 ) (δ21 + δ22 + δ31 + δ32 )(δ21 +δ22 +δ31 +δ32 ) , Subject to δ00  1, δ01 + δ02 + δ11 + δ12  δ00  1, − δ01 + δ21 + δ22  0, − δ01 + δ02 + δ22  0, − δ01 + δ02 + δ21  0, − δ11 + δ31 + δ32  0, − δ11 + δ12 + δ32  0, − δ11 + δ12 + δ31  0, Solving above set of linear equations, we have δ00  1, δ01 

1 1 1 1 1 1 1 1 , δ02  , δ11  , δ12  , δ21  , δ22  , δ31  , δ32  . 3 6 3 6 6 6 6 6

i.e.,  d(δ) 

λ−1 1

1

3(70 + α(80 − 70)) (260 − 56λ)

6.2(26 + α(30 − 26)) (260 − 56λ)

×

2.6 (10 + 5λ)

1 6

1 6

1 3

6.2(34 + α(40 − 34)) (260 − 56λ)

2.6 (10 + 5λ)

6 (10 + 5λ)

1 6

 

6 (10 + 5λ)  

1 4 4 4 6 4 6 6 . 6 6

1 6

1 6

3(80 + α(90 − 80)) (260 − 56λ)

1 3

206

8 Constrained Fuzzy Modified Geometric …

Table 8.4 Optimal solution of the application 8.3 Method

α

λ

x11

x12

x13

x21

x22

x23

z(x)

FMGP

0.5

0.812

2.10

0.69

0.69

4.41

0.53

0.53

202.84

MGP

–

–

2.89

0.92

0.45

2.24

1.04

0.53

215.95

We can determine λ by the aid of d(δ)  λ−1 . Then, the above equation is reduces to

1

1

1 3(70 + α(80 − 70)) 3 6.2(34 + α(40 − 34)) 6 3(80 + α(90 − 80)) 3 (260 − 56λ) (260 − 56λ) (260 − 56λ)

16

16

16 6.2(26 + α(30 − 26)) 2.6 6 (260 − 56λ) (10 + 5λ) (10 + 5λ)

16

16 ( 46 ) ( 46 ) 6 4 4 2.6  1. × 6 6 (10 + 5λ) (10 + 5λ)

Again from the relation between primal–dual variables δ01 (70 + α(80 − 70))  , δ00 (260 − 56λ)x11 x21 x31 δ02 2(34 + α(40 − 34))x21 x31  , δ00 (260 − 56λ) δ11 (80 + α(90 − 80))  δ00 (260 − 56λ)x12 x22 x32 2(26 + α(30 − 26))x22 x32 δ12  . δ00 (260 − 56λ) The optimal solution of the fuzzy model by max-min modified geometric programming at α  0.5 is presented in Table 8.4.

8.5 Conclusion In this chapter, we have discussed fuzzy constrained modified geometric programming (CMGP) technique with negative or positive integral degree of difficulty. Three different types of fuzzy CMGP techniques (CMGP problem with fuzzy parametric interval-valued function, fuzzy constrained parametric modified geometric programming, and CMGP problem with Zimmermann max-min operators) are defined here. This technique can be applied to solve the different decision-making problems (like in optimization engineering, inventory, and other areas). The parameters are generally estimated based on previous experiment and managerial judgment. Therefore, in real-life situation, fuzzy set theory is more realistic than crisp set theory or traditional probability theory.

References

207

References S. Islam, T.K. Roy, Modified Geometric programming problem and its applications. J. Appt. Math. Comput. 17(1–2), 121–144 (2005) S.T. Liu, Geometric programming with fuzzy parameters in engineering optimization. Int. J. Approximate Reasoning 46(3), 484–498 (2007) M.K. Maiti, Fuzzy inventory model with two warehouses under possibility measure on fuzzy goal. Eur. J. Oper. Res. 188, 746–774 (2008) W.A. Mandal, S. Islam, Fuzzy unconstrained Parametric Geometric programming problem and its application. J. Fuzzy Set Val. Anal. 2016(2), 125–139 (2016). http://dx.doi.org/10.5899/2016/ jfsva-00301 J. Rajgopal, D.L. Bricker, Solving posynomial geometric programming problems via generalized linear programming. Comput. Optim. Appl. 21, 95–109 (2002) T.K. Roy (2008) Fuzzy geometric programming with numerical examples. Springer Optimization and Its Applications 16: Fuzzy Multi-Criteria Decision Making Theory and Applications with Recent Developments, pp. 567–587 J.H. Yang, B.Y. Cao, Posynomial fuzzy relation geometric programming. Lecture Notes in Computer Science, vol. 4529 (Springer, Berlin, Heidelberg, 2007), pp. 563–572

Chapter 9

Fuzzy Signomial Geometric Programming Problem

9.1 Introduction and History Fuzzy set theories were presented by L. A. Zadeh and D. Klaua in 1965. The term “signomial” was introduced by Richard J. Duffin and Elmor L. Peterson in their original joint work on “general algebraic optimization,” published in the late 1960s and mid-1970s. An ongoing introductory exposition is optimization problems. In spite of the fact that nonlinear optimization problems (NLOP) with constraints and/or objectives defined by fuzzy signomials are typically harder to solve than those defined by posynomials (in light of the fact that not at all like posynomials, signomials are not ensured to be all globally convex). A fuzzy signomial optimization problem often gives a considerably more exact mathematical representation of real-world nonlinear optimization problems. Duffin and Peterson (1966) presented duality theory for geometric programming. Duffin and Peterson (1966) also discussed geometric programming with signomials. Avriel and Williams (1971) extended the geometric programming with applications in engineering optimization. Liu (2006) developed posynomial geometric programming with parametric uncertainty. Cao and Yang (2007) worked on fuzzy geometric programming briefly. Yang and Cao (2010) briefly discussed fuzzy geometric programming and its application. Definition 9.1 A fuzzy signomial function is of the form gk (x) 

k  i1

σi c˜i

m 

a

x jij ,

j1

where c˜i is the absolute value of fuzzy coefficient and σi is the sign of coefficient (+1 or −1).

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_9

209

210

9 Fuzzy Signomial Geometric Programming Problem

˜ 1 x2 For this fuzzy signomial ˜ 12 x2−1 − 5x Example 9.1 gk (x)  2x σ1  +1, σ2  −1, ˜ ˜ c˜2  5. c˜1  2,

9.2 Unconstrained Proble A problem with fuzzy coefficients and without any restrictions is called unconstrained problem, i.e., a problem of the form Minimize g˜ 0 (x1 , x2 , . . . , xm ) Subject to x j > 0, j  1, 2, . . . , m, is called unconstrained problem.

9.2.1 Unconstrained Fuzzy Signomial GP Problem Primal problem A primal unconstrained fuzzy signomial GP programming problem is of the form Minimize g˜ 0 (x1 , x2 , . . . , xm ) Subject to x j > 0, j  1, 2, . . . , m,

(9.1)

 k a where g˜ 0 (x)  i1 σi c˜i mj1 x j i j . ai j are real numbers and coefficients c˜i are fuzzy triangular, as c˜i   1Here ci , ci2 , ci3 . Using nearest interval approximation method, transform all triangular fuzzy numbers into interval number, i.e., [ciL , ciU ]. Then, the fuzzy signomial geometric programming problem is of the following form: Min

gˆ 0 (x) 

k  i1

Subject to x j > 0,

σi cˆi

m  j1

a

x jij

(9.2)

j  1, 2, . . . , m,

 where cˆi denotes the interval counter parts, i.e., cˆi ∈ ciL , ciU · ciL > 0, ciU > 0, for all i. Using parametric interval-valued functional form, the problem (9.2) reduces to

9.2 Unconstrained Proble

211

g0 (x, s) 

Min

k  i1

Subject to x j > 0,

m  1−s  U s  a ci σi ciL x jij j1

(9.3)

j  1, 2, . . . , m.

This is a parametric geometric programming (PGP) problem. Dual-signomial GP problem Dual GP problem of the given primal GP problem is ⎡

 1−s  s σi δi ⎤ζ0 ciL ciU ⎦ Maximize ζ0 ⎣ δ i i1 n 

Subject to k  i1 k 

σi δi  ζ0 , σi ai j δi  0,

j  1, 2, . . . , m

(9.4)

i1

δi > 0.

Case I n > m + 1 (i.e., DD > 0), so the DP presents a system of linear equations for the dual variables. Here the number of linear equations is less than the number of dual variables. More solutions of dual-variable vector exist. In order to find an optimal solution of DP, we need to use some algorithmic methods. Case II n < m + 1 (i.e., DD < 0), so the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is greater than the number of dual variables. In this case, generally no solution vector exists for the dual variables. However, using least squares (LS) or min-max (MM) method, one can get an approximate solution for this system. Furthermore, the primal–dual relation is m  L 1−s  U s  ∗a ci ci x j i j  ζ0 δi∗ v(δ ∗ , s ∗ ).

(9.5)

j1

Note 9.1 A weak duality theorem would say that g0 (x, s) ≥ v(δ), for any primalfeasible x and dual-feasible δ but this is not true of the pseudo-dual fuzzy signomial GP problem. Corollary 9.1 When the value of σi is 1, then a signomial geometric programming problem transforms to ordinary geometric programming problem. Theorem 9.1 When σi is 1, then go (x, s) ≥ v(δ) (Primal–Dual Inequality).

212

9 Fuzzy Signomial Geometric Programming Problem

Proof The expression for g0 (x, s) can be written as  L 1−s  U s m αk j ci ci j1 x j g0 (x, s)  δk . δk i1 n 

Here, the weights are δ1 , δ2 , . . . , δn and positive terms are   α α L 1−s (c2 ) (c2U )s mj1 x j 2 j (cnL )1−s (cnU )s mj1 x j n j , . . . , . δ2 δn Now applying A.M.-G.M. inequality, we get

(c1L )1−s (c1U )s

m j1

α1 j

xj

δ1

⎞(δ01 +δ02 +···+δn ) ⎛  1−s  s          m x α1 j + c L 1−s cU s m x α2 j + · · · + c L 1−s cU s m x αn j cL c1U n n j1 j j1 j j1 j 2 2 ⎟ ⎜ 1 ⎠ ⎝ (δ1 + δ2 + · · · + δn ) ⎛⎛ 

L ⎜⎜ c1 ≥⎜ ⎝⎝

⎞δ1 ⎛  1−s  s  ⎞δ2 1−s  s  m x α1 j m x α2 j cL cnU c2U j1 j j1 j ⎟ ⎜ 2 ⎟ ⎠ ⎝ ⎠ δ1 δ2

⎞ δn ⎞ ⎛  1−s  s  m x αn j cnU cL j1 j ⎟ ⎟ ⎜ n ···⎝ ⎠ ⎟ ⎠ δn

or 

g0 (x, s) n i1 δi

n  δi i1

  L 1−s  U s m αn j δi  n  ci ci j1 x j ≥ δk  1 as δi i1 i1 n 

or  1−s  s  δi m  n αi j δi i1 ciL ciU g0 (x, s) ≥ x j i1 δk j1 n

or  1−s  s δi m  n αi j δi ciL ciU x j i1 g0 (x, s) ≥ δ i i1 j1 

     δ i 1−s s T0 n   ciL ciU  α0k j δok  0 as δi i1 k1 n 

 v(δ), i.e., g0 (x, s) ≥ v(δ).

,

9.2 Unconstrained Proble

213

Application 9.1 ˜ y − 10x  + 5x  3y Min z˜ (x)  10x Subject to x, y > 0.

(9.6)

Solution When the input data is taken as triangular fuzzy number, i.e., 5˜  (3, 5, 7) and   (8, 10, 12). Using nearest interval approximation method, the corresponding 10 interval number and interval-valued function, i.e., 5˜  (3, 5, 7) ≈ [4, 6] ⇒ 5ˆ  (4)1−s (6)s ∈ [4, 6].   (8, 10, 12) ≈ [9, 11] ⇒ 10  (9)1−s (11)s ∈ [9, 11], where s ∈ [0, 1]. 10 

Then the above problem transforms to Min z˜ (x, s)  (9)1−s (11)s x + (4)1−s (6)s x y − (9)1−s (11)s x 3 y Subject to x, y > 0.

(9.7)

Assume σ00  1, Corresponding dual of (9.7) is Maximize v(δ, s) 



(9)1−s (11)s δ1

δ1 

(4)1−s (6)s δ2

δ2 

(9)1−s (11)s δ3

−δ3

Subject to δ1 + δ2 − δ3  1, δ1 + δ2 − 3δ3  0, δ2 − δ3  0, δ1 , δ2 , δ3 > 0

(9.8)

Solving above equations, we get δ1  1, δ2  21 and δ3  21 . Putting the value of δ1 , δ1 and δ3 in (9.8), the corresponding optimal dual value, i.e.,  v(δ, s) 

(9)1−s (11)s 1

1 

(4)1−s (6)s 1/2

1/2 

(9)1−s (11)s 1/2

−1/2 (9.9)

So for primal decision variables, the following equations are found: (9)1−s (11)s x  δ1 v(δ, s)  1 × v(δ, s), (4)1−s (6)s x y  δ2 v(δ, s)  0.5 × v(δ, s), (9)1−s (11)s x 3 y  δ3 v(δ, s)  0.5 × v(δ, s). Solving the above equations for different values of s, we get the following results (Table 9.1). A diagram of the optimal values Z (x, y) for different values of s is given below:

214

9 Fuzzy Signomial Geometric Programming Problem

Table 9.1 Optimal solution of Application 9.1

Table 9.2 Input data of Application 9.2

S. No.

s

x

y

v(δ, s)

Z(x, s)

1

0.1

0.674

1.101

6.185

6.185

2

0.3

0.687

1.059

6.571

6.571

3

0.5

0.702

1.015

6.982

6.982

4

0.7

0.716

0.975

7.418

7.418

5

0.9

0.731

0.936

7.881

7.881

 c0  20

 ch  50

 cs  50

D

(16, 20, 24)

(40, 50, 60)

(40, 50, 60)

10

Application 9.2 Minimize  T ac(Q, S)  subject to Q, S > 0,

c0 D Q

+

ch (Q−S)2 2Q

+

cs (S)2 2Q

(9.10)

with input values (Table 9.2). Using nearest approximation method,   (16, 20, 24) ≈ [18, 22] ≈ 181−s 22s ∈ [18, 22]; 20   (40, 50, 60) ≈ [45, 55] ≈ 451−s 55s ∈ [45, 55]; s ∈ [0, 1]. 50 Then the problem is Min. T ac(Q, S, s)  Sub. Q, S > 0.

181−s 22s .10 Q

+

451−s 55s (Q−S)2 2Q

Min. T ac(Q, S, s)  Sub. Q, S > 0.

181−s 22s .10 Q

+

451−s 55s (S)2 Q

+

451−s 55s (S)2 2Q

(9.11)

i.e., +

451−s 55s Q 2

−

451−s 55s S 1

This is primal problem and corresponding dual problem is  v(δ, s)  Subject to

10.161−s 20s δ1

δ1 

451−s 55s δ2

δ2 

451−s 55s 2δ3

δ3 

451−s 55s δ4

−δ4

9.2 Unconstrained Proble

215

δ1 + δ2 + δ3 − δ4  1, −δ1 − δ2 + δ3  0, 2δ2 − δ4  0.

(9.12)

Solving above equations, we have δ4  2δ2 , δ3  1 − δ1 − δ2 + δ4  1 − δ1 − δ2 + 2δ2  1 − δ1 + δ2 , δ1 + δ2  1 − δ1 + δ2 , 1 1 δ1  , δ3  + δ2 . 2 2 i.e.,  v(δ, s) 

10.161−s 24s 1/2

1/2 

451−s 55s δ2

δ2 

451−s 55s 2(0.5 + δ2 )

(0.5+δ2 ) 

451−s 55s 2δ2

−2δ2 . (9.13)

Taking log on both sides of (9.13) and then partially differentiating with respect to δ2 and using the conditions of finding optimal solution, we get the following equation:     (log 2451−s 55s − log2δ2 ) + log451−s 55s − log2(0.5 + δ2 − 2 log451−s 55s − log2δ2  0   ⇒ log 2.451−s 55s /451−s 55s − log2δ2 (1 + 2δ2 )  0 ⇒ δ2 (1 + 2δ2 )  1.

From primal–dual relation, 10.161−s 24s  δ1 v(δ), Q 451−s 55s S 2  δ2 v(δ), 2Q 451−s 55s Q  δ3 v(δ), 2 451−s 55s  δ4 v(δ). Solving above relations with difference values of weight, we get the list of values as shown in Table 9.3.

216

9 Fuzzy Signomial Geometric Programming Problem

Table 9.3 Optimal solution of Application 9.2 Optimal values objectives s 0.1

1−s

Optimal dual variables

0.9

δ1∗

 0.5,

δ2∗

 0.5,

δ3∗



1, δ4∗

 1.

Optimal primal variables

v(δ)

T ac(Q, S)

S ∗  1.905

87.464

87.464

92.929

92.929

98.736

98.736

104.906

104.906

111.462

111.462

Q ∗  3.810 0.3

S ∗  1.944

δ1∗  0.5, δ2∗  0.5, δ3∗  1, δ4∗  1.

0.7

Q ∗  3.889 0.5

S ∗  1.984

δ1∗  0.5, δ2∗  0.5, δ3∗  1, δ4∗  1.

0.5

Q ∗  3.969 0.7

S ∗  2.026

δ1∗  0.5, δ2∗  0.5, δ3∗  1, δ4∗  1

0.3

Q ∗  4.051 0.9

S ∗  2.068

δ1∗  0.5, δ2∗  0.5, δ3∗  1, δ4∗  1

0.1

Q ∗  4.135

9.2.2 Unconstrained Modified Fuzzy Signomial GP Problem An unconstrained modified signomial function differs from a posynomial function in which the coefficient needs not be positive. g˜ k (x) 

k n  

σli c˜li

l1 i1

m 

a

xl jli j ,

j1

where c˜li is the absolute value of coefficient and σli is the sign of coefficient (+1 or −1). 2 −1 2 −1 Example 9.2 g0 (x)  2x11 x12 − 5x11 x12 + 4x21 x22 − 3x21 x22 For this signomial,

σ11  +1, σ12  −1, σ21  +1, σ22  −1, c11  2, c12  5, c21  2, c22  5. Primal problem A primal unconstrained modified signomial GP programming problem is of the form   Minimize g˜ 0 xl j Subject to xl j > 0, where g˜ 0 (x) 

n  k  l1 i1

σli c˜li

m  j1

a

xl jli j .

j  1, 2, . . . , m,

(9.14)

9.2 Unconstrained Proble

217

Using nearest interval approximation (NIA) method, transform all triangular fuzzy numbers into interval number, i.e., [cliL , cliU ]. Then, the fuzzy signomial geometric programming problem is of the following form: Minimize gˆ 0 (x1 , x2 , . . . , xm ) Subject to x j > 0, j  1, 2, . . . , m,

(9.15)

where gˆ 0 (x) 

n  k  l1 i1

σli cˆli

m 

a

x j li j ,

j1

 where cˆli denotes the interval counter parts, i.e., cˆli ∈ cliL , cliU .cliL > 0, cliU > 0, for all i. Using parametric interval-valued functional form, the problem (9.15) reduces to Minimize g0 (x1 , x2 , . . . , xm , s) Subject to x j > 0, j  1, 2 . . . , m,

(9.16)

 L 1−s  U s m n k ali j where g0 (x, s)  l1 cli i1 σli cli j1 x j . This is a parametric geometric programming (PGP) problem. Dual-signomial GP problem Dual GP problem of the given primal GP problem is ⎡

 1−s  s σli δli ⎤ζ0 cliL cliU ⎦ Maximize ζ0 ⎣ δli l1 i1 n  k 

Subject to n  k  i1 i1 n  k 

σli δli  ζ0 , σli ali j δli  0,

j  1, 2, . . . , m

(9.17)

l1 i1

δli > 0.

Case I nk ≥ nm + n (i.e., DD > 0). So the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is less than the number of dual variables. More solutions of dual-variable vector exist. In order to find an optimal solution of DP, we need to use some algorithmic methods. Case II nk < nm + n (i.e., DD < 0). So the DP presents a system of linear equations for the dual variables. Here, the number of linear equations is greater than the number of dual variables. In this case, generally no solution vector exists for the dual

218

9 Fuzzy Signomial Geometric Programming Problem

variables. However, using least square (LS) or min-max (MM) method, one can get an approximate solution for this system. Furthermore, the primal–dual relation is 

cliL

1−s 

U cli

m s   ali j xl j  ζ0 δli∗ n v(δ ∗ )., (l  1, 2, . . . , k; i  1, 2, . . . , n), s ∈ [0, 1].

(9.18)

j1

  √ Note 9.2 A weak duality theorem would say that g0 xl j , s ≥ n n v(δ), for any primal-feasible x and dual-feasible δ but this is not true for the pseudo-dual fuzzy modified signomial GP problem. Corollary 9.2 When the value of σli is 1, then a fuzzy modified signomial geometric programming (GP) problem transform to ordinary modified geometric programming problem. √ Theorem 9.2 When σi is 1, then g0 (xi j , s) ≥ n n v(δ) (Primal–Dual Inequality). Proof The expression for g0 (xi j , s) can be written as g0 (xi j , s) 

 L 1−s  U s m αli j cli cli j1 x i j δik . δik i1

n  k  l1

Here, the weights are δl1 , δl2 , . . . , δlk and positive terms are   α α L 1−s (cl2 ) (cliU )s mj1 x j l2 j (cliL )1−s (cliU )s mj1 x j ln j , . . . , . δl2 δlk Now applying A.M.-G.M. inequality, we get

(cl1L )1−s (cl1U )s

m j1

δl1

n   ⎛ n  1−s  s m  L 1−s  U s m  1−s  U s m αl1 j αl2 j αlk j ⎞ (δl1 +δl2 +···+δlk ) U L i1 + cl2 + · · · + cliL cl1 cl2 cli l1 cl1 j1 x i j j1 x i j j1 x i j ⎠ ⎝ n l1 (δl1 + δl2 + · · · + δlk )

⎛⎛ ⎛     ⎞δl1 ⎛      ⎞δl2 ⎞δlk ⎞  L 1−s  U s m αi1 j αl2 j αlk j 1−s U s m m n L 1−s cU s  cliL cl2 cl1 cli j1 x i j j1 x i j j1 x i j l2 ⎜⎝ cl1 ⎝ ⎠ ⎝ ⎠ ⎠ ⎟ ≥ ... ⎝ ⎠ δl1 δl2 δlk l1

or



g0 xi j , s n k l1



i1 δli

k n 

 δli l1 i1

≥

n  k 

 L 1−s  U s m αli j δik cli cli j1 x i j

l1 i1

δik

or

  1−s  s  δlk m k    n n k   i1 αli j δli  l1 g0 xi j , s cliL cliU ≥ xi j δli  1 as n δlk l1 j1 i1 k

αl1 j

xj

,

9.2 Unconstrained Proble

219

 1−s  s δli m k  i1 αli j δli cliL cliU  xi j δli l1 i1 j1 k n  

or

  1−s  s δli    n k n  k   g0 xi j , s cliL cliU ≥ αli j δli  0 as n δli l1 i1 i1  v(δ)

i.e.,  g0 (xi j , s) ≥ n n v(δ). Application 9.3  1 +  13 y1 +   2 y2 −  Min Z (x)  10x 5x1 y1 − 10x 6x2 + 10x 5x23 y2

(9.19)

Subject to x1 , y1 , x2 , y2 > 0

Solution When the input data is taken as triangular fuzzy number, i.e., 5˜  (3, 5, 7), 6˜    (8, 10, 12). Using nearest interval approximation method, the (4, 6, 8) and 10 corresponding interval number and interval-valued function, i.e., 5˜  (3, 5, 7) ≈ [4, 6] ⇒ 5ˆ  (4)1−s (6)s ∈ [4, 6]. 6˜  (4, 6, 8) ≈ [5, 7] ⇒ 6ˆ  (5)1−s (7)s ∈ [5, 7].   (8, 10, 12) ≈ [9, 11] ⇒ 10  (9)1−s (11)s ∈ [9, 11], where s ∈ [0, 1]. 10 

Then, problem reduces to Z (x, y)  (9)1−s (11)s x1 + (4)1−s (6)s x1 y1 − (9)1−s (11)s x13 y1 +(5)1−s (7)s x2 + (9)1−s (11)s x2 y2 − (4)1−s (6)s x23 y2 Subject to x1 , y1 , x2 , y2 > 0.

Min

Assume σ00  1. Corresponding



dual of (9.20)

Maximize v(δ)  Subject to

91−s 11s δ1

δ1

41−s 6s δ2

δ2

91−s 11s δ3

−δ3

51−s 7s δ01

δ

01



91−s 11s δ02

δ

02



41−s 6s δ03

(9.20)

−δ

03

220

9 Fuzzy Signomial Geometric Programming Problem δ1 + δ2 − δ3  1, δ1 + δ2 − 3δ3  0, δ2 − δ3  0, δ01 + δ02 − δ03  1, δ02 − δ03  0, δ1 , δ2 , δ3 , δ01 , δ02 , δ03 > 0

(9.21)

Solving above linear equations, we have δ1  1, δ2  21 , δ3  21 , δ01  1, δ02  and δ03  21 . Putting the value of δ1 , δ2 , δ3 , δ01 , δ02 and δ03 in (9.21), the corresponding optimal dual values, i.e.,







1 , 2

v(δ, s) 

91−s 11s 1

1

41−s 6s 1/2

1/2

91−s 11s 1/2

−1/2

51−s 7s 1

1

91−s 11s 1/2

1/2

41−s 6s 1/2

−1/2

(9.22) So for primal decision variables, the following equations are found:  91−s 11s x1  δ1 v(δ, s),  41−s 6s x1 y1  δ2 v(δ, s),  91−s 11s x13 y1  δ3 v(δ, s),  51−s 7s x2  δ01 v(δ, s),  91−s 11s x2 y2  δ02 v(δ, s),  41−s 6s x23 y2  δ03 v(δ, s). Solving above equations for different values of s, we get the following results. Application 9.4

Minimize  T ac(Q i , Si )  subject to Q i , Si > 0,

n  i1

c 0i Di Qi

+

2 c hi (Q i −Si ) 2Q i

+

2 c si (Si ) 2Q i

(9.23)

with input values. Using nearest approximation (NIA) techniques,   (6, 10, 14) ≈ [8, 12] ≈ 81−s 12s ∈ [8, 12]; 10   (16, 20, 24) ≈ [18, 22] ≈ 181−s 22s ∈ [18, 22]; 20   (21, 25, 29) ≈ [23, 27] ≈ 231−s 27s ∈ [23, 27]; 25   (40, 50, 60) ≈ [45, 55] ≈ 451−s 55s ∈ [45, 55]; 50 125  (105, 125, 145) ≈ [115, 135] ≈ 1151−s 135s ∈ [115, 135]; 4s ∈ [0, 1]. Then, the problem is

9.2 Unconstrained Proble

221

2 1−s s 181−s 22s .10 + 45 552Q(Q1 1 −S1 ) Q1 2 1−s s 55 (Q 2 −S2 ) 55 (S2 ) + 45 2Q 2Q 2 2

Min. T ac(Q, S, s)  1−s

2

s

+

451−s 55s (S1 )2 2Q 1

+

181−s 22s .10 Q2

+ 45 Sub. Q 1 , Q 2 , S1 , S2 > 0.

(9.24)

i.e., 1−s s 1−s s 1−s s (S1 )2 181−s 22s .10 + 45 55 + 45 255 Q 1 − 45 155 S1 Q1 Q1 1−s s 1−s s 1−s s +231−s 27s )(S2 )2 + (115 135 2Q + 115 2135 Q 2 − 115 1135 S2 . 2

Min. T ac(Q, S, s) 

+

81−s 12s .15 Q2

Sub. Q 1 , Q 2 , S1 , S2 > 0.

(9.25) This is primal problem, and the corresponding dual problem is

δ

δ

δ

−δ

δ 14 15.81−s 12s 21 10.161−s 20s 11 451−s 55s 12 451−s 55s 13 451−s 55s δ11 δ12 2δ13 δ14 δ21

δ

δ

−δ 24 (1151−s 135s + 231−s 27s ) 22 1151−s 135s 23 1151−s 135s × 2δ22 2δ23 δ24

v(δ, s) 

Subject to δ11 + δ12 + δ13 − δ14  1, δ21 + δ22 + δ23 − δ24  1, −δ11 − δ12 + δ13  0, −δ21 − δ22 + δ23  0, 2δ12 − δ14  0, 2δ22 − δ24  0.

Solving above equations, we have δ14  2δ12 δ13  1 − δ11 − δ12 + δ14  1 − δ11 − δ12 + 2δ12  1 − δ11 + δ12 , δ11 + δ12  1 − δ11 + δ12 1 δ11  , 2 1 δ13  + δ12 . 2 And δ24  2δ22 , δ23  1 − δ21 − δ22 + δ24  1 − δ21 − δ22 + 2δ22  1 − δ21 + δ22 , δ21 + δ22  1 − δ21 + δ22

(9.26)

222

9 Fuzzy Signomial Geometric Programming Problem

δ21 

1 1 , δ23  + δ22 . 2 2

i.e.,

1/2

(0.5+δ )

1/2

δ

−2δ 2 2 10.161−s 24s 451−s 55s 2 451−s 55s 451−s 55s 15.81−s 12s 1/2 δ2 2(0.5 + δ2 ) 2δ2 1/2

(0.5+δ )

δ

−2δ 2 2 1151−s 135s 1151−s 135s + 231−s 27s 2 1151−s 135s × . (9.27) 2δ2 2(0.5 + δ2 ) 2δ2

v(δ, s) 

Taking log on both sides of (9.27) and then partially differentiating with respect to δ12 and δ22 , respectively, and using the conditions of finding optimal solution, we get   (log 2.451−s 55s − log2δ12 ) − 0.5 + log451−s 55s − log2(0.5 + δ12   − 0.5 − 2 log451−s 55s − log2δ12 + 1  0   ⇒ log 2.451−s 55s /451−s 55s − log2δ12 (1 + 2δ12 )  0 ⇒ δ12 (1 + 2δ12 )  1, and   (log 2.451−s 55s − log2δ22 ) − 0.5 + log451−s 55s − log2(0.5 + δ22   − 0.5 − 2 log451−s 55s − log2δ22 + 1  0   ⇒ log 2.451−s 55s /451−s 55s − log2δ22 (1 + 2δ22 )  0 ⇒ δ22 (1 + 2δ22 )  1. From primal–dual relation, 10.161−s 24s  δ11 v(δ), Q1  451−s 55s S12  δ12 v(δ, s), Q1  451−s 55s Q 1  δ13 v(δ, s), 2  451−s 55s S1  δ14 v(δ, s),  15.81−s 12s  δ21 v(δ, s), Q2  (1151−s 135s + 231−s 27s )S22  δ22 v(δ, s), 2Q 2  1151−s 135s Q 2  δ23 v(δ, s), 2

9.2 Unconstrained Proble

223

Table 9.4 Optimal solution of Application 9.3 S. No.

s

x1

y1

x2

y2

v(δ, s)

Z(x, y, s)

1

0.1

0.750

1.102

1.333

0.281

47.483

13.621

2

0.3

0.761

1.057

1.315

0.289

52.868

14.392

3

0.5

0.771

1.016

1.297

0.298

58.864

15.209

4

0.7

0.782

0.974

1.279

0.306

65.540

16.067

5

0.9

0.792

0.936

1.262

0.314

72.973

16.975

Table 9.5 Input data of Application 9.4 i

 c 0i  20

 c hi  50

 c si  50

Di

i1

(16, 20, 24)

(40, 50, 60)

(40, 50, 60)

10

i2

(6, 10, 14)

(105, 125, 145)

(21, 25, 29)

15

Table 9.6 Optimal solution of Application 9.4 Optimal values objectives s

1−s

0.1

0.9

0.3

0.7

0.5

0.7

0.9

0.5

0.3

0.1

Optimal dual variables

Optimal primal variables

∗ δ11 ∗ δ13 ∗ δ21 ∗ δ23

∗  0.5,  0.5, δ12 ∗  1,  1, δ14 ∗  0.5,  0.5, δ22 ∗  1.  1, δ24

S1∗  1.971 Q ∗1  3.912 S2∗  0.774 Q ∗2  1.549

8187.095

195.347

∗ δ11 ∗ δ13 ∗ δ21 ∗ δ23

∗  0.5,  0.5, δ12 ∗  1,  1, δ14 ∗  0.5,  0.5, δ22 ∗  1.  1, δ24

S1∗  2.008

9205.062

206.989

∗ δ11 ∗ δ13 ∗ δ21 ∗ δ23

∗  0.5,  0.5, δ12 ∗  1,  1, δ14 ∗  0.5,  0.5, δ22 ∗  1.  1, δ24

10349.600

219.326

∗ δ11 ∗ δ13 ∗ δ21 ∗ δ23

∗  0.5,  0.5, δ12 ∗  1,  1, δ14 ∗  0.5,  0.5, δ22 ∗  1.  1, δ24

11636.450

232.372

∗ δ11 ∗ δ13 ∗ δ21 ∗ δ23

∗  0.5,  0.5, δ12 ∗  1,  1, δ14 ∗  0.5,  0.5, δ22 ∗ 1  1, δ24

13083.300

246.191

v(δ, s)

T ac(Q, S)

Q ∗1  4.015 S2∗  0.795 Q ∗2  1.590 S1∗  2.045 Q ∗1  4.090 S2∗  0.816 Q ∗2  1.633 S1∗  2.083 Q ∗1  4.166 S2∗  0.838 Q ∗2  1.677 S1∗  2.122 Q ∗1  4.244 S2∗  0.861 Q ∗2  1.722

 1151−s 135s S2  δ24 v(δ, s). Solving above relations with difference values of “s”, we get the list of values in Table 9.4 (Tables 9.5 and 9.6).

224

9 Fuzzy Signomial Geometric Programming Problem

9.3 Constrained GP Problem A constrained GP programming problem is of the form Minimize g˜ 0 (x1 , x2 , . . . , xm ) g˜ (x , x , . . . , xm ) ≤ ζk , k  1, 2, . . . , p Subject to k 1 2 x j > 0, j  1, 2, . . . , m.

9.3.1 Constrained Fuzzy Signomial GP Problem Primal problem A primal constrained signomial GP programming problem is of the form Minimize g˜ 0 (x1 , x2 , . . . , xm ) Subject to g˜ k (x1 , x2 , . . . , xm ) ≤ ζk , k  1, 2, . . . , p x j > 0, j  1, 2 . . . , m,

(9.28)

 k a σi c˜i mj1 x j i j , and ζk  ±1. where g˜ k (x)  i1 Here, ai j are real  numbers and coefficients c˜i are triangular fuzzy number (TFN), as c˜i  ci1 , ci2 , ci3 . Using nearest interval approximation (NIA) method, transform all triangular fuzzy numbers into interval number, i.e., [ciL , ciU ]. Then, the fuzzy signomial geometric programming problem is of the following form: Minimize gˆ 0 (x1 , x2 , . . . , xm ) Subject to gˆ k (x1 , x2 , . . . , xm ) ≤ ζk , k  1, 2 . . . , p x j > 0, j  1, 2, . . . , m,

(9.29)

 k a σi cˆi mj1 x j i j , and ζk  ±1. where gˆ k (x)  i1  Here, cˆi denotes the interval counterparts, i.e., cˆi ∈ ciL , ciU .ciL > 0, ciU > 0, for all i. Using parametric interval-valued functional form, the problem (9.29) reduces to Minimize g0 (x1 , x2 , . . . , xm , s) Subject to gk (x1 , x2 , . . . , xm , s) ≤ ζk , k  1, 2, . . . , p x j > 0, j  1, 2, . . . , m,  1−s  U s m k ai j ci where gk (x, s)  i1 σi ciL j1 x j , and ζk  ±1. This is a parametric geometric programming (PGP) problem.

(9.30)

9.3 Constrained GP Problem

225

Dual-signomial GP problem Dual GP problem of the given primal GP problem is  Maximize ζ0

ζ0  σi δi p n   ζk λk (ciL )1−s (ciU )s λk δi

i1

k1

(9.31)

Subject to k 

σi δi  ζk λk , k  0, 1, . . . , p

i1 k 

σi ai j δi  0,

j  1, 2, . . . , m

i1

δi > 0, λ0  1. For every locally minimum x ∗ of the primal signomial GP, there exist a dualfeasible solution (δ ∗ , λ∗ ) and sign ζ0 such that   g0 x ∗  v(δ ∗ , λ∗ ). Furthermore, the primal–dual relation is m  L 1−s  U s  ∗a ci ci x j i j  ζ0 δi∗ v(δ ∗ , λ∗ ) for i  0, j1

and m  L 1−s  U s  ∗a ci ci x j i j  δi∗ /λ∗k for i [k], k ≥ 1.

(9.32)

j1

Application 9.5 ˜ 2 z 4 − 4x ˜ 2 Min Z (x, y, z)  2y 2 −2 −1 ˜ y − 1y ˜ Subject to 2x z ≤ −1 x, y, z > 0.

(9.33)

Solution When the input data is taken as triangular fuzzy number, i.e., 1˜  (0.6, 1, 1.4), 2˜  (1, 2, 3) and 4˜  (2, 4, 6). Using nearest interval approximation method, the corresponding interval number and interval-valued function, i.e., 1˜  (0.6, 1, 1.4) ≈ [0.8, 1.2] ⇒ 1ˆ  (0, 8)1−s (1.2)s ∈ [0.8, 1.2] 2˜  (1, 2, 3) ≈ [1.5, 2.5] ⇒ 2ˆ  (1.5)1−s (2.5)s ∈ [1.5, 2.5].

226

9 Fuzzy Signomial Geometric Programming Problem

4˜  (2, 4, 6) ≈ [4, 6] ⇒ 4ˆ  (3)1−s (5)s ∈ [3, 5], where s ∈ [0, 1]. Then, the problem reduces to Min Z (x, y, z)  (1.5)1−s (2.5)s y 2 z 4 − (4)1−s (6)s x 2 Subject to (1.5)1−s (2.5)s x 2 y −2 − (0.8)1−s (1.2)s y −1 z ≤ −1 x, y, z > 0.

(9.34)

Assume ζ0  1. Corresponding dual of (9.34) δ1  1−s s −δ2  δ01 (3) (5) (1.5)1−s (2.5)s (1.5)1−s (2.5)s Maximize v(δ, s)  δ1 δ2 δ01   −δ 02 (0, 8)1−s (1.2)s 1 λ−λ 1 δ02 Subject to 

δ1 − δ2  1, −2δ2 + 2δ01  0, 2δ1 − 2δ01 + δ02  0, 4δ1 − δ02  0, λ1  −δ02 + δ02 δ1 , δ2 , δ01 , δ02 > 0

(9.35)

Solving above linear equations, we have δ1  − 21 , δ2  − 23 , δ01  −3 , and 2 δ02  −2, but δi > 0. i.e., ζ0  −1. Corresponding dual of (9.34)  δ1  1−s s −δ2  δ01 (1.5)1−s (2.5)s (3) (5) (1.5)1−s (2.5)s Maximizev(δ, s)  − δ1 δ2 δ01  −1  −δ02 (0, 8)1−s (1.2)s 1 λ−λ 1 δ02 Subject to δ1 − δ2  1, −2δ2 + 2δ01  0, 2δ1 − 2δ01 + δ02  0, 4δ1 − δ02  0, λ1  −δ01 + δ02 δ1 , δ2 , δ01 , δ02 > 0.

(9.36)

9.3 Constrained GP Problem

227

Solving above linear equations, we have δ1  21 , δ2  23 , δ01  23 , δ02  2, and λ1  21 . Putting the value of δ1 , δ2 and δ01 in (9.36), the corresponding optimal dual values, i.e., ⎡ v ∗ (δ, s)  −⎣

(1.5)1−s (2.5)s 1/2

1/2

(3)1−s (5)s 3/2

−3/2

(1.5)1−s (2.5)s 3/2

3/2

(0, 8)1−s (1.2)s 2

−2

⎤−1 (0.5)(−0.5) ⎦

.

From primal–dual relation, (1.5)1−s (2.5)s y 2 z 4  ζ0 δ1 v∗  (3)1−s (5)s x 2  ζ0 δ2 v∗  (1.5)1−s (2.5)s x 2 y −2  (0.8)1−s (1.2)s y −1 z 

−v∗ , 2

−3v∗ , 2

δ01  3, λ1

δ02  4. λ1

Solving above equations with different values of s, we get the following results (Table 9.7).

9.3.2 Constrained Modified Fuzzy Signomial GP Problem Primal problem A primal modified fuzzy signomial GP programming problem is of the form   Minimize g˜ 0 xl j Subject to g˜ lk (x) ≤ ζlk , k  1, 2, . . . , p xl j > 0, j  1, 2, . . . , m, where g˜ lk (x) 

n k l1

i1

σli c˜li

m j1

(9.37)

a

xl jli j , and ζlk  ±1.

Table 9.7 Optimal solution of Application 9.5 S. No.

s

x

y

z

v ∗ (δ, s)

Z(x, y, z)

1

0.1

0.362

0.263

1.262

−0.276

−0.269

2

0.3

0.364

0.278

1.231

−0.309

−0.288

3

0.5

0.365

0.293

1.196

−0.345

−0.313

4

0.7

0.367

0.310

1.167

−0.385

−0.333

5

0.9

0.369

0.328

1.138

−0.431

−0.336

228

9 Fuzzy Signomial Geometric Programming Problem

ali j are real numbers and coefficients c˜li are fuzzy triangular, as c˜li   1Here, cli , cli2 , cli3 . Using nearest interval approximation method, transform all triangular fuzzy numbers into interval number, i.e., [cliL , cliU ]. Then, the fuzzy signomial geometric programming problem is of the following form: Minimize gˆ 0 (x1 , x2 , . . . , xm ) Subject to gˆ lk (x1 , x2 , . . . , xm ) ≤ ζk , k  1, 2, . . . , p x j > 0, j  1, 2, . . . , m,

(9.38)

m n k ali j where gˆ lk (x)  l1 i1 σi cˆli j1 x j , and ζlk  ±1. Here, cˆli denotes the interval counterparts, i.e., cˆli ∈ cliL , cliU .cliL > 0, cliU > 0, for all i. Using parametric interval-valued functional form, the problem (9.38) reduces to Minimize g0 (x1 , x2 , . . . , xm , s) Subject to glk (x1 , x2 , . . . , xm , s) ≤ ζlk , k  1, 2, . . . , p x j > 0, j  1, 2, . . . , m,

(9.39)

 L 1−s  U s m n k ali j where glk (x, s)  l1 cli i1 σli cli j1 x j , and ζlk  ±1. This is a parametric geometric programming (PGP) problem. Dual-signomial GP problem Dual GP problem of the given primal GP problem is Maximize ζ0

 n k   p   cli σli δli  l1 i1

δli

ζ0 ζ λ λkk k

k1

Subject to k  i1 k 

σi δi  ζk λk , k  0, 1, . . . , p σi ai j δi  0,

j  1, 2, . . . , m

(9.40)

i1

δi > 0, λ0  1.

For every locally minimum x ∗ of the primal signomial GP, there exist a dualfeasible solution (δ ∗ , λ∗ ) and sign ζ0 such that   g0 x ∗  v(δ ∗ , λ∗ ). Furthermore, the primal–dual relation

9.3 Constrained GP Problem n  k 

229

σli cli

l1 i1

m 

 a xl jli j  ζ0 δi∗ n v(δ ∗ , λ∗ ). for i  0,

j1

and k n   l1 i1

σli cli

m 

a

xl jli j 

j1

δli λli

for i [k], k ≥ 1.

(9.41)

Application 9.6 3 ˜ 3 x22 ˜ 21 − 5x  11  11 − 10x x12 + 6x Min Z (x)  10x 21 Subject to x11 x12 + x21 x22 ≤ 1 x11 , x12 , x21 , x22 > 0.

(9.42)

When the input data is taken as triangular fuzzy number, i.e., 5˜  (3, 5, 7), 6˜    (8, 10, 12). Using nearest interval approximation method, the (4, 6, 8) and 10 corresponding interval number and interval-valued function, i.e., 5˜  (3, 5, 7) ≈ [4, 6] ⇒ 5ˆ  (4)1−s (6)s ∈ [4, 6]. 6˜  (4, 6, 8) ≈ [5, 7] ⇒ 6ˆ  (5)1−s (7)s ∈ [5, 7].   (8, 10, 12) ≈ [9, 11] ⇒ 10  (9)1−s (11)s ∈ [9, 11], where s ∈ [0, 1]. 10 

Then problem transformed to 3 Z (x)  (9)1−s (11)s x11 − (9)1−s (11)s x11 x12 1−s s 1−s s 3 +(5) (7) x21 − (4) (6) x21 x22 Subject to x11 x12 + x21 x22 ≤ 1 x11 , x12 , x21 , x22 > 0.

Min

Taking ζ0  1. Then, the corresponding dual problem is δ  −δ02  1−s s δ11  1−s s −δ12 5 7 4 6 91−s 11s 01 91−s 11s d(δ, s)  δ01 δ02 δ11 δ12  δ21  δ22 1 1 (δ21 + δ22 )(δ21 +δ22 ) δ21 δ22 Subject to 

230

9 Fuzzy Signomial Geometric Programming Problem

δ01 − δ02  1, δ11 − δ12  1, δ01 − 3δ02 + δ21  0, −δ02 + δ21  0, δ11 − 3δ12 + δ22  0, −δ12 + δ22  0. Approximate solutions of this system of linear equations are δ01  2, δ02  1, δ21  1, and δ11  2, δ12  1, δ22  1. Then the objective function is  d(δ, s) 

91−s 11s 2

2 

91−s 11s 1

−1 

51−s 7s 2

2 

41−s 6s 1

−1  1  1 1 1 (2)(2) . 1 1

From the primal–dual relation, the following system of equations gives optimal primal variables:   91−s 11s x11  δ01 d(δ, s)  2. d(δ, s),   3 91−s 11s x11 x12  δ02 d(δ, s)  1. d(δ, s),   51−s 7s x21  δ11 d(δ, s)  2. d(δ, s),   3 41−s 6s x21 x22  δ12 d(δ, s)  1. d(δ, s), 1 x11 x12  , 2 1 x21 x22  . 2 Solving above equations with different values of s, we get the following results (Table 9.8).

9.4 Conclusion Previously, we expected the parameters involved with an optimization problem as a crisp values or i random variables. Yet, in reality, the parameters change every now and then. Once more, calculating these factors by the probability distribution is hard

9.4 Conclusion

231

Table 9.8 Optimal solution of Application 9.6 S. No.

s

x11

x12

x21

x22

d(δ, s)

Z(x)

1

0.1

0.836

0.598

1.485

0.337

14.736

7.390

2

0.3

0.842

0.594

1.455

0.344

16.183

7.423

3

0.5

0.847

0.590

1.425

0.351

17.771

7.459

4

0.7

0.853

0.586

1.396

0.358

19.516

7.488

5

0.9

0.859

0.582

1.367

0.366

21.431

7.506

because of the absence of recorded information. The cost parameters for the most part are evaluated in view of managerial judgment and past experiment. Hence, in this chapter, we have presented signomial geometric programming (SGP) technique under fuzzy environment.

References M. Avriel, A.C. Williams, An extension of geometric programming with applications in engineering optimization. J. Eng. Math. 5(2), 187–194 (1971) B.-Y. Cao, J.-H. Yang, Advances in fuzzy geometric programming. Fuzzy Inf. Eng. 40, 497–502 (2007) R.J. Duffin, E.L. Peterson, Duality theory for geometric programming. SIAM J. Appl. Math. 14, 1307–1349 (1966) R.J. Duffin, E.L. Peterson, Geometric programming with signomials. J. Optim. Theory Appl. 11, 3–35 (1973) S. Liu, Posynomial geometric programming with parametric uncertainty. Eur. J. Oper. Res. 168, 345–353 (2006) J.-H. Yang, B.-Y. Cao, Fuzzy geometric programming and its application. Fuzzy Inf. Eng. 2, 101–112 (2010)

Chapter 10

Goal Geometric Programming

10.1 Introduction Goal programming (GP) is one of the vital branches of multi-objective optimization, which thusly is a part of multi-criteria decision analysis (MCDA). Goal programming (GP) is one of the best optimization techniques. It may be very well thought of as an expansion of linear programming or nonlinear programming to deal with various, typically clashing objective measures. Every one of these measures is given an (goal) esteem to be achieved. Unwanted deviations from this set of target esteems are then minimized in an achievement work. The idea of goal programming was first introduced by Charnes et al. in 1955; however, the actual name originally used by Charnes and Cooper in 1961. Schniederjans introduced a large number of articles identifying with goal programming in pre1995. Jones and Tamiz gave a comprehensive overview of the condition of the goal programming. Goal programming is a mathematical model with linear or nonlinear functions and continuous or discrete variables, in which all functions have been transformed into goals. After World War II, industrial world faced a problem at that point developed the goal programming. Goal programming models can be arranged into two major subsets. In the first, the unwanted deviations are allocated weights as per their relative significance to the decision-making (DM) and minimized as an Archimedean sum. This type is known as weighted goal programming (WGP). In the other real subset of goal programming, the deviational variable is assigned into various priority structures where the goals in a similar priority level are recognized by putting relative weights. The goals with higher priority level will be achieved before lower priority goals and so on. In the field of GP, the priority-based GP is the most powerful strategy for solving multiple and conflicting goals in MODM environment. Generally, a goal programming (GP) uses three types of analysis as follows:

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_10

233

234

10 Goal Geometric Programming

(1) Determine the required resource to achieve a desired set of objectives. (2) Determine the degree of attainment of the goals with the available resources. (3) Providing the best satisfying solution under a varying amount of resources and priorities of the goals.

10.2 Strength and Weakness of Goal Programming The major strength of goal programming (GP) is that it is very simple and easy to use. Linear goal programming can be solved using linear programming software as either a single linear programming or a series of connected linear programming. The main weakness of goal programming is that solutions are not Pareto efficiency. This violated the fundamental concept of decision theory. To deal with multi-objective decision-making problems, there are two prominent approaches available in the literature as (a) min-sum GP and (b) priority-based GP. (a) Min-sum goal programming This is also of two types as given below: (i) General goal programming, (ii) Weighted goal programming. (i) General goal programming model Charnes and Cooper (1977) presented general goal programming problem as follows: Minimize Z 

m   +  di + di− i1

Subject to n 

ai j x j − di+ + di−  bi , i  1, 2, . . . , m. ⎛ ⎞ ≤ n  System constraints: ai j x j bi ⎝  ⎠bi , i  m + 1, m + 2, . . . , m. j1 ≥ Goal constraint:

j1

With di+ , di+ , x j ≥ 0, for i  1, 2, . . . , m and j  1, 2, . . . , n, where Z ai j xj bi di+ di−

objective function. the coefficient with the variable j in the ith goal. the jth decision variable. the associate right-hand side value. positive deviation variable from the ith goal. negative deviation variable from the ith goal.

(10.1)

10.2 Strength and Weakness of Goal Programming

235

(ii) Weighted goal programming (WGP) model Weighted goal programming (WGP) makes it more interesting and challenging to take a decision of a decision-making problem. Charnes and Cooper (1977) presented the weighted goal programming (WGP) model as follows: Minimize Z 

m   + +  wi di + wi− di− i1

Subject to n 

ai j x j − di+ + di−  bi , i  1, 2, . . . , m. ⎛ ⎞ ≤ n  System constraints: ai j x j bi ⎝  ⎠bi , i  m + 1, m + 2, . . . , m. j1 ≥

Goal constraints:

j1

(10.2)

With di+ , di+ , x j ≥ 0, for i  1, 2, . . . ., m and j  1, 2, . . . , n, where wi+ and are nonnegative weight function representing the relative weight to be assigned to the respective positive and negative deviation variables. wi−

(b) Priority-based goal programming In the priority-based GP procedure, the goals are ranked according to their priorities for achievement of the respective aspiration levels in the decision-making situation. The general priority-based GP model is presented as Find X (x1 , x2 , . . . , xn ) so as to       

 Minimum D  P1 d − , d + , P2 d − , d + , . . . , Pk d − , d + , . . . , PK d − , d + And satisfy

n 

ai j x j − di+ + di−  bi , i  1, 2, . . . , m.

j1

With di+ , di+ , x j ≥ 0, for i  1, 2, . . . , m  and j  1, 2, .. . −, n. +D  is − + d is of the form P d  , d , d the K priority achievement function, P k k  m  + + − − w ; k  1, 2, . . . , K ; and P d + w d ≫P · · · ≫P · · ·  P , here 1 2 k K i1 ik ik ik ik “≫” implies much greater than the previous priority level. − + wik , wik ≥ 0; dik+ , dik− ≥ 0; with dik+ · dik−  0, i  1, 2, . . . , m;

j  1, 2, . . . , n; k  1, 2, . . . , K . It is to be noted that the min-sum goal programming is actually considered as a special case of priority-based goal programming where no priority preference is given to the goals.

236

10 Goal Geometric Programming

10.3 Importance of Weighted Goal Programming (WGP) (a) WGP is used when the decision-maker is able to provide relative preference between goals. (b) WGP can be used when the decision-maker wishes to examine trade-offs between goals via different weight sets. (c) The achievement function consists of a weighed, normalized sum of unwanted deviations. Example of WGP w1− d1− w2+ d2+ + + 10w2− d2− 6 4 Subject to 4x1 + 3x2 − d1+ + d1−  20, 5x1 + 7x2 − d2+ + d2−  35. Min  5w1+ d1+ +

10.4 Chebyshev Goal Programming Model Chebyshev goal programming model was demonstrated by Flavell in 1976. It uses to minimize the maximum undesirable deviation as opposed to the total of deviations. Thus, Chebyshev goal programming is sometimes called min-max goal programming. This uses the Chebyshev distance metric, which underlines justice and balance as opposed to ruthless optimization. The preemptive lexicographic GP display in (10.1) and the non-preemptive weighted GP demonstrate in (10.2) can be seen as the two extreme types of goal programming models in which essentially all goal programming modeling are determined.

10.5 Multi-objective Problem A multi-objective problem can be written as Find X  (x1 , x2 , . . . , xn )T So as to Minimize f 10 (x)  Minimize f 20 (x) 

p10 

c10i

n

11

k1

p20 

n

11

c20i

k1

xkαk10i with target c10 , xkαk20i with target c20 ,

10.5 Multi-objective Problem

237

··· Minimize f m0 (x) 

pm0 

cm0i

n

xkαkm0i with target cm0 ,

11 pr

Subject to fr (X ) 

k1 n  α cri xk kri 11 k1

≤ cr , r  1, 2, . . . , q,

xk > 0, k  1, 2, . . . n,

(10.3)

where c j0i are positive real numbers for all j  1, 2 . . . m; i  1, 2 . . . , pr . αk j0i and αkri are real numbers for all k  1, 2, . . . , n; j  1, 2, . . . , m; i  1, 2, . . . , pr . number of terms present in j0 th objective function. p j0 number of terms present in rth constraint. pr boundary value of the rth constraint. cr In the above multi-objective nonlinear programming model, there are m number of minimizing objective functions, q number of inequality type constraints, and n number of strictly positive decision variables.

10.5.1 Multi-objective Goal Geometric Programming (MOGP) Problem One of the leading techniques for multi-objective decision analysis (MODA) is goal programming (GP). It (GP) is a “powerful tool which draws upon the highly developed and tested technique of linear programming problem (LPP), but provides a simultaneous solution to a complex system of competing objectives.” Goal programming (GP) can deal with decision problems having a single goal with numerous subgoals. The method was initially presented by Charnes and Cooper (1961), and additionally developed by Jaaskelainen (1969), Lee and Bird (1970), Lee (1972) and Ignizio (1976). At that point, numerous researchers and mathematician, for example, Kwak and Schniederjans (1979, 1985), Ignizio (1987, 1989), Hallefjord and Jornsten (1988), Reeves and Hedin (1993), Hemaida and Kwak (1994), Bryson (1995), Easton and Rossin (1996), and so on, reviewed contextual analysis and applications of goal programming and multiple criteria decision-making (MCDM) problems, and focus their perspectives for multi-objective decision-making (MODM) programming problems. In any case, the classification of MCDM techniques exhibited by Zanakis and Gupta (1985), Steuer (1986), Romero (1991), Tamiz and Jones (1995), and so on is usual practice to separate strategies in light of the classifications of the problem. MCDM is essential discipline that arrangements with decision-making problem with multiple goals (Objectives).

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10 Goal Geometric Programming

GP technique gives preferable outcomes over some other standard multi-objective programming technique. To formulate the goal geometric programming problem (GGPP) from the multi-objective nonlinear programming problem (MONLPP), positive or negative deviations are minimized relying on objective functions and constraints, e.g., for maximizing the objective function negative deviation, for minimizing the objective function positive deviation, for “≥” type requirement negative deviation is limited, and for “≤” type constraint positive deviation. From the multi-objective programming problem (10.3), goal formulation is given below: Minimize

m 

q 

dr+ r 1 + f j0 (x) + d − j0 − d j0  c j0 , − + fr (x) + dr − dr  cr , r

j1

subject to

d +j0 +

xk > 0, k  − + + d j0 , d j0 , dr , dr− > 0, + − d +j0 .d − j0 > 0, dr .dr > 0. d +j0 d− j0 dr+ dr− c j0 , cr

1, 2, . . . , n.

j  1, 2, . . . , m.  1, 2, . . . , q.

(10.4)

Positive deviation of the minimizing objective function. Negative deviation of the minimizing objective function. Positive deviation of “≤” constraint. Negative deviation of “≤” constraint. are boundary values of objective function and constraint.

The above model (10.4) can be transformed into Minimize

m 

q 

dr+ r 1 f j0 (x) − d +j0 ≤ c j0 , fr (x) − dr+ ≤ cr , r

j1

subject to

d +j0 +

j  1, 2, . . . , m,  1, 2, . . . , q xk > 0, k  1, 2, . . . , n. d +j0 , dr+ > 0.

The single solution like (x∗, d +j0 , dr+ ), j  1, 2, . . . , m; r  1, 2, . . . , q minimize the objective (10.4) and satisfy the constraints. But there are many cases where much more minimized value is required for any particular objectives or/and constraints. Then, we usually tackle this situation by introducing weights. Give biggest weight (priority) for that deviation of the objective function or constraint for which we want to get more minimized value.

10.5 Multi-objective Problem

239

10.5.2 Multi-objective Weighted Goal Programming (MOWGP) Formula The multi-objective weighted goal GP formula is given as follows: Minimize

m  j1

w j0 d +j0 +

q  r 1

wr dr+

subject to f j0 (x) − d +j0 ≤ c j0 , j  1, 2, . . . , m fr (x) − dr+ ≤ cr , r  1, 2, . . . , q, xk > 0, k  1, 2, . . . , n.

(10.5)

And m 

w jo +

q 

wr  1.

r 1

j1

w jo > 0, wr > 0.

10.5.3 Formulation of Goal GP Programming Using Weight A weighted goal geometric programming is given as follows: Minimize subject to

m 

w j0 d +j0 +

q 

wr dr+

r 1 j1 d +j0 f j0 (x) − c j0 ≤ 1, c j0 d+ fr (x) − crr ≤ cr , cr

j  1, 2, . . . , m,

(10.6)

r  1, 2, . . . , q, xk > 0, k  1, 2, . . . , n. And m  j1

w jo +

q 

wr  1.

r 1

w j0 > 0, wr > 0.

10.5.4 Dual Form of Goal GP Problem The given problem in (10.6) is simple signomial geometric programming problem. That is, dual form of (10.6) is

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10 Goal Geometric Programming

⎛

  m p j0  m δ j pl0+1 q m w j0 δ j0 wr δr c j0i δ ji 1 δ j0 δr c j0 δ ji c j0 δ j p j0+1 r 1 j1 j1 i1 j1 ⎞ξ δr pr +1 q pr q q m cri δri 1 λ j (δ)λ j (δ) λr (δ)λr (δ) ⎠ . c δ c δ r ri r r p r +1 r 1 i1 r 1 r 1 j1

v(δ)  ξ ⎝

Subject to q m   δ j0 + δr  ξ, j1

r 1

ξ  ±1, δ jo − δ j p j0+1  0, j  1, 2, . . . , m, δr − δ j pr +1  0, r  1, 2, . . . , q, p j0 q  pr m    ak j0i δ ji + akri δri  0, k  1, 2, . . . , n, j1 i1

where λ j (δ)  λr (δ) 

p j0  i1 pr  r 1

(10.7)

r 1 i1

δ ji − δ j p j0+1 , j  1, 2, . . . , m, δri − δr pr +1 , r  1, 2, . . . , q.

Application 10.1 Minimize f 1 (x1 , x2 )  x1−1 x2−2 with target value 4, Minimize f 2 (x1 , x2 )  2x1−2 x2−3 with target value 50, Subject to x1 + x2 ≤ 1, x1 , x2 > 0. Solution The above multi-objective goal programming problem is converted into single objective goal geometric programming problem using deviations and giving the weights (priorities). The formulation is given below: Minimize w1 d1+ + w2 d2+ Subject to

x1−1 x2−2 d+ − 41 ≤ 1, 4 2x1−2 x2−3 d+ − 502 ≤ 1, 50

x1 + x2 ≤ 1, x1 , x2 , d1+ , d2+ > 0. Here, the degree of difficulty  8 − (4 + 1)  3.

(10.8)

10.5 Multi-objective Problem

241

And corresponding dual programming is      δ21 w1 δ01 w2 δ02 1 δ11 1 δ12 2 v(δ)  ξ δ01 δ02 4δ11 4δ12 50δ21 ξ

δ22 δ31 δ32 3 1 1 1 λi (δi ) λi (δi ) 50δ22 δ31 δ32 i1 Subject to δ01 + δ01  ξ, δ01 − δ12  0, δ02 − δ22  0, −δ11 − 2δ21 + δ31  0, −2δ11 − 3δ21 + δ32  0, λ1 (δ1 )  δ11 − δ12 , λ2 (δ2 )  δ21 − δ22 , λ3 (δ3 )  δ31 + δ32 .

(10.9)

If ξ  −1, then from above, δ02  −1 − δ01. But δ01 > 0, and therefore according to the relation δ02 is negative which contradicts the positivity condition of dual variables. Hence, let ξ  1, then we get the following equations: δ02  1 − δ01 , δ12  δ01 , δ22  1 − δ01 , δ31  δ11 + 2δ21 , δ32  2δ11 + 3δ21 , λ1 (δ1 )  δ11 − δ01 , λ2 (δ2 )  δ21 + δ01 − 1, λ3 (δ3 )  3δ11 + 5δ21 . That is, δ

δ −δ δ −(1−δ ) (δ +2δ ) 1−δ01 11 01 21 01 11 21 w2 1 1 2 1 1 1 − δ01 4δ11 4δ01 50δ21 50(1 − δ01 ) δ11 + 2δ21    2δ +3δ  11 21   1  (δ11 − δ01 )(δ11 −δ01 ) δ21 + δ01 − 1 (δ21 +δ01 −1) (3δ11 + 5δ21 )(3δ11 +5δ21 ) . ×  2δ11 + 3δ21

v(δ) 

w1 δ01

δ 01

(10.10)

Taking log on both sides of (10.10) and then partially differentiating with respect to δ01 , δ11 , and δ21 , respectively, and using the conditions of finding optimal solution, we get these sets of equation

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10 Goal Geometric Programming

log(4w1 ) − log(50w2 ) − log(δ11 − δ01 ) + log(δ21 + δ01 − 1)  0,

(10.11)

− log(4δ11 ) + log(δ11 − δ01 ) − log(δ11 + 2δ21 ) − log(2δ11 + 3δ21 ) + log(3δ11 + 5δ21 )  0, log 2 − log(50δ21 ) + log(δ21 + δ01 − 1) − 2 log(δ11 + 2δ21 ) − 3 log(2δ11 + 3δ21 ) + 5 log(3δ11 + 5δ21 )  0.

(10.12)

(10.13)

From primal–dual relation, w1 d1+  δ01 d(δ), w2 d2+  δ02 d(δ)  (1 − δ01 )d(δ) d1+ δ01 δ12  ,  4 δ11 − δ12 δ11 − δ01 d2+ 1 − δ01 δ22   , 50 δ21 − δ22 δ21 + δ01 − 1 δ31 δ11 + 2δ21  , x1  δ31 + δ32 3δ11 + 5δ21 δ32 2δ11 + 3δ21 x2   . δ31 + δ32 3δ11 + 5δ21 Solving above with difference values of weight and putting in (10.11), (10.12), and (10.13), we get the list of values as shown in Table 10.1.

10.6 Goal Programming with Logarithmic Deviational Variables Ghosh and Roy (2011) introduced goal geometric programming with logarithmic deviational variables. Consider a goal programming problem as follows:

10.6 Goal Programming with Logarithmic Deviational Variables

243

Table 10.1 Optimal solution of Application 10.1 Optimal value objectives w1

w2

Optimal dual variables

0.1

0.9

∗  0.032, δ ∗  0.968 δ01 02 ∗ δ11 ∗ δ21 ∗ δ31

0.3

0.7

0.5

0.3

  

∗ 1.887, δ12 ∗ 0.032, δ22 ∗ 1.968, δ32

 2.886  0.968  2.968

∗  0.229, δ ∗  0.032 δ01 21 ∗ δ02 ∗ δ12 ∗ δ31

0.7

 2.968  0.968  2.968

∗  0.113, δ ∗  0.886 δ01 02 ∗ δ11 ∗ δ21 ∗ δ31

0.5

  

∗ 1.968, δ12 ∗ 0.032, δ22 ∗ 1.968, δ32

  

∗ 0.770, δ11 ∗ 2.771, δ22 ∗ 1.968, δ32

 1.770  0.968  2.968

0.1

1st objective f 1 (x1 , x2 )

2nd objective f 2 (x1 , x2 )

x1∗  0.399

6.938

57.971

6.930

57.932

6.892

57.935

x2∗

 0.601

x1∗  0.395 x2∗

 0.605

x1∗  0.390 x2∗

 0.610

∗  0.407, δ ∗  0.592 δ01 02

x1∗  0.381

6.850

58.105

∗  0.721, δ ∗  0.279 δ01 02

x1∗  0.359 x2∗  0.641

6.781

58.898

∗  1.593, δ ∗  2.593 δ11 12 ∗  0.032, δ ∗  0.968 δ21 22 ∗  1.968, δ ∗  2.968 δ31 32

0.9

Optimal primal variables

∗ δ11 ∗ δ21 ∗ δ31

  

∗ 1.279, δ12 ∗ 0.032, δ22 ∗ 1.968, δ32

 2.279  0.968  2.968

x2∗  0.619

q  d +j0 + dr+ r 1 j1   subject to log f j0 (x) + d +j0 − d − j  1, 2, . . . , m, j0 ≤ log c j0 , log( fr (x)) + dr+ − dr− ≤ log cr , r  1, 2, . . . , q, xk > 0, k  1, 2, . . . , n,

Minimize

m 

where d +j0 d− j0 dr+ dr− c j0 , cr

Positive deviation of the minimizing objective function. Negative deviation of the minimizing objective function. Positive deviation of “≤” constraint. Negative deviation of “≤” constraint. are boundary values of objective function and constraint.

(10.14)

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10 Goal Geometric Programming

  For the logarithmic change of deviational variables, let d +j0  log u +j0 , d − j0     +  − − + − log u j0 , dr  log vr , and dr  log vr , then the model (1.8) becomes   q m   + + Minimize log u j0 vr r 1 j1  f (x)u −  j0 j0 (10.15) ≤ log c j0 , j  1, 2, . . . , m, subject to log +  u j0 − +  log fr (x)vr /vr ≤ log cr , r  1, 2, . . . , q, xk > 0, u +j0 , u −j0 , vr+ , vr− > 1, k  1, 2, . . . , n. Without loss of generally, the optimal decision variables (ODV) remain same in the above model which is written as follows:   q m   + + Minimize u j0 vr subject to

r 1 j1 f j0 (x)u −j0 ≤ c j0 , j  1, 2, . . . , m, u +j0 fr (x)vr− ≤ cr , r  1, 2, . . . , q, vr+ xk > 0, u +j0 , u −j0 , vr+ , vr− > 1, k 

(10.16) 1, 2, . . . , n.

The above model (10.16) can be transformed into   q m   + + Minimize u j0 vr subject to

r 1 j1 f j0 (x) ≤ c j  1, 2, . . . m, + j0 , u j0 fr (x) ≤ cr , r  1, 2, . . . q, vr+ xk > 0, u +j0 , vr+ > 1, k  1, 2, . . . , n.

10.6.1 Weighted Goal Programming Problem with Logarithmic Deviational Variables The above goal programming (GP) problem can be reduced into the weighted goal programming (GP) with logarithmic variable as follows:

10.6 Goal Programming with Logarithmic Deviational Variables



w j0  q   m   w u +j0 vr+ r

Minimize subject to

245



r 1 j1 f j0 (x) ≤ c j0 , j  1, 2, . . . , m, u +j0 fr (x) ≤ cr , r  1, 2, . . . , q, vr+ q m   xk > 0, u +j0 , vr+ > 1; w j0 + wr r 1 j1

(10.17)  1, w j0 > 0, wr > 0.

k  1, 2, . . . , n. Dual program The given problem in (10.17) is simple signomial geometric programming (SGP) problem. That is, dual form of (10.17) is ⎛

⎞ξ δ ji p j0 q q pr m m cri δri c j0i d(δ)  ξ ⎝ λ j (δ)λ j (δ) λr (δ)λr (δ) ⎠ c δ c δ j0 ji r ri r 1 i1 r 1 j1 i1 j1 Subject to q m   δ j0 + δr  ξ, j1

r 1

ξ  ±1, p j0 q  pr m    ak j0i δ ji + akri δri  0, k  1, 2, . . . , n, r 1 i1

j1 i1

where λ j (δ)  λr (δ) 

p j0  i1 pr  r 1

δ ji ,

j  1, 2, . . . , m,

δri , r  1, 2, . . . , q.

Application 10.2 Minimize u w1 v w2 Subject to x1−1 x2−2 u −1 ≤ 4; 2x1−2 x2−3 v −1 ≤ 50; x1 + x2 ≤ 1; x1 , x2 > 0; u, v > 1; w1 , w2 > 0. Here, f 1 (x)  x1−1 x2−2 u −1 with target value 4. And f 2 (x)  2x1−1 x2−2 v −1 with target value 50.

(10.18)

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10 Goal Geometric Programming

Solution The dual program of the give primal problem is   δ21 δ31 2 1 1 δ01 1 δ11 Maximize v(δ)  δ01 4δ11 50δ21 δ31

δ32 1 λ1 (δ)λ1 (δ) λ2 (δ)λ2 (δ) λ3 (δ)λ3 (δ) δ31 Subject to

δ01  1, w1 δ01 − δ11  0, w2 δ01 − δ21  0, −δ11 − 2δ21 + δ31  0, −2δ11 − 3δ21 + δ32  0, λ1 (δ)  δ11 , λ2 (δ)  δ21 , λ3 (δ)  δ31 + δ32 .

(10.19)

Solving above sets of equations, δ01  1, δ11  w1 , δ21  w2 , δ31  w1 + 2w2 , δ32  2w1 + 3w2 , λ1 (δ)  w1 , λ2 (δ)  w2 and λ3 (δ)  3w1 + 5w2 . Putting the values of dual variable   w2 2w1 +3w2 w1 +2w2

1 1 w1 2 1 1 1 v(δ)  1 4w1 50w2 w1 + 2 2w1 + 3w2 w1w1 w2w2 (3w1 + 5w2 )(3w1 +5w2 ) . Again from primal–dual relation, x1−1 x2−2 u −1  1, 4 2x1−2 x2−3 v −1  1, 50 w1 + 2w2 x1  , 3w1 + 5w2 2w1 + 3w2 x2  . 3w1 + 5w2 Optimal solution of the given problem is shown in Table 10.2.

(10.20)

10.6 Goal Programming with Logarithmic Deviational Variables

247

Table 10.2 Optimal solution of Application 10.2 Optimal value objectives w1

w2

Optimal dual variables

0.1

0.9

∗ ∗ ∗ δ01  1, δ11  0.1, δ21  0.9

x1∗  0.396 x2∗  0.604 u 2.994 v 1.158

4.001

49.983

∗ ∗ ∗ δ01  1, δ02  0.3, δ11  0.7

x1∗  0.387 x2∗  0.614 u 1.713 v 1.154

4.001

49.991

∗ ∗ ∗ δ01  1, δ02  0.5, δ11  0.5

x1∗  0.375 x2∗  0.625 u 1.707 v 1.165

3.999

50.004

∗ ∗ ∗ δ01  1, δ02  0.7, δ11  0.3

x1∗  0.361 x2∗  0.639 u 1.696 v 1.176

4.000

50.016

x1∗  0.344 x2∗  0.656 u 1.689 v 1.197

3.999

50.016

∗ ∗  1.9, δ32  2.9 δ31

0.3

0.7

∗ δ12

0.5

0.5



∗ 1.7, δ21

 2.7

∗ ∗  1.5, δ21  2.5 δ12

0.7

0.3

∗ δ12

0.9

0.1



∗ 1.3, δ21

 2.3

∗ ∗ ∗ δ01  1, δ02  0.9, δ11  0.1 ∗ ∗  1.1, δ21  2.1 δ12

Optimal primal variables

1st objective f 1 (x1 , x2 )

2nd objective f 2 (x1 , x2 )

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10 Goal Geometric Programming

10.7 Fuzzy Goal Programming Problem Fuzzy goal programming (FGP) depends on fuzzy set theories. Fuzzy sets are utilized to depict uncertain goals. These goals are typically connected with objective functions and are utilized to reflect both weighting (with values from zero to one) and scope of goal achievement possibilities. The connection between the weighting and the profit function can be linear or nonlinear in nature. In particular, this system permits the decision-maker who cannot accurately characterize goals to in any problem express them utilizing a weighting structure that is not limited. This makes fuzzy programming approach when utility function type goals are to be utilized in the goal programming technique. Fazlollahtabar et al. (2013) derived “a fuzzy goal programming model for optimizing service industry market by using virtual intelligent agent.” Kumar et al. (2004) approached “a fuzzy goal programming (FGP) technique for vendor selection problem in a supply chain.” Mekidiche et al. (2013) presented “a weighted additive fuzzy goal programming aggregate production planning.” Yimmee and Phruksaphanrat (2011) presented “fuzzy goal programming for aggregate production and scientists.” A fuzzy multi-objective problem can be written as Find X  (x1 , x2 , . . . , xn )T So as to Minimize f 10 (x)  Minimize f 20 (x) 

p10 

c10i

n

11

k1

p20 

n

c20i

11

xkαk10i with target c10 and tolerance t10 , xkαk20i with target c20 and tolerance t20 ,

k1

... Minimize f m0 (x) 

pm0 

cm0i

n

xkαkm0i with target cm0 and tolerance tm0 ,

11 pr

Subject to fr (X ) 

k1 n  α cri xk kri 11 k1

xk > 0, k  1, 2, . . . , n.

≤ cr , r  1, 2, . . . , q. (10.21)

There are various kinds of membership functions such that linear, piecewise linear, hyperbolic, exponential, etc. Now the corresponding membership function of the objective functions is

10.7 Fuzzy Goal Programming Problem

μ( f i0 (x)) 

⎧ ⎪ ⎨ 1,

ci0 +ti0 − f i0 (x) , ti0

⎪ ⎩ 0,

249

if f i0 (x) ≤ ci0 , if f i0 (x) < f i0 (x) ≤ ci0 + ti0 , if f i0 (x) > ci0 .

(10.22)

For i  1, 2, . . . , m. Now a fuzzy multi-objective goal programming can be transformed into a crisp goal programming by substituting the membership functions. Then, the problem (10.21) is transformed to crisp problem as follows: Maximize z(μ) 

m 

wi μ( f i0 (xk ))

i1

Subject to fr (xk ) ≤ cr , r  1, 2, . . . , q, μ( f i0 (xk )) ≤ 1, m  wi  1, xk > 0, k  1, 2, . . . , n. i1

Application 10.3 Minimize f 1 (x1 , x2 )  x1−1 x2−2 with target value 4 tolerance 0.02. Minimize f 2 (x1 , x2 )  2x1−2 x2−3 with target value 50 tolerance 0.05. Subject to x1 + x2 ≤ 1, x1 , x2 > 0. Solution Considered membership function of the problem

μ( f 10 (x)) 

μ( f 20 (x)) 

⎧ ⎪ ⎨ 1,

4.02−x1−1 x2−2 , 0.02

⎪ ⎩ 0,

⎧ ⎨ 1, ⎩

50.05− f i0 (x) , 0.05

0,

Using fuzzy additive method,

if x1−1 x2−2 ≤ 4,

if 4 < x1−1 x2−2 ≤ 4.02 , if x1−1 x2−2 > 4.02.

if 2x1−2 x2−3 ≤ 50, if 50 < 2x1−2 x2−3 ≤ 50.05 , if 2x1−2 x2−3 > 50.05.

(10.23)

(10.24)

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10 Goal Geometric Programming

 Max w1

4.02 − x1−1 x2−2 0.02



 + w2

50.05 − 2x1−2 x2−3 0.05



Subject to x1 + x2 ≤ 1, x1 , x2 > 0.

(10.25)

Which is equivalent to  Min w1

x1−1 x2−2 0.02



 + w2

2x1−2 x2−3 0.05

 − 201w1 − 1001w2

Subject to x1 + x2 ≤ 1, x1 , x2 > 0.

(10.26)

If we ignore −201w1 − 1001w2 , the problem becomes  Min w1

x1−1 x2−2 0.02



 + w2

2x1−2 x2−3 0.05



Subject to x1 + x2 ≤ 1, x1 , x2 > 0.

(10.27)

Which is primal problem, now dual of (10.27) is

w1 v(δ)  0.02δ1 Subject to

δ1

w1 0.05δ2

δ2 (

1 δ01 1 δ02 ) ( ) δ01 δ02

δ1 + δ2  1, −δ1 − 2δ2 + δ01  1, −2δ1 − 3δ2 + δ02  1.

(10.28)

The solution of the above model using geometric programming technique is shown in Table 10.3.

10.7.1 Fuzzy Multi-objective Goal Programming (FMOGP) in Parametric Form When coefficients and exponents are fuzzy in nature, then a multi-objective goal programming problem is called fuzzy multi-objective goal programming problem. That is, an FMOGP problem is of the following form:

10.7 Fuzzy Goal Programming Problem

251

Table 10.3 Optimal solution of Application 10.3 Optimal value objectives w1

w2

Optimal dual variables

0.1

0.9

∗ ∗ δ01  0.032, δ02  0.968 ∗ ∗  1.968δ12  2.968 δ11

0.3

0.7

0.9

1st objective f 1 (x1 , x2 )

2nd objective f 2 (x1 , x2 )

x1∗  0.399

6.938

57.971

6.930

57.932

6.892

57.935

6.850

58.105

6.781

58.898

x2∗

 0.601

∗ ∗ δ01  0.113, δ02  0.886 ∗ ∗  1.887, δ12  2.886 δ11

x1∗ x2∗

 0.395

0.5

∗ δ01 ∗ δ11

 2.771

x1∗ x2∗

 0.390

0.3

∗ δ01 ∗ δ11

 0.592

x1∗  0.381

0.7

0.5

Optimal primal variables

0.1

   

∗ 0.229, δ02 ∗ 1.770, δ12 ∗ 0.407, δ02 ∗ 1.593, δ12

 0.770

 2.593

∗ ∗ δ01  0.721, δ02  0.279 ∗ δ11



∗ 1.279, δ12

 2.279

x2∗ x1∗ x2∗

 0.605  0.610  0.619  0.359  0.641

q  d +j0 + dr+ r 1 j1  subject to log  f j0 (x) + d +j0 − d − c j0 , j  1, 2, . . . , m, j0 ≤ log   + − log  fr (x) + dr − dr ≤ log c˜r , r  1, 2, . . . , q,

Minimize

m 

(10.29)

xk > 0, k  1, 2, . . . , n, where d +j0 d− j0 dr+ dr− c˜ j0 ,c˜r

Positive deviation of the minimizing objective function. Negative deviation of the minimizing objective function. Positive deviation of “≤” constraint. Negative deviation of “≤” constraint. are boundary values of objective function and constraint, which are fuzzy in umber in nature.

10.7.2 Fuzzy Goal Programming Problem with Logarithmic Deviational Variables   For the logarithmic change of deviational variables, let d +j0  log u +j0 , d − j0        log u −j0 , dr+  log vr+ , and dr−  log vr− , then the model () becomes

252

10 Goal Geometric Programming

 Minimize log

m 

j1

u +j0

 f j0 (x)u −j0 u +j0

q  r 1



 vr+

≤ log c˜ j0 , j  1, 2, . . . , m,  log  fr (x)vr− /vr+ ≤ log c˜r , r  1, 2, . . . , q,

subject to log



(10.30)

xk > 0, u +j0 , u −j0 , vr+ , vr− > 1, k  1, 2, . . . , n. Without loss of generally, the optimal decision variables (ODV) remain same in the above model which is written as follows:   q m   + + Minimize u j0 vr subject to

r 1 j1  f j0 (x)u −j0 ≤ c˜ j0 , j  1, 2, . . . , m, u +j0 −  fr (x)vr ≤ c˜r , r  1, 2, . . . , q, vr+ xk > 0, u +j0 , u −j0 , vr+ , vr− > 1, k

(10.31)  1, 2, . . . , n.

The above model (10.31) can be transformed into   q m   Minimize u +j0 vr+ subject to

r 1 j1  f j0 (x) ≤ c˜ j0 , j  1, 2, . . . , m, u +j0  fr (x) ≤ c˜r , r  1, 2, . . . , q, vr+ xk > 0, u +j0 , vr+ > 1, k  1, 2, . . . , n.

(10.32)

10.7.3 Fuzzy Weighted Goal Programming Problem with Logarithmic Deviational Variables The above goal programming (GP) problem can be reduced into the weighted goal programming (GP) with logarithmic variable as follows:

10.7 Fuzzy Goal Programming Problem

 Minimize subject to

m  

u +j0

w j0  q   w vr+ r

253



r 1 j1  f j0 (x) ≤ c˜ j0 , j  1, 2 . . . m, u +j0  fr (x) vr+ ≤ c˜r , r  1, 2 . . . q, q m   xk > 0, u +j0 , vr+ > 1; w j0 + wr r 1 j1

 1, w j0 > 0, wr > 0. k  1, 2, . . . , n.

(10.33) Dual program The given problem in (10.33) is simple signomial geometric programming (SGP) problem. That is, dual form of (10.33) is ⎞ξ δ ji δri p j0 q q pr m m c  c  j0i ri λ j (δ)λ j (δ) λr (δ)λr (δ) ⎠ d(δ)  ξ ⎝ c  δ c ˜ δ j0 ji r ri r 1 i1 r 1 j1 i1 j1 ⎛

Subject to q m   δ j0 + δr  ξ, j1

r 1

ξ  ±1, p j0 q  pr m    ak j0i δ ji + akri δri  0, k  1, 2, . . . , n, j1 i1

r 1 i1

where λ j (δ) 

p j0 

δ ji ,

j  1, 2, . . . , m,

i1 pr

λr (δ) 



δri , r  1, 2, . . . , q.

r 1

Application 10.4

Minimize u w1 v w2 Subject to ˜ x1−1 x2−2 u −1 ≤ 4; −2 −3 −1  ˜ 1 x2 v ≤ 50; 2x ˜ x1 + x2 ≤ 1;

x1 , x2 > 0; u, v > 1; w1 , w2 > 0.

(10.34)

254

10 Goal Geometric Programming

Here, f 1 (x)  x1−1 x2−2 u −1 with target value 4. And f 2 (x)  2x1−1 x2−2 v −1 with target value 50. Solution Let the coefficients are fuzzy triangular number as 1˜  (0.6, 1, 1, 4), 4˜    (40, 50, 60). (2, 4, 6) and 50 Then, using fuzzy parametric interval method, 1˜  (0.6, 1, 1, 4) ≈ [0.8, 0.12] ≈ (0.8)1−s .(1.2)s ∈ [0.8, 0.12], 2˜  (1.6, 2, 2.4) ≈ [1.8, 2.2] ≈ (1.8)1−s .(2.2)s ∈ [1.8, 2.2], 4˜  (2, 4, 6) ≈ [3, 5] ≈ (3)1−s .(5)s ∈ [3, 5],   (40, 50, 60) ≈ [45, 55] ≈ (45)1−s .(55)s ∈ [45, 55], s ∈ [0, 1]. 50 Then, given problem reduces to Minimize u w1 v w2 Subject to x1−1 x2−2 u −1 ≤ (3)1−s .(5)s (1.8)1−s .(2.2)s x1−2 x2−3 v −1 ≤ (45)1−s .(55)s x1 + x2 ≤ (0.8)1−s .(1.2)s ; x1 , x2 > 0; u, v > 1; w1 , w2 > 0.

(10.35)

The dual program of the give primal problem is δ21  δ11  δ31 1 1 1 δ01 (1.8)1−s .(2.2)s δ01 (3)1−s .(5)s δ11 (0.8)1−s .(1.2)s δ31 (45)1−s .(55)s δ21 δ32

1 × λ1 (δ)λ1 (δ) λ2 (δ)λ2 (δ) λ3 (δ)λ3 (δ) (0.8)1−s .(1.2)s δ31

Maximize v(δ) 

Subject to δ01  1, w1 δ01 − δ11  0, w2 δ01 − δ21  0, − δ11 − 2δ21 + δ31  0, − 2δ11 − 3δ21 + δ32  0, λ1 (δ)  δ11 , λ2 (δ)  δ21 , λ3 (δ)  δ31 + δ32 .

(10.36)

10.7 Fuzzy Goal Programming Problem

255

Solving above sets of equations, δ01  1, δ11  w1 , δ21  w2 , δ31  w1 + 2w2 , δ32  2w1 + 3w2 , λ1 (δ)  w1 , λ2 (δ)  w2 and λ3 (δ)  3w1 + 5w2 . Putting the values of dual variable, we get

d(δ) 

 w w +2w 2w +3w

1   2 1 2 1 2 w w 1 w1 2 1 1 1 w1 1 w2 2 (3w1 + 5w2 ) (3w1 +5w2 1 4w1 50w2 w1 + 2w2 2w1 + 3w2

(10.37)

Again from primal–dual relation, x1−1 x2−2 u −1  1, (3)1−s (5)s (1.8)1−s (2.2)s x1−2 x2−3 v −1  1, (45)1−s (55)s x1 w1 + 2w2  , 1−s s 3w (0.8) (1.2) 1 + 5w2 x2 2w1 + 3w2  . 3w1 + 5w2 (0.8)1−s (1.2)s The optimal solution of the given problem is shown in Table 10.4.

10.8 Conclusion By utilizing goal geometric programming (GGP), we can take care of multi-target objective programming issue (MOGPP) as indicated by our necessities or objectives. Goal geometric programming (GGP) is extremely helpful for some real-valued (practical) situations where the conditions are nonlinear. We demonstrate the effectiveness of this strategy with weighted sum deviations where weights (priorities) can be changed according to prerequisite. The variety of result as indicated by weights (priorities) likewise demonstrates the flawlessness of this technique. Comparing and other nonlinear optimization techniques (Kuhn–Tucker conditions), this technique gives better outcome. In the field of operations research, engineering, applied mathematics, sciences, etc., this technique has the great applications. Here, we have also discussed the goal geometric programming (GPP) in fuzzy environment and given some numerical precedents to confirm results. It is obvious to us that fuzzy objective geometric programming gives better outcome, i.e., this technique fulfills every one of our objectives and prerequisite according to our opinion. Rather than weighted sum method one can utilize weighted product method, min-max method, etc. We can apply this strategy in uncertain condition like intuitionistic fuzzy, neutrosophic fuzzy, and interval number also.

256

10 Goal Geometric Programming

Table 10.4 Optimal solution of Application 10.4 Optimal value objectives w1

w2

s

0.1

0.9

0.1

0.5

Optimal dual variables

Optimal primal variables

∗ 1 δ01

x1∗  0.330

∗ δ11 ∗ δ21 ∗ δ31 ∗ δ32

 0.1  0.9  1.9  2.9

1st objective f 1 (x1 , x2 )

2nd objective f 2 (x1 , x2 )

3.605

45.916

x1∗  0.388

3.875

49.736

x1∗  0.456

4.750

53.960

3.157

45.916

3.873

49.741

x1∗  0.445

4.766

53.962

x1∗  0.312

3.157

45.918

3.873

49.742

4.749

53.964

x2∗

 0.503 u  3.322 v  2.886 x2∗  0.592 u  1.898 v  1.281

0.9

x2∗

 0.696 u  0.953 v  0.570 0.3

0.7

0.1

0.5

∗ 1 δ01 ∗ δ02 ∗ δ11 ∗ δ12 ∗ δ21

 0.3  0.7  1.7  2.7

x1∗  0.322 x2∗

 0.511 u  3.767 v  2.891 x1∗  0.379 x2∗

 0.601 u  1.886 v  1.283 0.9

x2∗  0.707 u  0.946 v  0.571 0.5

0.5

0.1

0.5

∗ 1 δ01 ∗ δ02 ∗ δ11 ∗ δ12 ∗ δ21

 0.5  0.5  1.5  2.5

x2∗

 0.521 u  3.740 v  2.906 x1∗  0.367 x2∗

 0.612 u  1.878 v  1.296 0.9

x1∗  0.432 x2∗

 0.702 u  0.989 v  0.619 (continued)

References

257

Table 10.4 (continued) Optimal value objectives w1

w2

s

0.7

0.3

0.1

0.5

Optimal dual variables

Optimal primal variables

∗ 1 δ01

x1∗  0.301

∗ δ02 ∗ δ11 ∗ δ12 ∗ δ21

 0.7  0.3  1.3  2.3

1st objective f 1 (x1 , x2 )

2nd objective f 2 (x1 , x2 )

3.157

45.919

x1∗  0.354

3.873

49.744

x1∗  0.416

4.751

53.965

x2∗

 0.532 u  3.718 v  2.932 x2∗  0.626 u  1.861 v  1.301

0.7

x2∗

 0.736 u  0.934 v  0.580 0.9

0.1

0.1

0.3

0.9

∗ 1 δ01 ∗ δ02 ∗ δ11 ∗ δ12 ∗ δ21

 0.9  0.1  1.1  2.1

x1∗  0.286 x2∗  0.547 u  3.701 v  2.988

3157

45.921

x1∗  0.337 x2∗  0.643 u  1.853 v  1.324

3.873

49.745

x1∗  0.396 x2∗  0.756 u  0.930 v  0.590

4.751

53.967

References N. Bryson, A goal programming method for generating priority vectors. J. Oper. Res. Soc. 46, 641–648 (1995) A. Charnes, W.W. Cooper, R. Ferguson, Optimal estimation of executive compensation by linear programming. Manage. Sci. 1, 138–151 (1955) A. Charnes, W.W. Cooper. Goal programming and multiple objective optimization. Eur. J. Oper. Res, 1, 39–71 (1977). http://dx.doi.org/10.1016/S0377-2217(77)81007-2 A. Charnes, W.W. Cooper, Management models and industrial applications of linear programming (Wiley, New York, 1961) Easton and Rossin, A stochastic goal program for employee scheduling. Decis. Sci. 27(3), 541–568 (1996) H. Fazlollahtabar, I. Mahdavi, A. Mohajeri, Applying fuzzy mathematical programming approach to optimize a multiple supply network in uncertain condition with comparative analysis. Appl. Comp. J. 13(1), pp. 550–562, (2013) J.H. Flavell, Metacognitive aspects of problem solving, in The nature of intelligence, ed. by L.B. Resnick (pp. 231–235). (Hillsdale, NJ: Lawrence Erlbaum, 1976)

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P. Ghosh, T.K. Roy, A goal geometric programming problem (G 2 P 2) with logarithmic deviational variables and its applications on two industrial problems. J. Ind. Eng. Int. 9(1), pp. 1–9 (2013) A. Hallefjord, K. Jornsten, A critical comment on integer Goal Programming. J. Oper. Res. Soc. 39 101–104, (1988) R.S. Hemaida, N.W. Kwak A linear goal programming model for transshipment problems with flexible supply and demand constraints. J. Oper. Res. Soc. 45(2), 215–224 (1994) J.P. Ignizio, Goal programming and extensions (Lexington Books, Lexington, MA, 1976) J.P. Ignizio, A reply to Comments on an algorithm for solving the linear goal—programming problem by solving its dual. J. Oper. Res. Soc. 38, 1149–1154 (1987) J.P. Ignizio, On the merits and demerits of integer goal programming. J. Oper. Res. Soc. 40(8), 781–785. (Hampshire, UK, 1989) V. Jääskeläinen, Accounting and Mathematical Programming (contact author; Helsinki, 1969) M. Kumar, P. Vrat, R. Shankar, A fuzzy goal programming approach for vendor selection problem in a supply chain, Computers & Industrial Engineering, vol. 46, (2004), pp. 69–85 N.K. Kwak and M.J. Schniederjans, A goal programming model for improving transportation problem solving, OMEGA, vol. 7, (1979), pp. 367–370 N.K. Kwak, M.J. Schniederjans, An alternative solution method for solving goal programming problems: The Lexicographic Goal Programming Case. Socio-economic Planning Sciences, 19(2), 101–107 (April, 1985) S.M. Lee, M. Bird, A goal programming model for sales effort allocation. Bus. Perspect. 6 17–21 (1970) S.M. Lee, Goal programming for decision analysis (Auerback, Philadelphia, 1972) M. Mekidiche, M. Belmokaddem, Z. Djemmaa, Weighted additive fuzzy goal programming approach to aggregate production planning. Int. J Intell. Syst. Appl. 4, 20–29 (2013) C. Reeves, S.R. Hedin, A generalized interactive goal programming procedure, Computers and Oper. Res. 20 (1993) 747–753 Romero, A survey of generalized goal programming 1970–1982. Eur. J. Oper. Res. 25. 183–191 (1986) C. Romero, Handbook of critical issues in goal programming (Pergamon Press, Oxford, 1991) R.E. Steuer, Multiple criteria optimization: Theory, computation, and application (Wiley, New York, 1986) M. Tamiz, D.F. Jones, E. El-darzin, A review of goal programming and its applications. Ann. Oper. Res. 58, 39–53 (1995) R. Yimme, B. Phruksaphanrat, Fuzzy goal programming for aggregate production and logistics planning. International MultiConference of Engineers and Computer Scientist vol II. Hongkong. Halaman 16–18 (2011) S.H. Zanakis, S.K. Gupta, A categorized bibliographic survey of goal programming. Omega, 13(13), 211–222 (1985)

Chapter 11

Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

11.1 Introduction A nonlinear programming problem (NLP) may be taken as Find t  (t1 , t2 , . . . , tn )T of the following NLP Min g0 (t) subject to g j (t) ≤ b j ( j  1, 2 , . . . , m) t ≥0

(11.1)

In real-life problem, it is possible to soften the rigid requirements of the decisionmaker (DM) to strictly minimize the objective function and strictly satisfy the constraints. In this position, above NLP may be taken as following Fuzzy Nonlinear programming Problem.

11.2 Fuzzy Nonlinear Programming (FNLP) ˜ Min g0 (t) subject to g j (t) ≤ b j ( j  1, 2, . . . ., m) ∼

t ≥0

(11.2)

˜ Here the symbol “Min” denotes a relaxed or fuzzy version of “Min”. It implies the objective function should be minimized as well as possible. Similarly the symbol “≤” denotes a fuzzy version of “≤”. It also implies that the constraints should be ∼

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_11

259

260

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

µ j ( g j (t ))

Fig. 11.1 Linear membership function

well satisfied. These  fuzzy requirements may be quantified by eliciting membership functions μ j g j (t) (j  0, 1, 2, …, m) from the DM for all functions gj (t) (j  0, 1, 2,…, m). By taking into account the rate of increased membership   satisfaction, the DM must determine the subjective membership function μ j g j (t) . It is in general a strict monotone decreasing linear or nonlinear function u j g j (t) with respect to gj (t) (j  0, 1, 2,…, m). Here for simplicity, linear membership functions are taken. They are   μ j g j (t)  1 if g j (t) ≤ g 0j   gj −g j(t) u j g j (t)  g −g0 if g 0j ≤ g j (t) ≤ g j 0

j

j

if g j (t) ≥ g j

for j  0, 1, 2,…, m. As shown in Fig. 11.1,

  g 0j ≡ the value of gj (t) such that the grade of membership function μ j g j (t) is 1.   g j ≡ the value of gj (t) such that the grade of membership function μ j g j (t) is 0. g¯ j ≡ the intermediate value of gj (t) between g 0j and g j (i.e. g¯ j ∈ (g 0j , g j )) such that the grade of membership function say α ∈ (0, 1)

If we follow the fuzzy decision on fuzzy objective and constraint goals of  Bellman and Zadeh (1970) then using above-said membership functions μ j g j (t) (j  0, 1, 2, …, m), the problem of finding the maximizing decision to choose optimal t (i.e. t*) such that (i)      μ D t ∗  max minμ j g j (t) (max-min operator)    g j − g j(t) subject to u j g j (t)   ( j  1, 2, . . . , m) g j − g 0j

11.2 Fuzzy Nonlinear Programming (FNLP)

261

t >0

(11.3)

(ii) ⎛ ⎞ m   ∗   μ D t  max⎝ λ j μ j g j (t) ⎠ (max-addition operator) j0

  g j − g j(t) subject to u j g j (t)   ( j  1, 2, . . . , m) g j − g 0j t >0

(11.4)

(iii) ⎛ ⎞ m  λj  ∗  μ D t  max⎝ (μ j g j (t) ) ⎠ (max-product operator) j0

  g j − g j(t) subject to u j g j (t)   ( j  0, 1, 2, . . . , m) g j − g 0j t ≥0

(11.5) λj

Note 11.1 For normalized weights with

m j0 λ j  1 and λ j ∈ [0, 1].

(j  0, 1, 2, …, m),



T j a c ji rn1 tr jir If gj (t) (j  0, 1, 2, …, m) be posynomial function as g j (t)  i1 (cji (>0) and ajir be any real number for j  0, 1, 2, …, m; I  1, 2, …, T j ; r  1, 2, …, n) in max-addition operator (11.4) reduces to Max V (t) 

m 

λj

g j −

n a jir r 1 tr i1 c ji  0 gj − gj

T j

j0

subject to t > 0

(11.4a)

So optimal decision variable t * with optimal objective value V * (t * ) can be obtained

λ j g by V * (t * )  mj0 g −gj 0 − U ∗ (t ∗ ) where t * is optimal decision variable of the unconj j strained geometric programming problem Min U (t) 

m 

λj

Tj 

j0

g j − g 0j

i1

subject to t > 0

c ji

n

a

tr jir

r 1

(11.4b)

262

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

11.3 Multi-objective Mathematical Programming (MOMP) Problem A general multi-objective mathematical programming problem may be taken in the following form: Minimize k nonlinear objective functions f 1 (x1 , x2 , x3 , . . . , xn ), f 2 (x1 , x2 , x3 , . . . , xn ), ..., f k (x1 , x2 , x3 , . . . , xn ),

(11.6)

subject to the inequality constraints g1 (x1 , x2 , x3 , . . . , xn ) ≤ b1 , g2 (x1 , x2 , x3 , . . . , xn ) ≤ b2 , ... gm (x1 , x2 , x3 , . . . , xn ) ≥ bm , li ≤ xi ≤ u i (i  1, 2, .., n).

(11.7)

or in a vector-minimization problem (VMP) form Minimize

f (x)  [ f 1 (x), f 2 (x), . . . f k (x)]T

subject to x ε X  {xε R n : g j (x) ≤ or  or ≥ b j for j  1, 2, . . . , m; x ≥ 0}. and li ≤ xi ≤ u i (i  1, 2, .., n). It should be noted that if the convexity conditions of the objective functions and/or the feasible region are not satisfied, the MOMP (11.6) becomes a non-convex MOMP problem. In practice, however because only local optimal solutions are guaranteed in solving a single-objective programming problem by any available nonlinear programming technique, unless the problem is convex. A direct application of the optimality for single-objective nonlinear programming to the MOMP (11.6) leads us to the following complete optimality concept: Definition 11.1 (Complete Optimal Solution) x * is said to be a complete optimal solution to the MOMP (2.10) if and only if there exists x * ε X such that f r (x * ) ≤ f r (x), for r  1, 2, …, k and for all x ε X. However, when the objective functions of the MOMP conflict with each other, a complete optimal solution does not always exist and hence the Pareto-optimality concept arises and it is defined as follows .

11.3 Multi-objective Mathematical Programming (MOMP) Problem

263

Definition 11.2 (Pareto-Optimal Solution) x * is said to be a Pareto-optimal solution to the MOMP (11.6) if and only if there does not exist another x ε X such that f r (x * ) ≤ f r (x) for all r  1, 2, …, k and f j (x)  f j (x * ) for at least one j, j ∈ {1, 2, …, k}. Definition 11.3 (Local (Global) Optimal Solution) x * ∈ X (the feasible set of constrained decisions) is said to be a local (global) Pareto-optimal solution to the MOMP (2.10) iff x * is Pareto-optimal in X ∩ N(x * , δ) where N(x * , δ) denotes the δ neighborhood of x * defined by {x ∈ R n : x − x ∗ < δ, δ ∈ R + }.

11.4 Multi-objective Optimization In recent there has been a great increase the research interest on multi-objective optimization (MOO) techniques. Decisions with multi-objectives are very fruitful in military, government and other organizations. Researchers and Scientists from a mass variety of disciplines, for example, management science, mathematics, engineering, financial matters and others have added to the solution techniques for multiobjective optimization (MOO) problems. The circumstance is detailed as a multiobjective optimization (MOO) in which the goal is to maximize or minimize not a single-objective function but rather a several objective functions simultaneously. The reason for multi-objective problems in the mathematical programming structure is to optimize the distinctive target problems, (say “k” in number) simultaneously subject to set of constraints. For example, Minimize f (t)  [ f 1 (t), f 2 (t), . . . , f k (t)]T subject to g j (t) ≤ b j t >0

j  1, 2, . . . , m (11.8)

Here now we shall describe the fuzzy optimization technique (through GP) to solve the above multi-objective problem. Multi-Objective Geometric Programming Problem Using Fuzzy Technique Geometric programming (GP) is a special technique to solve a class of nonlinear programming problems (NLPs). Generally we utilize this technique to solve some problems like maximize volume, minimize cost and/or weight as well as effective and so on. Mainly, a management science and an engineering design problem have multi objective functions. For this situation it is not appropriate to use any

264

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

single-objective programming technique to find an optimal solution with compromise. We can utilize fuzzy programming to determine such a compromise solution. Pascal and Ben Israel deal with multi-objective geometric programming (MOGP) in 1971. Biswal (1992), Verma (1990) developed fuzzy geometric programming method to solve multi-objective geometric programming (MOGP) problem. Here we have talked about another fuzzy geometric programming (FGP) technique to solve MOGP problem. Das et al. (2000) presented multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: a geometric programming approach. Mandal et al. (2005) defined multi-objective fuzzy inventory model with three constraints under geometric programming approach. Islam and Roy (2006) developed a new fuzzy multi-objective entropy based geometric programming and its application of transportation problems. Panda et al. (2008a, b) developed multi-item EOQ model with hybrid cost parameters under fuzzy geometric programming (FGP) approach. Mahapatra and Roy (2009) presented single and multi container maintenance model under fuzzy geometric programming approach. Islam and Roy (2010) worked on multiobjective geometric programming (MOGP) problem and its application. Ojha and Biswal (2010) presented multi-objective geometric programming (MOGP) problem with weighted mean method. A multi-objective geometric programming (MOGP) problem can be stated as Find t  (t1 , t2 , . . . , tn )T so as to 0

Min f 1 (t) 

T1 

0 c1i

T20



a0

tt 1ir

r 1

i1

Min f 2 (t) 

n

0 c2i

n

a0

tt 2ir

r 1

i1

... 0

Min f k (t) 

Tk 

0 cki

i1

n

a0

tt kir

r 1

subject to gk (t) 

Tk  s1

t >0

cks

n

traksr ≤ 1 k  1, 2, . . . , m

r 1

(11.9)

where c0ji (>0), cks (>0), a 0jir , a jir are all real numbers for j  1, 2, …, k; i  1, 2, …, T j0 ; k  1, 2,…, m; s  1, 2, …,.Tk . To solve this multi-objective geometric programming, we use the Zimmermann’s (1978) solution procedure. This procedure consists of the following steps:

11.4 Multi-objective Optimization

265

Step 1: Solve the MOGP as a single-objective GP problem using only one objective at a time and ignoring the others. These solutions are known as ideal solution. Step 2: From the results of Step 1, determine the corresponding values for every objective at each solution derived. With the values of all objectives at each ideal solution, pay-off matrix can be formulated as follows: f (t) f (t) . . . f k (t) ⎤ ⎡ ∗11 2 1 t1 f 1 (t ) f 2 (t ) . . . f k (t 1 ) ∗ 2 2 2 ⎥ t2 ⎢ ⎢ f 1 (t ) f 2 (t ) . . . f k (t ) ⎥ ... ⎣ ... ... ... ... ⎦ k k t f 1 (t ) f 2 (t k ) . . . f k∗ (t k ) 1 2 Here t , t , …, t k are the ideal solutions of the objectives f 1 (t), f 2 (t), …, f k (t) respectively. So U r  max {f r (t 1 ), f r (t 2 ), …, f r (t k )} and L r  fr∗ (t r ) for r  1, 2, …, k. [L r and U r be lower and upper bounds of the rth objective function f r (t) for r  1, …, k]. Step 3: Using aspiration levels of each objective of the MOGP (29) may be written as follows: Find t so as to satisfy fr (t) ≤ L r (r  1, 2, . . . , k) ∼

subject to gk (t) 

Tm  i1

cki

n

a

tr mi j ≤ 1 k  1, 2 . . . , m

r 1

t >0

(11.10)

Here objective functions of the problem (11.9) are considered as fuzzy constraints. This type of fuzzy constraints can be quantified by eliciting a corresponding membership function ⎫ μr ( fr (t))  0 if fr (t) ≥ Ur ⎬  u r (t) if L r ≤ fr (t) ≤ Ur (r  1, 2, . . . , k) ⎭ 1 if fr (t) ≤ L r .

(11.11)

Here u r (t) is a strictly monotonic decreasing function with respect to f r (t). (Fig. 11.2). Having elicited the membership functions (as in Eq. (11.11)) μr (f r (t)) for r  1, 2, …, k, a general aggregation function μ D˜ (t)  μ D˜ (μ1 ( f 1 (t)), μ2 ( f 2 (t)), . . . , μk ( f k (t))) is introduced. If we follow the fuzzy decision on fuzzy objective and constraint goals of  Bellman and Zadeh (1970) then using above-said membership functions μ j g j (t) (j  0, 1, 2, …, m), the problem of finding the maximizing decision to choose optimal t (i.e. t*) such that

266

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

µr (fr (t)

1 u r (t) Lr

Ur

fr (t)

Fig. 11.2 Membership function for minimization problem

(i)      μ D t ∗  max minμ j f j (t) (max-minoperator)   U j − f j (t) subject to u j f j (t)  ( j  0, 1, 2, . . . ., m) Uj − Lj gk (t) 

Tm 

cki

i1

n

a

tr mi j ≤ 1 k  1, 2 . . . , m

r 1

t >0

(11.12)

(ii) ⎛ ⎞ m   ∗   μ D t  max⎝ λ j μ j f j (t) ⎠ (max-addition operator) j0



 U j − f j (t) subject to u j f j (t)  ( j  0, 1, 2, . . . , m) Uj − L j gk (t) 

Tm 

cki

n

a

tr mi j ≤ 1 k  1, 2 . . . , m

r 1

i1

t >0

(11.13)

(iii) ⎛ ⎞ m  ∗   λj μ D t  max⎝ (μ j f j (t) ) ⎠ (max-product operator) j0



 U j − f j (t) subject to u j f j (t)  ( j  0, 1, 2, . . . , m) Uj − L j

11.4 Multi-objective Optimization

gk (X ) 

Tm 

cki

i1

267 n

a

tr mi j ≤ 1 k  1, 2 . . . , m

r 1

t >0

(11.14)

(11.13) reduces to Max V (t) 

m 

λj

g j −

n a jir r 1 tr i1 c ji  0 gj − gj

T j

j0

subject to t > 0

(11.15)

So optimal decision variable t * with optimal objective value V * (t * ) can be obtained

λ j g by V * (t * )  mj0 g −gj 0 − U ∗ (t ∗ ) where t * is optimal decision variable of the unconj j strained geometric programming problem Min U (t) 

m 

λj

Tj 

j0

g j − g 0j

i1

c ji

n

a

tr jir

r 1

subject to t > 0

(11.16)

Pareto-Optimality Test A numerical test of Pareto-optimality for x * can be performed by solving the following problem: Maximize

k 

εr

r 1

  subject to fr (x) + εr  fr x ∗ (for r  1, 2, . . . , k) x ε X, ε  (ε1 , ε2 , . . . , εk )T ≥ 0 Let x¯ be an optimal solution to this problem. If εr  0, for all r  1, 2, …, k, then x * is a Pareto-optimal solution. If at least one εr > 0, not x * but x¯ is a Pareto-optimal solution of the MOMP. Application 11.1

Min {Z 1 (X ), Z 2 (X )} subject to Y1 (X ) ≤ 1 x1 , x2 > 0 where Z 1 (X )  x1−1 x2−2 , Z 2 (X )  2x1−2 x2−3 and Y1 (X )  x1 + x2

268

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

This is multi-objective primal geometric programming (MOPGP) Problem. In order to solve this problem, we shall solve the sub-problems

Min Z 1 (X ) subject to Y1 (X ) ≤ 1 (PGP-1) x1 , x2 > 0 and

Min Z 2 (X ) subject to Y1 (X ) ≤ 1 (PGP-2) x1 , x2 > 0 Solving the above problems by GP technique we have   1 2 1 , Z 1 (X )  6.75 For(PGP-1)x  3 3   2 3 , For(PGP-2)x 2  Z 2 (X )  57.8703 5 5 Now the pay-off matrix is given below Z Z  1 2  x 1 6.75 60.75 x 2 6.94 57.87 From pay-off matrix the lower and upper bound of Z 1 (X ) and Z 2 (X ) be 6.75 ≤ Z 1 (X ) ≤ 6.94 and 57.87 ≤ Z 2 (X ) ≤ 60.75 Let μ1 (X ), μ2 (X ) be the fuzzy membership function of the objective function Z 1 (X ) and Z 2 (X ) respectively and they are defined as μ1 (X ) 

⎧ ⎨1 ⎩

6.94−Z 1 (X ) 0.19

0

if Z 1 (X ) ≤ 6.75 if 6.75 ≤ Z 1 (X ) ≤ 6.94 if Z 1 (X ) ≥ 6.94

Figure 11.3 illustrated the graph of the membership function μ1 (X ) and

11.4 Multi-objective Optimization

269

Fig. 11.3 Membership function for Z 1 (x)

µ 2 (X )

1

0

57.87

60.75

Z 2 (X )

Fig. 11.4 Membership function for Z 2 (x)

μ2 (X ) 

⎧ ⎨1 ⎩

60.75−Z 2 (X ) 2.88

0

if Z 2 (X ) ≤ 57.87 if 57.87 ≤ Z 2 (X ) ≤ 60.75 if Z 2 (X ) ≥ 60.75

Now Fig. 11.4 illustrated the membership function The crisp form of the problem Max (μ1 (X ) + μ2 (X ))    Z 1 (X ) Z 2 (X ) + i.e., Max 57.61 − 0.19 2.88 subject to x1 + x2 ≤ 1 x1 , x2 > 0 1 (X ) 2 (X ) For maximizing the above problem, we minimize Z0.19 + Z2.88 subject to x1 + x2 ≤ 1. so our new problem is to solve   Z 1 (X ) Z 2 (X ) + Min g(X )  0.19 2.88   −1 −2 i.e Min g(X )  5.269x1 x2 + 0.699x1−2 x2−3

270

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

subject to x1 + x2 ≤ 1 x1 , x2 > 0 Degree of Difficulty  (4 − (2 + 1))  1. The dual problem of the above is 

       5.269 w01 0.699 w02 1 w11 1 w12 (w11 + w12 )w11 +w12 w01 w02 w11 w12 s.t w01 + w02  1 − w01 − 2w02 + w11  0

Max v(w) 

− w01 − 3w02 + w12  0 w01 , w02 , w11 , w12 > 0 i.e. 

Max v(w01 ) 

    2−w  3−2w 01 01 0.699 1−w01 1 1 5.269 w01 (5 − 3w01 )5−3w01 w01 1 − w01 2 − w01 3 − 2w01

subject to 0 < w01 < 1. 01 ) Solving the equation dv(w  0 by Newton–Raphson method we ultimately dw01 ∗ ∗ ∗ ∗  0.0065, w12  0.0113 and v(w∗ )  get w01  0.63745, w02  0.36254, w11 56.8389. Therefore, x1∗  0.36577, x2∗  0.63422 with g(X ∗ )  56.8389. So, Z 1 (X ∗ )  6.796 and Z 2 (X ∗ )  58.599.

Application 11.2: GP problem (Gravel-Box problem) 80 m3 of gravel is to be ferried across a river on a barrage. A box (with open top) is to be built for this purpose. After the entire grave has been ferried, the box is to be discarded. The transport cost-per-round trip of barrage of box is Rs. 1 and the cost of materials of sides and bottom of box are Rs. 10/m2 and Rs. 80/m2 and ends of box Rs. 20/m2 . Find the dimension of the box that is to be building for this purpose and total optimal cost.

11.4 Multi-objective Optimization

271

Let us assume the gravel box has length  t1 m width  t2 m height  t3 m ∴ The area of the end of the gravel box  t2 t3 m2 The area of the side of the gravel box  t1 t3 m2 The area of the bottom of the gravel box  t1 t2 m2 ∴ The volume of the gravel box  t1 t2 t3 m2 Cost function 80 m3  Rs. 80t1−1 t2−1 t3−1 , t1 t2 t3 m3 /trip   Material cost : End of box: 2 Rs. 20/m2 t2 t3 m2  Rs. 40t2 t3   Sides of box: 2 Rs. 10/m2 t1 t3 m2  Rs. 20t1 t3   Bottom: Rs. 80/m2 t1 t2 m2  Rs. t1 t2

Transport cost : (Rs. 1/trip)

The total cost (Rupees) g(t)  80t1−1 t2−1 t3−1 + 40t2 t3 + 20t1 t3 + 80t1 t2 , It is a posynomial function. As stated, this problem can be formulated as an unconstrained GP problem Minimize g(t)  80t1−1 t2−1 t3−1 + 40t2 t3 + 20t1 t3 + 80t1 t2 subject to t1 , t2 , t3 > 0 The optimal dimensions of the box are t *1  1 m, t *2  1/2 m, t *3  2 m and minimum total cost of this problem is Rs. 200. Application 11.3 Suppose that we now consider the following variant of the above problem (similar discussion have done Duffin et al. (1967) in their book). It is required that the sides and bottom of the box should be made from scrap material but only 4 m2 of this scrap material are available. This variation of the problem leads us to the following constrained posynomial GP problem: ⎧ 80 ⎪ ⎨ Minimize g0 (t)  t1 t2 t3 + 40t2 t3 subject to g1 (t) ≡ 2t1 t3 + t1 t2 ≤ 4, ⎪ ⎩ where t > 0, t > 0, t > 0. 1 2 3

272

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

Table 11.1 Input data of Problem 11.4 Objective goal Constraint goal

Non-fuzzy (Rs.)

Fuzzy (Rs.)

95.24

μ1

μ0

90

98

4

6

4

Solving this constrained GP problem, we have the minimum total cost Rs. 95.24 and optimal dimensions of the box are t *1  1.58 m, t *2  1.25 m, t *3  0.63 m. Application 11.4 Instead of giving rigid numerical values for the right-hand side constant as in the conventional problem 11.3, assume that the DM has the fuzzy goal and the fuzzy constraint as shown in Table 11.1. Assuming the linear membership functions from μ  0 and μ  1 for these ˜ g0 (t) i.e. g0 (t) ≤ 90 and g1 (t) ≤ 4) as equally important, this fuzzy inequalities (Min ∼

∼

problem is a fuzzy problem as follows:

Find t  (t1 , t2 , t3 )T so as to satisfy g0 (t) ≤ 90 and g1 (t) ≤ 4, ∼

∼

t >0 For treating the above fuzzy inequalities, we propose the following linear membership functions: μg0 (t)  1  98−g8 0 (t) 0 μg1 (t)  1  6−g21 (t) 0

if g0 (t) ≤ 90 if 90 ≤ g0 (t) ≤ 98 if g0 (t) ≥ 98, if g1 (t) ≤ 4 if 4 ≤ g1 (t) ≤ 6 if g1 (t) ≥ 6

Here 8(=98 − 90) and 2(=6 − 4) are subjectively chosen constants expressing the limit of the admissible violations of the inequalities. It is assumed that the membership function μg0 (t) should be 1 if the objective goal is well satisfied, 0 if the objective goal is violated beyond its limit 8(=98 − 90) and linear from 0 to 1. Such a linear membership function is illustrated in Fig. 11.5. Similarly linear membership function of fuzzy constraint g1 (t) ≤ 4 is illustrated ∼

in Fig. 11.6. Following the fuzzy decision on max-addition operator (24), the said problem can be transformed into the following equivalent conventional nonlinear programming problem as:

11.4 Multi-objective Optimization

273

µ g0 (t)

1

0

90

98

g0 (t)

Fig. 11.5 Linear membership function for g0 (x)

µg1 (t )

1

0

4

6

g1 (t)

Fig. 11.6 Membership function for g1 (x)

98 − g0 (t) 6 − g1 (t) + 8 2 subject to t > 0.

Max V (t) 

So optimal decision variable t * can be obtained by solving the following unconstrained GP problems: g0 (t) g1 (t) + 8 2 subject to t > 0

Min U (t) 

i.e., 1 Min U (t)  10t1−1 t2−1 t3−1 + 5t2 t3 + t1 t3 + t1 t2 2 subject to t1 , t2 , t3 > 0.

274

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

Solving this unconstrained GP we have t1∗  4.78, t2∗  0.96 and t3∗  0.48 ∗ ∗ and optimal objective goal g0∗ (t ∗ )  t ∗80 ∗ ∗ + 40t2 t3  97.75 and constraint goal 1 t2 t3 g1∗ (t ∗ )  2t1∗ t3∗ + t1∗ t2∗  5.178. Application 11.5 Not only minimizing total cost (=total transportation cost + material cost for two ends of the box) of the problem 11.3 but there is also another objective function which is to minimize the total number of trips.

Here no. of trips 

80 . t1 t2 t3

So, the problem is to determine dimensions of the box, i.e., find t  (t1 , t2 , t3 )T so as to satisfy ⎧ ⎪ Minimize g0 (t)  t180 + 40t2 t3 ⎪ t2 t3 ⎪ ⎨ Minimize g1 (t)  t180 t2 t3 ⎪ subject to 2t t + t t 1 3 1 2 ≤ 4, ⎪ ⎪ ⎩ where t1 , t2 , t3 > 0.

(11.17)

It may be written as a multi-objective geometric programming (MOGP) ⎧ ⎪ Minimize g0 (t)  t180 + 40t2 t3 ⎪ t2 t3 ⎪ ⎨ Minimize g1 (t)  t180 t2 t3 (11.18) 1 ⎪ subject to g t t + 14 t1 t2 ≤ 1, ≡ (t) 2 ⎪ 2 1 3 ⎪ ⎩ where t1 , t2 , t3 > 0. Here, two sub-problems are ⎧ 80 ⎪ ⎨ Minimize g0 (t)  t1 t2 t3 + 40t2 t3 1 subject to g2 (t) ≡ 2 t1 t3 + 41 t1 t2 ≤ 1, ⎪ ⎩ where t1 , t2 , t3 > 0.

(11.19)

and ⎧ ⎪ ⎨

Minimize g1 (t)  t180 t2 t3 subject to g2 (t) ≡ 21 t1 t3 + 41 t1 t2 ≤ 1, ⎪ ⎩ where t1 , t2 , t3 > 0.

(11.20)

(11.19), (11.20) are two GP with DD  −1, 0 respectively. Solving this MOGP (11.18) we have t1∗  2.93, t2∗  1.17 and t3∗  0.43 and optimal objective goals g0∗ (t ∗ )  86.78 and g1∗ (t ∗ )  3.3.

11.4 Multi-objective Optimization Table 11.2 Input data for the Application 11.6

275

Boxes (i)

ai (Rs./m2 )

bi (Rs./m2 )

ci (Rs./m2 )

d i (m3 )

1

80

10

20

80

2

50

15

25

90

3

40

5

30

70

Application 11.6: MGP Problem (Multi-gravel-Box Problem) Suppose that to shift gravel in a finite number (say n) of open rectangular boxes of lengths t 1i meters, widths t 2i meters, and heights t 3i meters (i  1, 2, …, n). The bottom, sides and the ends of the each box cost Rs. ai , Rs. bi , and Rs. ci /m2 . It costs Rs. 1 for each round trip of the boxes. Assuming that boxes  3 will have no salvage  the n value, find the minimum cost of transporting d  i1 di m of the gravels.

As stated, this problem can be formulated as an unconstrained MGP problem ⎧ n  ⎨ Minimize g(t) 

di + ai t1i t2i + 2bi t1i t3i + 2ci t2i t3i t1i t2i t3i i1 ⎩ where t1i > 0, t2i > 0, t3i > 0 (i  1, 2, . . . , n). In particular, here we assume that the transporting d m3 of gravels by the three different open rectangular boxes whose bottom, sides, and the ends of the each box costs are given in Table 11.2. It is solved and optimal results are shown in Table 11.3. Application 11.7 Suppose that we now the following variant of the above problem. It is required that the sides and bottom of the boxes should be made from scrap material but only w m2 of these scrap materials are available.

276

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

Table 11.3 Optimal results of the Application 11.6

Table 11.4 Input data for the Application 11.7

Table 11.5 Optimal results of the Application 11.7

Boxes (i)

t 1i (m)

t 2i (m)

t 3i (m)

g* (t * ) (Rs.)

1

1.00

0.5

2.0

572.23

2

1.2

0.72

1.2

3

2.16

0.36

1.44

Boxes (i)

ci (Rs./m2 )

d i (m3 )

w (m2 )

1

40

80

15

2

30

90

3

20

70

Boxes (i)

t 1i (m)

t 2i (m)

t 3i (m)

g* (t * ) (Rs.)

1

2.33

1.14

0.57

221.44

2

2.0

1.32

0.66

3

1.49

1.47

0.74

This variation of the problem leads us to the following constrained modified geometric program: ⎧ n 

di ⎪ ⎪ Minimize g(t)  + 2ci t2i t3i ⎪ t1i t2i t3i ⎪ ⎨ i1 n

subject to (2t1i t3i + t1i t2i ) ≤ w, ⎪ ⎪ ⎪ i1 ⎪ ⎩ where t1i > 0, t2i > 0, t3i > 0(i  1, 2, . . . , n). In particular, here we assume transporting d m3 of gravels by the three different open rectangular boxes. The end of each box cost is Rs. ci /m2 and amount of the

3 di m3 . Input transporting gravels by three open rectangular boxes are d  i1 data of this problem is given in Table 11.4. It is a constrained posynomial MGP problem. Optimal results of this problem are shown in Table 11.5.

11.5 Fuzzy Multi-objective Mathematical Programming (FMOMP) Problem Accepting that the decision-maker (DM) has fuzzy goals (targets value) for each of the objective functions in the MOMP problem, like fuzzy multi-objective linear programming problem presented by Zimmermann (1978), it is conceivable to soften the rigid requirements of the MOMP problem to strictly minimize the k objective functions under the given requirements. In such a circumstance, the MOMP prob-

11.5 Fuzzy Multi-objective Mathematical Programming (FMOMP) Problem

277

lem might be mellowed into the accompanying fuzzy version (called fuzzy multiobjective mathematical programming (FMOMP) problem): ˜ Minimize f (x)  [ f 1 (x), f 2 (x), . . . f k (x) ]T ! " subject to x ε X  x ε R n : g j (x)K ≤ b j , j  1, . . . , m; x ≥ 0

(11.21)

˜ Here the symbol Minimize denotes fuzzy version of “minimize” with the interpretation that the k-th objective function should be minimized under the given conditions or constraints. In this way, the problem (11.21) is transformed to following fuzzy optimization problem: Find x so as to satisfy fr (x) ≤ fr0 for r  1, 2, . . . , k ∼

x ∈ X.

(11.22)

Those fuzzy requirements ( fr (x) ≤ fr0 for r  1, 2,…, k) can be quantified by ∼

eliciting membership functions μr (f r (x)). To derive a membership function μr (f r (x)) from the decision-maker (DM) for each of the objective function f r (x) of the FMOMP (11.21), one can suggest the following procedure: In the first, ascertain the individual minimum L r and maximum U r of each f r (x) under the given constraints. At that point by assessing the individual maximum and minimum of every objective function (objective goal function) together with the rate of increase of membership of satisfaction, the DM is utilized to choose a membership function (MF) in an subjective way from among the various types of functions (e.g., linear, piecewise linear, hyperbolic, hyperbolic inverse, exponential, and so on). The parameter esteems are determined through the connection with the DM. So membership function μr (f r (x)) may be taken as follows: μr ( fr (x))  0 or → 0 if fr (x) ≥ fr1  vr (x) if fr0 ≤ fr (x) ≤ fr1  1 or → 1 if fr (x) ≤ fr0 (for r  1, 2, . . . , k)

(11.22)

Here vr (x) is a strictly monotonic decreasing function with respect to f r (x). This membership function is determined by asking the DM to specify the two points fr0 , fr1 within L r and U r (i.e., L r ≤ f 0r < f 1r ≤ Ur ). For deciding the membership function for each of the objective functions we propose Bellman and Zadeh’s (1970) fuzzy decision and afterward utilizing Zimmermann’s approach (1978) and Tewari’s approach (1987). Then the FMOMP problem (11.21) is transformed to the following crisp problems: (Based on minimum operator)

278

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

Maximize α subject to μr ( fr (x)) ≥ α for r  1, 2, . . . , k xε X 0 ≤ α ≤ 1.

(11.23)

and (based on additive operator) Maximize

k 

μr ( fr (x))

r 1

subject to x∈X 0 ≤ μr ( fr (x)) ≤ 1

(11.24)

Here “min ˜ fr (x)” denotes the fuzzy goal of the decision-maker (DM), such as, “f r (x) be substantially less than or equal to f r ”. Thus, the thought of Pareto-optimal solutions, defined as far as objective functions cannot be applied. In this way, the idea of M-Pareto-optimal solutions that are defined in terms of membership functions (MF) rather than objective functions is presented (here the word M refers to membership function). Definition 11.4 (M-Pareto-optimal solutions) “x * ∈ X is said to be a M-Paretooptimal solution if and only if there does not exist another x ∈ X such that μr (f r (x)) ≥ μr (f r (x * )) for all r  1, 2, …, k, but μj (f j (x))  μj (f (x * )) for at least one j”. After ascertaining the MF for all fuzzy goals of the DM, the compromise solution of the DM can be derived by choosing and solving one of the two types of problems, one is FNLP and other is FAGP according to DM’s decisions. M-Pareto-Optimality Test A numerical test of M-Pareto-optimality for x * can be performed by solving the following problem: Maximize

k 

εr

r 1

   subject to μr ( fr (x)) + εr  μr fr x ∗ (r  1, 2, . . . , k) x ε X, ε  (ε1 , ε2 , . . . , εk )T ≥ 0

(11.25)

Let x¯ be an optimal solution to this problem. If εr  0, for all r  1, 2, …, k, then x * is a M-Pareto-optimal solution. If at least one εr > 0, not x * but x¯ is an M-Pareto-optimal solution of the FMOMP.

11.6 Inventory Models Through Geometric Programming Approach Fig. 11.7 The economic production quantity model

279

qi (t)

ri qi Ti

t

11.6 Inventory Models Through Geometric Programming Approach Assumptions The demand rate Di (i  1, 2, …, n) is uniform over time, Shortages are not allowed, The time horizon is infinite, Total cost of interest and depreciation per production cycle is inversely related to a setup cost (S i ) and directly related to production process reliability (r i ) according to the following equation f i (Si , ri )  ai Si−bi rici where ai , bi , ci (≥ 0) are shape parameters. (v) The unit production cost is a continuous function of demand Di and takes the −β following form pi (Di )  αi Di i βi (> 1) and αi (> 0) are shape parameters.

(i) (ii) (iii) (iv)

Model Formulation The situation of this inventory model is illustrated in Fig. 11.7. If qi (t) is the inventory level of the i-th item (i  1, 2, …, n) at time t over the time period (0, Ti ), then dqi (t)  −Di for 0 ≤ t ≤ Ti dt with initial and boundary conditions qi (0)  ri qi , qi (Ti )  0 The solution of this differential equation is qi (t)  ri qi −Di t for 0 ≤ t ≤ Ti and Ti  (ri qi )/Di . #Ti Now inventory carrying cost  Hi

qi (t) dt  0

[Here H i  inventory carrying cost per item per unit time.]

Hi ri2 qi2 . 2Di

280

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

Total inventory related cost per production cycle of the i-th item  setup cost + production cost + inventory carrying cost + interest as well as depreciation cost  Si + pi (Di )qi + Hi qi2 ri2 /2Di + f (Si , ri ). The total average cost of the inventory system for thei-th item   Hi ri2 qi2 Di −βi −bi ci Si + αi Di qi + + ai Si ri  r i qi 2Di Hi qi ri 1−β + ai Di Si−bi qi−1 ri(ci −1) .  Di Si qi−1 ri−1 + αi Di i ri−1 + 2 So, total average cost of the inventory system for n items TC(D, S, q, r ) 

n  

Di Si qi−1 ri−1 + αi Di

1−βi

ri−1 +

i1

and required storage area SA(r, q) 

n i1

 Hi qi ri (c −1) , + ai Di Si−bi qi−1 ri i 2

wi ri qi .

Model I (Modified GP model with DD  −1) Min TC(D, S, q, r ) 

n  

Di Si qi−1 ri−1 + αi Di

i1

1−βi

ri−1 +

Hi qi ri (c −1) + ai Di Si−bi qi−1 ri i 2



subject to Di , Si , qi , ri > 0 ( i  1, 2, . . . , n)

This is a modified geometric programming (MGP) problem with negative DD. The problem is illustrated by the following numerical example. A manufacturing company makes a machine. It is given that the stock carrying cost of the machine is $10.5 per unit per year. The production cost of the machine varies inversely with the machine demand. From the past experience, the production cost of the machine is 12,000D−3.6 , where D is the demand rate of the machine. The total cost of interest and depreciation per machine production cycle is 1500S −1.6 r, where S and r are setup cost per batch and machine production process reliability respectively. Decide the production quantity (q), demand rate (D), production process reliability (r), setup cost (S), and optimum total average cost (TC) of the given production system. Here the company produces two types of machines. Lots of machines are produced by the company. The demand rate of each machine is uniform over the time and can be thought to be deterministic. The appropriate information for the machines is given in the accompanying Table 11.6. Determine the demand rates (D1 , D2 ), production quantity (q1 , q2 ), setup cost (S 1 , S 2 ), production process reliability (r 1 , r 2 ) of each types of machines and the minimum total average cost (TC) of the machine production system. The optimal solutions are given in Table 11.7.

11.6 Inventory Models Through Geometric Programming Approach

281

Table 11.6 Input data of the Model I Types of machines (i)

Production cost

Interest and depreciation cost

Carrying cost H i ($)

1

12,000D1−3.6

1500S1−1.6 r1

10.5

2

12,500D2−3.8

1550S2−1.8 r2

12.9

Table 11.7 Optimal solutions of the Model I Method

D1∗

S1∗ ($)

q1∗

r1∗

D2∗

S2∗ ($)

q2∗

r2∗

TC* ($)

NLP

12.88

19.05

9.83

0.89

11.29

16.46

7.34

0.91

210.9305

GP (modified)

12.77

19.22

9.61

0.91

11.79

15.77

8.27

0.81

210.9302

Table 11.8 Optimal solutions for Model II Methods

D1∗

S1∗ ($)

q1∗

r1∗

D2∗

S2∗ ($)

q2∗

r2∗

TC* ($)

NLP

2108

4.77

96.3

0.89

2936

2.36

163.4

0.76

677.3

GP (modified)

2624.9 5.31

153

0.92

1547.3 4.12

85.2

0.93

668.8

Model II (Modified GP model with DD  0) Min TC(D, S, q, r ) 

n  

Di Si qi−1 ri−1 + αi Di

i1

subject to

n 

1−βi

ri−1 +

Hi qi ri (c −1) + ai Di Si−bi qi−1 ri i 2



wi ri qi ≤ w,

i1

Di , Si , qi , ri > 0 (i  1, 2, . . . , n).

In addition to the above model I the principle managing director of the company has to be restricted to the available storage space is W  2000 sq. ft. Then the optimal solutions of the model I are given in Table 11.8. Model III (Multi-objective GP Model) Mathematical Formulation Notations (i) (ii) (iii) (iv)

C 1  Holding cost per unit per unit time C 3  Setup cost per order Q  Order quantity, i.e., number of units ordered per order (units) D  Demand rate, units per unit time.

282

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

Fig. 11.8 The economic production quantity model

Assumption The following basic assumptions about the model are made: (i) (ii) (iii) (iv)

Production is instantaneous, No back order is allowed, Demand for the product exceeds supply, The unit production cost C p is inversely related to the demand rate D according to the following equation C p (D)  α D −β , where α(> 0), β(> 1) are constants real numbers selected to provided the best fit of the estimated cost function. The situation of the inventory is illustrated in Fig. 11.8.

If q(t) is the inventory level at time t then d q(t)  −D for 0 ≤ q(t) ≤ Q dt With initial condition q(0)  Q and boundary condition q(T )  0. So, we get q(t)  Q − Dt for 0 ≤ t ≤ T and Q  DT . #T Holding cost  C1

q(t)dt  C1

Q2 . 2D

0

Therefore the total items infinite replacement problem is  average cost of single C1 Q 2 D TAC(D, Q)  C3 + C p (D)Q + 2D Q  CQ3 D + α D 1−β + C12Q , Here total number of orders No (D, Q)  QD and total purchasing and holding cost PHC(D, Q)  α D −β Q + C12Q . So, multi-objective inventory model is minimizing total average cost as well as number of orders under limited purchasing and holding cost. Min (TAC(D, Q), N o(D, Q)) subject to PHC(D, Q) ≤ C, D, Q > 0

11.6 Inventory Models Through Geometric Programming Approach

283

C3 D C1 Q D + + α D 1−β , No(D, Q)  Q 2 Q Q C 1 PHC(D, Q)  α D −β Q + . 2

where TAC (D, Q) 

The above problem can be written as a multi-objective geometric programming problem

Min (TAC(D, Q), No(D, Q)) subject to A1 D −β Q + B1 Q ≤ 1, D, Q > 0 D C3 D C1 Q + + α D 1−β , No(D, Q)  . where TAC(D, Q)  Q 2 Q Step 1 In order to solve the above problem, we shall solve the sub-problems

Min TAC(D, Q) subject to A1 D −β Q + B1 Q ≤ 1, D, Q > 0 Min No (D, Q) and subject to A1 D −β Q + B1 Q ≤ 1, D, Q > 0 Solve the sub-problems by GP technique we have    D1∗ , Q ∗1  (1.001, 0.5207) TAC D1∗ , Q ∗1  149.2995  ∗ ∗   D2 , Q 2  (0.9935, 0.5185) No D2∗ , Q ∗2  1.9145 

Step 2     Now evaluate all these objective functions at D1∗ , Q ∗1 & D2∗ , Q ∗2 . We have formulated a pay-off matrix as follows:

TAC(D, Q) No(D, Q)  ∗ ∗   D , Q 149.2995 1.9224 1 1  ∗ ∗ D2 , Q 2 146.4783 2.4825

284

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

Step 3 From the pay-off matrix, we get

148.4205 ≤ TAC(D, Q) ≤ 149.2995 and 1.9145 ≤ No(D, Q) ≤ 1.9224 Step 4 Let μTAC (TAC(D, Q)), μNo (No(D, Q)) TAC(D, Q), No(D, Q) as follows: ⎧ ⎪ ⎨1 TAC(D,Q) μTAC (TAC(D, Q))  149.2995−0.879 ⎪ ⎩0

be the membership functions of if TAC(D, Q) ≤ 5.0722 if 148.4205 ≤ TAC(D, Q) ≤ 149.2995 if TAC(D, Q) ≥ 149.2995

and

μNo (No(D, Q)) 

⎧ ⎪ ⎨1

1.9224−No(D,Q) 0.0079

⎪ ⎩0

if No(D, Q) ≤ 1.9145 if 1.9145 ≤ No(D, Q) ≤ 1.9224 if No(D, Q) ≥ 1.9224

The problem can be formulated as   149.2995 − TAC(D, Q) 1.9224 − No(D, Q) + Max V (D, Q)  0.879 0.0079 subject to A1 D −β Q + B1 Q ≤ 1 D > 0, Q > 0. So, it is sufficient to derive the next minimization problem   TAC(D, Q) No(D, Q) + Min U (D, Q)  0.879 0.0079 subject to A1 D −β Q + B1 Q ≤ 1, D > 0, Q > 0, 1.9224 where V (D, Q)  149.2995 + 0.0079 − U (D, Q). 0.879 The optimal solutions are D*  1.054, Q*  0.481 and optimal objective values TAC(D* , Q* )  147.345, No(D* , Q* )  2.043.

11.7 Conclusion

285

11.7 Conclusion Here, we have talked about multi-objective geometric programming (MOGP) problem in light of fuzzy programming technique through geometric programming (GP) method. We have also given an example of gravel-box design problem and solved the problem by fuzzy programming technique. geometric programming (GP) procedure is utilized to determine the optimal solutions for various preferences on objective functions. The multi-objective inventory models may also be solved by fuzzy geometric programming (FGP) method. Today, a large portion of this real-world decision-making problems or real-life problems in financial, social, technical areas and environmental areas are multi-objectives and multi-dimensional ones. Multiobjective optimization problems (MOOPs) contrast from single-objective optimization problems (SOOPs). It is significant to understand that multiple objectives are frequently non-commensurable and in conflict with one another in optimization problems. Nevertheless, it is possible for one to express the attractive quality of achieving an aspiration level in an imprecise interval around it. An objective within exact target value is named as a fuzzy goal. Along these lines, a multi-objective model with fuzzy objectives or fuzzy coefficients is more sensible than deterministic of it.

References R.E. Bellman, L.A. Zadeh, Decision making in a fuzzy environment. Manage. Sci. 17, 141–164 (1970) M.P. Biswal, Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy Sets Syst. 51(1), 67–71 (1992) K. Das, T.K. Roy, M. Maiti, Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: a geometric programming approach. Prod. Plann. Control 11(8), 781–788 (2000) R.J. Duffin, E.L. Peterson, C. Zener, Geometric Programming Theory and Applications (Wiley, New York, 1967) S. Islam, T.K. Roy, A new fuzzy multi-objective programming: entropy based geometric programming and its application of transportation problems. Eur. J. Oper. Res. 173(2), 387–404 (2006) S. Islam, T.K. Roy, Multi-objective geometric-programming problem and its application. Yugoslav J. Oper. Res. 20, 213–227 (2010) N.K. Mandal, T.K. Roy, M. Maiti, Multi-objective fuzzy inventory model with three constraints: a geometric programming approach. Fuzzy Sets Syst. 150(1), 87–106 (2005) G.S. Mahapatra, T.K. Roy, Single and multi container maintenance model: a fuzzy geometric programming approach. J. Math. Res. 1(2), 47–60 (2009) A.K. Ojha, K.K. Biswal, Multi-objective geometric programming problem with weighted mean method. Int. J. Comput. Sci. Inf. Secur. 7(2) (2010) D. Panda, S. Kar, M. Maiti, Multi-item EOQ model with hybrid cost parameters under fuzzy/fuzzystochastic resource constraints: a geometric programming approach. Comput. Math Appl. 56, 2970–2985 (2008a) D. Panda, S. Kar, M. Maiti, Multi-item EOQ model with hybrid cost parameters under fuzzy/fuzzy stochastic resource constraints: a geometric programming approach. Comput. Math Appl. 56(11), 2970–2985 (2008b)

286

11 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem

A.L. Peressini, F.E. Sullivan, J.J. Jr. Uhl, The Mathematics Nonlinear Programming (SpringerVerlag, New York, 1993) R.K. Verma, Fuzzy geometric programming with several objective functions. Fuzzy Sets Sys. 35(1), 115–120 (1990) H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets Sys. 1, 45–55 (1978)

Chapter 12

Geometric Programming Problem Under Uncertainty

12.1 Introduction Uncertainty theory is one of the relatively new branches of uncertain mathematics presented by Liu (2015) and in this way studied by many researchers. Now uncertainty theory has become a part of mathematics for modeling belief degrees. Liu displayed some research problems in the environment of uncertainty theory. Peng and Yao (2011) proposed “a new uncertain stock model and some option price formulas.” Li et al. developed “the uncertain premium principle based on the distortion function.” In addition, the uncertain reliability analysis, the uncertain risk analysis, and the uncertain control were introduced by Liu, Liu and Zhu, respectively. Madjid et al. (2015) developed “the geometric programming (GP) technique with normal, linear and zigzag uncertainty.” Mandal and Islam (2018) presented “multi-objective geometric programming problem under uncertainty.” Mandal (2018) also “developed two-bar truss structural model under uncertainty.” In this chapter, we have discussed three types of geometric programming methods under uncertainty, that is (1) Uncertain Chance-Constrained Geometric Programming Model (2) Geometric Programming Approach under expected, variance, 2-ND moment and entropy-based zigzag uncertainty distribution (3) Multi-Objective Geometric Programming Problem under Uncertainty

12.2 Uncertain Chance-Constrained Geometric Programming (UCCGP) Model Uncertain programming is a type of mathematical programming involving uncertain variables. Assume that x is a decision vector, and ξ˜ is an uncertain vector. Since an uncertain objective function f (x, ξ˜ ) cannot be directly minimized, we may minimize its expected value, i.e., © Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_12

287

288

12 Geometric Programming Problem Under Uncertainty

min E[ f (x, ξ˜ )].

(12.1)

x

In addition, since the uncertain constraints g j (x, ξ˜ ) ≤ 0, j  1, 2, …, p do not define a crisp feasible set, it is generally preferred that the uncertain constraints hold with the confidence levels α1 , α2 , . . . , α P . Then, we have a set of chance constraints   M g j (x, ξ˜ ) ≤ 0 ≥ α j , j  1, 2, . . . , p. (12.2) In order to obtain a decision with minimum expected objective value subject to a set of chance constraints, Liu (2009) proposed the following uncertain programming model: ⎧   ⎨ min E f (x, ξ˜ ) x   (12.3) ⎩ subject to M g j (x, ξ˜ ) ≤ 0 ≥ α j , j  1, 2, . . . , p. A standard geometric programming (GP) technique allows solving a minimization problem, where the objective function is a posynomial whose variables can take only positive values and are constrained by a finite number of inequality constraints. GP primal problem: Primal Geometric Programming (PGP): T0

Minimize g0 (t) 

c0k

m

k1

Subject to gr (t) 

Tr k1+Tr −1

t j > 0,

α

t j 0k j

j1

cr k

m

α

t j rkj ≤ 1

j1

j  1, 2, . . . , m.

(12.4)

where crk (>0) and α rkj (k  1, 2, …, 1 + T r −1 , …, T r ; r  0, 1, 2, …, l; j  1, 2, …, m) are real numbers. It is a constrained posynomial geometric programming (PGP) problem. The number of terms in each posynomial constraint function varies and it is denoted by T r for each r  0, 1, 2, …, l. Let T  T 0 + T 1 + T 2 + ··· + T l be the total number of terms in the primal program. Then the Degree of Difficulty(DD)  T − (m + 1). Dual Program: The dual programming of (12.4) is as follows: Maximize v(δ) 

Tr  l

cr k δr k r 0 k1

δr k

⎛ ⎝

Tr s1+Tr −1

⎞δr k δr s ⎠

12.2 Uncertain Chance-Constrained Geometric Programming (UCCGP) Model

289

Subject to T0 

δ0k k1 Tr l   r 0 k1

 1,

(Normality condition)

αr k j δr k  0, ( j  1, 2, . . . , m)

(Orthogonality conditions)

δr k > 0, (r  0, 1, 2, . . . , l; k  1, 2, . . . , Tr ). (Positivity conditions) (12.5)

Madjid et al. (2015) developed an uncertain GP model, whose associated chanceconstrained version admits an equivalent crisp formulation. First, they transform the  conventional GP problem in Eq. (12.4) in an uncertain GP problem, where c 0k , c rk are UVs. Then the new model is Min g0 (t) 

T0

c˜ok

k1

c˜r k

k1+Tr −1

t j > 0,

α

t j 0k j

j1

Tr

Sub gr (t) 

m

m

α

t j rkj ≤ 1

j1

j  1, 2, . . . , m.

(12.6)

Based on the model in Eq. (12.6), Madjid et al. (2015) formulated the following generic GP models, under the linear, normal, and zigzag-based uncertainty distributions, respectively, which is a variant of uncertain chance-constrained geometric programming (UCCGP) model as follows: ⎤ ⎡ T0 m

α c˜ok t j 0k j ⎦, Min E(g0 (t))  E ⎣ ⎛ Sub M ⎝

k1 Tr

k1+Tr −1

c˜r k

m

α t j rkj

j1

⎞

≤ 1⎠ ≥ α.

(12.7)

j1

12.2.1 UCCGP Model with Linear Uncertainty Distributions Let the coefficients (12.7) be independent positive linear UVs. That  a b  c˜0k , c˜r k in Eq.   a b , c0k , with 0 < c0k < c0k and c˜r k : L crak , crbk , with 0 < crak < crbk . is, c˜0k : L c0k By Lemma 4.3, we obtain a deterministic objective for the proposed UCCGP problem in Eq. (12.7), that is

290

12 Geometric Programming Problem Under Uncertainty

⎡ E⎣

T0

c˜0k

k1

m

⎤ a t j 0k j ⎦



T0

j1

E(c˜0k )

k1

m

a t j 0k j

j1

T0  a m b

c0k + c0k a  t j 0k j . 2 k1 j1

Moreover, by Lemma 4.2, the constraints in Eq. (12.7) admit the following deterministic equivalent form: ⎛

⎞ m

ar k j c˜r k tj ≤ 1⎠ ≥ α ⇔

Tr

∀i  1, . . . , n, M ⎝

k1+Tr −1

j1

Tr



(1 − α)crak + αcrak

k1+Tr −1

m 

ar k j tj ≤ 1. j1

Thus, when the coefficients are UVs endowed with linear distributions, the model in Eq. (12.7) is equivalent to T0  a m b

c0k + c0k a Min E(g0 (t))  t j 0k j 2 k1 j1

Sub gr (t) 

Tr m  

a t j r k j ≤ 1. (1 − α)crak + αcrak k1+Tr −1

(12.8)

j1

The corresponding dual problem is as follows: T0  a Tr  l

b δ0k

c0k + c0k (1 − α)crak + αcrak δr k Max E(d(δ))  2δ0k δr k r 0 k1 k1 Subject to Constraints of the Model (12.4).

(12.9)

12.2.2 UCCGP Model with Normal Uncertainty Distributions Let the coefficients c˜ j0i , c˜ri in Eq. (12.7) be independent positive normal UVs, that is to say, c˜ j0i : N c j0i , σ j0i , and c˜ri : N (cri , σri ), where c j0i , cri , σ j0i and σri are all positive real values. By Lemma 4.5, we obtain a deterministic objective for the proposed UCCGP problem in Eq. (12.7), that is ⎤ ⎡ T0 T0 T0 m m m





a0k j a a c˜0k tj ⎦  E(c˜0k ) t j 0k j  c0k t j 0k j . E⎣ k1

j1

k1

j1

k1

j1

Moreover, by Lemma 4.4, the constraints in Eq. (12.7) admit the following deterministic equivalent form:

12.2 Uncertain Chance-Constrained Geometric Programming (UCCGP) Model ⎛ ∀i  1, . . . , n, M ⎝

Tr

⎞

m

c˜r k

k1+Tr −1

≤ 1⎠ ≥ α ⇔

a t j rkj

j1

291



  Tr m σri α a cri + t j r k j ≤ 1. log π 1−α

k1+Tr −1

j1

Thus, when the coefficients are UVs endowed with linear distributions, the model in Eq. (12.7) is equivalent to Min E(g0 (t)) 

T0

m

c0k

k1

Sub gr (t) 

Tr k1+Tr −1

a

t j 0k j

j1

 

m α σri a cri + log t j r k j ≤ 1. π 1−α j1

(12.10)

The corresponding dual problem is as follows: Max E(d(δ)) 

 T0  Tr l

cri + c0k δ0k

k1

δ0k

r 0 k1

σri π

 α  δr k log 1−α

δr k

Subject to Constraints of the Model (12.4).

(12.11)

12.2.3 UCCGP Model with Zigzag Uncertainty Distributions Let the parameters be independent positivezigzag UVs. That  a b c˜0kc ,c˜r k in Eq. (12.7) a b c , c0k , c0k with 0 < c0k < c0k < c0k and c˜r k : Z crak , crbk , crck with is, c˜0k : Z c0k a b c 0 < cr k < cr k < cr k . By Lemma 4.7, we obtain a deterministic objective for the proposed UCCGP problem in Eq. (12.7) that is ⎤ ⎡ T0 T0 T0  a m m m b c





c0k + 2c0k + c0k a0k j a0k j a ⎦ ⎣ c˜0k tj E(c˜0k ) tj  t j 0k j .  E 4 k1 j1 k1 j1 k1 j1 Moreover, by Lemma 4.6, the constraints in Eq. (12.7) admit the following deterministic equivalent form: ⎛ ∀i  1, . . . , n, M ⎝ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

k1+Tr −1 Tr 

k1+Tr −1

⎞ m

ar k j ˜ c˜r k tj ≤ λi ⎠ ≥ α ⇔ j1

   m a rkj tj ≤ 2αλib + (1 − 2α)λic , (1 − 2α)crak + 2αcrbk

k1+Tr −1  Tr ⎪ 

⎪ ⎪ ⎪ ⎩

Tr

j1

(2α − 1)crck + 2(1 − α)crbk

if α ∈ ]0, 0.5[;

  m a rkj tj ≤ (2α − 1)λia + 2(1 − α)λib , if α ∈ ]0.5, 1[. j1

292

12 Geometric Programming Problem Under Uncertainty

Thus, when the coefficients are UVs endowed with zigzag distributions, the model in Eq. (12.7) is equivalent to For α < 0.5, we have Min E(g0 (t))  Sub gr (t) 

T0  a m b c

c0k + 2c0k + c0k a t j 0k j 4 k1 j1

Tr m  

a a b t j r k j ≤ 2αλib + (1 − 2α)λic . (12.12) (1 − 2α)cr k + 2αcr k k1+Tr −1

j1

The corresponding dual problem is as follows:  δr k T0  a Tr l

b c δ0k

1 − 2α)crak + 2αcrck c0k + 2c0k + c0k   Max E(d(δ))  4δ0k δr k 2αλib + (1 − 2α)λic r 0 k1 k1 Subject to Constraints of the Model (12.4).

(12.13)

For α ≥ 0.5 we have  m T0  a b + cc

a0k j c0k + 2c0k 0k Min E(g0 (t))  tj 4 k1

Sub gr (t) 

Tr

j1



(2α − 1)crck + 2(1 − α)crbk

m 

ar k j tj ≤ (2α − 1)λia + 2(1 − α)λib .

k1+Tr −1

(12.14)

j1

The corresponding dual problem is as follows: ⎞δr k δ0k l T ⎛ T0  a b + cc r



c0k + 2c0k (2α − 1)crck + 2(1 − α)crbk 0k ⎝  ⎠ . Max E(d(δ))  a b 4δ0k r 0 k1 δr k (2α − 1)λi + +2(1 − α)λi k1

(12.15) Constraints of the Model (12.4).

12.2.4 Numerical Examples Here a numerical example is given to show the efficacy of the above uncertain GP models. Example 12.1 Let us consider the following instance of uncertain GP problem − Min θ1 x1−1 x2 2 x3−1 + θ2 x1 x3 + θ1 x1 x2 x3 1

−1

Sub. θ1,1 x1−2 x2−2 + θ1,2 x2 2 x3−1 ≤ 1, x1 , x1 , x1 > 0.

(12.16)

12.2 Uncertain Chance-Constrained Geometric Programming (UCCGP) Model

293

where all coefficients are assumed to be UVs. Case 1: Linear Uncertainty Distributions θ˜1 : L(30, 50), θ˜2 : L(25, 45), θ˜3 : L(35, 65), θ˜1,1 : L



 1 2 2 4 ˜ , , θ1,2 : L , . 3 3 3 3

By Eq. (12.8), the problem of Eq. (12.16) becomes the following deterministic GP: −1

Min 40x1−1 x2 2 x3−1 + 35x1 x3 + 50x1 x2 x3   1 1 2 2 4 −2 −2 Sub. (1 − α) + α x1 x2 + (1 − α) + α x22 x3−1 ≤ 1, 3 3 3 3 (12.17) x1 , x2 , x3 > 0. Using Eq. (12.9), we can move the problem to its dual form as follows: 

40 Max δ1

δ1 

35 δ2

δ2 

50 δ3

δ3 

(1 − α) 13 + 23 α δ1,1

δ1,1 

(1 − α) 23 + 43 α δ1,2

δ1,2 ρρ

s.t. δ1 + δ2 + δ3  1, − δ1 + δ2 + δ3 − 2δ1,1  0, − δ1 + δ2 + δ3 − 2δ1,2  0, δ1,1 + δ1,2  ρ, −δ1 δ1,2 + δ3 − 2δ1,1 −  0, 2 2 δ1 , δ2 , δ3 , δ1,1 , δ1,1 > 0.

(12.18)

The optimal values of decision variables and objective are presented in Table 12.1. In particular, it can be noted that the optimal objective values increase as α increases. Case 2: Normal Uncertainty Distributions θ˜1 : N (10, 1), θ˜2 : N (15, 3), θ˜3 : N (20, 5), θ˜1,1 : N



 1 4 , 0.1 , θ˜1,2 : N , 0.2 . 3 3

By Eq. (12.10), the problem of Eq. (12.16) becomes the following deterministic GP: −1

Min 10x1−1 x2 2 x3−1 + 15x1 x3 + 20x1 x2 x3   √ √     1 1 0.1 3 4 0.2 3 α α −2 −2 + ln + ln Sub. x1 x2 + x22 x3−1 ≤ 1, 3 π 1−α 3 π 1−α

294

12 Geometric Programming Problem Under Uncertainty

Table 12.1 Solving the GP problem of Example with linear uncertainty distributions α

Objective value

Corresponding primal solutions

Corresponding dual solutions

0.25

118.6675

x1∗  0.8461, x2∗  1.3214, x3∗  0.7159

δ1∗  0.4841, δ2∗  0.1787, ∗  0.0159, δ3∗  0.3372, δ1,1 ∗ δ1,2  0.0318

0.50

119.7881

x1∗  0.8295, x2∗  1.4765, x3∗  0.6881

δ1∗  0.4815, δ2∗  0.1668, ∗  0.0185, δ3∗  0.3518, δ1,1 ∗  0.0371 δ1,2

0.75

120.8784

x1∗  0.8153, x2∗  1.6225, x3∗  0.6648

δ1∗  0.4770, δ2∗  0.1577, ∗  0.0231, δ3∗  0.3657, δ1,1 ∗  0.0462 δ1,2

Table 12.2 Solving the GP problem of Example with normal uncertainty distributions α

Objective value

Corresponding primal solutions

Corresponding dual solutions

0.25

38.4250

x1∗  0.5912, x2∗  1.5301, x3∗  0.7379

δ1∗  0.4823, δ2∗  0.1703, ∗  0.0177, δ3∗  0.3474, δ1,1 ∗  0.0354 δ1,2

0.50

37.7849

x1∗  0.5911, x2∗  1.6918, x3∗  0.6988

δ1∗  0.4926, δ2∗  0.1640, ∗  0.0207, δ3∗  0.3699, δ1,1 ∗  0.0413 δ1,2

0.75

39.1313

x1∗  0.5894, x2∗  1.8445, x3∗  0.6680

δ1∗  0.4779, δ2∗  0.1509, ∗  0.0221, δ3∗  0.3712, δ1,1 ∗  0.0442 δ1,2

x1 , x2 , x3 > 0.

(12.19)

Using Eq. (12.11), we can move the problem to its dual form as follows: 

10 Max δ1

δ1 

15 δ2

δ2 

20 δ3

δ3  1 3

+

√ 0.1 3 π

ln



α 1−α

 δ1,1  4 3

δ1,1

+

√ 0.2 3 π

ln

δ1,2

s.t. Constraints of Model (12).



α 1−α

 δ1,2 ρρ (12.20)

The optimal values of decision variables and objective are presented in Table 12.2. In particular, it can be noted that the optimal objective values increase as α increases. Case 3: Zigzag Uncertainty Distributions θ˜1 : Z (10, 20, 40), θ˜2 : Z (15, 25, 35), θ˜3 : Z (35, 65, 75), θ˜1,1 : Z



 4 1 2 2 . , , 1 , θ˜1,2 : Z , 1, 3 3 3 3

By Eqs. (12.12) or (12.14), the problem of Eq. (12.16) becomes the following deterministic GP:

12.2 Uncertain Chance-Constrained Geometric Programming (UCCGP) Model

295

Table 12.3 Solving the GP problem of Example with zigzag uncertainty distributions α

Objective value

Corresponding primal solutions

Corresponding dual solutions

0.25

92.4646

x1∗  0.7941, x2∗  1.5423, x3∗  0.5255

δ1∗  0.4695, δ2∗  0.1128, ∗  0.0305, δ3∗  0.4176, δ1,1 ∗ δ1,2  0.0609

0.50

94.3989

x1∗  0.7999, x2∗  1.7681, x3∗  0.4798

δ1∗  0.4670, δ2∗  0.1016, ∗  0.0329, δ3∗  0.4313, δ1,1 ∗  0.0659 δ1,2

0.75

96.1164

x1∗  0.7996, x2∗  1.9773, x3∗  0.4476

δ1∗  0.4651, δ2∗  0.0931, ∗  0.0349, δ3∗  0.4418, δ1,1 ∗  0.0697 δ1,2

⎫ −1 ⎪ Min 22.5x1−1 x2 2 x3−1 + 25x1 x3 + 60x1 x2 x3 ⎬   −2 −2   21 −1 1 2 2 Sub. (1 − 2α) 3 + 3 · 2α x1 x2 + (1 − 2α) 3 + 2α x2 x3 ≤ 1 ⎪ if α < 0.5, ⎭ x1 , x2 , x3 > 0. Or

⎫ −1 ⎪ ⎪ Min 22.5x1−1 x2 2 x3−1 + 25x1 x3 + 60x1 x2 x3 ⎬ 1     Sub. (2α − 1) + 23 · 2(1 − α) x1−2 x2−2 + (2α − 1) 43 + 2(1 − α) x22 x3−1 ≤ 1 ⎪ if α ≥ 0.5, ⎪ ⎭ x1 , x2 , x3 > 0.

(12.21) Using Eq. (12.13), we can move the problem to its dual form as follows: ⎧       δ1 δ2 δ3 (1−2α) 1 + 2 ·2α δ1,1  (1−2α) 2 +2α δ1,2 ⎪ 25 60 3 3 3 ⎨ 22.5 ρρ , if α < 0.5, δ δ δ δ1,1 δ1,2 max  1 δ1  2 δ2  3 δ3    δ1,2 δ 2 4 1,1 ·2(1−α) +2(1−α) (2α−1)+ (2α−1) ⎪ 25 60 3 3 ⎩ 22.5 ρ ρ , if α ≥ 0.5. δ1 δ2 δ3 δ1,1 δ1,2 s.t. Constraints of Model (12).

(12.22)

The optimal values of decision variables and objective are presented in Table 12.3. In particular, it can be noted that the optimal objective values increase as α increases.

12.3 Geometric Programming Approach Under Expected, Variance, 2-ND Moment, and Entropy-Based Zigzag Uncertainty Distribution Mandal and Islam (2018) assumed the uncertain variables to have expected, variance, 2-ND moment, and entropy-based zigzag uncertainty distribution, respectively, and shown that the corresponding uncertain chance constraints Two-Bar Truss Structural Model can be changed into the crisp problem to calculate the optimal values or objective values. A standard geometric programming (GP) problem permits solving a

296

12 Geometric Programming Problem Under Uncertainty

minimization problem, where the objective function is a posynomial whose variables can take just positive values and are constrained by a finite number of inequality constraints. GP primal problem: Primal Geometric Programming (PGP): T0

Minimize g0 (t) 

c0k

k1

cr k

k1+Tr −1

t j > 0,

α

t j 0k j

j1

Tr

Subject to gr (t) 

m

m

α

t j rkj ≤ 1

j1

j  1, 2, . . . , m,

(12.23)

where crk (>0) and α rkj (k  1, 2, …, 1 + T r −1 , …, T r ; r  0, 1, 2, …, l; j  1, 2, …, m) are real numbers. It is a constrained posynomial geometric programming (PGP) problem. The number of terms in each posynomial constraint function varies and it is denoted by T r for each r  0, 1, 2, …, l. Let T  T 0 + T 1 + T 2 + ··· + T l , be the total number of terms in the primal program. Then the Degree of Difficulty (DD)  T − (m + 1). Dual Program: The dual programming of (12.23) is as follows: Maximize v(δ) 

Tr  l

cr k δr k r 0 k1

δr k

⎛

Tr

⎝

⎞δr k δr s ⎠

s1+Tr −1

Subject to T0 

δ0k k1 Tr l   r 0 k1

 1,

(Normality condition)

αr k j δr k  0, ( j  1, 2, . . . , m)

(Orthogonality conditions)

δr k > 0, (r  0, 1, 2, . . . , l; k  1, 2, .., Tr ). (Positivity conditions) (12.24)

When coefficients (parameters) of the conventional GP problem (12.23) are uncertain in nature then transform the conventional GP problem in an uncertain GP problem as follows: Min g0 (t) 

T0 k1

c˜0k

m

j1

α

t j 0k j

12.3 Geometric Programming Approach Under Expected, Variance, … Tr

Sub gr (t) 

c˜r k

m

k1+Tr −1

t j > 0,

297

α

t j rkj ≤ 1

j1

j  1, 2, . . . , m,

(12.25)

 where c 0k , c r k are UVs. Based on the model in Eq. (12.25), Mandal and Islam (2018) formulated the following generic GP models, under the expected, variance, 2-ND moment, and entropy-based uncertainty value (UV), respectively, which is a variant of uncertain chance-constrained geometric programming (UCCGP) model as follows: For expected UVs ⎡ ⎤ T0 m

α0k j c˜ok tj ⎦ Min E(g0 (t))  E ⎣ k1

⎛

Tr

Sub M ⎝

c˜r k

k1+Tr −1

t j > 0,

m

j1

α t j rkj

⎞

≤ 1⎠ ≥ α

j1

j  1, 2, . . . , m.

(12.26)

For variance UVs ⎡ Min V (g0 (t))  V ⎣

T0

c˜ok

k1

⎛

Tr

Sub M ⎝

c˜r k

k1+Tr −1

t j > 0,

m

m

j1

⎤ α

t j 0k j ⎦ ⎞

α

t j r k j ≤ 1⎠ ≥ α

j1

j  1, 2, . . . , m.

(12.27)

For 2-ND moment UVs ⎡ ⎤ T0 m

α0k j c˜ok tj ⎦ Min Mo(g0 (t))  Mo⎣ ⎛ Sub M ⎝

Tr

k1+Tr −1

t j > 0,

c˜r k

m

k1

j1

α t j rkj

≤ 1⎠ ≥ α

⎞

j1

j  1, 2, . . . , m.

(12.28)

298

12 Geometric Programming Problem Under Uncertainty

For entropy UVs ⎡ Min H (g0 (t))  H ⎣

T0

c˜ok

k1

⎛

Tr

Sub M ⎝

c˜r k

k1+Tr −1

t j > 0,

m

m

j1

α t j rkj

⎤ α t j 0k j ⎦

⎞

≤ 1⎠ ≥ α

j1

j  1, 2, . . . , m.

(12.29)

12.3.1 Solution Procedure Under Expected-Based UVs Let the parameters be independent positivezigzag UVs. That  a b c˜0kc ,c˜r k in Eq. (12.25) a b c , c0k , c0k with 0 < c0k < c0k < c0k and c˜r k : Z crak , crbk , crck with is, c˜0k : Z c0k a b c 0 < cr k < cr k < cr k . By Lemma 4.7, we obtain a deterministic objective for the proposed UCCGP problem in Eq. (12.26) that is ⎤ ⎡ T0 T0 T0  a m m m b c





c0k + 2c0k + c0k a a a c˜0k t j 0k j ⎦  E(c˜0k ) t j 0k j  t j 0k j . E⎣ 4 k1 j1 k1 j1 k1 j1 Moreover, by Lemma 4.6, the constraints in Eq. (12.26) admit the following deterministic equivalent form: ⎛ ∀i  1, . . . , n, M ⎝ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

k1+Tr −1 Tr 



k1+Tr −1  Tr ⎪ 

⎪ ⎪ ⎪ ⎩

Tr

k1+Tr −1

⎞ m

ar k j c˜r k tj ≤ λ˜ i ⎠ ≥ α ⇔ j1

(1 − 2α)crak + 2αcrbk

  m a rkj tj ≤ 2αλib + (1 − 2α)λic , j1

(2α − 1)crck + 2(1 − α)crbk

if α ∈ ]0, 0.5[;

  m a rkj tj ≤ (2α − 1)λia + +2(1 − α)λib , if α ∈ ]0.5, 1[. j1

Thus, when the coefficients are UVs endowed with zigzag distributions, the model in Eq. (12.26) is equivalent to For α < 0.5 we have T0  a m b c

c0k + 2c0k + c0k a Min E(g0 (t))  t j 0k j 4 k1 j1

Sub gr (t) 

Tr  k1+Tr −1

(1 − 2α)crak + 2αcrbk

m 

j1

a

t j r k j ≤ 2αλib + (1 − 2α)λic . (12.30)

12.3 Geometric Programming Approach Under Expected, Variance, …

299

The corresponding dual problem is as follows:  δr k T0  a Tr l

b c δ0k

(1 − 2α)crak + 2αcrck c0k + 2c0k + c0k   Max E(d(δ))  b c 4δ 2αλ δ + − 2α)λ (1 0k r k i i r 0 k1 k1 Subject to T0 

δ0k k1 Tr l   r 0 k1

 1,

(Normality condition)

αr k j δr k  0, ( j  1, 2, . . . , m)

(Orthogonality conditions)

δr k > 0, (r  0, 1, 2, . . . , l; k  1, 2, .., Tr ). (Positivity conditions) (12.31)

For α ≥ 0.5 we have Min E(g0 (t)) 

 m T0  a b + cc

a0k j c0k + 2c0k 0k tj 4

k1

Sub gr (t) 

j1

m  

ar k j tj ≤ (2α − 1)λia + +2(1 − α)λib . (2α − 1)crck + 2(1 − α)crbk

Tr k1+Tr −1

j1

(12.32) The corresponding dual problem is as follows: ⎞δr k δ0k l T ⎛ T0  a b + cc r



c0k + 2c0k (2α − 1)crck + 2(1 − α)crbk 0k ⎝  ⎠ Max E(d(δ))  a b 4δ0k r 0 k1 δr k (2α − 1)λi + +2(1 − α)λi k1

(12.33)

Subject to Constraints of the model (12.31).

12.3.2 Solution Procedure Under Variance-Based UVs Let the parameters be independent positivezigzag UVs. That  a b c˜0kc ,c˜r k in Eq. (12.27) a b c , c0k , c0k with 0 < c0k < c0k < c0k and c˜r k : L crak , crbk , crck with is, c˜0k : Z c0k a b c 0 < cr k < cr k < cr k . By Lemma 4.8, we obtain a deterministic objective for the proposed UCCGP problem in Eq. (12.27) that is ⎡ ⎤      c   c   m T0 T0 T0 m m b − ca 2 + 6 cb − ca b b 2





a0k j 5 c0k a a 0k 0k 0k c0k − c0k + 5 c0k − c0k V⎣ c˜0k t j 0k j ⎦  E(c˜0k ) t j 0k j  tj . 48 k1

j1

k1

j1

k1

j1

Moreover, by Lemma 4.6, the constraints in Eq. (12.27) admit the following deterministic equivalent form:

300

12 Geometric Programming Problem Under Uncertainty ⎛

Tr

∀i  1, . . . , n, M ⎝ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

k1+Tr −1 Tr 



k1+Tr −1  Tr ⎪ 

⎪ ⎪ ⎪ ⎩

k1+Tr −1

⎞ m

ar k j ˜ c˜r k tj ≤ λi ⎠ ≥ α ⇔ j1

(1 − 2α)crak + 2αcrbk

  m a rkj tj ≤ 2αλib + (1 − 2α)λic , j1

(2α − 1)crck + 2(1 − α)crbk

if α ∈ ]0, 0.5[;

  m a rkj tj ≤ (2α − 1)λia + +2(1 − α)λib , if α ∈ ]0.5, 1[. j1

Thus, when the coefficients are UVs endowed with zigzag distributions, the model in Eq. (12.27) is equivalent to For α < 0.5 we have ⎛   2    2 ⎞ b − ca b − ca c − cb + 5 cc − cb T0 m + 6 c 5 c0k c 0k 0k 0k 0k 0k 0k 0k ⎟ a0k j ⎜ Min E(g0 (t))  tj ⎠ ⎝ 48 k1

Sub gr (t) 

j1

m  

ar k j tj ≤ 2αλib + (1 − 2α)λic . (1 − 2α)crak + 2αcrbk

Tr k1+Tr −1

(12.34)

j1

The corresponding dual problem is as follows:     c   c  δ0k  b T0 b a 2 a b b 2

5 c0k c0k − c0k + 5 c0k − c0k + 6 c0k − c0k − c0k Max E(d(δ))  48δ0k k1   δr k Tr l

(1 − 2α)crak + 2αcrck   δr k 2αλib + (1 − 2α)λic r 0 k1 Subject to Constraints of the model (12.31).

(12.35)

For α ≥ 0.5 we have ⎛  2    2 ⎞  b − ca b − ca c − cb + 5 cc − cb T0 m c0k + 6 c0k 5 c0k 0k 0k 0k 0k 0k ⎜ ⎟ a0k j Min E(g0 (t))  tj ⎝ ⎠ 48 k1

Sub gr (t) 

Tr k1+Tr −1

j1



(2α − 1)crck + 2(1 − α)crbk

m 

ar k j tj ≤ (2α − 1)λia + 2(1 − α)λib .

(12.36)

j1

The corresponding dual problem is as follows:     c   c  δ0k  b T0 b a 2 a b b 2

5 c0k c0k − c0k + 5 c0k − c0k + 6 c0k − c0k − c0k Max E(d(δ))  48δ0k k1   δ rk Tr l

(2α − 1)crck + 2(1 − α)crbk   δr k (2α − 1)λia + 2(1 − α)λib r 0 k1 Subject to Constraints of the model (12.31).

(12.37)

12.3 Geometric Programming Approach Under Expected, Variance, …

301

12.3.3 Solution Procedure Under 2-ND Moment-Based UVs Let the parameters be independent positivezigzag UVs. That  a b c˜0kc ,c˜r k in Eq. (12.25) a b c , c0k , c0k with 0 < c0k < c0k < c0k and c˜r k : L crak , crbk , crck with is, c˜0k : Z c0k a b c 0 < cr k < cr k < cr k . By Lemma 4.9, we obtain a deterministic objective for the proposed UCCGP problem in Eq. (12.28) that is ⎡ Mo⎣

T0

c˜0k

k1

⎤

m

a t j 0k j ⎦



j1

T0

E(c˜0k )

k1

m

a

t j 0k j 

j1

 T0 ca

0k

k1

2

a cb + 2cb 2 + cb cc + cc 2 + c0k 0k 0k 0k 0k 0k 6



m

a

t j 0k j .

j1

Moreover, by Lemma 4.6, the constraints in Eq. (12.28) admit the following deterministic equivalent form: ⎛ ∀i  1, . . . , n, M ⎝ ⎧ ⎪ ⎪ ⎪ ⎨

Tr

c˜r k

k1+Tr −1 Tr 

k1+Tr −1

⎞ ≤ λ˜ i ⎠ ≥ α ⇔

a t j rkj

j1

m a   t j r k j ≤ 2αλib + (1 − 2α)λic , (1 − 2α)crak + 2αcrbk

k1+Tr −1 Tr  ⎪ 

⎪ ⎪ ⎩

m

j1

(2α − 1)crck + 2(1 − α)crbk

m  j1

if α ∈ ]0, 0.5[;

a

t j r k j ≤ (2α − 1)λia + 2(1 − α)λib , if α ∈ ]0.5, 1[.

Thus, when the coefficients are UVs endowed with zigzag distributions, the model in Eq. (12.28) is equivalent to For α < 0.5, we have Min Mo(g0 (t)) 

T0  a c

2

0k

k1

Sub gr (t) 

Tr

a b b 2 b c c 2 + c0k c0k + 2c0k + c0k c0k + c0k 6

((1 − 2α)crak + 2αcrbk )

m

k1+Tr −1



m

a

t j 0k j

j1

a

t j r k j ≤ 1.

(12.38)

j1

The corresponding dual problem is as follows: Max Mo(d(δ)) 

 T0  a 2 Tr  l

a cb + 2cb 2 + cb cc + cc 2 δ0k

1 − 2α)crak + 2αcrck δr k c0k + c0k 0k 0k 0k 0k 0k 6δ0k

k1

r 0 k1

δr k

(12.39)

Subject to Constraints of the model (12.31).

For α ≥ 0.5, we have Min Mo(g0 (t)) 

T0  a c

0k

k1

2

a b b 2 b c c 2 + c0k c0k + 2c0k + c0k c0k + c0k 6



m j1

a

t j 0k j

302

12 Geometric Programming Problem Under Uncertainty

Sub gr (t) 

Tr

m  

a ((2α − 1)crck + 2 1 − α)crbk t j r k j ≤ 1.

k1+Tr −1

(12.40)

j1

The corresponding dual problem is as follows: T0  a

c

a b b 2 b c c 2 + c0k c0k + 2c0k + c0k c0k + c0k Max Mo(d(δ))  6δ0k k1  δ Tr l

(2α − 1)crck + 2(1 − α)crbk r k δr k r 0 k1 2

δ0k

0k

Subject to Constraints of the model (12.31).

(12.41)

12.3.4 Solution Procedure Under Entropy-Based UVs Let the parameters be independent positivezigzag UVs. That  a b c˜0kc ,c˜r k in Eq. (12.25) a b c , c0k , c0k with 0 < c0k < c0k < c0k and c˜r k : L crak , crbk , crck with is, c˜0k : Z c0k a b c 0 < cr k < cr k < cr k . By Lemma 4.10, we obtain a deterministic objective for the proposed UCCGP problem in Eq. (12.29) that is ⎤ ⎡ T0 T0 T0  c m m m a





c0k − c0k a0k j a0k j a ⎦ ⎣ c˜0k tj E(c˜0k ) tj  t j 0k j .  H 2 k1 j1 k1 j1 k1 j1 Moreover, by Lemma 4.6, the constraints in Eq. (12.29) admit the following deterministic equivalent form: ⎛ ∀i  1, . . . , n, M ⎝ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

k1+Tr −1 Tr 

k1+Tr −1

⎞ m

ar k j ˜ c˜r k tj ≤ λi ⎠ ≥ α ⇔ j1

   m a rkj tj ≤ 2αλib + (1 − 2α)λic , (1 − 2α)crak + 2αcrbk

k1+Tr −1  Tr ⎪ 

⎪ ⎪ ⎪ ⎩

Tr

j1

(2α − 1)crck + 2(1 − α)crbk

if α ∈ ]0, 0.5[;

  m a rkj tj ≤ (2α − 1)λia + 2(1 − α)λib , if α ∈ ]0.5, 1[. j1

Thus, when the coefficients are UVs endowed with zigzag distributions, the model in Eq. (12.29) is equivalent to For α < 0.5, we have Min H (g0 (t)) 

T0  c m a

c0k − c0k a t j 0k j 2 k1 j1

12.3 Geometric Programming Approach Under Expected, Variance, …

303

Tr m  

a t j r k j ≤ 2αλib + (1 − 2α)λic . (12.42) (1 − 2α)crak + 2αcrbk

Sub gr (t) 

k1+Tr −1

j1

The corresponding dual problem is as follows:  δr k T0  c Tr l

a δ0k

1 − 2α)crak + 2αcrbk c0k − c0k   Max H (d(δ))  2δ0k δr k 2αλib + (1 − 2α)λic r 0 k1 k1 Subject to Constraints of the model (12.31).

(12.43)

For α ≥ 0.5 we have Min H (g0 (t)) 

T0  c m a

c0k − c0k a0k j tj 2

k1

Sub gr (t) 

Tr

j1



(2α − 1)crck + 2(1 − α)crbk

k1+Tr −1

m 

ar k j tj ≤ (2α − 1)λia + 2(1 − α)λib .

(12.44)

j1

The corresponding dual problem is as follows:  δr k T0  c Tr l

a δ0k

c0k − c0k (2α − 1)crck + 2(1 − α)crbk   Max H (d(δ))  2δ0k δr k (2α − 1)λia + 2(1 − α)λib r 0 k1 k1 Subject to Constraints of the model (12.31).

(12.45)

12.3.5 Applying the Proposed UCCGP to a Two-Bar Truss Structural Model In this section, an application of the proposed UCCGP model to a Two-Bar Truss Structural Model under uncertainty is considered. Nasseri and Alizadeh (2014) developed a symmetric two-bar truss, which is shown in Fig. 12.1 is considered in this paper. The aim is to minimize the weight of two-bar truss subject to the stress σ constraints of each bar.   √ The weight of the two-bar truss structure is ρ 2dπ t b2 + h 2 and stress is √ P b2 +h 2 . dπth

Then the two-bar truss structural model can be written as   √ Min W T (d, h)  ρ 2dπ t b2 + h 2 Subject to

√ P b2 + h 2 σ (d, h)  ≤ σ0 dπ th

304

12 Geometric Programming Problem Under Uncertainty

2P

d A

A

b

A

A

H

b

Fig. 12.1 A two-bar truss structural model

d, h > 0.

(12.46)

√ Let b2 + h 2  y ⇒ b2 + h 2  y 2 . Hence, the new constraint is b2 + h 2 ≤ y 2 ⇒ b2 y −2 + h 2 y −2 ≤ 1. So the two-bar truss structural model is Min W T (d, h)  2ρdπ t y Subject to P yh −1 ≤ σ0 dπ t b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0. σ (d, h) 

(12.47)

where 2P, t, d, 2b are applied load, thickness of the bar, mean diameter of the bar (decision variable), and the distance between two hinges, respectively. When t, P, and σ0 are UVs then the problem (12.47) becomes Min W T (d, h)  2ρdπ t˜y Subject to P˜ yh −1 σ (d, h)  ≤ σ˜ 0 dπ t˜ b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.48)

12.3 Geometric Programming Approach Under Expected, Variance, …

305

      Let t˜ : Z t a , t b , t c , P˜ : Z P a , P b , P c and σ0 : Z σ0a , σ0b , σ0C , and all the parameters be positive. Using the zigzag UDs, the uncertain GP problem of Eq. (12.48) takes the following form.

12.3.5.1

Expected-Based UCCGP Model with Zigzag Expected-Based UDs

For α < 0.5 we have  Min E(W T (d, h))  2ρdπ

t a + 2t b + t c y 4

Subject to

  (1 − 2α)P a + 2α P b yh −1  ≤ 2ασ0b + (1 − 2α)σ0c  σ (d, h)  dπ (1 − 2α)t a + 2αt b

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.49)

And the dual programming is ⎛ Max d(δ)  ⎝ 

b2 δ21

δ21 

 2ρπ

t a +2t b +t c 4

 ⎞δ01 

δ01

1 δ22

δ01

⎠

(1 − 2α)P a + 2α P b    δ11 π (1 − 2α)t a + 2αt b 2ασ0b + (1 − 2α)σ0c

δ11

(δ21 + δ22 )(δ21 +δ22 )

Subject to δ01  1 δ01 + δ11 − 2δ21 − 2δ22  0 − δ11 + 2δ22  0 δ01 − δ11  0 − δ1 + 2δ2 + δ01  0 δ01 , δ11 , δ21 , δ22 > 0.

(12.50)

Solving the above linear equations, we get the optimal values as δ01  1, δ11  1, δ21  0.5, and δ22  0.5. Putting the optimal values in the objective function of the problem (12.50), we get ⎛ Max d ∗ (δ)  ⎝

 2ρπ

t a +2t b +t c 4

1

 ⎞1  ⎠

(1 − 2α)P a + 2α P b   δ11 π (1 − 2α)t a + 2αt b 2ασ0b + (1 − 2α)σ0c 

From primal–dual relation, we have

1 

b2 0.5

0.5 

1 0.5

0.5 .

306

12 Geometric Programming Problem Under Uncertainty

t a + 2t b + t c

y  δ01 2ρdπ × d ∗ (δ), 4  

(1 − 2α)P a + 2α P b yd −1 h −1 δ11    

, δ11 π (1 − 2α)t a + 2αt b 2ασ0b + (1 − 2α)σ0c δ

b2 y −2  21 , δ21 + δ22 δ

h 2 y −2  22 . δ21 + δ22 

The optimal solution of the model through the geometric programming approach is given by & y∗  & ∗

h 



b2 (δ21 + δ22 ) ,

δ21

b2 δ22

, δ21

(1 − 2α)P a + 2α P b  × d∗   π (1 − 2α)t a + 2αt b 2ασ0b + (1 − 2α)σ0c

&



b2 (δ21 + δ22 ) ×

δ21

&

b2 δ22

. δ21

For α ≥ 0.5, we have  Min E(W T (d, h))  2ρdπ

t a + 2t b + t c y 4

Subject to

  (2α − 1)P a + 2(1 − α)P b yh −1  ≤ (2α − 1)σ0a + 2(1 − α)σ0b  σ (d, h)  dπ (2α − 1)t a + 2(1 − α)t b

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.51)

And the dual programming is ⎛ Max d(δ)  ⎝ 

b2 δ21

δ21 

 2ρπ

1 δ22

t a +2t b +t c 4

δ01 δ01

 ⎞δ01  ⎠

(2α − 1)P a + 2(1 − α)P b )    δ11 π (2α − 1)t a + 2(1 − α)t b 2α − 1)σ0a + 2(1 − α)σ0b

(δ21 + δ22 )(δ21 +δ22 )

Such that Constraints of the model (12.50).

δ11

12.3 Geometric Programming Approach Under Expected, Variance, …

12.3.5.2

307

Variance-Based UCCGP Model with Zigzag Variance Based UD

For α < 0.5, we have   2    2   5 tb − ta + 6 tb − ta tc − tb + 5 tc − tb Min V (W T (d, h))  2ρdπ y 48 Subject to

  (1 − 2α)P a + 2α P b yh −1  ≤ 2ασ0b + (1 − 2α)σ0c  σ (d, h)  dπ (1 − 2α)t a + 2αt b

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.52)

And the dual programming is 

⎛ ⎜ Max d(δ)  ⎜ ⎝

2ρπ



5(t b −t a ) +6(t b −t a )(t c −t b )+5(t c −t b ) 48 2

δ01

(1 − 2α)P a + 2α P b    δ11 π (1 − 2α)t a + 2αt b 2ασ0b + (1 − 2α)σ0c  2 δ21  δ01 b 1 (δ21 + δ22 )(δ21 +δ22 ) δ21 δ22 Such that Constraints of the model (12.50).

2

⎞δ01 ⎟ ⎟ ⎠

δ11

For α ≥ 0.5, we have   2    2   5 tb − ta + 6 tb − ta tc − tb + 5 tc − tb Min V (W T (d, h))  2ρdπ y 48 Subject to

  (2α − 1)P a + 2(1 − α)P b yh −1   ≤ (2α − 1)σ0a + 2(1 − α)σ0b σ (d, h)  dπ (2α − 1)t a + 2(1 − α)t b

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0. And the dual programming is

(12.53)

308

12 Geometric Programming Problem Under Uncertainty



⎛ ⎜ Max d(δ)  ⎜ ⎝

2ρπ

5(t b −t a ) +6(t b −t a )(t c −t b )+5(t c −t b ) 48 2

2

⎞δ01

δ01

⎟ ⎟ ⎠



(2α − 1)P a + 2(1 − α)P b )    δ11 π (2α − 1)t a + 2(1 − α)t b 2α − 1)σ0a + 2(1 − α)σ0b  2 δ21  δ01 b 1 (δ21 + δ22 )(δ21 +δ22 ) δ21 δ22 Such that Constraints of the model (12.50).

12.3.5.3

δ11

2-ND Moment UCCGP Model with Zigzag 2-ND Moment Based UD

For α < 0.5 we have 

  2 (t a )2 + t a t b + 2 t b + t b t c + (t c )2 Min Mo(W T (d, h))  2ρdπ y 6 Subject to

 (1 − 2α)P a + 2α P b yh −1  ≤ 2ασ0b + (1 − 2α)σ0c  σ (d, h)  dπ (1 − 2α)t a + 2αt b 

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.54)

And the dual programming is 

⎛ ⎜ Max d(δ)  ⎜ ⎝ 

2ρπ

(t a )2 +t a t b +2(t b ) +t b t c +(t c )2 6 2

δ01

⎞δ01 ⎟ ⎟ ⎠

(1 − 2α)P a + 2α P b    δ11 π (1 − 2α)t a + 2αt b 2ασ0b + (1 − 2α)σ0c  2 δ21  δ01 b 1 (δ21 + δ22 )(δ21 +δ22 ) δ21 δ22 Such that Constraints of the model (12.50). For α ≥ 0.5, we have

δ11

12.3 Geometric Programming Approach Under Expected, Variance, …

309



  2 (t a )2 + t a t b + 2 t b + t b t c + (t c )2 Min Mo(W T (d, h))  2ρdπ y 6 Subject to

  (2α − 1)P a + 2(1 − α)P b yh −1  ≤ (2α − 1)σ0a + 2(1 − α)σ0b  σ (d, h)  dπ (2α − 1)t a + 2(1 − α)t b

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.55)

And the dual programming is 

⎛ ⎜ Max d(δ)  ⎜ ⎝ 

b2 δ21

δ21 

2ρπ

 2 (t a )2 +t a t b +2 t b +t b t c +(t c )2 6

δ01

1 δ22

δ01

⎞δ01 ⎟ ⎟ ⎠



(2α − 1)P a + 2(1 − α)P b )   δ11 π (2α − 1)t a + 2(1 − α)t b 2α − 1)σ0a + 2(1 − α)σ0b

δ11



(δ21 + δ22 )(δ21 +δ22 )

Such that Constraints of the model (12.50).

12.3.5.4

Entropy-Based UCCGP Model with Zigzag Entropy Based UD

For α < 0.5 we have tc − ta y Min H (W T (d, h))  2ρdπ 2 Subject to   (1 − 2α)P a + 2α P b yh −1  ≤ 2ασ0b + (1 − 2α)σ0c  σ (d, h)  dπ (1 − 2α)t a + 2αt b 

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.56)

And the dual programming is δ11  c a  δ01  2ρπ t −t (1 − 2α)P a + 2α P b 2    Max d(δ)  δ01 δ11 π (1 − 2α)t a + 2αt b 2ασ0b + (1 − 2α)σ0c  2 δ21  δ01 b 1 (δ21 + δ22 )(δ21 +δ22 ) δ21 δ22 Such that Constraints of the model (12.50). 

For α ≥ 0.5 we have

310

12 Geometric Programming Problem Under Uncertainty

Global minimum

Fig. 12.2 Local minimum and global minimum

Local minimum tc − ta y Min H (W T (d, h))  2ρdπ 2 Subject to   (2α − 1)P a + 2(1 − α)P b yh −1  ≤ (2α − 1)σ0a + 2(1 − α)σ0b  σ (d, h)  dπ (2α − 1)t a + 2(1 − α)t b 

b2 y −2 + h 2 y −2 ≤ 1 d, h, y > 0.

(12.57)

And the dual programming is  c a  ⎞δ01 ⎛ ⎞δ11 a + 2(1 − α)P b ) 2ρπ t −t − 1)P (2α 2 ⎠ ⎝ ⎠ Max d(δ)  ⎝   δ01 δ11 π (2α − 1)t a + 2(1 − α)t b 2α − 1)σ0a + 2(1 − α)σ0b  δ21  b2 1 δ01 (δ21 + δ22 )(δ21 +δ22 ) δ21 δ22 ⎛

Such that Constraints of the model (12.50).

12.3.6 Numerical Example and Solution A Two-bar Truss problem shown in Fig. 12.2 is subject to a vertical load 2P and is to be designed for minimum weight. The members have a tubular section with mean tube diameter d, distance between two hinges 2b, and wall thickness t and maximum permissible stress in each member σ0 . Determine the values of h, d and y using chance constraints geometric programming approach for the following numerical data: P˜ : Z (32, 000, 33, 000, 34, 000) lbs, t˜ : Z (0.09, 0.1, 0.11) in, σ˜ 0 : Z (56, 000, 60, 000, 64, 000) psi, ρ  0.3 lbs/in3 , and 2b  60 in. Tables 12.1, 12.2, 12.3, and 12.4 show the objective values for, expected, variance, 2-ND moment, and entropy cases at α  0.1, α  0.5 and α  0.9, respectively. For each value of α, the objective values (minimum weight) differ substantially across the specifications assumed for the uncertainty distributions. While the objective values based on the expected, variance 2-ND moment, and entropy-based zigzag uncertain distributions, the objective values (minimum weight) fluctuate.

12.3 Geometric Programming Approach Under Expected, Variance, …

311

Table 12.4 Output values for expected zigzag uncertain distribution C

α

d ∗ (in)

h ∗ (in)

y ∗ (in)

E(Min W T (d, h))(lbs)

C

0.1

2.491962

30

42.42641

19.93671

G

0.5

2.474874

30

42.42641

19.80000

P

0.9

2.772746

30

42.42641

22.18310

GP

–

2.474874

30

42.42602

19.74

Schmit (1981)

–

2.47

30

–

19.8

Table 12.5 Output values for variance based zigzag distribution α

d ∗ (in)

h ∗ (in)

y ∗ (in)

V (Min W T (d, h))(lbs)

0.1

2.491962

30

42.42641

0.6645571E−02

0.5

2.474874

30

42.42641

0.6600001E−02

0.9

2.772746

30

42.42641

0.7394366E−02

Table 12.6 Output values for 2-ND moment based zigzag distribution α

d ∗ (in)

h ∗ (in)

y ∗ (in)

Mo(Min W T (d, h))(lbs)

0.1

2.491962

30

42.42641

1.237176

0.5

2.474874

30

42.42641

1.228693

0.9

2.772746

30

42.42641

1.376576

Table 12.7 Output values for entropy-based zigzag distribution α

d ∗ (in)

h ∗ (in)

y ∗ (in)

H (Min W T (d, h))(lbs)

0.1

2.491962

30

42.42641

1.993671

0.5

2.474874

30

42.42641

1.980000

0.9

2.772746

30

42.42641

2.218310

Here a numerical result of the expected minimum weight of a Two-Bar Truss Structural problem for different values of α under uncertainty distribution is presented. And the result with other (GP, Schmit (1981)) techniques are compared (Table 12.5). The variance is the difference between each number in a set and the mean squaring the differences and dividing the sum of the squares by the number of values in the set. Variance-based uncertainty distribution on Two-Bar Truss Structural problem is important to assist in managing weights by controlling budgets versus actual weight. A low variance and high variance indicate that the input points tend to be very close to mean and spread out from the mean, respectively (Table 12.6). The second moment of the area of a beam in the field of structural engineering is a very important property used in the calculation of the beam’s deflection and the calculation of the stress caused by a moment applied to the beam (Table 12.7).

312

12 Geometric Programming Problem Under Uncertainty

Sometimes, entropy is referred to as a measure of the amount of “disorder” in a system. High entropy  lots of disorder, while low entropy  order. By studying of entropy-based uncertainty distribution on Two-Bar Truss Structural problem, one can easily understand the input values. Low entropy implies the rigid results and high entropy referred the disorder of input values.

12.4 Multi-objective Geometric Programming Problem Under Uncertainty Multi-objective geometric programming (MOGP) is an efficient optimization technique, generally utilized for solving various nonlinear optimization problems (NLOP). Usually, the parameters of a multi-objective geometric programming (MOGP) problem (model) are thought to be deterministic and fixed. Yet, coefficients or parameters in real-world MOGP problems are often uncertain and subject to vacillations. Consequently, we utilize MOGP model within an uncertainty-based structure and propose a MOGP model whose coefficients (parameters) are uncertain in nature. We expect the uncertain variables (UVs) to have normal, linear, or zigzag uncertainty distributions and show that “the corresponding uncertain chanceconstrained multi-objective geometric programming (UCCMOGP) problems can be transformed into conventional MOGP problems to calculate the objective values.”

12.4.1 Uncertain Multi-objective Programming It has been gradually acknowledged that many real decision-making problems include multiple, noncommensurable, and contradictory objectives, which should be measured simultaneously. In order to optimize multiple objectives, multi-objective programming has been well developed and applied widely. For modeling multiobjective decision-making problems with uncertain parameters, Liu-Chen (2015) presented the following uncertain multi-objective programming:        ' E f 1 (x, ξ˜ ) , E f 2 (x, ξ˜ ) , . . . , E f m (x, ξ˜ ) min x (12.58) ( ) subject to M g j (x, ξ ) ≤ 0 ≥ α j , j  1, 2, . . . , n. where f i (x, ξ˜ ) are objective functions for i  1, 2, …, m, g j (x, ξ˜ ) are constraint functions, and α j are confidence levels for j  1, 2, …, n. Since the objectives are usually in conflict, there is no optimal solution that simultaneously minimizes all the objective functions. In this case, we have to introduce the concept of Pareto solution, which means that it is impossible to improve any one objective without sacrificing on one or more of the other objectives. A multi-objective geometric programming (MOGP) problem can be written as

12.4 Multi-objective Geometric Programming Problem Under Uncertainty

Find X  (x1 , x2 , . . . , xn )T

313

(12.59)

So as to p10

Minimize f 10 (x)  Minimize f 20 (x) 

c10i

n

i1

k1

p20

n

c20i

i1

xkαk10i xkαk20i

k1

... Minimize f m0 (x) 

pm0

cm0i

i1

Subject to fr (X ) 

pr i1

n

xkαkm0i

k1

cri

n

xkαkri ≤ cr , r  1, 2, . . . , q,

k1

xk > 0, k  1, 2, . . . , n, where c j0i are positive real numbers for all j  1, 2 …, m; i  1, 2, …, pr , αk j0i and αkri are real numbers for all k  1, 2, …, n; j  1, 2, …, m; i  1, 2, …, pr . number of terms present in the j0 th objective function. p j0 number of terms present in the r-th constraint. pr boundary value for the r-th constraint. cr In the above multi-objective nonlinear programming model, there are m minimizing objective functions, q inequality-type constraints, and n strictly positive decision variables. Mandal (2018) developed an MOGP model under uncertainty, whose associated chance-constrained version admits an equivalent crisp formulation. First, Mandal (2018) transformed the conventional MOGP problem in Eq. (12.59) into an MOGP problem under uncertainty, where c˜ j0i , c˜ri (j  1, 2, …, m; i  1, 2, …, pr ) are UVs. The model is Find X  (x1 , x2 , . . . , xn )T So as to Minimize f 10 (x)  Minimize f 20 (x) 

p10

c˜10i

n

i1

k1

p20

n

i1

c˜20i

k1

xkαk10i xkαk20i

(12.60)

314

12 Geometric Programming Problem Under Uncertainty

... Minimize f m0 (x) 

pm0

c˜m0i

n

xkαkm0i

i1 pr

Subject to fr (X ) 

k1 n α c˜ri xk kri i1 k1

≤ cr , r  1, 2, . . . , q,

xk > 0, k  1, 2, . . . , n, where c˜ j0i Uncertain positive real numbers for all j  1, 2, …, m; i  1, 2, …, pr , c˜ri Uncertain boundary value for the r-th constraint. In the above multi-objective nonlinear geometric programming model, there are m minimizing objective functions, q inequality-type constraints, and n strictly positive decision variables. Based on the model defined by Eq. (12.60) and the related constraints, we can formulate the following generic multi-objective GP model, which is a variant of the Uncertain Chance-Constrained Multi-objective Geometric Programming (UCCMOGP) model: Find X  (x1 , x2 , . . . , xn )T

(12.61)

So as to Minimize E( f 10 (x))  E

*p 10

c˜10i

i1 *p 20

Minimize E( f 20 (x))  E



c˜20i

i1

... Minimize E( f m0 (x))  E

*p m0

Subject to M( fr (x))  M

i1

k1 n

+ xkαk10i

, +

xkαk20i

,

k1

c˜m0i

i1  p r



n

c˜ri

n

+ xkαkm0i

k1 n

xkαkri

k1

xk > 0, k  1, 2, . . . , n, α ∈ ]0, 1[.

, 

≤ cr

≥ α, r  1, 2, . . . , q,

12.4 Multi-objective Geometric Programming Problem Under Uncertainty

315

12.4.2 UCCMOGP Model with Linear Uncertainty Distributions Let the coefficients c˜ joi , c˜ri in linear UVs. That  Eq. (12.61) be independent positive   is to say, c˜ j0i : L caj0i , cbj0i with 0 < caj0i < cbj0i and c˜ri : L cria , crib with 0 < cria < crib . From Lemma 4.3, we obtain the following deterministic objective function for the UCCGP problem proposed in Eq. (12.61): ⎡p ⎤ p ⎛ ⎞ p j0 j0 j0 n n n cajoi + cbj0i





αk j0i αk j0i αk j0i ⎣ ⎦ ⎝ ⎠ E c˜ j0i xk E(c˜ j0i ) xk  xk ,  2 i1

k1

i1

k1

i1

j  1, 2, . . . , m.

k1

Moreover, from Lemma 4.2, the constraints in Eq. (12.61) admit the following equivalent deterministic form: ∀i  1, . . . , n, M

 pr i1

c˜ri



n

xkαkri ≤ 1 ≥ α ⇔

k1

pr

a a ((1 − α)cri + αcri )

i1

n

xkαkri ≤ 1.

k1

Thus, when the coefficients are UVs endowed with linear distributions, the model corresponding to Eq. (12.61) is equivalent to Find X  (x1 , x2 , . . . , xn )T So as to p10  a n b

c1oi + c10i xkαk10i 2 i1 k1 p20  a n b

c2oi + c20i Minimize E( f 20 (x))  xkαk20i 2 i1 k1

Minimize E( f 10 (x)) 

... Minimize E( f m0 (x))

pm0  a c

moi

i1

Subject to

b + cm0i 2

pr

  (1 − α)cria + αcria

n

i1

k1



n

xkαkm0i

k1

xkαkri ≤ 1, r  1, 2, . . . , q,

xk > 0, k  1, 2, . . . , n, α ∈ ]0, 1[.

(12.62)

316

12 Geometric Programming Problem Under Uncertainty

12.4.2.1

Solution of MOGP Problem by the Weighted-Sum Method

 Let w  (w j , : w ∈ Rn , w j > 0, mj1 w j  1) be a set of nonnegative weights. Using the weighted-sum technique, the above multi-objective model can be written as Minimize E( f (x)) 

m

w j E( f j0 (x)) 

j1

m

wj

 n p j0  a c joi + cbj0i

j1

2

i1

α

xk k j0i .

k1

Hence, the multi-objective optimization problem under uncertainty reduces to a single-objective crisp geometric programming problem as follows: Minimize E( f (x)) 

m

wj

j1

 n p j0  a c joi + cbj0i

2

i1

α

xk k j0i

k1

pr n  

a a Subject to xkαkri ≤ 1, (1 − α)cri + αcri i1

k1

xk > 0, α ∈ ]0, 1[ (r  1, 2, . . . , q; k  1, 2, . . . , n).

(12.63)

Definition 12.1 A feasible solution x ∗ is said to be a Pareto solution to the multiobjective programming problem under uncertainty (12.63), if there is no feasible solution x such that ,  E[ f (x)] ≤ E f x ∗ , ,  and E[ f (x)] < E f x ∗ , for at least one index i. Definition 12.2 A feasible solution x ∗ is said to be a weak Pareto solution to the multi-objective programming problem under uncertainty (12.63), if there is no solution x such that ,  E[ f (x)] < E f x ∗ . Theorem 12.1 The solution of the MOGP problem (12.62) generated by the weighted-sum method (12.63) is Pareto-optimal if w j > 0 for all j  1, 2, . . . , m. Proof Let x ∗ be the solution of the MOGP problem (12.63) obtained by minimizing  p j0  cajoi +cbj0i  n m m αk j0i . the function ( f (x))  j1 w j E( f j0 (x))  j1 w j k1 x k i1 2 Obviously, it follows that E f (x ∗ ) ≤ E( f (x)), ∀x ∈ X , which implies that m j1

wj

 n p j0  a c joi + cbj0i

i1

2

k1

xk∗ αk j0i

≤

m j1

wj

 n p j0  a c joi + cbj0i

i1

2

k1

α

xk k j0i

12.4 Multi-objective Geometric Programming Problem Under Uncertainty

⇔

m

wj

j1

 n p j0  a c joi + cbj0i

 2

i1

317

 α xk k j0i − xk∗ αk j0i ≥ 0.

(12.64)

k1

Suppose the solution x ∗ of the problem (12.62) is not Pareto-optimal.   Then there exists some solution x  of the problem (12.62) satisfying E f j0 x  ≤ E f j0 (x ∗ ), which implies that     E f j0 x  − E f j0 x ∗ < 0 for all j  1, 2, . . . , m. ⇒

 n p j0  a c joi + cbj0i

2

i1

⇒

−

k1

 n p j0  a c joi + cbj0i

2

i1

xk αk j0i

 n p jo  a c joi + cbj0i

2

i1

xk∗ αk j0i < 0

k1

(xk − xk∗ )αk j0i < 0.

k1

By summing these inequalities and considering the assumption of the theorem that the weights w j are all positive, we obtain m

wj

j1

 n p j0  a c joi + cbj0i

 i1

2

 xk αk j0i − xk∗ αk j0i < 0.

k1

This inequality stands in contradiction to statement (12.64). Therefore, the solution x ∗ is a Pareto solution for w j > 0. Theorem 12.2 If x ∗ is a Pareto-optimal solution of a convex multi-objective optimization problem, then there exists a nonzero positive weight vector w such that x ∗ is a solution of the problem given by (12.63). Proof See Miettinen’s (1998) book on Nonlinear Multi-objective Optimization for the proof.

12.4.3 UCCMOGP Model with Normal Uncertainty Distributions Let the coefficients  c˜ j0i , c˜riin Eq. (12.61) be independent positive normal UVs, that is to say, c˜ j0i : N c j0i , σ j0i and c˜ri : N (cri , σri ), where c j0i , cri , σ j0i and σri are all positive real values. From Lemma 4.5, we obtain the following deterministic objective function for the proposed UCCGP problem given by Eq. (12.61): E

* p j0 i1

c˜ j0i

n

k1

+ α xk k j0i



p j0 i1

E(c˜ j0i )

n

k1

α

xk k j0i 

p j0 i1

c j0i

n

k1

α

xk k j0i . j  1, 2, . . . , m.

318

12 Geometric Programming Problem Under Uncertainty

Moreover, from Lemma 4.4, the constraints in Eq. (12.61) admit the following equivalent deterministic form: ⎛ ⎞ √  

pr pr  n n

α σri 3 α α kri log ∀i  1, . . . , n, M ⎝ c˜ri xk ≤ 1⎠ ≥ α ⇔ xk kri ≤ 1. cri + π 1−α i1

k1

i1

k1

Thus, when the coefficients are UVs endowed with normal distributions, the model corresponding to Eq. (12.61) is equivalent to Find X  (x1 , x2 , . . . , xn )T

(12.65)

So as to Minimize E( f 10 (x))  Minimize E( f 20 (x)) 

p10

(c10i )

n

i1

k1

p20

n

(c20i )

i1

xkαk10i xkαk20i

k1

... Minimize E( f m0 (x)) 

pm0 n

xkαkm0i (cm0i ) i1

k1

 √  

n α σri 3 log Subject to xkαkri ≤ 1, r  1, 2, . . . , q, cri + π 1 − α i1 k1 pr

xk > 0, k  1, 2, . . . , n, α ∈ ]0, 1[.

12.4.3.1

Solution of the MOGP Problem by the Weighted-Sum Method

 Let w  (w j , : w ∈ Rn , w j > 0, mj1 w j  1) be a set of nonnegative weights. Using the weighted-sum technique, the above multi-objective model can be written as Minimize

m j1

wj

p j0  i1

c j0i

n 

α

xk k j0i .

k1

Hence, this multi-objective optimization problem reduces to a single-objective crisp geometric programming problem as follows:

12.4 Multi-objective Geometric Programming Problem Under Uncertainty

Minimize

m

wj

j1

319

p j0 n  

α c j0i xk k j0i i1

k1

 √  

n α σri 3 log Subject to xkαkri ≤ 1, cri + π 1 − α i1 k1 pr

xk > 0, α ∈ ]0, 1[ (r  1, 2, . . . , q; k  1, 2, . . . , n).

(12.66)

12.4.4 UCCMOGP Model with Zigzag Uncertainty Distributions Let the coefficients  (12.61) be independent positive zigzagUVs. That is  c˜ j0i , c˜ri in Eq.  a b c to say, c˜ j0i : Z c j0i , c j0i , c j0i with 0 < caj0i < cbj0i < ccj0i and c˜ri : Z cria , crib , cric , with 0 < cria < crib < cric . From Lemma 4.7, we obtain the following deterministic objective function for the proposed UCCGP problem given by Eq. (12.61): ⎡p ⎤ p p j0  n j0 j0 n n



αk j0i αk j0i a + 2b + c αk j0i ⎣ ⎦ E c˜ j0i xk E(c˜ j0i ) xk  xk , 4 i1

k1

i1

k1

i1

j  1, 2, . . . , m.

k1

Moreover, from Lemma 4.6, the constraints in Eq. (12.61) admit the following equivalent deterministic form: ∀i  1, . . . , n, ⎛ M⎝

pr i1

c˜ri

n

k1

⎧ pr    n  α a + 2αcb ⎪ ⎪ xk kri ≤ 1, if α ∈ ]0, 0.5[; (1 − 2α)cri ⎨ ri i1 k1 ⎠ ≤1 ≥α⇔   p n r   αkri ⎪ c + 2(1 − α)cb ⎪ ⎩ xk ≤ 1, if α ∈ ]0.5, 1[. (2α − 1)cri ri ⎞

α

xk kri

i1

k1

Thus, when the coefficients are UVs endowed with zigzag distributions, the model corresponding to Eq. (12.61) is equivalent to For α < 0.5 we have Find X  (x1 , x2 , . . . , xn )T So as to p10  a n b c

c1oi + 2c10i + c10i xkαk10i 4 i1 k1 p20  a n b c c + 2c + c

2oi 20i 20i Minimize E( f 20 (x))  xkαk20i 4 i1 k1

Minimize E( f 10 (x)) 

(12.67)

320

12 Geometric Programming Problem Under Uncertainty

... Minimize E( f m0 (x)) 

pm0  a c

moi

i1

Subject to

b c + 2cm0i + cm0i 4

pr

  (1 − 2α)cria + 2αcrib

n

i1

k1



n

xkαkm0i

k1

xkαkri ≤ 1, r  1, 2, . . . , q. k  1, 2, . . . , n,

xk > 0, α ∈ ]0, 1[. For α > 0.5 we have Find X  (x1 , x2 , . . . , xn )T

(12.68)

So as to Minimize E( f 10 (x)) 

p10  a c

b c 1oi + 2c10i + c10i

4

 n

α xk k10i

i1 k1  n p20  a b + cc

α c2oi + 2c20i 20i Minimize E( f 20 (x))  xk k20i 4 i1 k1

...

 n pm0  a b + cc

α cmoi + 2cm0i m0i xk km0i Minimize E( f m0 (x))  4 i1

Subject to

pr 

k1

C + 2(1 − α)cb (2α − 1)cri ri

i1

n 

α

xk kri ≤ 1, (r  1, 2, . . . , q; k  1, 2, . . . , n)

k1

xk > 0, α ∈ ]0, 1[.

12.4.4.1

Solution of the MOGP Problem by the Weighted-Sum Method

 Let w  (w j , : w ∈ Rn , w j > 0, mj1 w j  1) be a set of nonnegative weights. Using the weighted-sum technique, the above multi-objective model can be written as Minimize

m j1

wj

 n p j0  a c

c joi + 2cbj0i + cm0i i1

4

α

xk k j0i .

k1

Hence, this multi-objective optimization problem under uncertainty reduces to a single-objective crisp geometric programming problem:

12.4 Multi-objective Geometric Programming Problem Under Uncertainty

321

For α < 0.5 we have Minimize

m

wj

j1

Subject to

 n p j0  a c

c joi + 2cbj0i + cm0i i1

4

α

xk k j0i

k1

pr

  (1 − 2α)cria + 2αcrib

n

i1

k1

xkαkri ≤ 1,

xk > 0, α ∈ ]0, 1[ (r  1, 2, . . . , q; k  1, 2, . . . , n).

(12.69)

For α > 0.5 we have Minimize

m j1

wj

 n p j0  a c

c joi + 2cbj0i + cm0i i1

4

α

xk k j0i

k1

pr n  

Subject to xkαkri ≤ 1, (2α − 1)cric + (1 − 2α)crib i1

k1

xk > 0, α ∈ ]0, 1[ (r  1, 2, . . . , q; k  1, 2, . . . , n).

(12.70)

12.4.5 Numerical Examples Local and Global Optima: Optimization is the process of finding the point that minimizes an appropriately defined function. More specifically: • A local minimum of a function is a point where the value of the function is smaller than or equal to the value at nearby points, but possibly greater than at a distant point. • A global minimum is a point where the value of a function is smaller than or equal to the value at all other feasible points (the numerical examples which are given here give the global optimal) We now give some numerical examples to show the efficacy of the proposed MOGP models. Problem: c˜101 + c˜102 x2 x3 x1 x2 x3 c˜201 min f 20 (x)  x1 x2 x3 such that c˜11 x1 x2 + c˜12 x1 x3 ≤ 4, x1 , x2 , x3 > 0.

min f 10 (x) 

(12.71)

322

12 Geometric Programming Problem Under Uncertainty

12.4.5.1

For Linear Uncertainty Distributions

c˜101 : L(30, 50), c˜102 : L(30, 50), c˜201 : L(700, 900), c˜11 : L(0.8, 1.2), c˜12 : L(1.6, 2.4).

Thus, the UCCMOGP problem is 40 + 40x2 x3 x1 x2 x3 800 min f 20 (x)  x1 x2 x3 such that (0.8(1 − α) + 1.2α)x1 x2 + (01.6(1 − α) + 2.4α)x1 x3 ≤ 4, x1 , x2 , x3 > 0. (12.72) min f 10 (x) 

From Eq. (12.63), the problem given by Eq. (12.72) becomes the following deterministic weighted-sum MOGP:  40 800 40w1 + 800w2 Min f (x)  w1 + 40x2 x3 + w2  + 40w1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 Such that (0.8(1 − α) + 1.2α)x1 x2 + (01.6(1 − α) + 2.4α)x1 x3 ≤ 4, x1 , x2 , x3 > 0. (12.73) Here, DD  4 − (3 + 1)  0. The DMOGPP (dual multi-objective geometric programming problem) corresponding to (12.73) is   40w1 + 800w2 δ01 40w1 δ02 (0.8(1 − α) + 1.2α) δ11 δ01 δ02 4δ11  δ12 (01.6(1 − α) + 2.4α) (δ11 + δ12 )(δ11 +δ12 ) 4δ12 (12.74) 

Max d(δ) 

Such that δ01 + δ02  1, −δ01 + δ11 + δ12  0 −δ01 + δ02 + δ11  0,

12.4 Multi-objective Geometric Programming Problem Under Uncertainty

323

−δ01 + δ02 + δ12  0, w1 + w2  1, δ01 , δ02 , δ11 , δ12 > 0. Solving the above normal and orthogonal conditions, we have δ01 

2 1 1 1 , δ02  , δ11  , δ12  . 3 3 3 3

From the primal–dual relation, we obtain 40w1 + 800w2  δ01 d(δ), x1 x2 x3 40w1 x2 x3  δ02 d(δ), δ11 (0.8(1 − α) + 1.2α)x1 x2  , 4 δ11 + δ12 δ12 (1.6(1 − α) + 2.4α)x1 x3  . 4 δ11 + δ12 And the corresponding optimal solution is (Table 12.8)  x3 

12.4.5.2

w1 + 20w2 4(1.6(1 − α) + 2.4α))w1

13

, x2  2x3 , x1 

2 . (1.6(1 − α) + 2.4α))x3

For Normal Uncertainty Distributions

c˜101 : N (40, 4), c˜102 : N (40, 4), c˜201 : N (800, 80), c˜11 : N (1, 0.1), c˜12 : N (2, 0.2). Then the UCCMOGP problem is 40 + 40x2 x3 x1 x2 x3 800 min f 20 (x)  x1 x2 x3   √ √     α α 0.1 3 0.2 3 log log subject to 1 + x1 x2 + 2 + x1 x3 ≤ 4. π 1−α π 1−α (12.75) min f 10 (x) 

324

12 Geometric Programming Problem Under Uncertainty

Table 12.8 Optimal solution under linear UDs α

0.2

0.4

0.6

0.8

Weights

Optimal values of the primal variables

Optimal values of the objective functions

x1∗

x2∗

x3∗

∗ (x) f 01

∗ (x) f 02

w1  0.1, w2  0.9

0.38

5.90

2.95

702.25

120.96

w1  0.5, w2  0.5

0.79

2.88

1.44

178.10

244.18

w1  0.9, w2  0.1

1.48

1.54

0.77

70.22

455.84

w1  0.1, w2  0.9

0.36

5.74

2.87

665.70

134.90

w1  0.5, w2  0.5

0.74

2.80

1.40

170.59

275.79

w1  0.9, w2  0.1

1.39

1.50

0.75

70.58

511.59

w1  0.1, w2  0.9

0.34

5.58

2.79

630.28

151.14

w1  0.5, w2  0.5

0.71

2.72

1.36

163.20

304.60

w1  0.9, w2  0.1

1.32

1.46

0.73

71.06

568.64

w1  0.1, w2  0.9

0.33

5.44

2.72

600.06

163.84

w1  0.5, w2  0.5

0.67

2.66

1.33

158.39

337.51

w1  0.9, w2  0.1

1.26

1.42

0.71

71.82

629.76

From Eq. (12.66), the problem given by Eq. (12.75) becomes the following deterministic weighted-sum MOGP:  40 800 40w1 + 800w2 Min f (x)  w1 + 40x2 x3 + w2  + 40w1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3    √ √    α α 0.1 3 0.2 3 log log Such that 1 + x1 x2 + 2 + x1 x3 ≤ 4, π 1−α π 1−α x1 , x2 , x3 > 0.

(12.76)

Here DD  4 − (3 + 1)  0. The DMOGPP (dual multi-objective geometric programming problem) corresponding to (12.76) is ⎛  40w1 + 800w2 δ01 40w1 δ02 ⎝ 1 + Max d(δ)  δ01 δ02   √ ⎛  α  ⎞δ12 2 + 0.2π 3 log 1−α ⎝ ⎠ (δ11 + δ12 )(δ11 +δ12 ) 4δ12 

√ 0.1 3 π

log

4δ11



α 1−α

 ⎞δ11 ⎠

12.4 Multi-objective Geometric Programming Problem Under Uncertainty

325

Such that δ01 + δ02  1, − δ01 + δ11 + δ12  0, − δ01 + δ02 + δ11  0, − δ01 + δ02 + δ12  0, w1 + w2  1, δ01 , δ02 , δ11 , δ12 > 0. Solving the above normal and orthogonal conditions, we have δ01 

2 1 1 1 , δ02  , δ11  , δ12  . 3 3 3 3

From the primal–dual relation, we obtain 40w1 + 800w2  δ01 d(δ), x1 x2 x3 40w1 x2 x3  δ02 d(δ),  √  α  1 + 0.1π 3 log 1−α x1 x2  2+

√ 0.2 3 π

4 log 4



α 1−α



x1 x3



δ11 , δ11 + δ12



δ12 . δ11 + δ12

And the corresponding optimal solution is (Table 12.9) ⎞ 13 w + 20w 1 2 x3  ⎝  √  α  ⎠ , x2  2x3 , x1   0.2 3 2+ 4 2 + π log 1−α w1 ⎛

12.4.5.3

√

0.2 3 π

2 log



α 1−α

For Zigzag Uncertainty Distributions

c˜101 : Z (30, 40, 60), c˜102 : Z (30, 40, 50), c˜201 : Z (700, 800, 1000), c˜11 : Z (0.8, 1.0, 1.2), c˜12 : Z (1.6, 2.2.4). For α < 0.5, we have The UCCMOGP problem is min f 10 (x) 

42.5 + 40x2 x3 x1 x2 x3



. x3

326

12 Geometric Programming Problem Under Uncertainty

Table 12.9 Optimal solution under normal UDs Weights

α

0.2

0.4

0.6

0.8

Optimal values of the primal variables

Optimal values of the objective functions

x1∗

x2∗

x3∗

∗ (x) f 01

∗ (x) f 02

w1  0.1, w2  0.9

0.37

5.80

2.90

679.23

128.55

w1  0.5, w2  0.5

0.76

2.84

1.42

174.36

261.02

w1  0.9, w2  0.1

1.42

1.52

0.76

70.59

487.69

w1  0.1, w2  0.9

0.38

5.70

2.85

656.28

129.59

w1  0.5, w2  0.5

0.78

2.78

1.39

167.84

265.42

w1  0.9, w2  0.1

1.46

1.48

0.74

68.49

500.32

w1  0.1, w2  0.9

0.39

5.62

2.81

638.18

129.89

w1  0.5, w2  0.5

0.79

2.74

1.37

163.64

269.77

w1  0.9, w2  0.1

1.48

1.46

0.73

67.99

507.17

w1  0.1, w2  0.9

0.39

5.52

2.76

616.14

134.64

w1  0.5, w2  0.5

0.80

2.70

1.35

159.52

274.35

w1  0.9, w2  0.1

1.50

1.44

0.72

67.19

514.40

825 x1 x2 x3 subject to (0.8(1 − 2α) + 1.0(2α))x1 x2 + (1.6(1 − 2α) + 2.0(2α))x1 x3 ≤ 4, x1 , x2 , x3 > 0. (12.77) min f 20 (x) 

From Eq. (12.69), the problem given by Eq. (12.77) becomes the following deterministic weighted-sum MOGP:  42.5 825 42.5w1 + 825w2 + 40x2 x3 + w2  + 40w1 x2 x3 Min f (x)  w1 x1 x2 x3 x1 x2 x3 x1 x2 x3 Such that (0.8(1 − 2α) + 1.0(2α))x1 x2 + (1.6(1 − 2α) + 2.0(2α))x1 x3 ≤ 4, x1 , x2 , x3 > 0.

(12.78)

Here, DD  4 − (3 + 1)  0. The DMOGPP (dual multi-objective geometric programming problem) corresponding to (12.78) is  Max d(δ) 

42.5w1 + 825w2 δ01

δ01 

40w1 δ02

δ02 

(0.8(1 − 2α) + 1.0(2α)) 4δ11

δ11

12.4 Multi-objective Geometric Programming Problem Under Uncertainty



(01.6(1 − 2α) + 2.0(2α)) 4δ12 Such that δ01 + δ02  1,

δ12

327

(δ11 + δ12 )(δ11 +δ12 )

− δ01 + δ11 + δ12  0, − δ01 + δ02 + δ11  0, − δ01 + δ02 + δ12  0, w1 + w2  1, δ01 , δ02 , δ11 , δ12 > 0. Solving the above normal and orthogonal conditions, we have δ01 

2 1 1 1 , δ02  , δ11  , δ12  . 3 3 3 3

From the primal–dual relation, we obtain 42.5w1 + 825w2  δ01 d(δ), x1 x2 x3 40w1 x2 x3  δ02 d(δ), δ11 0.8(1 − 2α) + 1.0(2α)  , 4 δ11 + δ12 1.6(1 − 2α) + 2.0(2α) δ12  . 4 δ11 + δ12 And the corresponding optimal solution is  x3 

42.5w1 +825w2 40

4(1.6(1 − 2α) + 2.0(2α))w1

 13 , x2  2x3 , x1 

2 . (1.6(1 − 2α) + 2.0(2α))x3

For α > 0.5, we have The UCCMOGP problem is 42.5 + 40x2 x3 x1 x2 x3 825 min f 20 (x)  x1 x2 x3 such that (1.2(2α − 1) + 2(1 − α)1.0)x1 x2 + (2.4(2α − 1) + 2(1 − α)2.0)x1 x3 ≤ 4, min f 10 (x) 

x1 , x2 , x3 > 0.

(12.79)

328

12 Geometric Programming Problem Under Uncertainty

From Eq. (12.70), the problem given by Eq. (12.79) becomes the following deterministic weighted-sum MOGP:  42.5 825 42.5w1 + 825w2 Min f (x)  w1 + 40x2 x3 + w2  + 40w1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 Such that (1.2(2α − 1) + 2(1 − α)1.0)x1 x2 + (2.4(2α − 1) + 2(1 − α)2.0)x1 x3 ≤ 4, x1 , x2 , x3 > 0. (12.80) Here, DD  4 − (3 + 1)  0. The DMOGPP (dual multi-objective geometric programming problem) corresponding to (12.80) is  Max d(δ)  

42.5w1 + 825w2 δ01

δ01 

(2.4(2α − 1)) + 2(1 − α)2.0) 4δ12 Such that δ01 + δ02  1,

40w1 δ02

δ12

δ02 

(1.2(2α − 1) + 2(1 − α)1.0) 4δ11

δ11

(δ11 + δ12 )(δ11 +δ12 )

− δ01 + δ11 + δ12  0, − δ01 + δ02 + δ11  0, − δ01 + δ02 + δ12  0, w1 + w2  1, δ01 , δ02 , δ11 , δ12 > 0. Solving the above normal and orthogonal conditions, we have δ01 

2 1 1 1 , δ02  , δ11  , δ12  . 3 3 3 3

From the primal–dual relation, we obtain 42.5w1 + 825w2  δ01 d(δ), x1 x2 x3 40w1 x2 x3  δ02 d(δ), δ11 1.2(2α − 1) + 2(1 − α)1.0  , 4 δ11 + δ12 δ12 2.4(2α − 1) + 2(1 − α)2.0  . 4 δ11 + δ12 And the corresponding optimal solution is (Table 12.10)  x3 

42.5w1 +825w2 40

4(2.4(2α − 1) + 2(1 − α)2.0)w1

 13 , x2  2x3 , x1 

2 . (2.4(2α − 1) + 2(1 − α)2.0)x3

12.5 Conclusion

329

Table 12.10 Optimal solution under zigzag UDs α

0.2

0.4

0.6

0.8

Weights

Optimal values of the primal variables

Optimal values of the objective functions

x1∗

x2∗

x3∗

∗ (x) f 01

∗ (x) f 02

w1  0.1, w2  0.9

0.38

5.96

2.98

716.73

122.24

w1  0.5, w2  0.5

0.78

2.90

1.45

181.16

251.53

w1  0.9, w2  0.1

1.46

1.56

0.78

72.60

464.39

w1  0.1, w2  0.9

0.36

5.80

2.90

679.82

136.25

w1  0.5, w2  0.5

0.74

2.82

1.41

173.49

280.38

w1  0.9, w2  0.1

1.37

1.52

0.76

73.06

521.29

w1  0.1, w2  0.9

0.34

5.64

2.82

644.05

152.56

w1  0.5, w2  0.5

0.70

2.76

1.38

168.29

309.43

w1  0.9, w2  0.1

1.30

1.48

0.74

73.66

579.45

w1  0.1, w2  0.9

0.35

5.50

2.75

613.03

155.84

w1  0.5, w2  0.5

0.72

2.68

1.34

160.08

319.07

w1  0.9, w2  0.1

1.34

1.44

0.72

69.94

552.58

12.5 Conclusion In this chapter, we have considered geometric programming (GP) problem under uncertainty with positive or negative integral degree of difficulty. The benefit of geometric programming (GP) technique is that we can find straightforwardly optimal solution of the objective function without solving two-level mathematical programs. This technique is simple and takes minimal time. Three different types of uncertainty chance-constrained GP techniques (Uncertain chance-constrained geometric programming model, Geometric programming approach under expected, variance, 2-ND moment and entropy-based zigzag uncertainty distribution, Multi-objective geometric programming problem under uncertainty) are discussed here. This method can be applied to solve the various decision-making problems (like in engineering problem, multi-objective optimization, inventory control theory, and many other areas).

330

12 Geometric Programming Problem Under Uncertainty

References B. Liu, Some research problems in uncertainty theory. J. Uncertain Syst. 3(1), 3–10 (2009) B. Liu: Uncertainty Theory, 4th Edition. (Springer-Verlag, Berlin, 2015). B. Liu, X. Chen, Uncertain multiobjective programming and uncertain goal programming. J. Uncertain. Anal. Appl. 3, 10 (2015). https://doi.org/10.1186/s40467-015-0036-6 T. Madjid et al., Solving geometric programming problems with normal, linear and zigzag uncertainty. J. Optim. Theory Appl. (2016) 170, 243–265 (2015) W.A. Mandal, Int. J. Interact. Des. Manuf. (2018). https://doi.org/10.1007/s12008-018-0477-5 W.A. Mandal, S. Islam, Multi-objective geometric programming problem under uncertainty. Oper. Res. Decis. 27(4), 85–109 (2018) K.A. Miettinen, Posteriori methods, in Nonlinear Multiobjective Optimization. Boston, MA: Springer (1998). ISBN 978-1-4613-7544-9. https://doi.org/10.1007/978-1-4615-5563-6. S.H. Nasseri, Z. Alizadeh, Optimized solution of a two-bar truss nonlinear problem using fuzzy geometric programming. J. Nonlinear Anal. Appl. 1–9 (2014) https://doi.org/10.5899/2014/jnaa00230 J. Peng, K. Yao, A new option pricing model for stocks in uncertainty markets. Int. J. Oper. Res. 8(2), 18–26 (2011) L.A. Schmit, Structural synthesis- its genesis and development. AIAA J. 119(10), 1249–1263 (1981).

Chapter 13

Intuitionistic and Neutrosophic Geometric Programming Problem

13.1 Introduction In 1965, Zadeh first introduced the concept of fuzzy set. In recent time, fuzzy set theory has been widely developed and its generalized form have appeared. In 1986, Atanassov extended the concept of fuzzy set and introduced the idea of intuitionistic fuzzy set theory, which consider both the degree of acceptance and degree of rejection such that the sum of both these values is less than one. Later, in 1995 neutrosophic set was introduced by Samarandache, which was actually a generalization of fuzzy sets and intuitionistic fuzzy sets (IFS) and was characterised by a truth membership. Kheiri and Cao (2016) introduced posynomial geometric programming with intuitionistic fuzzy coefficients. Ghosh and Roy (2014) described the fuzzy goal geometric programming method in intuitionistic environment. Kundu and Islam (2018) developed neutrosophic goal geometric programming problem and its application to multi-objective reliability optimization model. In this chapter we have discussed three types of geometric programming methods under uncertainty, that is (1) Intuitionistic Fuzzy Posynomial Geometric Programming Problem (2) Intuitionistic Fuzzy Goal Programming Model (3) Neutrosophic Goal Geometric Programming Problem and Its Application.

13.2 Intuitionistic Fuzzy Posynomial Geometric Programming Problem A posynomial geometric programming problem is of the following form

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4_13

331

332

13 Intuitionistic and Neutrosophic Geometric Programming Problem

Minimize g0 (t)  subject to gr (t) 

T0 

m 

c0 k

k1 Tr 

j1

cr k

k1+Tr −1

α

t j 0 kj m  j1

α

t j rkj ≤ 1

(13.1)

t j > 0, j  1, 2, . . . , m. where crk (>0) and α rkj (k  1, 2,…, 1 + Tr −1 ,…, T r ; r  0, 1, 2,…, l j  1, 2,…, m) are real numbers. Starting from the geometric programming (13.2) that is a standard PGP, we form a perturbed PGP, by replacing the number one with parameter br which are all positive constants, on the right-hands side of each constraint. T0 

Minimize g0 (t) 

c0 k

k1

subject to gr (t) 

Tr 

cr k

k1+Tr −1

m  j1

m  j1

α t j rkj

α

t j 0 kj (13.2)

≤ br

t j > 0, j  1, 2, . . . , m. where crk (>0) and α rkj (k  1, 2,…, 1 + T r −1 ,…, T r ; r  0, 1, 2,…, l j  1, 2,…, m) are real numbers. When bi  1, this reduces to the standard PGP, otherwise, the constraints need some amendment to be standard PGP equation. To solve the standard PGP can use the dual problem of the PGP. The corresponding posynomial GP dual problem is to. Dual Program The dual programming of (13.1) is as follows: Maximize v(δ) 

⎛ Tr  c δ l   rk rk ⎝ r 0 k1

δr k

Tr 

r 0 k1

rk

δr s ⎠

s1+Tr −1

subject to T0  δ0k  1, k1 Tr l  

⎞δ

αr k j δr k  0, ( j  1, 2, . . . , m)

δr k > 0, . . . (r  0, 1, 2, . . . , l; k  1, 2, . . . , Tr )

Normality condition Orthogonality conditions(one per primal variable) Positivity conditions

(13.3) The quantity T − (m + 1) is termed a degree of difficulty in geometric programming. In the case of a constrained geometric programming problem, T denotes the total number of terms in all the posynomials and m represents the number of primal variables. The posynomial function contains T 0 terms in the objective function and Tr terms in the inequality constrains where T  T 0 + T 1 +···+Tp .

13.2 Intuitionistic Fuzzy Posynomial Geometric Programming Problem

333

13.2.1 Posynomial Geometric Programming with Intuitionistic Fuzzy Coefficient Kheiri and Cao (2016) has developed the posynomial geometric programming (PGP) with intuitionistic fuzzy coefficient and described its solution procedure. Let c˜rIk and b˜rI (r  0, 1, . . . , l) denote the intuitionistic fuzzy number that can be GTrIFN, GTIFN or STrIFN. The posynomial geometric programming problem with intuitionistic fuzzy number coefficients is of the following form: T0 

Minimize g0 (t) 

k1

subject to gr (t)  t j > 0.

Tr 

k1+Tr −1

c˜rIk

c˜0I k

m  j1

m  j1

α

t j 0k j

α t j r k j ≤ b˜rI

(13.4)

where c˜rIk (>0) and α rkj (k  1, 2,…, 1 + T r −1 ,…, T r ; r  0, 1, 2,…, l j  1, 2,…, m) are real numbers. By using (α, β)-cut of the intuitionistic fuzzy coefficients and parameter bi and according to Eq. (13.4), the model is reduced to min

T0 

[C L 0k , C R 0k ]

k1

sub. gr (t)  t j > 0,

Tr  k1+Tr −1

m  j1

α

t j 0 kj

m

 

α C L r k , C Rr k t j r k j ≤ b L r , b Rr

(13.5)

j1

where

c L r k  max c L r k (α), c L r k (β) , (r  0, 1, 2, . . . , l), c Rr k  max c Rr k (α), c Rr k (β)

, (r  0, 1, 2, . . . , l), b L r  max b L r (α), b L r (β) , (r  0, 1, 2, . . . , l), b Rr  max b Rr (α), b Rr (β) , (r  0, 1, 2, . . . , l). In model (13.5), we denote g0 (t)   m αr k j Tr

. j1 t j k1 c L r k , c Rr k

 m α0 k j T0

, and gr (t)  j1 t j k1 c L 0k , c R0k

The above model is the posynomial geometric programming problem with interval coefficients. This model can be transformed into the following parametric form: min g0 (t) 

T0 m   (1−ρ)  ρ  α c L 0k c R0k t j 0k j , k1

j1

334

13 Intuitionistic and Neutrosophic Geometric Programming Problem

sub. gr (t) 

Tr m   (1−ρ)  ρ  (1−ρ)  ρ  α t j r k j ≤ bLr cLr k c Rr k b Rr , k1

j1

t j > 0.

(13.6)

The following theorem shows that model (13.5) can be transformed to a parametric posynomial geometric programming that is model (13.6). Theorem 13.1 Kheiri and Cao (2016). The interval posynomial geometric programming problem min

T0

m   α c L 0k , c R0k t j 0 kj

k1

gr (t)  t j > 0,



Tr 

j1

c L r k , c Rr k

m 

k1+Tr −1

j1



α t j r k j ≤ b L r , b Rr

(13.7)

is equivalent to following parametric posynomial geometric programming min g0 (t, ρ)  sub. gr (t, ρ) 

T0  m α0 (1−ρ)  ρ   c L 0k c R0k t j kj ,

k1 Tr 

k1

j1

m αr  (1−ρ)  ρ   (1−ρ)  ρ cLr k c Rr k b Rr , t j k j , ≤ bLr

(13.8)

j1

t j > 0. Proof Let Q 1 and Q 2 be the sets of all feasible solutions to (13.7) and (13.8), respectively. Then x ∈ Q 1 if and only if: Tr m 



  α t j r k j ≤ b L r , b Rr c L r k , c Rr k k1+Tr −1

(13.9)

j1





Then, for any k, we take dk ∈ c L r k , c Rr k and q ∈ b L r , b Rr , problem (13.9) is substituted by the following crisp problem: Tr  k1+Tr −1

dk

m 

α

t j r k j ≤ q.

(13.10)

j1

From of Interval-valued function, the interval-valued functions of 



the definition C  c L r k , c Rr k and B  b L r , b Rr for any fixed r, are obtained respectively as  (1−ρ)  ρ h C (ρ)  c L r k c Rr k , for ρ ∈ [0, 1],  (1−ρ)  ρ h B (ρ)  b L r b Rr , for ρ ∈ [0, 1].

13.2 Intuitionistic Fuzzy Posynomial Geometric Programming Problem

335

According to Lemma 4.14, h C (ρ) and h B (ρ) are strictly monotone increasing continuous functions. We obtain dk ∈ h C (ρ) and q ∈ h B (ρ), then for ρ ∈ [0, 1], problem (13.10) reduces to m Tr   (1−ρ)  ρ   (1−ρ)  ρ α cLr k c Rr k b Rr t j r k j ≤ bLr k1

(13.11)

j1

Then x ∈ Q 2 , hence Q 1  Q 2 . Now we suppose t0  (t01 , . . . , t0m )T to be an optimal feasible solution to (13.7), then for all x ∈ Q 1 , we have: g0 (x) ≥ g0 (x0 ) T0

 m α0k j T0

 m α0k j c L 0k , c R0k c L 0k , c R0k ⇔ tj ≥ t0 j . j1

k1

j1

k1



We take ϕk ∈ c L 0k , c R0k , for any k, then above problems turn to T0 

ϕk

k1

m 

T0 

α

t j 0k j ≥

k1

ϕk

m 

k1

α

t0 j0k j .

(13.12)

j1

From of interval-valued function, the interval-valued function of

the definition  D  c L 0k , c R0k , is obtained (1−ρ)  ρ  h D (ρ)  c L 0k c R0k , for ρ ∈ [0, 1]. Then ϕk ∈ h D (ρ), problem (13.12) reduces to T0 T0 m m    (1−ρ)  ρ   (1−ρ)  ρ  α α c L 0k c R0k c L 0k c R0k t j 0 kj ≥ t0 j0 k j . k1

j1

k1

(13.13)

j1

We conclude that t0 is an optimal feasible solution to (13.18). We turn the inner program of model (13.6) to the following standard posynomial geometric program form: min g0 (t, ρ)  sub. gr (t, ρ) 

T0  m (1−ρ)  ρ   α c L 0k c R0k t j 0 kj ,

k1 Tr 

k1

(c L r k )

(1−ρ)

(c Rr k )

ρ

(bLr )(1−ρ) (b Rr )ρ

m 

j1

j1

α

t j r k j ≤ 1, (r  0, 1, 2, . . . , l),

(13.14)

t j > 0. We drive standard posynomial geometric program problem, and can solve by the dual problem of the PGP.

336

13 Intuitionistic and Neutrosophic Geometric Programming Problem

For ρ  0 and ρ  1 the lower bound and upper bound of the interval value of the parameter is used to find the optimal solution respectively. These two values yield the lower and upper bounds of the optimal solution. Although, one can gain the intermediate optimal result by using a proper value of ρ.

13.2.2 Numerical Example (Design of a Two-Bar Truss) (Kheiri and Cao 2016) Consider a simple mechanical design problem, the two-bar truss is subjected to a vertical load 2p and is to be designed for minimum weight. The members have a tubular section with mean diameter d and wall thickness t and the maximum permissible stress in each member (σ0 ) is approximately equal to 60,000 psi. Determine the values of h and d using geometric programming for the following data: p  33,000 Ib, t  0.1 in., b  30 in, σ0  60,000 psi, and ρ (density)  0.3 lb/ in3 . √ min 2ρπ√dt b2 + h 2 2 2 sub. π pdt b h+h ≤ σ0 , d, h ≥ 0.

(13.15)

The optimal solution is h ∗  30, d ∗  2.474874 and optimal objective value is 19.74. Now we consider the intuitionistic fuzzy optimization model of the two-bar truss as following: √ min 2ρ˜ I π√dt b2 + h 2 I 2 2 sub. πp˜dt b h+h ≤ σ˜ 0I , d, h ≥ 0.

(13.16)

I

 where p˜ I  33, 000 Ib  {(33000, 10, 30; 0.5); (33000, 40, 50; 0.25)}, t  0.1 in., I I  000  {(60000, 20, 30; 0.75); (60000, 30, 70; 0.2)}, psi, and b  30 in, σ˜ 0  60,  I lb/ in3  {(0.3, 0.04, 0.08; 0.75); (0.3, 0.06, 0.14; 0.2)}are the ρ˜ I (density)  0.3 TIFNs. It can be seen that the √ objective and constraint functions are not posynomials, due to the presence of the b2 + h 2 . The functions can be converted to posynomials by introducing a new variable y as y  b2 + h 2 . Thus the intuitionistic fuzzy optimization problem can be stated as:  I dy 21 min0.628 × 0.3 I

I

1   sub. 33,000 y 2 d −1 h −1 ≤ 60, 000 , 0.314 −1 −1 2 900y + y h ≤ 1, y, d, h ≥ 0.

(13.17)

13.2 Intuitionistic Fuzzy Posynomial Geometric Programming Problem

337

According to the intuitionistic fuzzy set theory, achieve the following model: 1

min{(0.188, 0.025, 0.050; 0.75); (0.188, 0.037, 0.087; 0.2)}dy 2 sub.{(105095.54, 31.84, 95.54; 0.5); (105095.54, 127.38, 159.23; 0.25)}d −1 h −1 ≤ {(60000, 20, 30; 0.75); (60000, 30, 70; 0.2)}, 900y −1 + y −1 h 2 ≤ 1 y, d, h ≥ 0. (13.18) by using (α, β)-cut of the intuitionistic fuzzy coefficients and parameter σ˜ 0I that are α  0.2, β  0.5 the model (13.18) is reduced to 1

min(0.174, 0.220)dy 2 1 sub.(105076.43, 105148.61)y 2 d −1 h −1 ≤ (59988.75, 60022), 900y −1 + y −1 h 2 ≤ 1, y, d, h ≥ 0.

(13.19)

This interval optimization problem can be transformed into the parametric form (13.20), min(0.174)(1−ρ) (0.220)(ρ) dy 2 1 sub.(105076.43)(1−ρ) (105148.61)(ρ) y 2 d −1 h −1 ≤ (59988.75)(1−ρ) (60022)(ρ) , 900y −1 + y −1 ≤ 1, y, d, h ≥ 0, ρ ∈ [0, 1]. (13.20) 1

Turn this parametric perturbed PGP to standard geometric program form: min(0.174)(1−ρ) (0.220)(ρ) dy 2 (1−ρ) (105148.61)(ρ) 21 −1 −1 sub. (105076.43) y d h ≤ 1, (59988.75)(1−ρ) (60022)(ρ) 1

900y −1 + y −1 h 2 ≤ 1, y, d, h ≥ 0. ρ ∈ [0, 1].

(13.21)

This model is parametric standard PGP, for ρ ∈ [0, 1], numerical solutions of this problem are presented in following Table 13.1. For ρ  0, the lower bound of interval value of the coefficient is used to find the optimal solution. And ρ  1, present the upper bound of the interval coefficients is used for the optimal solution. These two values yield the lower and upper bounds of I I   000 the optimal solution. We change 33, 000 to 32990, 33030, 32960, 33050, 60,  I to 0.26, 0.38, 0.24, 0.44, in model (13.6) to 59980, 66030, 65970, 66070, and 0.3

338

13 Intuitionistic and Neutrosophic Geometric Programming Problem

Table 13.1 Numerical result ρ

Optimal objective value

d∗

y∗

h∗

0.0

18.28673

2.477140

1800.000

30

0.2

19.16559

2.477205

1800

30

0.4

20.08669

2.477271

1800

30

0.6

21.05206

2.477337

1800

30

0.8

22.06382

2.477402

1800

30

1.0

23.13703

2.478841

1799.999

30

respectively. Then finding optimal objective values, 17.13109, 25.02297, 15.85812, 28.91133. Thus from the above discussion we see that the optimal objective values for ρ ∈ [0, 1], are between 15.85812 and 28.91133.

13.3 Intuitionistic Fuzzy Goal Programming Model Here an intuitionistic fuzzy goal programming problem of minimization type objective function is considered. Mathematically, the problem can be stated as: I

 f i (X ) with target value ai and acceptance tolarence ti , Minimize rejection tolarence ti0 , i  1, 2, . . . ., m subject to g j (X ) ≤ b j , i  1, 2, . . . ., m,  T X  x1 , x2 , . . . , xq > 0.

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(13.22)

In addition with degree of acceptance (membership), when degree of nonacceptance (non membership) is taken into consideration without complementing each other, then an intuitionistic fuzzy set can be used as a general tool for decision  making under uncertainty. Let μ x , x f 1 2 , . . . , x q be the degree of acceptance (mem i bership) and ν fi x1 , x2 , . . . , xq be the degree of non-acceptance (non-membership) of X to i-th intuitionistic fuzzy set. Then in optimization problem membership function should be maximized and non-membership function should be minimized. Hence, the above model (2.1) can be written in crisp programming as: ⎫     Maximize μ fi x1 , x2 , . . . , xq , Minimize ν fi x1 , x2 , . . . , xq , i  1, 2, . . . , m ⎪ ⎪ ⎪ T  ⎬ subject to g j (X ) ≤ b j , j  1, 2, . . . , n;X  x1 , x2 , . . . , xq > 0, ⎪ 0 ≤ μ fi x1 , x2 , . . . , xq  + ν fi x1 , x2 , . . . , xq ≤ 1,  ⎪ ⎪ ⎭ 0 ≤ μ fi x1 , x2 , . . . , xq ≤ 1, 0 ≤ ν fi x1 , x2 , . . . , xq ≤ 1 (13.23)

13.3 Intuitionistic Fuzzy Goal Programming Model

339

Fig. 13.1 Linear membership and non-membership functions

The linear membership and non-membership functions for Intuitionistic fuzzy objective goals are ⎧   ⎪ 1, f i x1 , x2 , . . . , xq ≤ ai   ⎨   q )−ai μ fi x1 , x2 , . . . , xq  1 − fi (x1 ,x2 ,...,x x ≤ ai + ti , a ≤ f , x , . . . , x i i 1 2 q t i ⎪   ⎩ 0, f i x1 , x2 , . . . , xq ≥ ai + ti and 

ν fi x1 , x2 , . . . , xq



⎧   f i x1 , x2 , . . . , xq ≤ ai ⎪ ⎨ 0,   f i (x1 ,x2 ,...,xq )−ai  x ≤ ai + ti0 , a ≤ f , x , . . . , x 0 i i 1 2 q ti ⎪   ⎩ 0, f i x1 , x2 , . . . , xq ≥ ai + ti0

with ti ≤ ti0 , for i  1, 2, . . . , m (Fig. 13.1). Definition 13.1 (M-N Pareto optimal solution) x ∗ is said to be a M-N Pareto optimal solution if and only if there does not exist another x ∈ X such that μ fi (x) ≥ μ fi (x ∗ ) and ν fi (x) ≤ ν fi (x ∗ ) for all i and strict inequality holds for at least one i. Theorem 13.2 (Ghosh and Roy 2014) x ∗ ∈ X is M-N Pareto optimal solution of (13.22) if and only if x ∗ is Pareto optimal solution of  Minimize f i (X ), i  1, 2, . . . , m T  (13.24) Subject to g j (X ) ≤ b j , j  1, 2, . . . , n; X  x1 , x2 , . . . , xq > 0. solution then there does Proof Let x∗ ∈ X is M-N Pareto optimal   of (13.22),  ∗ not exist any x ∈ X such that μ fi x1 , x2 , . . . , xq ≥ μ fi x1 , x2 , . . . , xq and

340

13 Intuitionistic and Neutrosophic Geometric Programming Problem

   ∗ ν fi x1 , x2 , . . . , xq ≤ ν fi x1 , x2 , . . . , xq for all i and strictly inequality holds for at least one i. So from the expression of membership function we have ∗ f x ,x ,...,x −a f x ,x ,...,x −a 1 − i ( 1 2 ti q ) i ≥ 1 − i ( 1 2 ti q ) i i.e.,    ∗ f i x1 , x2 , . . . , xq ≤ f i x1 , x2 , . . . , xq .

Also for non-membership function ∗ f x ,x ,...,x −a ≥ i ( 1 2 t 0 q ) i i.e.,   i ∗ f i x1 , x2 , . . . , xq ≥ f i x1 , x2 , . . . , xq .

f i (x1 ,x2 ,...,xq )−ai 0  ti

with strict inequality holding for at least one i. So x∗ is Pareto optimal solution of (2.3). of (13.24). Then there On the other hand, let x ∗ be a Pareto optimal  solution  ∗ does not exist x ∈ X such that f i x1 , x2 , . . . , xq ≤ f i x1 , x2 , . . . , xq with strict inequality holding for at least one i.    ∗ So f i x1 , x2 , . . . , xq − ai ≤ f i x1 , x2 , . . . , xq − ai , ∗ f x ,x ,...,x −a f x ,x ,...,x −a , i.e., i ( 1 2 q ) i ≤ i ( 1 2 q ) i ti0

ti0

   ∗ tells that ν fi x1 , x2 , . . . , xq ≤ ν fi x1 , x2 , . . . , xq and    ∗ f i x1 , x2 , . . . , xq − ai f i x1 , x2 , . . . , xq − ai 1− ≥1− ti ti    ∗ tells that μ fi x1 , x2 , . . . , xq ≥ μ fi x1 , x2 , . . . , xq , with strict inequality holding for at least one i. Hence, x∗ is M-N Pareto optimal solution of (13.22).

13.3.1 Goal Geometric Programming Model Model (13.23) is a standard crisp programming model, which is derived from the intuitionistic fuzzy goal programming problem (13.22). The crisp programming model (13.23) can be formulated as:

13.3 Intuitionistic Fuzzy Goal Programming Model

341

Maximize α, Minimize β f x ,x ,...,x −a f x ,x ,...,x −a subject to 1 − i ( 1 2 ti q ) i ≥ α, i ( 1 2 t 0 q ) i ≤ β, i  1, 2, . . . , m i   g j x1 , x2 , . . . , xq ≤ b j , j  1, 2, . . . , n T  X  x1 , x2 , . . . , xq > 0, α + β ≤ 1, 0 ≤ α, β ≤ 1. (13.25) It is easily seen that Minimize β is equivalent to Maximize (1 − β) as 0 ≤ β ≤ 1. Therefore using arithmetic mean method (Sakawa et al. 2011) the above model (13.25) can be written as ⎫ Maximize α+1−β ⎪ 2     ⎪ ⎪ f i (x1 ,x2 ,...,xq )−ai 1 ⎬ α+1−β ai 1 1 1 ≤ 1 + − subject to + + , i  1, 2, . . . , m, 0 0 2 ti 2 ti 2 ti ti   ⎪ g j x1 , x2 , . . . , xq ≤ b j , j  1, 2, . . . , n, ⎪ ⎪  T ⎭ X  x1 , x2 , . . . , xq > 0, α + β ≤ 1, 0 ≤ α, β ≤ 1. (13.26) Let us consider u  α+1−β , then the above model (13.26) reduces to following 2 primal geometric programming form as ⎫ ⎪ Minimize u −1 ⎪   ⎪ ⎪ f i (x1 ,x2 ,...,xq )−ai ⎪ 1 1     ⎪ ≤ 1, i  1, 2, . . . , m, subject to + 0 ⎬ t ti ai i 1 1 2 1+ 2 t + 0 −u i ti (13.27) ⎪ g j (x1 ,x2 ,...,xq ) ⎪ ⎪ ≤ 1, j  1, 2, . . . , n, ⎪ ⎪ ⎪ b j T ⎭ X  x1 , x2 , . . . , xq > 0, u > 0. which can be solved by geometric programming technique with u (>0) as parameter.

13.3.2 Illustrative Numerical Example (Ghosh and Roy 2014) Every manufacturing unit wants to minimize the expenditure like loading unloading cost, cutting cost, tool cost and tool changing cost, while machining a 150 mm long and 25 mm in diameter cylindrical bar with cutting speed v m/min and feed rate f mm/rev. Decision maker of the manufacturing unit sets some flexible target of 1.8$ of total expenditure. The maximum feed to be used to control the surface finish is 0.115 mm/rev with some flexibility. Uncertainty of total expenditure mainly depends upon tool life T which is related to the cutting condition via Taylor’s equation T  Cv− n f − m 1

1

342

13 Intuitionistic and Neutrosophic Geometric Programming Problem

In spite of some fixed cutting conditions, tool life T may change due to nonhomogeneity of the machined and cutting material. The required data for material removal economics case study are: • • • • • • • • • • • • •

R0  operator rate ($/min)  0.60 $/min Rm  machine rate ($/min)  0.40 $/min Ct  tool cost ($/cutting edge)  2.00 $/edge tl  machine loading and unloading time (min)  1.5 min tch  tool changing time (min)  0.8 min 1/m  feed rate exponent  1.25 (m  0.80) 1/n  cutting speed exponent  4.00 (n  0.25) C  Taylor’s Modified Tool Life Constant (min)  2.46 × 108 min Q  fraction of cutting path that tool is cutting material  1.0 (for turning) B  cutting path surface factor of tool  11.77286 (mm-m). Loading unloading cost k00  (R0 + Rm ) tl  1.50, Cutting cost  k01 f −1 v−1  R0 + Rm ) B f −1 v−1  1.78 f −1 v−1 , 1 1 Tool cost and tool changing cost  k02 f ( m −1) v( n −1)  [(R0 + Rm )tch + Ct ]Q BC −1 f ( m −1) v( n −1) 1

1

 1.34 × 10−7 f 0.25 v3 Hence the total expenditure Cu  1.50 + 11.78 f −1 v−1 + 1.34 × 10−7 f 0.25 v3 along with the feed rate f is to be minimized with some flexible targets. The intuitionistic fuzzy goal programming problem is: I

 C  1.50 + 11.78 f −1 v−1 +1.34 × 10−7 f 0.25 v3 with target value 1.8 and • Minimize u acceptance tolerance 0.3, rejection tolerance 0.5.  I  f with target value 0.115 and acceptance tolerance 0.24, rejection • Minimize f ed

tolerance 0.26 subject to f, v > 0. Membership and non-membership functions for intuitionistic fuzzy objective functions are: ⎧ cu ( f, v) ≤ 1.8 ⎨ 1, μcu ( f, v)  1 − cu ( f,v)−1.8 , 1.8 ≤ cu ( f, v) ≤ 2.1 0.3 ⎩ 0, cu ( f, v) ≥ 12.3 ⎧ cu ( f, v) ≤ 1.8 ⎨ 0, νcu ( f, v)  cu ( f,v)−1.8 , ai ≤ cu ( f, v) ≤ 2.3 0.5 ⎩ 0, cu ( f, v) ≥ 2.3 And

13.3 Intuitionistic Fuzzy Goal Programming Model

343

⎧ f ed ( f, v) ≤ 0.115 ⎨ 1, μ fed ( f, v)  1 − fed ( f,v)−0.115 , 0.115 ≤ f ed ( f, v) ≤ 0.355 0.24 ⎩ 0, f ed ( f, v) ≥ 0.355 ⎧ f ed ( f, v) ≤ 0.115 ⎨ 0, ν fed ( f, v)  fed ( f,v)−0.115 , 0.115 ≤ f ed ( f, v) ≤ 0.375 0.26 ⎩ 0, f ed ( f, v) ≥ 0.375 Now the crisp model is ⎫ ⎪ Maximize μcu ( f, v), Minimize νcu ( f, v) ⎪ ⎪ ⎪ ⎪ Maximize μ fed ( f, v), Minimize ν fed ( f, v) ⎬ subject to 0 ≤ μcu ( f, v), μ fed ( f, v), νcu ( f, v), ν fed ( f, v) ≤ 1, ⎪ ⎪ ⎪ 0 ≤ μcu ( f, v) + νcu ( f, v) ≤ 1, ⎪ ⎪ ⎭ 0 ≤ μ fed ( f, v) + ν fed ( f, v) ≤ 1.

(13.28)

Now we have got the below model of goal geometric programming problem by following expressions (13.25) and (13.26) as ⎫  −1 ⎪ α+1−β ⎪ Minimize ⎪ 2 ⎪ ⎪ 5.333×11.78 −1 −1 5.333×1.34 ⎪ −7 0.25 3 f v ×10 f v ⎬ 2 2 subject to 1+ 5.333×0.3 + ≤ 1, α+1−β α+1−β 5.333×0.3 − 2 1+ − 2 (13.29) 2 2 9.109731 ⎪ f ⎪ 2 ⎪ ≤ 1, ⎪ 1+ 9.109731×0.115 − α+1−β ⎪ 2 2 ⎪ ⎭ f, v > 0, α + β ≤ 1, 0 ≤ α, β We solve the above geometric programming model where degree of difficulty is 3 − (2 + 1)  0. Dual of the above model is ⎛ ⎞δ ⎛ ⎞δ 11 12 δ 5.333×11.78 5.333×1.34 × 10−7 01 2 2 2 ⎝ ⎠ ⎝ ⎠   d(δ)  (α + 1 − β)δ01 − α+1−β − α+1−β 1 + 5.333×0.3 δ11 1 + 5.333×0.3 δ12 2 2 2 2 ⎞δ ⎛ 

9.109731

2 ⎠  ×⎝ − α+1−β 1 + 9.109731×0.115 δ21 2 2

21





δ (δ11 + δ12 ) δ11 +δ12 δ2121

(13.30)

such that δ01  δ11 + δ12  1, −δ11 + 0.25δ12 + δ21  0, −δ11 + 3δ12  0. . Solving them we have δ01  1, δ11  43 , δ12  41 , δ21  2.75 4 Hence from primal dual relation 

⎛ ⎞3 ⎛ ⎞1    4 4 5.333×11.78 5.333×1.34 × 10−7 α + 1 − β −1 α + 1 − β −1 ⎝ 2 2 ⎠ ⎝ ⎠      α+1−β α+1−β 5.333×0.3 3 5.333×0.3 1 2 2 − − 1+ ×4 1+ ×4 2 2 2 2 ⎛ ×⎝

  ⎞ 2.75  2.75 4  9.109731 2.75 4 2 ⎠  , 4 − α+1−β 1 + 9.109731×0.115 × 2.75 2 2 4

(13.31)

344

13 Intuitionistic and Neutrosophic Geometric Programming Problem

Table 13.2 Numerical result Method Primal Variables

IFG2 P2

f∗  0.257297 mm/rev v∗  112.4525 m/min

Optimal objective functions

α∗ · β ∗

Membership and nonmembership

Sum of membership and nonmembership

cu∗  2.04285$ ∗  f ed 0.25729 mm/rev

α ∗ ∈ [0, 0.64801] β ∗ ∈ [0, 0.35198]

μcu ( f, v)  0.1904979 vcu ( f, v)  0.4857013 μ fed ( f, v)  0.407095 v fed ( f, v)  0.547296

μcu ( f, v) + vcu ( f, v)  0.6761992 μ fed ( f, v) + v fed ( f, v)  0.954392

5.333×11.78 −1 −1 f v δ11 2 ,  α+1−β 5.333×0.3 (δ11 + δ12 ) 1+ − 2 2 5.333×1.34 × 10−7 f 0.25 v3 δ21 2 .  α+1−β 5.333×0.3 (δ 1+ − 2 11 + δ12 ) 2

(13.32) (13.33)

Considering u  α+1−β in primal dual relation, the optimal values of decision 2 variables and objective functions are obtained solving equation number (13.31), (13.32) and (13.33). The equations are solved using Lindo-Lingo software and the lists of values are given in Table 13.2.

13.4 Neutrosophic Goal Geometric Programming Problem Kundu and Islam (2018) has considered neutrosophic goal geometric programming problem (NGGPP) as a generalization of intuitionistic fuzzy goal geometric programming problem (IFGGPP). In NGGPP Kundu and Islam (2018) has introduced the degree of indeterminacy of neutrosophic goal geometric programming (NGGP) objectives together with the degree of rejection and degree of acceptance.

13.4.1 Neutrosophic Multi-objective Goal Geometric Programming Problem A multi-objective non-linear neutrosophic goal geometric programming model with p objective functions can be taken as follows: Minimize ( f 01 (x), f 02 (x), . . . , f 0 p (x)) Subject to gr (x) ≤ br , r  1, 2, . . . , l.

(13.34)

13.4 Neutrosophic Goal Geometric Programming Problem

345

where, f 0i 

T0i 

C0ik

k1

m  j1

α

x j 0ik j , i  1, 2, . . . , p

and gr (x) 

Tr p  k1+T(r −1) p

Cr k

m 

α

x j r k j , r  1, 2, . . . , l, x  (x1 , x2 , . . . , xm ) > 0.

j1

Here we consider each of the objective functions f 0i (x) satisfying target goal achievement value b0i with the acceptance tolerance a0i , rejection tolerance r0i and indeterminacy tolerance d0i . Where we have C0ik > 0,(for k  1, 2, 3, . . . , T0i ; i  1, 2, . . . , p), Cr k > 0, for k  1 + T0 p , . . . , T1 p , T1 p + 1, . . . , Tlp α0ik j (k  1, 2, . . . , T0i ; i  1, 2, . . . , p; j  1, 2, . . . , m) and αr k j (k  1 + T0 p , . . . , T1 p , T1 p + 1, . . . , Tlp ; j  1, 2, . . . , m) are real numbers. The above Eq. (3.1) is a constrained posynomial geometric programming (PGP) model. Now using the concept of neutrosophic sets, construct the truth membership function μi ( f oi (x)), indeterminacy membership function σi ( f oi (x)) and falsity membership function ϑi ( f oi (x)) of the neutrosophic goal programming (NGP) objectives are given by as follows ⎧ ⎪ f 0i (x) ≤ b0i ; ⎨ 1, 0i , b (13.35) μi ( f oi (x))  1 − foi (x)−b 0i ≤ f 0i (x) ≤ b0i + a0i a0i ⎪ ⎩ 0, f 0i (x) ≥ b0i + a0i ; ⎧ ⎪ f 0i (x) ≤ b0i ; ⎨ 0, 0i , b ϑi ( f oi (x))  foi (x)−b (13.36) 0i ≤ f 0i (x) ≤ b0i + r 0i ; r0i ⎪ ⎩ 1, f 0i (x) ≥ b0i + r0i ; and ⎧ ⎪ ⎨ 1, σi ( f oi (x))  1 − ⎪ ⎩ 0,

f oi (x)−b0i d0i

f 0i (x) ≤ b0i ; , b0i ≤ f 0i (x) ≤ b0i + d0i ; f 0i (x) ≥ b0i + d0i ;

(13.37)

346

13 Intuitionistic and Neutrosophic Geometric Programming Problem

Fig. 13.2 Truth membership function, indeterminacy membership function and falsity membership function for the objective functions f 0i (x).

Now the above NGP model (13.34) can be reduced to a crisp model by maximizing the degree of acceptance, degree of indeterminacy and also minimizing the degree of falsity of NGP objective functions. Hence we have (Fig. 13.2). Maximize μi ( f oi (x)) for i  1, 2, . . . , p Minimize ϑi ( f oi (x)) for i  1, 2, . . . , p Maximize σi ( f oi (x)) for i  1, 2, . . . , p Subject to, gr (x) ≤ br ; r  1, 2, . . . , l 0 ≤ μi ( f oi ) + ϑi ( f oi ) + σi ( f oi ) ≤ 3, ϑi ( f oi ) ≥ 0, μi ( f oi ) ≥ ϑi ( f oi ), μi ( f oi ) ≥ σi ( f oi ), for i  1, 2, . . . , p and x  (x1 , x2 , . . . , xm ) > 0

(13.38)

Lemma 13.4.1 The ranges of truth, indeterminacy and falsity membership function of neutrosophic goal geometric programming problem will satisfy if r0i > 2d0i and a0i > d0i , where a0i , r0i and d0i are acceptance tolerance, rejection tolerance and indeterminacy tolerance respectively of the NGP objective functions. Proof From Eqs. (3.5) and (13.38) we have: μi ( f oi ) ≥ σi ( f oi ) implies 1 −

f oi (x)−b0i a0i

≥1−

f oi (x)−b0i d0i

or, ( f oi (x) − b0i )



1 d0i

−

1 a0i



≥0

(i)

13.4 Neutrosophic Goal Geometric Programming Problem

347

In the above mentioned neutrosophic goal programming problem, we consider each objective functions f oi (x) satisfying target achievement value b0i and also from the relation: ϑi ( f oi ) ≥ 0 0i or, foi (x)−b ≥0 r0i or, ( f oi (x) − b0i ) ≥ 0 Thus the relation (i) is true if   1 1 ≥ 0 i.e. a0i > d0i − d0i a0i

(ii)

(iii)

Hence from relation (iii), we have in neutrosophic goal geometric programming problem, acceptance tolerance a0i should be greater than indeterminacy tolerance d0i . Again from the relation μi ( f oi ) ≥ ϑi ( f oi ) and μi ( f oi ) ≥ σi ( f oi ) we have, f oi (x) − b0i f oi (x) − b0i ≥ a0i r0i

(iv)

f oi (x) − b0i f oi (x) − b0i ≥1− a0i d0i

(v)

1− and 1−

Adding the above inequalities (iv) and (v), we get: 1−

  f oi (x) − b0i 1 ( f oi (x) − b0i ) 1 1 ≥ + − a0i 2 2 r0i d0i

(vi)

Using the relation μi ( f oi ) ≥ ϑi ( f oi ) ≥ 0 and μi ( f oi ) + ϑi ( f oi ) + σi ( f oi ) ≤ 3 we get, σi ( f oi ) ≤ 3 0i ≤3 or 1 − foi (x)−b d0i or f oi (x) − b0i ≥ −2d0i 1 or foi (x)−b ≤ − 2d10i 0i Hence from μi ( f oi ) + ϑi ( f oi ) + σi ( f oi ) ≤ 3 using (vi) and (vii):   f oi (x) − b0i 1 ( f oi (x) − b0i ) 1 1 f oi (x) − b0i + − +1− ≤3 + 2 2 r0i d0i r0i d0i

(vii)

348

13 Intuitionistic and Neutrosophic Geometric Programming Problem

gives r0i > 2d0i . Thus from the above relation it is clear that in neutrosophic goal geometric programming problem half of the rejection tolerance r0i should be greater than the indeterminacy tolerance d0i . Definition 13.2 (Pareto Optimal solution) x ∗ is said to be a pareto optimal solution to the neutrosophic goal geometric programming problem (13.34) if there does not exist another x such that μi ( f oi (x)) ≥ μi ( f oi (x ∗ )), ϑi ( f oi (x)) ≤ ϑi ( f oi (x ∗ )) and σi ( f oi (x)) ≥ σi ( f oi (x ∗ )) for all i  1, 2, . . . , p with strict inequality holds for at least one i. Theorem 13.3 x ∗ is a pareto optimal solution to NGGPP (13.34) iff x ∗ is a pareto optimal solution to fuzzy goal geometric programming problem (FGGPP) which is of the form   Minimize f 01 (x), f 02 (x), . . . , f 0 p (x) Subject to, gr (x) ≤ br , r  1, 2, . . . , l, x ∈ R m &x > 0.

(13.39)

Proof If x ∗ be a pareto optimal solution of the FGGPP (3.6) and (13.39) then there does not exist any x such that f oi (x) ≤ f oi (x ∗ ) for all i  1, 2,…, p and f oi (x ∗ )  f oi (x) for at least one i. Then we have for all x  (x1 , x2 , . . . , xm ),   f oi (x) ≤ f oi x ∗

(13.40)

with strict inequality hold for at least one i. i.e. f oi (x) − b0i ≤ f oi (x ∗ ) − b0i ∗ 0i or, foi (x)−b ≤ foi (xa0i)−b0i a0i ∗ 0i or, 1 − foi (x)−b ≥ 1 − foi (xa0i)−b0i implies μi ( f oi (x)) ≥ μi ( f oi (x ∗ )). a0i Similarly from (13.40) we have

f oi (x)−b0i r0i

≤

f oi (x ∗ )−b0i r0i

which implies

ϑi ( f oi (x)) ≤ ϑi ( f oi (x ∗ )) ∗ 0i ≤ foi (xd0i)−b0i and also foi (x)−b d0i ∗ 0i or, 1 − foi (x)−b ≥ 1 − foi (xd0i)−b0i d0i or, σi ( f oi (x)) ≥ σi ( f oi (x ∗ )). Hence from the definition of pareto optimal solution to the NGGPP, we have x ∗ is the pareto optimal solution of (13.34). Conversely, let x ∗ is a pareto optimal solution to NGGPP (13.34), then from 0i ≥ the expression of membership function given in (13.35) we get 1 − foi (x)−b a0i ∗

1 − foi (xa0i)−b0i i.e. f oi (x) ≤ f oi (x ∗ ).

13.4 Neutrosophic Goal Geometric Programming Problem

349 ∗

0i Again using (13.36) we have foi (x)−b ≤ foi (xd0i)−b0i which implies f oi (x) ≤ d0i ∗ f oi (x ). ∗ 0i ≥ 1 − foi (xd0i)−b0i gives f oi (x) ≤ f oi (x ∗ ). Similarly, using (13.37), 1 − foi (x)−b d0i Thus we have f oi (x) ≤ f oi (x ∗ ) with strict inequality hold for at least one i, i ∈ {1, 2, . . . , p} and which shows that x ∗ is a pareto optimal solution of (13.39). Now (13.38) is equivalent to:

Maximize α Minimize β Maximize γ Subject to, μi ( f oi (x)) ≥ α ϑi ( f oi (x)) ≥ β σi ( f oi (x)) ≥ γ , for i  1, 2, . . . , p gr (x) ≤ br ; r  1, 2, . . . , l 0 ≤ α + β + γ ≤ 3, α ≥ β, α ≥ γ α, β, γ ∈ [0, 3], and x  (x1 , x2 , . . . , xm ) > 0.

(13.41)

Since, minimize β is equivalent to maximize (1 – β), so by arithmetic mean method the above model (13.41) is reduces to the following form:   Maximize α+1−β+γ 3      f (x) 1 b α+1−β+γ 1 1 Subject to, oi3 ≤ 1 + 30i a1 + r 1 + d1 − , for i  1, 2, . . . , p. a + r + d 3 0i 0i 0i 1 br gr (x) ≤ 1, r  1, 2, . . . , l.

0i

0i

0i

(13.42)

0 ≤ α + β + γ ≤ 3, α ≥ β, α ≥ γ α, β, γ ∈ [0, 3], and x  (x1 , x2 , . . . , xm ) > 0.

Now to make (13.43) in minimization form, we have Minimize v−1 Subject to, 1 br

1 3



1 1 1 a +r +d  0i 0i 0i b 1+ 30i a1 + r 1 + d1 0i 0i 0i

  f oi (x) −v

≤ 1, for i  1, 2, . . . , p.

gr (x) ≤ 1, r  1, 2, . . . , l. x > 0. where v 

(13.43)

α+1−β+γ 3

Let N be the total number of terms in the above primal problem. Then the degree of difficulty (DD) of the single objective geometric programming problem is N − (m + 1). From (13.43) we construct the dual programming model as:

350

13 Intuitionistic and Neutrosophic Geometric Programming Problem

 Maximize

1 vδ00

 δ00  p  T0i

l 

i1 k1 Tr p 

r 1 q1+T(r −1)P



C0ik 3



  b 1+ 30i a1

Crq br δorq

0i

 δ0ik + r 1 + d1 0i 0i   + r 1 + d1 −v δ0ik

1 a0i

T0i δorq 

(

k1

0i

0i

(

T 0i 

δ0ik ) k1

δ0ik )

!

Tr p 

Tr p "q1+T

δ0rq

(r −1) p

δ0rq

q1+T(r −1) p

Subject to, δ00  1 T0i  δ0ik  1, k1

Tr p 

δ0rq  1, (for i  1, 2, . . . , p; r  1, 2, . . . , l)

q1+T(r −1) p p  T0i 

l 

i1 k1

r 1 q1+(r −1) p

α0ik j δ0ik +

Tr p 

αr k j δ0rq  0.for j  1, 2, . . . , m.

where δ0ik > 0(fork  1, 2, . . . , T0i ; i  1, 2, . . . , p)  δ0rq > 0 forq  1 + T(r −1) p , . . . , Tr p ; r  1, 2, . . . , l (13.44) Case I: for N > (m + 1), a solution vector exists for the dual variables. Case II: N < (m + 1), generally no solution vector exists for the dual variables but we can get the approximate solution for this system using different methods. Now to find the solution of the geometric programming model (13.43), firstly we have to find out the optimal solution of the dual problem (13.44). Hence from the primal-dual relationship we can easily obtain the corresponding values of the primal variable vector x. The LINGO-13.0 software is used here to find optimal dual variables from the equations of (13.44).

13.4.2 Mathematical Model According to Kundu and Islam (2018), consider a series system of reliability with n components. Let Ri (i  1, 2, . . . , n) be the reliability of i-th component of the framework (system). Likewise assume, R S (R1 , R2 , . . . , Rn ) and C S (R1 , R2 , . . . , Rn ) represents the framework (system) reliability and framework (system) cost of the ncomponent series framework (system) (Fig. 13.3). We know that, in a single objective reliability optimization problem either we need to locate the maximum (max problem) system reliability subject to the restricted accessible resources consumption or to find out the minimum (min problem) resources subject to the lower limit system reliability. In day-to-day life situations, it isn’t always possible to get the limit resource consumption rightly. In this

Fig. 13.3 Series system with n component

13.4 Neutrosophic Goal Geometric Programming Problem

351

manner to create a highly reliable framework (system), it is always better to think about the multiple objective approaches to design a system. Thus Kundu and Islam (2018), has developed the following model as: n 

Maximize R S (R1 , R2 , . . . , , R N ) 

Ri  R 1 R 2 . . . R n

i1 n 

Minimize C S (R1 , R2 , . . . , , R N ) 

i1

Ci Riai

(13.45)

Subject to, 0 < Ri ≤ 1; i  1, 2, . . . , n; Since we are interested to solve the above model using geometric programming approach, the model should be in minimization form. Thus, the suitable form of optimization model (13.45) is taken as Minimize R S (R1 , R2 , . . . , , R N )  Minimize C S (R1 , R2 , . . . , , R N ) 

n  i1 n  i1

Ri−1  R1−1 R2−1 . . . R −1 N Ci Riai

(13.46)

Subject to, 0 < Ri ≤ 1; i  1, 2, . . . , n; Where Ci (> 0) and ai (> 0) are shape parameters f or i  1, 2,…, n.

13.4.3 Neutrosophic Goal Geometric Programming Technique on Reliability Optimization Model To solve the above defined model (4.2) using NGGP technique, let Minimize R S (R1 , R2 , . . . , , R N ) 

n 

Ri−1  R1−1 R2−1 . . . R −1 N

i1

satisfying target achievement value R0 with acceptance tolerance a R , rejection tolerance r R and indeterminacy tolerance d R .  n Ci Riai satisfying target achieveAlso, Minimize C S (R1 , R2 , . . . , , R N )  i1 ment value C0 with acceptance tolerance aC , rejection tolerance rC and indeterminacy tolerance dC . Now, construct the truth membership function, falsity membership function and indeterminacy membership function as follows: ⎧ ⎪ R S (R1 , R2 , . . . , R N ) ≤ R0 ⎨ 1, R S −R0 μ R S (R)  1 − a , R0 ≤ R S (R1 , R2 , . . . , R N ) ≤ R0 + a R (13.47) R ⎪ ⎩ 0,

R S (R1 , R2 , . . . , R N ) ≥ R0 + a R ;

352

13 Intuitionistic and Neutrosophic Geometric Programming Problem

ϑ R S (R) 

⎧ ⎪ ⎨ 0,

⎪ ⎩

R S −R0 , rR

1,

R S (R1 , R2 , . . . , R N ) ≤ R0 ; R0 ≤ R S (R1 , R2 , . . . , R N ) ≤ R0 + r R ; R S (R1 , R2 , . . . , R N ) ≥ R0 + r R ;

⎧ ⎪ R S (R1 , R2 , . . . , R N ) ≤ R0 ; ⎨ 1,

R − R 0 S σ R S (R)  1 − d , R0 ≤ R S (R1 , R2 , . . . , R N ) ≤ R0 + d R ; R ⎪ ⎩ 0, R S (R1 , R2 , . . . , R N ) ≥ R0 + d R ; ⎧ ⎪ C S (R1 , R2 , . . . , R N ) ≤ C0 ; ⎨ 1, μC S (R)  1 − C Sa−C0 , C0 ≤ C S (R1 , R2 , . . . , R N ) ≤ C0 + aC C ⎪ ⎩ 0, C S (R1 , R2 , . . . , R N ) ≥ C0 + aC ⎧ ⎪ C S (R1 , R2 , . . . , R N ) ≤ C0 ; ⎨ 0, C S −C0 ϑC S (R)  , C0 ≤ C S (R1 , R2 , . . . , R N ) ≤ C0 + rC ; r ⎪ ⎩ C 1, C S (R1 , R2 , . . . , R N ) ≥ C0 + rC ; ⎧ ⎪ C S (R1 , R2 , . . . , R N ) ≤ C0 ; ⎨ 1, C S −C0 σC S (R)  1 − d , C0 ≤ C S (R1 , R2 , . . . , R N ) ≤ C0 + dC ; C ⎪ ⎩ 0, C S (R1 , R2 , . . . , R N ) ≥ C0 + dC ;

(13.48)

(13.49)

(13.50)

(13.51)

(13.52)

Now using (13.38), the above model (13.46) reduces to the following form: Maximize μ R S Maximize μC S Maximize σ R S Maximize σC S Maximize ϑ R S Maximize ϑC S Subject to, 0 ≤ μ R S + ϑ R S + σ R S ≤ 3, 0 ≤ μC S + σC S + ϑC S ≤ 3, ϑ R S ≥ 0, ϑC S ≥ 0 μ R S ≥ ϑ R S , μC S ≥ ϑC S , μ R S ≥ σ R S , μC S ≥ σC S , 0 < Ri ≤ 1; i  1, 2, . . . , n;

(13.53)

The above model (13.53) is equivalent to Maximize α, Minimize β, Maximize γ Subject to, μ R S ≥ α, μC S ≥ α, ϑ R S ≤ β, ϑC S ≤ β, σ R S ≥ γ , σC S ≥ γ , 0 ≤ α + β + γ ≤ 3, α ≥ β, ≥ γ , 0 ≤ α, β, γ ≤ 1

(13.54)

13.4 Neutrosophic Goal Geometric Programming Problem

353

Using arithmetic mean method (13.54) becomes: Minimize v−1 1 3





1 1 1 a +r +d R R R  R 1+ 30 a1 + r1 + d1 −v R R R  1 1 1 1 n 3 aC + rC + dC   C0 1 1 1 1+ 3 a + r + d −v i1 C C C

Subject to,



n  i1

Ri−1 ≤ 1,

Ci Riai

(13.55)

≤ 1,

where we take v  α+1−β+γ > 0 as a parameter. The degree of difficulty (D.D) of 3 (13.55) is (n + 2) − (n + 1)  1(> 0). Now using (13.44), the above model (13.55) can be solved by geometric programming technique after finding its dual.

13.4.4 Numerical Example Here we consider the three component series system reliability optimization model for the numerical exposure. Thus the model (13.45) becomes (Table 13.3): Maximize R S  Minimize C S 

3 

Ri  R 1 R 2 R 3

i1 3  i1

(13.56)

Ci Riai

Subject to, 0 < Ri ≤ 1; i  1, 2, 3; The above table portrays the compare of results of objective functions for the dual problem and in addition the primal problem of the given neutrosophic objective geometric programming problem with the IFGGPP approach. It is obvious from Table 13.4 that NGGPP approach gives preferable result over the IFGGPP approach in context of system reliability. Be that as it may, in perspective of system cost the proposed approach gives somewhat higher value than the IFGGPP approach.

Table 13.3 The input data for the neutrosophic goal geometric programming problem (13.56) is given as follows C1

C2

C3

C0

aC

rC

dC

aR

rR

dR

R0

40

40

45

100

24

40

18

0.3

0.5

0.24

0.3

0.282631

0.280369

NGGPP

0.361125

0.426250

0.358506

0.291118 0.903778

0.676680

δ04 0.824809

0.771292 0.863345

0.875617

R2

R1

δ03

δ01

δ02

Primal variables

Dual variables

IFGGPP

Method

0.861773

0.775105

R3

1.95

1.80

v

0.613664

0.523472

R S∗

Objective functions

Table 13.4 Comparison of optimal solutions of NGGPP Eq. (13.56) and intuitionistic fuzzy goal geometric programming problem (IFGGPP)

80.372

78.766

C S∗

354 13 Intuitionistic and Neutrosophic Geometric Programming Problem

13.5 Conclusion

355

13.5 Conclusion In this chapter we have talked about intuitionistic fuzzy and neutrosophic fuzzy geometric programming problem respectively with positive or negative integral degree of difficulty. Three different type of GP techniques (Intuitionistic Fuzzy Posynomial Geometric Programming Problem, Intuitionistic Fuzzy Goal Programming Model, Neutrosophic Goal Geometric Programming Problem and Its Application) are discussed here. The new idea of the posynomial geometric programming (PGP) approach in an intuitionistic fuzzy number (IFN) environment, open a major researchable subject of optimization. Likewise the neutrosophic goal geometric programming problem (NGGPP) approach can be used to solve the multi-objective nonlinear programming problem and gives preferable outcome than the other technique (approach). This method can be connected to solve the different decision making optimization problems (like in reliability engineering, single-objective optimization, multi-objective optimization, inventory management and control and many other areas).

References K. Atanassov, Intuitionistic fuzzy sets. Fuzzy sets syst. 20, 87–96 (1986) P. Ghosh and T.K. Roy, Fuzzy goal geometric programming method in intuitionistic environment, Notes on Intuitionistic Fuzzy Sets 20(1), 63–78 (2014). ISSN 1310–4926 Z. Kheiri and B. Cao, Posynomial geometric programming with intuitionistic fuzzy coefficients. Fuzzy Systems & Operations Research and Management, Advances in Intelligent Systems and Computing 367 (2016) T. Kundu, and S. Islam, Int. J. Fuzzy Syst. (2018). https://doi.org/10.1007/s40815-018-0479-2 M. Sakawa, I. Nishizaki and H. Katagiri, Fuzzy Stochastic Multiobjective Programming, (Springer, New York, 2011). https://doi.org/10.1007/978-1-4419-8402-9 L.A. Zadeh, Fuzzy sets. Inf. Control 8, 338–353 (1965)

Index

A a-cut, 55 a-level, 55 B Basic fuzzy set theory, 50 Bell shaped fuzzy number, 76 Boundary, 55 C Cardinality, 57 Cartesian Product, 71 Chebyshev goal programming, 236 Complement, 59 Concentration, 65 Constrained fuzzy signomial GP, 210 Constrained geometric programming, 19 Constrained GP under max-min operator, 168 Constrained MGP problem, 186 Constrained MGP under max-min operator, 200 Constrained modified fuzzy signomial GP, 227 Constrained Modified Geometric Programming (CMGP), 21 Constrained NLP problem, 37 Containment, 58 Contrast intensification, 66 Convex combination, 102 Convex fuzzy sets, 54 Convexity of signomial functions, 28 Core, 55 Crisp Set, 51 Crossover, 53

D Decision-making in fuzzy environment, 100 Degree of containment, 59 Degree of equality, 58 Dilation, 66 Disjoint Sets, 61 Distance between two fuzzy sets, 61 Division, 78 Dual Problem, 5 E Empty fuzzy set, 53 Equality, 58 Extension principle and its application, 70 Extension principle on n-dimensional universe, 71 F First resolution theorem, 68 Fuzzification, 66 Fuzzy Goal Programming (FGP), 248 Fuzzy Goal Programming problem with logarithmic deviational variables, 251 Fuzzy NonLinear Programming (FNLP), 259 Fuzzy number, 75 Fuzzy optimization, 104 Fuzzy parametric geometric programming, 152 Fuzzy parametric modified geometric programming, 197 Fuzzy set, 50 G Gaussian fuzzy number, 88 Generalized Fuzzy Number (GFN), 91

© Springer Nature Singapore Pte Ltd. 2019 S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-13-5823-4

357

358 Generalized Trapezoidal Fuzzy Number (GTrFN), 91 Geometrical interpretation of fuzzy sets, 63 Geometric Programming (GP), 1 Goal programming with logarithmic deviational variables, 242 GP problem with fuzzy, 134 GP problem with Zimmermann max-min operators, 134 Grain-box problem, 11 Gravel-box problem, 270

Index Multi-Objective Goal Geometric Programming (MOGP), 237 Multi-Objective Weighted Goal Programming (MOWGP), 239 N Nearest interval approximation, 93 Neutrosophic multi-objective goal geometric programming, 344 Neutrosophic set, 121 Newton–Raphson method, 99 Normal, 54

H Hedges, 65 Height, 53 Hesitant Fuzzy Set (HFS), 122

O Operations on fuzzy sets, 58

I Integral value, 92 Intersection, 60 Interval number, 93 Intuitionistic fuzzy goal programming, 338 Intuitionistic fuzzy numbers, 117 Intuitionistic fuzzy posynomial geometric programming, 331 Inventory models, 279

P Parabolic flat fuzzy Number, 201 Parabolic Fuzzy Number (PrFN), 201 Parametric interval-valued function, 95 Pareto-optimal solution, 263 Pentagonal fuzzy number, 76, 81 Polynomial, 3 Posynomial, 3 Pricing model, 108 Primal-modified signomial GP, 40 Pythagorean fuzzy set, 126

J Job hiring policy, 107 Job selection strategy, 107 L Linear, 111 Local and global optima, 321 LS & MM Method, 8 M MGP problem with fuzzy, 156 MGP problem with Zimmermann max-min operators, 156 Min-sum goal, 234 M-N Pareto optimal solution, 339 Modified Geometric Programming (MGP), 12 Modified geometric programming problem, 156 Monomial, 1 M-Pareto optimal solutions, 278 Multi-Grain-box problem, 16 Multi-Gravel-box problem, 275 Multiobjective geometric programming problem using fuzzy technique, 263

R Reliability optimization, 331 S Scalar product, 68 Second resolution theorem,, 69 Sigmoidal fuzzy number, 90 Single-objective mathematical programming, 98 Support, 53 T Third resolution theorem, 69 Trapezoidal fuzzy number, 79 Triangular Fuzzy Number (TFN), 76 Two-bar truss, 287 Type-2 fuzzy set, 128 U Uncertain Chance-Constrained Geometric Programming (UCCGP), 287 Uncertainty and Imprecision, 48 Uncertainty theory, 114

Index Unconstrained fuzzy signomial GP, 210 Unconstrained modified fuzzy signomial GP, 216 Unconstrained NLP problem, 30 Union, 59 Universal fuzzy set, 52

359 W Weighted Goal Programming (WGP), 233 Z Zigzag, 116 Zimmerman approach, 111