This book presents the necessary and essential backgrounds of fuzzy set theory and linear programming, particularly a br

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*English*
*Pages XVIII, 232
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*Year 2019*

- Author / Uploaded
- Seyed Hadi Nasseri
- Ali Ebrahimnejad
- Bing-Yuan Cao

*Table of contents : Front Matter ....Pages i-xviii Preliminaries and Backgrounds (Seyed Hadi Nasseri, Ali Ebrahimnejad, Bing-Yuan Cao)....Pages 1-37 Fuzzy Linear Programming (Seyed Hadi Nasseri, Ali Ebrahimnejad, Bing-Yuan Cao)....Pages 39-61 Fuzzy Number Linear Programming (Seyed Hadi Nasseri, Ali Ebrahimnejad, Bing-Yuan Cao)....Pages 63-114 Linear Programming with Fuzzy Variables (Seyed Hadi Nasseri, Ali Ebrahimnejad, Bing-Yuan Cao)....Pages 115-176 Semi-fully Fuzzy Linear Programming (Seyed Hadi Nasseri, Ali Ebrahimnejad, Bing-Yuan Cao)....Pages 177-222 Application for the Flexible Linear Programming (Seyed Hadi Nasseri, Ali Ebrahimnejad, Bing-Yuan Cao)....Pages 223-232*

Studies in Fuzziness and Soft Computing

Seyed Hadi Nasseri Ali Ebrahimnejad Bing-Yuan Cao

Fuzzy Linear Programming: Solution Techniques and Applications

Studies in Fuzziness and Soft Computing Volume 379

Series Editor Janusz Kacprzyk, Polish Academy of Sciences, Systems Research Institute Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related ﬁelds. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover signiﬁcant recent developments in the ﬁeld, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Contact the series editor by e-mail: [email protected] Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/2941

Seyed Hadi Nasseri Ali Ebrahimnejad Bing-Yuan Cao •

•

Fuzzy Linear Programming: Solution Techniques and Applications

123

Seyed Hadi Nasseri Department of Mathematics University of Mazandaran Babolsar, Iran

Ali Ebrahimnejad Department of Mathematics Qaemshahr Branch, Islamic Azad University Qaemshahr, Iran

Bing-Yuan Cao Foshan University Foshan, Guangdong, China

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-030-17419-4 ISBN 978-3-030-17421-7 (eBook) https://doi.org/10.1007/978-3-030-17421-7 © Springer Nature Switzerland AG 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, expressed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To: Roya, Zahra & Nazi The second author: would like to present this book to his wife Zeynab and his Son Arvin

Preface

Fuzzy set theory has been applied to many disciplines such as control theory and management sciences, mathematical modeling and industrial applications. We usually face some difﬁculties when such real-world problems are formulated into a mathematical programming problem. One of the difﬁculties is caused by the uncertainty in knowledge, information and/or decision maker’s preferences. If the model’s coefﬁcients are determined as real numbers appropriately, the formulated problem has no feasible solution or yields improper solution. In these cases, we have to investigate the causes, change the determined real numbers, solve the reformulated problem and repeat this procedure until we obtain the satisfactory solution. Thus, this requires a formidable effort and plenty of time. In order to reflect this uncertainty, the model of the problem is often constructed with fuzzy data. On the other hand, linear programming is one of the most widely used decision-making tools for solving real-world problems. One of the main assumptions used in this technique is that the input data have complete accuracy. However, more often than not, real-world situations are characterized by imprecision (fuzziness) rather than exactness. Thus, the fuzzy programming approach is useful and efﬁcient to treat a programming problem under uncertainty. While classical and stochastic programming approach may require a lot of costs to obtain the exact coefﬁcient value or distribution, the fuzzy programming approach does not. From this fact, the fuzzy programming approach will be very advantageous when the coefﬁcients are not known exactly but vaguely speciﬁed by human expertise. Since the fuzziness may appear in a linear programming problem in many ways, the deﬁnition of fuzzy linear programming problem is not unique. Fuzzy linear programming models which appeared in the literature have been classiﬁed into the following seven groups: • Group 1: The FLP problems in this group involve fuzzy numbers for the coefﬁcients of the decision variables in the objective function (e.g., [1, 2, 3]). • Group 2: The FLP problems in this group involve fuzzy numbers for the coefﬁcients of the decision variables in the constraints and the right-hand side of the constraints (e.g., [4]).

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• Group 3: The FLP problems in this group involve fuzzy numbers for the coefﬁcients of the decision variables in the objective function, the coefﬁcients of the decision variables in the constraints and the right-hand side of the constraints (e.g., [5, 6, 7]). • Group 4: The FLP problems in this group involve fuzzy numbers for the decision variables and the right-hand side of the constraints (e.g., [1]). • Group 5: The FLP problems in this group involve fuzzy numbers for the decision variables, the coefﬁcients of the decision variables in the objective function and the right-hand side of the constraints (e.g., [8, 9]). • Group 6: The FLP problems in this group involve fuzzy numbers in the decision variables, the coefﬁcients of the decision variables in the constraints and the right-hand side of the constraints (e.g., [10]). • Group 7: The FLP problems in this group, so-called fully fuzzy linear programming (FFLP) problems, involve fuzzy numbers in the decision variables, the coefﬁcients of the decision variables in the objective function, the coefﬁcients of the decision variables in the constraints and the right-hand side of the constraints (e.g., [11]). Although the FFLP group is the general case of the FLP, it may not be suitable for all FLP problems with different assumptions and sources of fuzziness. On the other hand, for the problems in Group 6 similar to the problems in Group 7 there are not suitable approaches which are accepted by a wide range of the researchers, so we omit them in our discussion and hope to add the new version of the solving approaches which will be appeared in the literature in the next version of this book. In general, fuzzy linear programming problems are ﬁrst converted into equivalent crisp linear or nonlinear programs, which are then solved by standard methods. In effect, the most convenient methods are based on the concept of comparison of fuzzy numbers by use of ranking functions which is desired in this book. Instead of discussing the general type, we focus on the issues involved by three common classes of fuzzy linear programming. Therefore, the class of fuzzy linear programming problems can be broadly classiﬁed as: (i) Fuzzy linear programming problems in which all coefﬁcients are a kind of fuzzy numbers. (ii) Fuzzy linear programming problems in which the right-hand side vectors and the decision variables are a kind of fuzzy numbers. (iii) Fuzzy linear programming problems in which the cost coefﬁcients, the right-hand side vectors and the decision variables are a kind of fuzzy numbers. In this book, we generalize the well-known solution algorithms such as primal algorithm, dual algorithm and primal–dual algorithm in linear programming for fuzzy linear programming problems. Similar results for a different concept of the optimal solution of fuzzy linear programming based on ranking functions are obtained. Most of the theoretical results and associated algorithms are explored through some applications which are formulated in the form of a kind of fuzzy linear programming problems and illustrative examples.

Preface

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This book is organized in six chapters. Since the main scope of this book is established based on the concepts of fuzzy set theory, hence Chap. 1 represents some essential deﬁnitions and operation properties in fuzzy sets, such as acut sets and convex fuzzy sets. Besides, based on the elaborative fuzzy relation, it introduces a fuzzy operator related to this book, and it exhibits fuzzy functions. At the same time, it describes fuzzy mathematics of three mainstream theorems—expansion principle, decomposition theorem and representation theorem. Since ranking of fuzzy numbers is an important aspect in the study of fuzzy mathematical programming, we shall continue our discussion on this topic in continuation of this chapter. In the last part of this chapter, some applications of linear programming problems such as data envelopment analysis, network optimization and diet planning are dealt with in this chapter that will be useful for description as some real case problems which are modeled as a fuzzy mathematical problem. Chapter 2 is assigned to a brief review of some convenient categories of fuzzy mathematical programming which appeared in the literature. We also emphasize that this chapter will be useful for understanding the introduced models and their associated solving methods in the upcoming chapters. Chapter 3 is assigned to the linear programming problem with fuzzy coefﬁcients as one of the convenient models of fuzzy linear programs. In this way, we ﬁrst introduce a general form of fuzzy number linear programming (FNLP) problems and then give the fundamental concepts which are useful during this chapter. Then, some other standard methods and results such as primal, dual and primal–dual simplex algorithms and duality results will be discussed. In particular, some applications of FNLP problems in the real world such as data envelopment analysis (DEA), network flows (NFs) and diet problem (DP) are given. Finally, the last part of this chapter is allocated to the sensitivity analysis. In the real world environment, there are many problems which are concern to the linear programming models and sometimes it is necessary to formulate these models with parameters of uncertainty. Many numbers from these problems are linear programming problems with fuzzy variables. Chapter 4 is allocated to these kinds of problems. Recently, some authors considered linear programming problems with trapezoidal fuzzy data and/or variables and stated a fuzzy simplex algorithm to solve these problems. In particular, they developed the duality results in the fuzzy environment and presented a dual simplex algorithm for solving linear programming problems with trapezoidal fuzzy variables. It is seen that the presented dual simplex algorithm directly using the primal simplex tableau algorithm tenders the capability for sensitivity (or post-optimality) analysis using primal simplex tableaus. In Chap. 5, we introduce a kind of linear programming problems where the coefﬁcients in the objective function, the right-hand side vector and the decision variables are a type of fuzzy numbers, simultaneously. We name such problems as semi-fully fuzzy linear programming (SFFLP) problems and review fuzzy analogues of some important results and theorems of linear programming proposed by them. In this way, we ﬁrst focus on the symmetric form of the trapezoidal fuzzy

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numbers and then give the fundamental deﬁnitions, concepts and results concerning arithmetic and in particular fuzzy ordering. Finally, one of the most interesting models of the linear programming is the flexible linear programming in uncertainty environment problem. In Chap. 6, we introduce a kind of these problems. It is shown that by using a suitable membership function for their constraints we can obtain an equivalent linear programming problem with fuzzy variables (FVLP). Some methods have been developed for solving these problems by introducing and solving a certain auxiliary problem. In particular, we suggest the fuzzy primal simplex method to solve the flexible linear programming problems directly without solving any auxiliary problem. This method will be especially useful for sensitivity analysis using the primal simplex tableaus also. Thereafter, we shall illustrate this approach with some illustrative examples. Babolsar, Iran Qaemshahr, Iran Foshan, China

Seyed Hadi Nasseri Ali Ebrahimnejad Bing-Yuan Cao

References 1. Mahdavi–Amiri, N., Nasseri, S.H.: Duality results and a dual simplex method for linear programming with trapezoidal fuzzy variables. Fuzzy Sets Syst. 158, 1961–1978 (2007). 2. Maleki, H.R., Tata, M., Mashinchi, M: Linear programming with fuzzy variable. Fuzzy Sets Syst. 109, 21–33 (2000). 3. Wu, H.C.: Optimality conditions for linear programming problems with fuzzy coefﬁcients. Comput. Math. Appl. 55(12), 2807–2822 (2008). 4. Xinwang, L.: Measuring the satisfaction of constraints in fuzzy linear programming. Fuzzy Sets Syst. 122(2), 263–275 (2001). 5. Hatami–Marbini, A., Agrell, P., Tavana, M., Emrouznejad, A.: A stepwise fuzzy linear programming model with possibility and necessity relation. J. Intell. Fuzzy Syst. 25(1), 81–93 (2013). 6. Hatami–Marbini, A., Tavana, M.: An extension of the linear programming method with fuzzy parameters. Int. J. Math. Oper. Res. 3(1), 44–55 (2011). 7. Mahdavi–Amiri, N., Nasseri, S.H.: Duality in fuzzy number linear programming by use of a certain linear ranking function. Appl. Math. Comput. 180(1), 206–216 (2006). 8. Ganesan, K., Veeramani, P.: Fuzzy linear programs with trapezoidal fuzzy numbers. Ann. Oper. Res. 143(1), 305–315 (2006). 9. Nasseri, S.H., Mahdavi–Amiri, N.: Some duality results on linear programming problems with symmetric fuzzy numbers. Fuzzy Inf. Eng. 1, 59–66 (2009). 10. Saati, S., Hatami–Marbini, A., Tavana, M., Hajiahkondi, E.: A two–fold linear programming model with fuzzy data. Int. J. Fuzzy Syst. Appl. 2(3), 1–12 (2012). 11. Hosseinzadeh Lotﬁ, F., Allahviranloo, T., Alimardani Jondabeh, M., Alizadeh, L.: Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Appl. Math. Model. 33(7), 3151–3156 (2009).

Acknowledgements

Special thanks to: Salim Bavandi for his assistance to edit this book.

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About This Book

This book is established based on the concept of fuzzy set theory on linear programming models and solution methods and contains six chapters as follows: Preliminaries and Backgrounds, Fuzzy Linear Programming, Fuzzy Number Linear Programming, Linear Programming with Fuzzy Variables, Semi-Fully Fuzzy Linear Programming and Application for the Flexible Linear Programming. It can not only be used as teaching materials or reference books for undergraduates in higher education, master graduates and Ph.D. graduates in the courses of applied mathematics, computer science, management, decision science, fuzzy information process, operations research, system science and engineering, and the like, but also serve as a reference book for the researchers in these ﬁelds, particularly, for the researchers in soft science.

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1 Preliminaries and Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Operations in Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 aCuts and Convex Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 1.4.1 a–Cut Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Convex Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Three Mainstream Theorems in Fuzzy Mathematics . . . . . . . 1.5.1 Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . 1.5.2 Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Representation Theorem . . . . . . . . . . . . . . . . . . . . . 1.6 Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Deﬁnition and Properties of Interval Fuzzy Numbers 1.6.2 Type ð; cÞ; T; L R and Flat Fuzzy Numbers . . . . . 1.6.3 Trapezoidal and Triangular Fuzzy Number . . . . . . . 1.6.4 Order on Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . 1.7 Data Envelopment Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Basic DEA Models . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Cost Efﬁciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Network Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Minimum Cost Flow Problem . . . . . . . . . . . . . . . . . 1.8.2 Maximum Flow Problem . . . . . . . . . . . . . . . . . . . . 1.9 Diet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Deﬁnition of Problem . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Fuzzy Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Primal Simplex Algorithm . . . . . . . . . . . . . . . . . . 2.2.4 Starting Solution for the Primal Simplex Algorithm 2.2.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fuzzy Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Fuzzy Number Linear Programming . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Formulation of a Fuzzy Number Linear Programming . . . . 3.3 A Fuzzy Primal Simplex Method for FNLP Problem . . . . . 3.4 Duality in Fuzzy Number Linear Programming . . . . . . . . . 3.4.1 Formulation of the Dual Problem . . . . . . . . . . . . . 3.4.2 The Relationships Between FNLP and Dual FNLP 3.4.3 Complementary Slackness Property in FNLP . . . . . 3.5 A Fuzzy Dual Simplex Method for FNLP Problem . . . . . . . 3.5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . 3.6 Bounded Fuzzy Number Linear Programming . . . . . . . . . . 3.7 Applications of Fuzzy Number Linear Programming . . . . . . 3.7.1 Application in Input-Oriented CCR Model . . . . . . . 3.7.2 Application in Fuzzy Cost Efﬁciency . . . . . . . . . . . 3.7.3 Application in MCFP with Fuzzy Costs . . . . . . . . 3.8 Animal Diet Problem as an FNLP Model . . . . . . . . . . . . . . 3.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Animal Diet Formulation with Floating Price . . . . . 3.9 Sensitivity Analysis in FNLP Problems . . . . . . . . . . . . . . . 3.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2 Change in the Fuzzy Right-Hand-Side Vector . . . . 3.9.3 Change in the Constraint Matrix . . . . . . . . . . . . . . 3.9.4 Adding a New Constraint . . . . . . . . . . . . . . . . . . . 3.9.5 Change in Fuzzy Cost Vector . . . . . . . . . . . . . . . . 3.9.6 Adding a New Activity . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Linear Programming with Fuzzy Variables . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Formulation of a Linear Programming with Fuzzy Variables Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Basic Feasible Solution and Improving the Solution . . . . . . .

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Fuzzy Primal Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Main Steps of FVLP Simplex Algorithm . . . . . 4.4.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Duality in Linear Programming with Fuzzy Variables . . . . . . 4.5.1 Formulation of the Dual Problem . . . . . . . . . . . . . . 4.5.2 The Relationships Between FVLP and Dual FVLP . 4.6 A Fuzzy Dual Simplex Method for FVLP Problem . . . . . . . . 4.6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Fuzzy Primal-Dual Method for FVLP Problem . . . . . . . . . . . 4.7.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Tableau Form of the Primal-Dual Method for FVLP Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Fuzzy Primal-Dual Method for FNLP Problem . . . . . . . . . . . 4.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Tableau Form of the Primal-Dual Method for FNLP 4.9 Bounded Linear Programming with Fuzzy Variables . . . . . . . 4.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Basic Feasible Solutions and Optimality Conditions . 4.9.3 Improving a Fuzzy Basic Feasible Solution . . . . . . . 4.9.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Sensitivity Analysis in FVLP Problems . . . . . . . . . . . . . . . . 4.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.2 Change in the Fuzzy Right-Hand-Side Vector . . . . . 4.10.3 Change in the Constraint Matrix A . . . . . . . . . . . . . 4.10.4 Adding a New Constraint . . . . . . . . . . . . . . . . . . . . 4.10.5 Change in the Cost Vector . . . . . . . . . . . . . . . . . . . 4.10.6 Adding a New Activity . . . . . . . . . . . . . . . . . . . . . 4.11 Application of Bounded FVLP in Maximum Flow Problem . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Semi-fully Fuzzy Linear Programming . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Formulation of a Semi-fully Fuzzy Linear Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fuzzy Arithmetic and Ranking . . . . . . . . . . . . . . . . . . . 5.4 Fuzzy Primal Simplex Method for SFFLP Problem . . . .

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5.5

Duality in Semi-fully Fuzzy Linear Programming . . . . . . . . . . 5.5.1 Formulation of the Dual Problem for SFFLP . . . . . . . 5.5.2 The Relationships Between SFFLP and Dual SFFLP . 5.6 Fuzzy Dual Simplex Method for SFFLP Problem . . . . . . . . . . 5.6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Dual Feasibility and a Fuzzy Dual Algorithm . . . . . . 5.6.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Fuzzy Primal-Dual Method for SFFLP Problem . . . . . . . . . . . 5.7.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Tableau Form of the Primal-Dual Method for SFFLP . 5.8 Bounded Semi-fully Fuzzy Linear Programming . . . . . . . . . . . 5.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Formulation of Bounded SFFLP . . . . . . . . . . . . . . . . 5.8.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Sensitivity Analysis in SFFLP Problems . . . . . . . . . . . . . . . . 5.9.1 Change in the Fuzzy Right-Hand-Side Vector . . . . . . 5.9.2 Change in the Constraint Matrix . . . . . . . . . . . . . . . . 5.9.3 Adding a New Fuzzy Constraint . . . . . . . . . . . . . . . . 5.9.4 Adding a New Activity . . . . . . . . . . . . . . . . . . . . . . 5.9.5 Change in Fuzzy Cost Vector . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Application for the Flexible Linear Programming 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Flexible Linear Programming . . . . . . . . . . . . 6.2.1 Deﬁnition of NFLP Problem . . . . . . . 6.2.2 Numerical Example . . . . . . . . . . . . . 6.3 Application in Minimum Cost Flow Problem . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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182 182 183 186 186 186 189 192 192 196 198 201 201 202 209 212 215 216 217 218 219 222

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223 223 223 223 225 226 231

Chapter 1

Preliminaries and Backgrounds

1.1 Introduction Fuzzy sets theory has been applied to many disciplines such as control theory and management sciences, mathematical modeling, industrial applications and etc. We usually face some difficulties when a such real-world problems are formulated into a mathematical programming problem. One of the difficulties is caused by the uncertainty in knowledge, information and/or decision makers preferences. If the model’s coefficients are determined as real numbers appropriately, the formulated problem has no feasible solution or yields improper solution. In these cases, we have to investigate the causes, change the determined real numbers, solve the reformulated problem and repeat this procedure until we obtain the satisfactory solution. Thus, this requires a formidable effort and plenty of time. In order to reflect this uncertainty, the model of the problem is often constructed with fuzzy data. Thus, fuzzy programming approach is useful and efficient to treat a programming problem under uncertainty. While classical and stochastic programming approach may require a lot of cost to obtain the exact coefficient value or distribution, fuzzy programming approach does not. From this fact, fuzzy programming approach will be very advantageous when the coefficients are not known exactly but vaguely specified by human expertise. In this book we generalize the well-known solution algorithms such as bounded primal algorithm, bounded dual algorithm and primal-dual algorithm in linear programming for fuzzy linear programming problems. Similar results for different concept of optimal solution of fuzzy linear programming based on ranking functions, are obtained. Most of the theoretical results and associated algorithms are illustrated through small application and numerical examples. Since the main scope of this book is established based on the concepts of fuzzy sets theory, hence the next sections in this chapter represent definitions and operation properties in fuzzy sets, such as α−cut sets and convex fuzzy sets and etc. Besides, based on the elaborative fuzzy relation, it introduces a fuzzy operator related to this book, and it exhibits fuzzy functions. At the same time, describes fuzzy mathematics © Springer Nature Switzerland AG 2019 S. H. Nasseri et al., Fuzzy Linear Programming: Solution Techniques and Applications, Studies in Fuzziness and Soft Computing 379, https://doi.org/10.1007/978-3-030-17421-7_1

1

2

1 Preliminaries and Backgrounds

of three mainstream theorems—expansion principle, decomposition theorem and representation theorem. Finally, some applications of linear programming problems such as data envelopment analysis, network optimization, diet planning, and etc. are dealt with in this chapter that will be useful for descriptionof some real case problems which are modeled as a fuzzy mathematical problem.

1.2 Fuzzy Sets The purpose of this section is to review the basic definitions and results on fuzzy sets and related topics. Moreover, we present a very basic but brief discussion on fuzzy numbers and fuzzy arithmetic. Since ranking of fuzzy numbers is an important aspect in the study of fuzzy mathematical programming, we shall continue our discussion on this topic in later chapters as well. We give its mathematics description as follows. Definition 1.1 A so-called fuzzy subset A˜ in set universe X is a set ˜ A˜ = {(x, A(x))|x ∈ X }, ˜ where A(x) is a real number in interval [0, 1], called a membership degree from ˜ This function is defined in the interval [0, 1], A(·) ˜ point x to A. or ˜ :X −→ [0, 1], A(·) ˜ x → A(x) ˜ called a membership function in fuzzy sets A. Under the condition of misunderstanding unlikely, the fuzzy set A˜ with its mem˜ bership function A(x) fails to add the distinction. At the same time, fuzzy subsets are also often called fuzzy sets. From Definition 1.1 of the fuzzy sets, there exist few next conclusions, obviously: (1) The concept of fuzzy sets is an expansion concept of classical sets. If F(X ) means all fuzzy sets on X , i.e., ˜ A˜ is a fuzzy set on X }, F(X ) = { A| Then P(X ) ⊂ F(X ), where P(X ) is the power sets on X , i.e., P(X ) = {A|A is a classic set on X }, that is, if the membership function of fuzzy set A˜ takes only 0 and 1, two values, then A˜ is exuviated into the classic sets of X . (2) The concept of the membership function is the expansion of the characteristic function concept.

1.2 Fuzzy Sets

3

When A ∈ P(X ) is an ordinary subset in X , the characteristic function of A is 1, x ∈ A (membership degree of x for A is 1), χ A (x) = 0, x ∈ / A (membership degree of x for A is 0). ˜ This means in fuzzy sets, the nearer the membership degree A(x) in fuzzy set A˜ is to ˜ ˜ 1, the bigger x belonging to A degree is; whereas, the nearer A(x) is to 0, the smaller ˜ x belonging to A˜ degree is. If the value region of A(x) is {0, 1}, then fuzzy set A˜ is ˜ an ordinary set A, but membership function A(x) is a characteristic function χ A (x). (3) We call fuzzy sets in F(X )P(X ) true fuzzy sets. Several representation methods to fuzzy sets shows as follows. (a) A representation method to fuzzy sets by Zadeh If set X is a finite set, let university X = {x1 , x2 , . . . , xn }, the fuzzy set is n ˜ 2) ˜ n) ˜ i) ˜ 1) A(x A(x A(x A(x + + ··· + = , A˜ = x1 x2 xn xi i=1

˜ i )/xi is not a fraction; it here symbol “Σ” is no longer a numerical sum, A(x only has the sign meaning, that is, only membership degree of the point xi with ˜ i ). respect to fuzzy set A˜ is A(x If X is an infinite set, a fuzzy set on X is A˜ =

x∈X

˜ A(x) . x

Similarly, the sign “ ” is not an integral any more, only means an infinite logic ˜ sum. But the meaning of A(x)/x is in accordance with the finite case. (b) When the university X is a finite set, the fuzzy sets represented in Definition 1.1 is ˜ 1 )), (x2 , A(x ˜ 2 )), . . . , (xn , A(x ˜ n ))}. A˜ = {(x1 , A(x (c) When the universe X is a finite set, the fuzzy sets represented according to a vector form is ˜ 2 ), . . . , A(x ˜ n )). ˜ 1 ), A(x A˜ = ( A(x Remarkably, X and φ also can be seen as fuzzy set in X , if membership functions ˜ ˜ A(x) ≡ 1 and A(x) ≡ 0, then A˜ is a complete set X and an empty set φ, respectively. An element that the membership degree is 1 definitely belongs to this fuzzy set; an element that the membership degree is 0 does not belong to this fuzzy sets definitely. But the membership function value in (0, 1) forms a distinct boundary, also calling distinct subsets of fuzzy sets. When a fuzzy object is described by using the fuzzy set, choice of its membership function is a key. Now we give three membership functions basically:

4

1 Preliminaries and Backgrounds

1. Partial mini type ˜ A(x) =

[1 + (a(x − c))b ]−1 , when x c, 1, when x < c.

where c ∈ X is an arbitrary point, a > 0, b > 0 are two parameters. 2. Partial large-scale 0, when x c, ˜ −1 A(x) = 1 + (a(x − c))−b , when x > c, where x ∈ X is an arbitrary point, a > 0, b > 0 are two parameters. 3. Normal type x − a 2 − ˜ b , A(x) = e where a ∈ X is an arbitrary value, b > 0 is a parameter. Obviously, Type 1 and 2 is dual, and its meaning shows clear at a glance; Type 3 is ˜ which is “ sufficiently near to number set of a”, then this membership a fuzzy set A, function in A˜ is defined on a center type according to the definition. Example 1.1 Let X ⊆ R+ (non-negative real number sets). Regard age as university and take X =[0,100]. Zadeh gave “oldness” O˜ and “ youth” Y˜ , these two membership functions respectively are ⎧ ⎪ ⎪ ⎪ ⎨

˜ O(x) =

⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎨

Y˜ (x) =

⎪ ⎪ ⎪ ⎩

1+

1+

0, x − 50 5 1, 1,

x − 25 5 0,

−2 −1

0 x 50, , 50 < x 100, x > 100.

2 −1

0 x 25, , 25 < x 100, x > 100.

If some person’s age is 28, then his membership degree belongings to “youth” or “oldness” respectively is 1+

28 − 25 5

2 −1 = 0.735 and 0.

If some person’s age is 55, then his membership degree belongings to “youth” or “oldness” respectively is

1.2 Fuzzy Sets

1+

5

55 − 25 5

2 −1 = 0.027 and

1+

55 − 50 5

−2 −1

= 0.5.

According to the three-type membership functions mentioned above, we can, to a certain, calculate its membership degree by concrete object x. When its accuracy is not required high, for simple account, we can determine the membership degree by adopting evaluation. Example 1.2 Suppose X = {1, 2, 3, 4}, these four elements constitute a small number set. Obviously, element 1 is standardly a small number, it should belong to this sets, and its membership degree is 1; element 4 is not a small number, and it should not belong to this set, its membership degree being 0. Element 2 “also returns small” or make “eighty percent small”, its membership degree being 0.8; element 3 probably is “force small”, or makes “two percent small”; its membership degree being 0.2. ˜ its elements still is 1,2,3,4, at the same The fuzzy sets written in small numbers as A, time, and a membership degree of element in A˜ is given, denoted by 1 0.8 0.2 0 + + . Zadeh’s representation is A˜ = + 1 2 3 4 An order dual representation is A˜ = {(1, 1), (2, 0.8), (3, 0.2), (4, 0)}. A vector method simply shows as A˜ = (1, 0.8, 0.2, 0).

1.3 Operations in Fuzzy Sets Because the value region in membership function of fuzzy sets corresponding to clear-subset characteristic function is extend from {0, 1} to [0, 1]. Similar to the characteristic function to demonstrate the relation between a distinctive subset, we have the following definitions. ˜ B˜ ∈ F(X ). For an arbitrary x ∈ X , we have Definition 1.2 Let A, ˜ ˜ Inclusion: A˜ ⊆ B˜ ⇐⇒ A(x) B(x); ˜ ˜ Equality: A˜ = B˜ ⇐⇒ A(x) = B(x). ˜ That is to say, the inclusion From Definition 1.2, A˜ = B˜ ⇐⇒ A˜ ⊆ B˜ and B˜ ⊆ A. relation is a binary relation on fuzzy power set F(X ) with following properties, i.e., (1) A˜ ⊆ A˜ (reflection); (2) A˜ ⊆ B˜ and B˜ ⊆ A˜ =⇒ A˜ = B˜ (symmetry); (3) A˜ ⊆ B˜ and B˜ ⊆ C˜ =⇒ A˜ ⊆ C˜ (transitivity). Since relation “⊆” constitutes an order relation on F(X ), (F(X ), ⊆) stands for a partially ordered set. Again as φ, X ∈ F(X ), hence F(X ) contains maximum element X and minimum element φ.

6

1 Preliminaries and Backgrounds

˜ B˜ ∈ F(x). Then we define Definition 1.3 Let A, ˜ whose membership function is Union: A˜ B, ( A˜ Intersection: A˜

( A˜

˜ ˜ B)(x) = A(x)

˜ ˜ ˜ B(x) = max{ A(x), B(x)};

˜ whose membership function is B,

˜ ˜ B)(x) = A(x)

˜ ˜ ˜ B(x) = min{ A(x), B(x)};

Complement: A˜ c , whose membership function is ˜ A˜ c (x) = 1 − A(x). Their images show like Figs. 1.1, 1.2 and 1.3. Comparing operation of union, intersection and complement in distinctive sets, we discover immediately that the fuzzy sets operation is exactly a parallel definition ˜ B˜ is a maximum fuzzy set embodying A˜ and of the distinct sets operation, A ˜ A˜ B˜ is a minimum fuzzy set embodying A˜ and embodied embodied again in B. ˜ again in B.

Fig. 1.1 Union tilde A and tilde B

6 1

A

x

B A

0

Fig. 1.2 Intersection tilde A and tilde B

B

6 1

0

A

B B A

x

1.3 Operations in Fuzzy Sets

7

Fig. 1.3 Complement tilde A

6 1

0

c A

A c

A

x

According to the two kinds of cases, where the universe X is finite or infinite, the calculation formula of union, intersection and complement in fuzzy sets A˜ and B˜ can represent, respectively, the following: (1) The universe is X = {x1 , x2 , . . . , xn }, and also A˜ =

n n ˜ i) ˜ i) A(x B(x , B˜ = , xi xi i=1 i=1

then A˜ A˜

B˜ =

n ˜ i) ˜ i ) ∨ B(x A(x , xi i=1

B˜ =

n ˜ i) ˜ i ) ∧ B(x A(x , xi i=1

A˜ c =

n ˜ i) 1 − A(x . xi i=1

˜ ˜ (2) The universe setX is an infinite set, and A˜ = x∈X A(x)/x, B˜ = x∈X B(x)/x, then ˜ ˜ ˜ ˜ A(x) ∨ B(x)/x, A B= x∈X ˜ ˜ ˜ ˜ A(x) ∧ B(x)/x, A B= x∈X ˜ (1 − A(x))/x. A˜ c = x∈X

Example 1.3 Suppose X = {x1 , x2 , x3 , x4 }; A˜ = 0 0.2 0.8 + + , then x2 x3 x4

1 0.8 0.2 0 0 + + + ; B˜ = + x1 x2 x3 x4 x1

8

1 Preliminaries and Backgrounds

1 ∨ 0 0.8 ∨ 0.2 0.2 ∨ 0.8 0 ∨ 0 + + + B˜ = x1 x2 x3 x4 1 0.8 0.8 0 = + + + . x1 x2 x3 x4 1 ∧ 0 0.8 ∧ 0.2 0.2 ∧ 0.8 0 ∧ 0 A˜ B˜ = + + + x1 x2 x3 x4 0 0.2 0.2 0 = + + + . x1 x2 x3 x4 1 − 1 1 − 0.8 1 − 0.2 1 − 0 A˜ c = + + + x1 x2 x3 x4 0 0.2 0.8 1 = + + + . x1 x2 x3 x4

A˜

Example 1.4 Compute union, intersection and complement of the fuzzy sets Y˜ and O˜ in Example 1.1 in Sect. 1.2. From the definition, we have ˜ Y˜ (x) O(x)/x Y˜ O˜ = x∈X

−1 x − 25 2 1 /x 1+ = + 5 0x25 x 25 L(x).

xn →x ∗

Since xn ∈ cα and cα is closed, then x ∈ cα , such that c(x) ˜ = L(x) α, a contradiction.

1.6 Fuzzy Numbers

23

For the same reason, c(x) ˜ = R(x) is a continuously decreasing function on the left for x > c2 , with 0 R(x) < 1. Sufficiency. Let c˜ satisfy the condition in the theorem. (1) c˜ is obviously normal. (2) Prove cα = [c1α , c2α ], ∀α ∈ (0, 1]. ˜ = R(x) c(x) ˜ = L(x) for x < c1 , so we select c1α = min{x|L(x) α} and c(x) for x > c2 , such that we select c2α = max{x|R(x) α}. Obviously, cα ⊂ [c1α , c2α ]. Now, prove [c1α , c2α ] ⊂ cα , and we prove only [c1α , c1 ) ⊂ cα (because we can prove (c2 , c2α ] ⊂ cα for the same reason). Again, we prove only c1α ∈ cα due to the monotonicity of L(x). Select xn → c1α , then L(c1α ) = lim (L(xn )) α, such that c1α ∈ cα . xn →c1α

1.6.2 Type (·, c), T, L − R and Flat Fuzzy Numbers Definition 1.17 c˜ = (α, c) is defined as a type (·, c) fuzzy number on a product space α1 × α2 × · · · × α J ; its membership function is c(a) ˜ = min[c˜ j (a j )], j

⎧ ⎨1 − |α j − a j | , α − c a α + c j j j j j cj c(a ˜ j) = ⎩ 0, otherwise,

(1.5)

where α = (a1 , a2 , . . . , a J )T , c = (c1 , c2 , . . . , c J )T ; α denotes the center of c, ˜ c the extension of c, ˜ with c j > 0. Coming next are special cases. Definition 1.18 L is called a reference function of fuzzy numbers, if L satisfies (i) L(x) = L(−x); (ii) L(0) = 1; (iii) L(x) is a nonincreasing and piecewise continuous function at [0, +∞). Definition 1.19 Let L , R be reference functions of fuzzy numbers c, ˜ called a type L-R fuzzy number. If ⎧ c−x ⎪ ⎪ , x c, c > 0, ⎨L c c(x) ˜ = x −c ⎪ ⎪ , x c, c > 0, ⎩R c

(1.6)

we write c˜ = (c, c, c) L R , where c is a mean value; c and c are called the left and the right spreads of c, ˜ respectively. L is called a left reference and R a right reference. If take c to be variable x, then x˜ = (x, ξ, ξ) L R represents T -fuzzy variables.

24

1 Preliminaries and Backgrounds

Definition 1.20 If L and R are functions satisfying T (x) =

1 − |x|, if − 1 x 1, 0, otherwise,

(1.7)

then we call c˜ = (c, c, c)T T -fuzzy numbers, T (R) representing T -fuzzy numbers sets. If take c to be variable x, then x˜ = (x, ξ, ξ)T represents T -fuzzy variables. Definition 1.21 Let L , R be reference functions and the quadruple c˜ = (c− , c+ , σc− , σc+ ) L R is called a type L-R flat fuzzy numbers. Then, we have ⎧ − c −x ⎪ ⎪ L , x c− , σc− > 0, ⎪ ⎪ ⎨ σc− x − c+ c(x) ˜ = , x c+ , σc+ > 0, R ⎪ + ⎪ σ ⎪ c ⎪ ⎩ 1, otherwise,

(1.8)

satisfying ∃ : (c− , c+ ) ∈ R, c− < c+ , with c(x) ˜ = 1. Especially, c˜ = (c− , c+ , σc− , σc+ ) is said to be a flat fuzzy number, where ⎧ c− − x ⎪ − − − ⎪ 1 − ⎪ − , if c − σc x c , ⎪ σ ⎪ c ⎪ ⎨1, if c− < x < c+ , c(x) ˜ = + x −c ⎪ + + + ⎪ 1− ⎪ + , if c x c + σc , ⎪ ⎪ σ c ⎪ ⎩ 0, otherwise.

(1.9)

If take interval (c− , c+ ) to be fuzzy interval, ∀x˜ ∈ [c− , c+ ], then x˜ = ((x − , x + ), ξ, ξ) L R and x˜ = ((x − , x + ), ξ, ξ) represent L-R fuzzy variables and flat fuzzy ones, respectively. Definition 1.22 Suppose that “∗” represents an arbitrary ordinary binary operation in R, such that ∀c, ˜ d˜ ∈ F(R) and we define ˜ ˜ c˜ ∗ d = c(x) ˜ ∧ d(y)/x ∗ y, (1.10) x,y∈R

that is, ∀z ∈ R,

˜ c˜ ∗ d(z) =

˜ (c(x) ˜ ∧ d(y)),

(1.11)

x∗y=z

where “∗” represents arithmetic operations +, −, ·, ÷. Accordingly, we can define the operations of type L-R, T and flat fuzzy numbers.

1.6 Fuzzy Numbers

25

A. Operations properties in type L-R fuzzy number Let c˜ = (c, c, c) L R , d˜ = (d, d, d) L R , p˜ = ( p, p, p) R L be an L-R fuzzy number. Then, (1) c˜ + d˜ =(c + d, c + d, c + d) L R . (kc, kc, kc) L R , when k 0 (2) k · c˜ = (k ∈ R). (kc, −kc, −kc) R L , when k < 0 Let (−1)c˜ = −c˜ for k = −1. Then −c˜ = (−c, c, c) R L . (3) c˜ − p˜ = (c − p, c + p, c + p) L R for L = R. (4) c˜ · d˜ ≈ (cd, cd + dc, cd + dc) L R . c pc + c p pc + c p (5) c˜ ÷ p˜ ≈ ( , , ) L R , p = 0, c˜ and p˜ can not be divided for p p2 p2 L = R. ˜ ≈ (c ∧ d, c ∨ d, c ∧ d) L R . ˜ ≈ (c ∨ d, c ∧ d, c ∨ d) L R , min(c, ˜ d) (6) max(c, ˜ d) ˜ (7) c˜ d˜ ⇐⇒ c d, c d, c d; c˜ ⊆ d˜ ⇐⇒ c + c d − d, or c˜ = d. B. Operations properties in type T -fuzzy numbers If c˜1 = (c1 , c1 , c1 )T , c˜2 = (c2 , c2 , c2 )T , then ˜ c˜2 = (c1 + c2 , c1 + c2 , c1 + c2 )T ; (1) c˜1 + ˜ c˜2 = (c1 − c2 , c1 + c2 , c1 + c2 )T ; (2) c˜1 − (λc, λc, λc)T , (3) λc˜ = λ(c, c, c)T = (λc, −λc, −λc)T

∀λ ≥ 0, ∀λ < 0.

−2 −2 (4) c˜−1 = (c, c, c)−1 T ≈ (1/c, cc , cc )T .

C. Operation properties in type flat fuzzy numbers Let c˜ = (c− , c+ , σc− , σc+ ) and d˜ = (d − , d + , σd− , σd+ ) be flat fuzzy numbers. Then (1) c˜ + d˜ =(c− + d − , c+ + d + , σc− + σd− , σc+ + σd+ ). (kc− , kc+ , kσc− , kσc+ ), for k ≥ 0 (2) k · c˜ = + − − + (kc , kc , −kσc , −kσc ), for k < 0. By the definition of type L-R, T or flat fuzzy numbers, it is easy to prove their operation properties [6, 7]. We can deduce operation properties of type (·, c) fuzzy numbers since it is extended over flat fuzzy ones.

1.6.3 Trapezoidal and Triangular Fuzzy Number For an L R fuzzy number A˜ with the following membership function ⎧ L x−a ⎪ , f or a L − α ≤ x ≤ a L , ⎪ ⎪L α ⎪ ⎨ 1, f or a L ≤ x ≤ a U , μ A˜ (x) = U x−a ⎪R , f or a U ≤ x ≤ a U + β, ⎪ ⎪ β ⎪ ⎩ 0, other wise,

(1.12)

26

1 Preliminaries and Backgrounds

where L (.) and R (.) are piecewise continuous functions, L (.) is non-decreasing, R (.) is non-increasing and L (0) = R (0) = 1, the LR fuzzy number a˜ will be rep resented as a˜ = a L , a U , α, β L R . Here L and R are called as the left and right reference functions, a L and a U are respectively called starting and end points of the flat interval, and α is called the left spread and β is called the right spread. We denote the all L R fuzzy number by F (R). It needs to point out that when L (x) = 1 + x, R (x) = 1 − x and a L < a U , fuzzy number a˜ denotes Trapezoidal Fuzzy number. In fact, we have the following definition. Definition 1.23 A fuzzy number a˜ = a L , a U , α, β L R is called a Trapezoidal Fuzzy number, if its membership function is given by ⎧ x−(a L −α) ⎪ , ⎪ ⎪ ⎨ 1, α μa˜ (x) = (aU +β )−x ⎪ , ⎪ β ⎪ ⎩ 0,

f or a L − α ≤ x ≤ a L , f or a L ≤ x ≤ a U , f or a U ≤ x ≤ a U + β, other wise.

(1.13)

Remark 1.1 If a L = a U = a in Trapezoidal Fuzzy number a˜ = a L , a U , α, β , we obtain a Triangular Fuzzy Number, and we denote it as a˜ = (a, α,β). It can be verified that the arithmetic on L R fuzzy numbers a˜ = a L , a U , α, β L R L U and b˜ = b , b , γ, θ L R by use of Zadeh’s extension principle is as follows: x ≥ 0, x ∈ R; x a˜ = xa L , xa U , xα, xβ L R , x < 0, x ∈ R; x a˜ = xa U , xa L , −xβ, xα R L a˜ + b˜ = a L + b L , a U + bU , α + γ, β + θ L R

(1.14a) (1.14b) (1.14c)

Note that when a˜ = a L , a U , α, β L R and b˜ = b L , bU , γ, θ R L , then we have: a˜ − b˜ = a L − bU , a U − b L , α + θ, β + γ L R

(1.15)

1.6.4 Order on Fuzzy Numbers Ranking of fuzzy numbers is an important issue in the study of fuzzy set theory. Ranking procedures are also useful in various applications and one of them will be in the study of fuzzy mathematical programming in later chapters. There are numerous methods proposed in the literature for the ranking of fuzzy numbers, some of them seem to be good in a particular context but not in general. Here, we describe only three simple methods for the ordering of fuzzy numbers. The first approach is called the k-Preference index approach. This approach has been suggested by Adamo [8]. Let a˜ be the given fuzzy number and k ∈ [0, 1]. The

1.6 Fuzzy Numbers

27

k-Preference index of a˜ is defined as Fk (a) ˜ = max {x : μa˜ (x) ≥ k}. Now, using this ˜ A˜ b˜ with degree k ∈ [0, 1] if k-Preference index, for two fuzzy number a˜ and b, ˜ ˜ ≤ Fk b . and only if Fk (a) The second approach for ranking of fuzzy numbers is based on possibility theory. In fact, Dubois and Prade [9] studied the ranking of fuzzy numbers in the setting ˜ of possibility theory. To develop this suppose, we have two fuzzy number a˜ and b. Then, in accordance with the extension principle of Zadeh, the crisp inequality x ≤ y can be extended to obtain the truth value of the assertion that a˜ is less than or equal ˜ as follows: to b, (1.16) T a˜ b˜ = sup min μa˜ (x) , μb˜ (y) . x≤y

This truth value T a˜ b˜ is also called the grade of possibility of dominance of b˜ on a˜ and is denote by Poss a˜ b˜ . Now define a˜ b˜ if and only is Poss (a˜ a) ˜ ≤ Poss a˜ b˜ .

(1.17)

The third approach to order of fuzzy numbers is based on the concept of comparison of fuzzy numbers by the use of ranking functions, in which a ranking functions R : F (R) → R that maps each fuzzy number into the real line is defined for ranking the elements of F (R). Thus, using the natural order of the real numbers we can compare fuzzy numbers easily as follows: a˜ b˜ if and only if R (a) ˜ ≥ R b˜ , a˜ b˜ if and only if R (a) ˜ > R b˜ , a˜ ≈ b˜ if and only if R (a) ˜ = R b˜ .

(1.18a) (1.18b) (1.18c)

where a˜ and b˜ are in F (R). Also, we write a˜ b˜ if and only if b˜ a. ˜ ˜ if a˜ b, ˜ then −a˜ −b. ˜ Lemma 1.1 For trapezoidal fuzzy numbers a˜ and b, Several ranking functions have been proposed by researches to suit their requirements of the problems under consideration. We restrict our attentiontolinear ranking functions, that is, a ranking R such that R k a˜ + b˜ = k R (a) ˜ + R b˜ for any a˜ and b˜ belonging to F (R) and any k ∈ R. We consider the linear on ranking functions F (R) as R (a) ˜ = c L a L + cU a U + cα α + cβ β, where a˜ = a L , a U , α, β , and c L , cU , cα and cβare constants, at least one of which is nonzero. For a trapezoidal fuzzy number a˜ = a L , a U , α, β , some of these ranking functions are presented here.

28

1 Preliminaries and Backgrounds

(a) Yager’s [10] ranking function: 1 Y2 (a) ˜ = 2

1 0

β−α 1 L U int a˜ α + sup a˜ α dα = a +a + . 2 2

(1.19)

(b) Campos and Munoz’s [11] ranking function: C M1λ

λint a˜ α + (1 − λ) sup a˜ α dα 0 U α+β α L L − , =a + λ a − a + 2 2 1

˜ = (a)

α λint a˜ α + (1 − λ) sup a˜ α dα 0 U α+β α L L − . =a + λ a − a + 3 3

C M2λ (a) ˜ =

(1.20)

1

(1.21)

In the later chapters, the liner ranking functions will play its crucial role in ordering the trapezoidal fuzzy numbers being used for testing the optimality (inequality) conditions and making the decision for pivoting. Specially, we usually use the linear ranking function ˜ to illustrate our approaches. Then, for the trapezoidal fuzzy Y2 (a) numbers a˜ = a L , a U , α, β and b˜ = b L , bU , γ, θ , we have a˜ b˜ if and only if 1 β−α θ−γ 1 L U L U ˜ a +a + ≥R b = b +b + . R (a) ˜ = 2 2 2 2

(1.22)

Remark 1.2 For any trapezoidal fuzzy number a, ˜ the relation a˜ 0˜ holds, if there exist ε ≥ 0 and α ≥ 0 such that a˜ (−ε, ε, α, α). We realize that R (−ε, ε, α, α) = 0 (we also consider a˜ ≈ 0˜ if and only if R (a) ˜ = 0). Thus, without loss of generality, throughout the book we let 0˜ = (0, 0, 0, 0) as the zero trapezoidal fuzzy number. Remark 1.3 We realize that the results obtained throughout the book are independent of the choice of the linear ranking function. In fact, we can use any other linear ranking function, and although the solution obtained may be different but the results are still valid for the new solution. As for the types of the fuzzy data in the model and the assumption of fuzziness in the variables, the choice and compatibility of the ranking function for fuzzy linear programs should be the decision maker’s main concerns. For trapezoidal fuzzy numbers and variables, the linear ranking function used here is deemed to be appropriate.

1.7 Data Envelopment Analysis

29

1.7 Data Envelopment Analysis The problem of the evaluation of the function of the Decision Making Unit, (DMU), has permanently been under great consideration by managers of different fields since a long time ago, especially, since the second war world along the scientific approaches have commenced and promoted Data Envelopment Analysis (DEA) for the handling of this problems. Finally in 1956, Farell established a model for the evaluation of the efficiency of the DMUs with multiple inputs and single outputs for the first time. After two decades, in 1978, Charnes et al. [12] expanded this technique for the DMUs with multiple inputs and multiple outputs and called the model, data envelopment analysis. DEA is an extreme point method (frontier analysis) and compares each DMU with only the best DMUs. A fundamental assumption behind an extreme point method is that if a given DMU p , is capable of producing Y ( p) units of output with X ( p) inputs, then other DMUs should also be able to do the same if they were to operate efficiently. Similarly, if DMUq is capable of producing Y (q) units of output with X (q) inputs, then other DMUs should also be capable of the same production schedule. DMU p , DMUq and others can then be combined to form a composite DMU with composite inputs and composite outputs. Since this composite DMU does not necessarily exist, it is sometimes called a virtual DMU. The heart of the analysis lies in finding the best virtual DMU for each real DMU. If the virtual DMU is better than the original DMU by either making more output with the same input or making the same output with less input then the original DMU is inefficient. There are various ways that a DMU can be lie on efficient frontier. These ways can be called such as reduce in its inputs whilst keeping its outputs constant, increase in its outputs whilst keeping its inputs constant and do some combination of two the above relations. The procedure of finding the best virtual DMU can be formulated as a linear program. Analyzing the efficiency of n DMUs is then a set of n linear programming problems. In continue, we formulate some basic models in DEA.

1.7.1 Basic DEA Models Data envelopment analysis measures the relative efficiency of decision making units with multiple performance factors which are grouped into outputs and inputs. The units are assumed to operate under similar conditions. Based on information about existing data on the performance of the units and some preliminary assumptions, DEA forms an empirical efficient surface (frontier). If a DMU lies on the surface, it is referred to as an efficient unit, otherwise inefficient. In fact, once the efficient frontier is determined, inefficient DMUs can improve their performance to reach the efficient frontier by either increasing their current output levels or decreasing their current inputs levels. DEA also provides efficiency scores and reference set for inefficient DMU. The efficiency scores are used in practical applications as performance indicators of the DMUs. The reference set for inefficient units consists of efficient units and determines a virtual unit on the efficient surface. The virtual unit which can

30

1 Preliminaries and Backgrounds

be found in DEA by projecting an inefficient DMU radially to the efficient surface, is regarded as a target unit for the inefficient unit. Based on different empirical axioms and corresponding to different characteristics of the production possibility set and production frontiers, different DEA models, are developed and applied in practice. Some of these models is discussed here. We assume that there exist n DMUs to be evaluated. Each DMU consumes varying amounts of m different inputs to produce s different outputs. Specifically, DMU j consumes amounts X j = (xi j ) of inputs (i = 1, 2, ..., m) and produces amounts Y j = (yr j ) of outputs (r = 1, 2, ..., s). All data are assumed to be nonnegative but at least one component of every input and output vector is positive, that is; X j ≥ 0, X j = 0 and Y j ≥ 0, Y j = 0. X p = (x1 p , x2 p , ..., xmp ) and Y p = (y1 p , y2 p , ..., ysp ) are amounts of inputs and outputs of D MU p , which is evaluated. The DEA postulates that underlying the Production Possibility Set (P P S), denoted by T = {(X, Y ) : output vector Y ≥ 0 can be produced from input vector X ≥ 0} , possess the following properties: (P1) (Envelopment). The observed (X j , Y j ) ∈ T, j = 1, 2, ..., n. (P2) (Constant returns to scale). If (X, Y ) ∈ T , then (λX, λY ) ∈ T for all λ ≥ 0. (P3) (Convexity). T is a closed and convex set, i.e. If (X 1 , Y1 ) ∈ T and (X 2 , Y2 ) ∈ T then for all λ ∈ (0, 1), λ(X 1 , Y1 ) + (1 − λ)(X 2 , Y2 ) ∈ T. (P4) (Plausibility). If (X, Y ) ∈ T , X t ≥ X and Yt ≤ Y , then (X t , Yt ) ∈ T . (P5) (Minimum extrapolation). T is the smallest set that satisfies Postulates P1 − P4. The above-mentioned postulates define the following unique set: Tc = {(X, Y ) : X ≥ λX, Y ≤ λY, λ ≥ 0} . We now turn to efficiency estimation of DMUs from the P P S. The classic Farrell input efficiency measure, is defined as E(X p , Y p ) = min α p |(α p X p , Y p ) ∈ Tc

!

which by considering Tc , we have the following linear programming: min α p s.t.

n j=1 n j=1

λ j xi j ≤ α p xi p , i = 1, 2, ..., m, (1.23) λ j yr j ≥ yr p ,

λ j ≥ 0,

r = 1, 2, ..., s, j = 1, 2, ..., n.

1.7 Data Envelopment Analysis

31

This model is called input-oriented CCR envelopment model. In this model, an efficiency score is generated for a DMU by minimizing inputs with fixed outputs and for each observed DMU p an imaginary composite unit is constructed that outperforms DMU p . Also, λ j represents the proportion to which DMU j is used to construct the composite unit for DMU p ( j = 1, 2, ..., n). In model (1.23), the composite unit produces inputs that are at most equal to a proportion α of the inputs of DMU p with 0 < α p ≤ 1, where the α p is the efficiency score of DMU p , and consumes at least the same levels of outputs as DMU p . If α p < 1, DMU p is not efficient and the parameter α p indicates the extent by which DMU p has to decrease its inputs to become efficient. The dual problem of model (1.23) is as follows: max s.t.

s r =1 m i=1 s

u r yr p vi xi p = 1, u r yr j −

r =1

u r ≥ 0, vi ≥ 0,

m

(1.24) vi xi j ≤ 0, j = 1, 2, ..., n,

i=1

r = 1, 2, ..., s, i = 1, 2, ..., m.

This model is called the input-oriented CCR multiplier model. The multiplier model provides information on the weights of inputs and outputs. The weights are interpreted as prices in many applications. Moreover, the output efficiency measure is defined as the inverse of max{β p |(X p , β p Y p ) ∈ Tc }, which by considering Tc , the following linear programming is obtained: max β p n s.t. λ j xi j ≤ xi p , j=1 n j=1

i = 1, 2, ..., m, (1.25)

λ j yr j ≥ β p yr p , r = 1, 2, ..., s

λ j ≥ 0,

j = 1, 2, ..., n.

This model is called output-oriented CCR envelopmet model. In this model, an efficiency score is generated for a DMU by maximizing outputs with limited inputs and for each observed DMU p an imaginary composite unit is constructed that outperforms DMU p . Also, λ j represents the proportion to which DMU j is used to construct the composite unit for DMU p ( j = 1, 2, ..., n). In model (1.25), the composite unit consumes at most the same levels of inputs as DMU p and produces outputs that are

32

1 Preliminaries and Backgrounds

at least equal to a proportion β p of the outputs of DMU p with β p ≥ 1. The inverse of β p is the efficiency score of DMU p . If β p > 1, DMU p is not efficient and the parameter β p indicates the extent by which DMU p has to increase its outputs to become efficient. The dual problem of model (1.25), called the output-oriented CCR multiplier model, is as follows: min s.t.

s i=m s r =1 s

vi xi p u r yr p = 1, u r yr j −

r =1

u r ≥ 0, vi ≥ 0,

m i=1

(1.26) vi xi j ≤ 0, j = 1, 2, ..., n, r = 1, 2, ..., s, i = 1, 2, ..., m.

It needs to point out that the CCR models are built on the assumption of constant returns to scale of activities. However, in efficiency analysis, variable returns to scale can also be considered. In fact, Banker et al. [13] published the BCC model, whose PPS associated to variable returns to scale is defined by Tv = {(X, Y ) : X ≥ λX, Y ≤ λ, eλ = 1, Y, λ ≥ 0} , where e is a row vector with all elements unity. The BCC models differ from the CCR models only in the adjunction of the conn λ j = 1, which we also write eλ = 1. Together with the condition λ ≥ 0, dition j=1

this imposes a convexity condition on allowable ways in which the observation for the n DMUs may be combined.

1.7.2 Cost Efficiency Cost Efficiency (C E) evaluates the ability to produce current outputs at minimal cost, given its input prices. In this subsection, we assume that prices are fixed and known, although they may possibly be different between the DMUs. In order to obtain a measure of cost efficiency for DMUs with multiple inputs and outputs, the minimum cost for the production of current outputs, given the input prices, is obtained solving the following linear problem, as first formulated by Fare et al. [14]:

1.7 Data Envelopment Analysis

33

min c p x p = n

s.t.

m

p

ci p xi

i=1 p

λ j xi j = xi , i = 1, 2, ..., m, (1.27)

j=1

n

λ j yr j ≥ yr p , r = 1, 2, ..., s,

j=1

λ j ≥ 0, p xi ≥ 0,

j = 1, 2, ..., n, i = 1, 2, ..., m.

In the model above, ci p is the price of input i for the DMU p under assessment and p xi is a variable that at the optimal solution gives the amount of input i to be employed by DMU p in order to produce the current outputs at minimal cost, given the current prices at DMU p and the technological restrictions of the production possibility set. Note that, this model assumes that the input prices at each DMU (ci p , i = 1, ..., m) are fixed and known, although they can differ between DMUs. Cost efficiency is then obtained as the ratio of minimum cost with specific prices (the optimal solution to model (1.27) to the observed cost at DMU p , as follows: m

(C E) p =

i=1 m

ci p xi∗ p (1.28) ci p xi p

i=1

Alternatively, the measure of C E can be obtained with the inclusion of weight restrictions in the standard DEA model. The resulting cost efficiency model based on the standard DEA formulation with the addition of weights restrictions is as follows (For more details see Schaffnit et al. [15]): max q = s.t.

m i=1 s r =1

s

u r yr p

r =1

vi xi p = 1, u r yr j −

vi a −

ci a p vb ci b p i

u r ≥ ε,

m i=1

(1.29) vi xi j ≤ 0, j = 1, 2, ..., n,

= 0,

i a , i b = 1, 2, ..., m, r = 1, 2, ..., s,

where vi a and vi b are the input weights used for the cost efficiency assessment with the D E A model, and pi a p and pi b p are the input prices observed at DMU p for any two inputs i a and i b used by the DMU.

34

1 Preliminaries and Backgrounds

It can easily be shown the efficiency measure (q) obtained from (1.29) is equal to the cost efficiency measure obtained from (1.28).

1.8 Network Optimization Network flow is a problem domain that lies at the cusp of several fields of inquiry, including applied mathematics, computer science, engineering and operations research. Minimum Cost Flow Problem (MCFP) and Maximum Flow Problem (MFP) are two important problems in network optimization [16].

1.8.1 Minimum Cost Flow Problem In order to make mathematical model for minimum cost flow problem, we give a directed network G = (V, E), consisting of a finite set of nodes V = {1, 2, ..., n} and a set of arcs E. Each arc is denoted by an ordered pair (i, j), where (i, j) ∈ E. Moreover, two numbers li j and u i j are given as the lower and upper bounds capacities of arc (i, j). We associate with each i ∈ V a number bi representing its supply, demand or transient node depending on whether bi ≥ 0, or bi ≤ 0, bi = 0 respectively. The decision variables in MCFP are arc flows and we represent the flow on an arc (i, j) ∈ E by xi j . In addition, for every arc (i, j) in E, we consider a cost ci j denoting the cost per unit flow on that arc. MCFP requirements can be written in the following linear program: min z =

ci j xi j

(i, j)∈E

s.t.

{ j:(i, j)∈E}

xi j −

{ j:( j,i)∈E}

li j ≤ x i j ≤ u i j ,

where

x ji = bi ,

for all i ∈ V,

(1.30)

for all (i, j) ∈ E,

bi = 0.

i∈E

1.8.2 Maximum Flow Problem The aim of MFP is to sent as much flow as possible between two special nodes in the network, a source node and a sink node without exceeding the capacity of any arc. Again consider the network G = (V, E) with n nodes and m arcs. We associate with each arc (i, j), a lower bound on flow of li j and an upper bound on flow of

1.8 Network Optimization

35

u i j . There are no costs involved in the MFP. In such a network, we wish to find the maximum amount of fuzzy flow from node 1 to node n. Let f be the amount of flow in the network from node 1 to node n. Then, the maximum flow problem may be formulated as follows: max

f

⎧ ⎨ f, if i = 1, xi j − x ji = 0, if i = 1 or m, s.t. ⎩ − f, if i = m, { j:(i, j)∈A} { j:( j,i)∈A}

li j ≤ x i j ≤ u i j ,

(1.31)

for all (i, j) ∈ E.

We refer to a vector x = {xi j } satisfying (1.31) as a feasible flow and the corresponding value of f as the value of feasible flow.

1.9 Diet Problem The diet problem was the first of some about optimization. The goal was to find the cheapest way to feed the army but insuring at same time some determined nutrition levels. This kind of problem can be questioned in several ways such as minimizing purchase expenses, diet for the cattle, slimming diet that meets certain levels of calories, proteins, carbohydrates,….

1.9.1 Definition of Problem Domestic animals continue to make important contributions to global food supply and, as a result, animal feeds have become an increasingly critical component of the integrated food chain. Livestock products account for about 30% of the global value of agriculture and 19% of the value of food production, and provide 34% of protein and 16% of the energy consumed in human diets. Meeting consumer demand for more meat, milk, eggs and other livestock products is dependent to a major extent on the availability of regular supplies of appropriate, cost-effective and safe animal feeds. Few issues have generated as much public concern in recent times, however, as the protein supply in feeds for livestock production. The diet problem was one of the first optimization problems studied in the 1930s and 1940s. The problem was motivated by the Army’s desire to minimize the cost of feeding GIs in the field while still providing a healthy diet. One of the early researchers to study the problem was George Stigler, who made an educated guess of an optimal solution using a heuristic method. The goal of the diet problem is to select a set of foods that will satisfy a set of daily nutritional requirement at minimum cost. The problem is formulated as a linear

36

1 Preliminaries and Backgrounds

program where the objective is to minimize cost and the constraints are to satisfy the specified nutritional requirements. The diet problem constraints typically regulate the number of calories and the amount of vitamins, minerals, fats, sodium, and cholesterol in the diet. The diet problem was one of the first optimization problems studied in the 1930s and 1940s. The problem was motivated by the Army’s desire to minimize the cost of feeding GIs in the field while still providing a healthy diet. One of the early researchers to study the problem was George Stigler, who made an educated guess of an optimal solution using a heuristic method. The goal of the diet problem is to select a set of foods that will satisfy a set of daily nutritional requirement at minimum cost. The problem is formulated as a linear program where the objective is to minimize cost and the constraints are to satisfy the specified nutritional requirements. The diet problem constraints typically regulate the number of calories and the amount of vitamins, minerals, fats, sodium, and cholesterol in the diet.

1.9.2 Mathematical Formulation The Diet Problem can be formulated mathematically as a linear programming problem as shown below. Sets F = set of foods N = set of nutrients Parameters ai j = amount of nutrient j in food i, ∀i ∈ F, ∀ j ∈ N ci = cost per serving of food i, ∀i ∈ F Fmini = minimum number of required servings of food i, ∀i ∈ F Fmaxi = maximum allowable number of servings of food i, ∀i ∈ F Nmini = minimum required level of nutrient j, ∀ j ∈ N Nmaxi = maximum allowable level of nutrient j, ∀ j ∈ N Variables xi = number of servings of food i to purchase/consume, ∀i ∈ F Objective Function: Minimize the total cost of the food Minimi ze ci xi i∈F

Constraint Set 1: For each nutrient j ∈ N , at least meet the minimum required level. ai j xi ≥ N min j , ∀ j ∈ N i∈F

Constraint Set 2: For each nutrient j ∈ N , do not exceed the maximum allowable level. ai j xi ≤ N max j , ∀ j ∈ N i∈F

1.9 Diet Problem

37

Constraint Set 3: For each food i ∈ F, select at least the minimum required number of servings. xi ≥ Fmin i , ∀i ∈ F Constraint Set 4: For each food i ∈ F, do not exceed the maximum allowable number of servings. xi ≤ Fmaxi , ∀i ∈ F

References 1. Zadeh, L.A.: Fuzzy sets. Inform. Control. 8, 338–353 (1965) 2. Zadeh, L.A.: Fuzzy sets and systems. In: Polytechnic Press of Polytechnic Institute of Brooklyn, Proceedings of the Symposium on Systems Theory. NY (1965) 3. Lu, M., Wu, W.: Interval value and derivate of fuzzy-valued function. J. Fuzzy Syst. Math. 6, 182–184 (1992) 4. Luo, C.Z.: The extension principle and fuzzy numbers (I). Fuzzy Math. 4(3), 109–116 (1984) 5. Luo, C.Z.: The extension principle and fuzzy numbers (II). Fuzzy Math. 4(4), 105–114 (1984) 6. Diamond, P.: Least squares fitting of several fuzzy data. Proc. of IFSA Congress, Tokyo. I, 329–332 (1987) 7. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980) 8. Adamo, J.M.: Fuzzy decision trees. Fuzzy Sets Syst. 4, 207–219 (1980) 9. Dubois, D., Prade, H.: Ranking fuzzy numbers in the setting of possibility theory. Inf. Sci. 30, 183–224 (1980) 10. Yager, R.R.: A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24, 143–161 (1981) 11. Campos, L., Munoz, A.: A subjective approach for ranking fuzzy numbers. Fuzzy Sets Syst. 29, 145–153 (1989) 12. Charnes, A., Cooper, W., Rhodes, E.: Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2, 429–444 (1978) 13. Banker, R., Charnes, A., Cooper, W.: Some models for estimating technical and scale ineciencies in data envelopment analysis. Manag. Sci. 30, 1078–1092 (1984) 14. Fare, R., Grosskopf, S., Lovell, C.A.K.: The Measurement of Efficiency of Production. Kluwer Academic Publishers (1985) 15. Schaffnit, C., Rosen, D., Paradi, J.C.: Best practice analysis of bank branches: an application of DEA in a large Canadian bank. Eur. J. Oper. Res. 98, 269–289 (1997) 16. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs, NJ (1993)

Chapter 2

Fuzzy Linear Programming

2.1 Introduction For the linear programming problems in the crisp scenario, the aim is to maximize or minimize a linear objective function under linear constraints. But in many practical situations, such as those concerning management sciences, diet problem, network optimization, assignment problem, transportation, and etc., the decision maker may not be in a position to specify the objective and/or constraint functions precisely but rather can specify them in a fuzzy sense. In such situations, it is desirable to use some fuzzy linear programming type of modeling so as to provide more edibility to the decision maker. Since the fuzziness may appear in a linear programming problem in many ways, the definition of fuzzy linear programming problem is not unique. This chapter is assigned to briefly introduce various models of fuzzy linear programming problems which is discussed in the next chapters. For this aim, in the next section, we first introduce Linear Programming (LP) problems and then give the fundamental definitions and concepts of LP models and explore some convenient solution techniques such as primal and dual simplex algorithms, and in particular, we give the important results of the duality theory. Finally, Sect. 2.3 is assigned to a brief review of some convenient categories of fuzzy mathematical programming which are appeared in the literature. We also emphasize that this section will be useful for understanding the introduced models and their associated solving methods in the upcoming chapters.

2.2 Linear Programming 2.2.1 Introduction Linear Programming (LP) is a mathematical modeling technique useful for the optimal allocation of limited resources such as material, machines, money, energy, manpower, machine resources, time, space and etc to several competing activities © Springer Nature Switzerland AG 2019 S. H. Nasseri et al., Fuzzy Linear Programming: Solution Techniques and Applications, Studies in Fuzziness and Soft Computing 379, https://doi.org/10.1007/978-3-030-17421-7_2

39

40

2 Fuzzy Linear Programming

such as projects, services etc on the basis of given criteria of optimality. A typical linear programming problem consists of a linear objective function which is to be maximized or minimized subject to a finite number of linear constraints. The founders of LP are George B. Dantzig, who established the simplex method in 1947, John von Neumann, who developed the theory of the duality in the same year, and Leonid Kantorovich, a Russian mathematician who used similar techniques in economics before Dantzig and won the Nobel prize in 1975 in economics. The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger major theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior point method for solving linear programming problems. The basic components of linear programming are as follows: • Decision variables—These are the quantities to be determined. • Objective function—This represents how each decision variable would affect the cost, or, simply, the value that needs to be optimized. • Constraints—These represent how each decision variable would use limited amounts of resources. • Data—These quantify the relationships between the objective function and the constraints.

2.2.2 Formulations We now present examples of four general linear programming problems. Each of these problems has been extensively studied.

Problem 1 The Diet Problem Assume there are m different types of food, F1 , ..., Fm , that supply varying quantities of the n nutrients, N1 , ..., Nn , that are essential to good health. Let c j be the minimum daily requirement of nutrient, N j . Let bi be the price per unit of food, Fi . Let ai j be the amount of nutrient N j contained in one unit of food Fi . The problem is to supply the required nutrients at minimum cost. Let yi be the number of units of food Fi to be purchased per day. The cost per day of such a diet is (2.1) b1 y1 + b2 y2 + · · · + bm ym The amount of nutrient N j contained in this diet is a1 j y1 + a2 j y2 + · · · + am j ym

(2.2)

2.2 Linear Programming

41

for j = 1, ..., n. We do not consider such a diet unless all the minimum daily requirements are met, that is, unless a1 j y1 + a2 j y2 + · · · + am j ym ≥ c j , j = 1, ..., n

(2.3)

Of course, we cannot purchase a negative amount of food, so we automatically have the constraints (2.4) y1 ≥ 0, y2 ≥ 0, · · · , ym ≥ 0 The diet problem as an optimization problem is to minimize (2.1) subject to (2.3) and (2.4). This is exactly the standard minimum problem, too. Problem 2 The Transportation Problem Assume there are I ports, or production plants, P1 , ..., PI , that supply a certain commodity, and there are J markets, M1 , ..., M J , to which this commodity must be shipped. Port Pi possesses an amount si of the commodity ( j = 1, ..., I ), and market M j must receive the amount r j of the commodity ( j = 1, ..., J ). Let bi j be the cost of transporting one unit of the commodity from port Pi to market M j . The problem is to meet the market requirements at minimum transportation cost. Let yi j be the quantity of the commodity shipped from port Pi to market M j . The total transportation cost is I J yi j bi j (2.5) i=1 j=1

The amount sent from port Pi is

J

yi j , and since the amount available at port Pi is

j=1

si , we must have

J

yi j ≤ si , i = 1, ..., I

(2.6)

j=1

The amount sent to market M j is

I

yi j , and since the amount required there is r j ,

i=1

we must have

I

yi j ≥ r j , j = 1, ..., J

(2.7)

i=1

It is assumed that we cannot send a negative amount from Pi to M j , we have yi j ≥ 0, i = 1, ..., I, j = 1, ..., J

(2.8)

The transportation problem as an optimization problem is to minimize (2.5) subject to (2.6), (2.7) and (2.8).

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Problem 3 The Activity Analysis Problem Assume there are n activities, A1 , ..., An , that a company may employ, using the available supply of m resources, R1 , ..., Rm (labor hours, steel, etc.). Let bi be the available supply of resource Ri . Let ai j be the amount of resource Ri used in operating activity A j at unit intensity. Let c j be the net value to the company of operating activity A j at unit intensity. The problem is to choose the intensities which the various activities are to be operated to maximize the value of the output to the company subject to the given resources. Let x j be the intensity at which A j is to be operated. The value of such an activity allocation is n

cjxj

(2.9)

j=1

The amount of resource Ri used in this activity allocation must be no greater than the supply, bi ; that is n ai j x j ≤ bi , i = 1, ..., m (2.10) j=1

It is assumed that we cannot operate an activity at negative intensity; that is, x1 ≥ 0, x2 ≥ 0, ..., xn ≥ 0.

(2.11)

The activity analysis problem is to maximize (2.9) subject to (2.10) and (2.11). Problem 4 The Optimal Assignment Problem Assume there are I persons available for J jobs. The value of person i working 1 day at job j is ai j , for i = 1, ..., I , and j = 1, ..., J . The problem is to choose an assignment of persons to jobs to maximize the total value. An assignment is a choice of numbers, xi j , for i = 1, ..., I , and j = 1, ..., J where xi j represents the proportion of person is time that is to be spent on job j. Thus, J

xi j ≤ 1, i = 1, ..., I

(2.12)

xi j ≤ 1, j = 1, ..., J

(2.13)

j=1 I i=1

and xi j ≥ 0, i = 1, ..., I, j = 1, ..., J

(2.14)

Equation (2.12) reflects the fact that a person cannot spend more than 100% of his time working, (2.13) means that only one person is allowed on a job at a time, and

2.2 Linear Programming

43

(2.14) says that no one can work a negative amount of time on any job. Subject to (2.12), (2.13) and (2.14), we wish to maximize the total value, I J

ai j xi j .

i=1 j=1

This is a standard maximum problem with m = I + J and n = I J . Mathematically, the general LP problem may be formulated as follows [1]: max z = s.t.

n j=1

The function z =

n

n

cjxj

j=1

ai j x j ≤ bi , i = 1, 2, ..., m,

(2.15)

x j ≥ 0, j = 1, 2, ..., n.

c j x j to be maximized is called an objective function. The vector

j=1 (b1 , b2 , ..., bm )T

is called the right-hand-side vector or resources vector. The b= vector x = (x1 , x2 , ..., xn )T is called the decision variables vector. The vector c = (c1 , c2 , ..., cn ) is called the cost coefficients the coefficients of decision variables or in the objective function and matrix A = ai j m×n is called the constraints matrix or the matrix of technical coefficients. Using this notation, the LP problem (2.15) can be formulated in a vector-matrix form as follows: max z = cx s.t. Ax ≤ b, x ≥ 0.

(2.16)

In conventional LP problems, it is assumed that the data have precise values. This means that the elements of the problem are crisp numbers, inequality “≤” is defined in the crisp sense and “max” is a strict imperative.

2.2.3 Primal Simplex Algorithm Consider the linear programming problem which is defined in (2.16).Without loss of generality, we may assume that the general form of the linear programming problems is as follows: max z = cx s.t. Ax = b (2.17) x ≥ 0,

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Table 2.1 The linear programming simplex tableau

or equivalently,

Basis

xB

xN

R.H.S.

z

0

y00 = c B B −1 b

xB

I

z N − cN = c B B −1 N − cN Y = B −1 N

max z = c B x B + c N x N s.t. Bx B + N x N = b x B ≥ 0, x N ≥ 0,

y0 = B −1 b

(2.18)

where x B and x N are respectively the vector of basic and non-basic variables, and also c B and c N are respectively the coefficients vector of the basic and non-basic variables in the objective function, B and N are also basic and non-basic matrix, respec tively. Hence, we have x B + B −1 N x N = B −1 b. Therefore, z + c B B −1 N − c N x N = c B B −1 b. With x N = 0, we have x B = B −1 b = y0 , and z = c B B −1 b. Thus, we rewrite the above linear programming problem as in Table 2.1. Definition 2.1 Any x satisfying the set of constraints (2.17) of the LP problem is called a feasible solution. Let Q be the set of all feasible solutions of the LP problem. Then, we say that x0 ∈ Q is an optimal feasible solution for the LP problem if cx0 ≥ cx for all x ∈ Q. Consider the system Ax = b and x ≥ 0, where A is an m × n and b is an m vector. Suppose that rank (A, b) = rank (A) = m. Partition A, after possibly rearranging the columns of A, as [B, N ] where B, m × m, is nonsingular. It is apparent that x B = T T x B1 , ..., x Bm = B −1 b, x N = 0 is a solution of Ax = b. The point x = x BT , x NT , where x N = 0 is called a basic solution of the system. If x B ≥ 0, then x is called a basic feasible solution of the system and the corresponding fuzzy objective value is z = c B x B , where c B = c B1 , ..., c Bm . Now corresponding to every index j, 1 ≤ j ≤ n, define y j = B −1 a j and z j = c B y j . Observe that for any basic index j = Bi , 1 ≤ i ≤ m, we have z j − c j = c B B −1 a j − c j = c B ei − c j = c j − c j = 0,

(2.19)

where e j is the jth unit vector. Note that B is called the basic matrix and N is called the non-basic matrix. The corresponding of x B are called basic variables, and the components of x N are called nonbasic variables. The following theorem concerns the so-called non-degenerate LP problems, where every basic variable corresponding to every feasible basis B is positive. Theorem 2.1 Let the LP problem be non-degenerate. A basic feasible solution x B = B −1 b, x N = 0 is optimal to (2.17) if and only if z j ≥ c j , for all j = 1, ..., n.

2.2 Linear Programming

45

T Proof Suppose that x∗ = x BT , x NT is a basic feasible solution to (2.17), where x B = B −1 b, x N = 0. Then, z ∗ = c B x B = c B B −1 b. On the other hand, for every feasible solution x, we have b = Ax = Bx B + N x N . Hence, we obtain z = cx = c B x B + c N x N c B B −1 a j − c j x j = c B B −1 b −

(2.20)

j= Bi

Then z = z∗ −

z j − cj xj

(2.21)

j= Bi

The proof can now be completed using (2.18) and Theorem 2.3 given in the following. Remark 2.1 Table 2.1 gives all the information needed to proceed with the simplex method. The cost row in Tabel 2.1 is y0T = c B B −1 A − c, where y0 j = c B B −1 a j − c j = z j − c j , 1 ≤ j ≤ n with y0 j = 0 for j = Bi , 1 ≤ i ≤ m. According to the optimality condition (Theorem 2.1), we are at the optimal solution if y0 j ≥ 0 for all j = Bi , 1 ≤ i ≤ m. On the other hand, if y0k < 0, for some k = Bi , 1 ≤ i ≤ m, then the problem is either unbounded or an exchange of a basic variable x Br and the nonbasic variable xk can be made to increase the objective value (under non-degeneracy assumption). The following results help us to utilize the primal simplex algorithm. Theorem 2.2 If in an LP simplex tableau, there is a column k (not in basis) so that y0k = z k − ck < 0 and yik ≤ 0, i = 1, ..., m, then the LP problem is unbounded. Theorem 2.3 If in an LP simplex tableau, a non-basic index k exist such that y0k = z k − ck < 0 and there exist a basic index Bi such that yik < 0, then a pivoting row r can be found so that pivoting on yr k yields a feasible tableau with a corresponding non-decreasing (increasing under non-degeneracy) objective value. Remark 2.2 If k exists such that y0k < 0 and the problem is not unbounded, then r can be chosen so that yi0 yr 0 |yik > 0 = min (2.22) 1≤i≤m yik yr k in order to replace x Br in the basis by xk , resulting in a new basis Bˆ = a B1 , ..., a Br −1 , ak , a Br +1 , ..., am The new basis is primal feasible and the corresponding objective value is nondecreasing (increasing under non-degeneracy assumption). It can be shown that the new simplex tableau is obtained by pivoting on yr k , that is, doing Gaussian elimination on the kth column using the pivot row r , with the pivot yr k , to transform the kth column to the unit vector er . It is easily seen that the new objective value is yˆ00 = y00 − y0k yyrr 0k ≥ y00 , since y0k ≤ 0 and (if the problem is non-degenerate) yr 0 > 0, then yˆ00 > y00 . yr k

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2 Fuzzy Linear Programming

We now describe the pivoting strategy which will be used in the primal simplex. If xk enters the basis and x Br leaves the basis, the pivoting on yr k in the simplex tableau is carried out as follows (a) Divide row r by yr k . (b) For i = 1, ..., m and i = r , update the ith row by adding to it −yik times the new r th row. We now present the simplex algorithm for the LP problem. Algorithm 2.1 Primal simplex method for the LP problem Assumption: A basic feasible solution with basis B and the corresponding simplex tableau is at the hand. Step 1: The basic feasible solution is given by x B = y0 = B −1 b and x N = 0. The objective value is z = y00 = c B B −1 b. Step 2: Calculate y0 j = z j − cj = c B B −1 a j − c j , j = 1, ..., n, j = Bi , i = 1, ..., m. Let y0k = min j=1,...,n y0 j . If y0k ≥ 0, then stop; the current solution is optimal. Step 3: If yk ≤ 0, then stop; the problem is unbounded. Otherwise, determine an index r corresponding to a variable x Br leaving the basis as follows yr 0 = min 1≤i≤m yr k

yi0 |yik > 0 yik

Step 4: Pivot on yr k and update the simplex tableau. Go to step (2).

2.2.4 Starting Solution for the Primal Simplex Algorithm A. The initial basic feasible solution As in the last section we mentioned the primal simplex method starts with a basic feasible solution and moves to an improved basic feasible solution, until the optimal point is reached or else unboundedness of the objective function is verified. However, in order to initialize the primal simplex method, a basis B with y0 = B −1 b ≥ 0 must be available. We shall show that the primal simplex method can always be initiated with a very simple basis, namely, the identity. We describe two following cases. Case 1: Suppose that the constraints are of the form Ax ≤ b, x ≥ 0, where A is an m × n crisp matrix and b is an m × 1 nonnegative vector. By adding the fuzzy slack vector xa , the constraints can be put in the following standard form: Ax + xa = b, x ≥ 0, xa ≥ 0. Note that the new m × (m + n) constraint matrix (A|I ) has rank m, and a basic feasible solution of this system is at hand, by letting xa be the basic vector, and x˜ be the nonbasic vector. Hence, at this starting basic feasible solution, xa = b, x = 0, and the primal simplex method now can be applied.

2.2 Linear Programming

47

Case 2: In many situations, finding a starting basic feasible solution is not straightforward as the case just described. To illustrate, suppose that the constraints are of the form Ax ≤ b, x ≥ 0, such that at least one of the elements of b is negative. In this case, after introducing the slack vector xa , we cannot let x = 0, because xa = b violates the non negativity requirements. Another situation occurs when the constraints are of the form Ax ≥ b, x ≥ 0, where for at least an i, i = 1, ..., m we have bi > 0. After subtracting the slack vector xa we get Ax − xa = b, x ≥ 0, xa ≥ 0. Again, there is no obvious way of picking a basis B from matrix (A| − I ) with y0 = B −1 b ≥ 0. Again consider the linear programming problem as defined in (2.17). It is simple to see that any LP problem can be transformed into a problem of the following form max z = cx s.t. Ax = b x ≥0

(2.23)

where b ≥ 0 (if b ≤ 0, the ith row can be multiplied by −1). This can be accomplished by introducing slack variables and by simple manipulations of the constraints and variables. If A contains an identity matrix, then an immediate fuzzy basic feasible solution is at hand, by simply letting B = I , and since b ≥ 0, then b = B −1 b ≥ 0. Otherwise, something else must be done. B. Artificial variables After manipulating the constraints and introducing slack variables, suppose that the constraints are put in the format Ax = b, x ≥ 0, where A is an m × n matrix and b ≥ 0 is an m × 1 vector. Furthermore, suppose that A has no identity submatrix (if A has an identity submatrix then we have an obvious to get a starting basic feasible solution). In this case, we shall resort to the artificial variables to get a starting basic feasible solution, and then use the primal simplex method itself and get rid of these artificial variables. To illustrate, suppose that we change the restrictions by adding an artificial vector xa leading to the system Ax + xa = b, x ≥ 0, xa ≥ 0. Note that by construction, we created an identity matrix corresponding to the artificial vector. This gives an immediate basic feasible solution of the new system, namely, xa = b, and x = 0. Even though we now have a starting basic feasible solution and the primal simplex method can be applied, we have in effect changed the problem. In order to get back to our original problem, we must force these artificial variables to zero, because Ax = b if and only if Ax + xa = b, with xa = 0. In other words, artificial variables are only a tool to get the primal simplex method started; however, we must guarantee that these variables will eventually drop to zero, if at all possible. Remark 2.3 We emphasize that a slack variable is introduced to put the problem in equality form, and the slack variable can very well be positive, which means that the inequality holds as a strict inequality. Artificial variables, however, are not legitimate variables, and they are merely introduced to facilitate the initiation of the simplex

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2 Fuzzy Linear Programming

method. These artificial variables, however, must eventually drop to zero in order to attain feasibility in the original linear programming problem. C. Two-phase method There are various methods in the literature of linear programming that can be used to eliminate the artificial variables. One of these methods is to minimize the sum of the artificial variables. Here we propose this idea for linear programming problem. In fact, we define a new objective function which is the sum of the artificial variables in minimization form, subject to the constraints are Ax + xa = b, x ≥ 0 and xa ≥ 0. If the original linear programming problem has a feasible solution, then the optimal value of this problem is zero, where all the artificial variables drop to zero. More importantly, as the artificial variables drop to zero, they leave the basis, and legitimate variables enter instead. Eventually, all artificial variables leave the basis (this is not always the case, because we may have an artificial variable in the basis at level zero). The basis then consists of legitimate variables. In other words, we get a basic feasible solution for the original system Ax = b, x ≥ 0, and the simplex method can be stated with the original objective function cx. If, on the other hand, after solving this problem we have a positive artificial variable, then the original problem has no feasible solution. This procedure is called the two-phase method. In the first phase we reduce artificial variables to value zero or conclude that the original problem has no feasible solutions. In the former case, the second phase minimizes the original objective function starting with the basic feasible solution obtained at the end of Phase I. The two-phase method is outlined below. Phase I: Solve the following LP problem starting with the basic feasible x = 0 and xa = b min x0 = 1xa s.t. Ax + xa = b (2.24) x, xa ≥ 0 where 1 = (1, ..., 1) is 1 × m constant vector and xa = (x1a , . . . , xma )T . This problem is called a simplex Phase I problem. If at optimality xa = 0, then stop; the original LP problem has no feasible solutions. Otherwise, let the basic and nonbasic legitimate variables be x B and x N . Phase II: Solve the following LP problem starting with the basic feasible solution x B = B −1 b and x N = 0: max z = c B x B + c N x N s.t. x B + B −1 N x N = B −1 b xB, xN ≥ 0

(2.25)

The foregoing linear programming problem is of course equivalent to the original problem. Remark 2.4 In the solving process of problem (2.23), in fact, in Phase I we have artificially created a basic feasible solution, namely x = 0 and xa = b. There are

2.2 Linear Programming

49

three possible outcomes when we use the primal simplex method for solving problem (2.23) in Phase I. • (a) x0 is reduced to zero and no artificial variables remain in the optimal basis, i.e. we left with a basis consisting only of original variables. In this case, we simply delete the columns corresponding to the artificial variables, replace the objective function by the objective function of (2.23) after having expressed it in terms of the nonbasic variables and use Phase II. • (b) x0 > 0 at optimality. This means that the original linear programming problem (2.23) is infeasible. Indeed, if x is feasible in problem (2.23), then (x, xa = 0) is feasible in (2.24) with value x0 = 0. • (c) x0 is reduced to zero while some artificial variables remain in the basis. These m artificial variables must be at zero level since, for this solution, xi0 = 0. Suppose i=1

that the ith variable of the basis is artificial. We may pivot on any nonzero (not necessarily positive) element yi j of row i corresponding to a non-artificial variable x j . Since yi0 = 0, no change in the solution or in the objective value will result. We say that we are driving the artificial variables out of the basis. By original variables. We have thus reduced this case a. There is still one detail that needs considerations. We might be unsuccessful in driving one artificial variable out the basis, if yi j = 0 for j = 1, . . . , n. However, this means that we have arrived at a zero row in the original matrix by performing elementary row operations, implying that the constraint is redundant. We can delete this constraint and continue in Phase II with a basis of lower dimension. D. Big-M method As in the last part, we have been explained there are various methods that can be used to eliminate the artificial variables, when any initial basic feasible solution is not at hand. One of the convenient approach is bigM method. In fact, it is possible to combine the two phases of the two-phase method into a single procedure by bigM method. We are going to explain it briefly. Assume that the linear program in standard form as follows: max z = cx s.t. Ax = b (2.26) x ≥0 one forms the auxiliary problem as: max z = cx − M xa s.t. Ax + xa = b x, xa ≥ 0

(2.27)

where M = (M, . . . , M) is 1 × m constant vector and xa = (x1a , . . . , xma )T is a vector of artificial variables and M is a large constant. The term M xa serves as

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2 Fuzzy Linear Programming

a penalty term for nonzero xia s. Solve the problem (2.27) starting with the basic feasible x = 0 and xa = b: max z = cx − M xa s.t. Ax + xa = b (2.28) x, xa ≥ 0 Now if problem (2.27) is solved by the primal simplex method, we have the following results: (a) If an optimal solution is found and no artificial variables remain in the optimal basis (this means that all artificial variables are at zero level, that is, xa = 0 ), then the corresponding x is an optimal basic feasible solution to the original problem. (In this case, we simply delete the columns corresponding to the artificial variables). (b) If for every M > 0 an optimal solution is found with xa = 0, then the original problem is infeasible. Remark 2.5 For the objective function of problem (2.26) in the form of minimization, the auxiliary problem will be defined as follows: min z = cx + M xa s.t. Ax + xa = b x, xa ≥ 0

(2.29)

2.2.5 Duality In this subsection, we present the duality results which is one of the important tools in our discussion. As it is appeared in the literature of operations research, for every linear programming problem there is an associated problem which satisfies some important properties. We give this subsection in three parts. First, a formulation of the dual problem is given. Second, we explore the important relations between linear programming and its dual problem. Third, a dual simplex algorithm will be presented. For defining of the formulation of the dual problems, consider the linear programming (LP) problem as below: max z = cx s.t. Ax ≤ b (2.30) x ≥0 where c T = (c1 , . . . , cn )T ∈ Rn , b = (b1 , . . . , bm )T ∈ Rm , x = (x1 , . . . , xn )T ∈ Rn , A = ai j m×n ∈ Rm×n . We name this problem as primal linear problem. The above linear programming problem can be rewritten in the extended form as:

2.2 Linear Programming

51

max z = s.t.

n j=1

n

cjxj

j=1

ai j x j ≤ bi , i = 1, . . . , m

(2.31)

x j ≥ 0, j = 1, . . . , n,

now, define the dual of the linear programming problem as: min u = s.t.

m i=1

m

bi yi

i=1

ai j yi ≥ c j , j = 1, . . . , n

(2.32)

yi ≥ 0, i = 1, . . . , m,

or equivalently, in the matrix-vector form as: min u = yb s.t. y A ≥ c y≥0

(2.33)

We name the above problem as DLP problem. Note that there is exactly one dual variable (of the form ≥ 0) for each primal LP problem constraint of the form ≤ and exactly one dual constraint (of the form ≥) for each variable of the form ≥ 0 in LP problem. Now, we are going to discuss the relationships between the primal linear programming problem and its corresponding dual. Lemma 2.1 The dual of the problem (2.33) is the problem (2.30). Remark 2.6 Lemma 2.1 indicates that the duality results can be applied to any of the primal or dual problems posed as the primal problem. Theorem 2.4 (The weak duality property). If x0 and w0 are feasible solutions to LP and DLP problems, respectively, then cx0 ≤ w0 b. The following corollaries are immediate consequences of Theorem 2.4. Corollary 2.1 If x0 and w0 are feasible solutions to LP and DLP problems, respectively, and cx0 = w0 b, then x0 and w0 are optimal solutions to their respective problems. The following corollary relates unboundedness of one problem to infeasibility of the other. We use the definition below. Definition 2.2 We say LP problem (or DLP problem) is unbounded, if feasible solutions exist with arbitrary large (or small) value for the objective value.

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Corollary 2.2 If any one of the LP or DLP problem is unbounded, then the other problem has no feasible solution. We are now ready to present the strong duality result. Theorem 2.4 (Strong duality). If any one of the LP or DLP problem has an optimal solution, then both problems have optimal solutions and the two optimal values for the objective functions are equal. (In fact, if x∗ with x B∗ = B −1 b, and x N∗ = 0 is the optimal solution for the primal problem, then w∗ = c B B −1 , where B is the optimal basis, is an optimal solution of the dual problem.) Remark 2.6 We emphasize that the basic solution to the primal linear programming problem is a vector that is uniquely determined by the optimal basis B. The solution to the dual problem is a unique crisp solution, however, depending solely on the basis matrix B, with the objective value of the dual having the same value as the primal problem. We can now state the fundamental duality result. Theorem 2.5 (Fundamental theorem of duality). For any linear programming problem and its corresponding dual problem, exactly one of the following statements is true. 1. Both have optimal solutions x∗ and w∗ with cx∗ = w∗ b. 2. One problem is unbounded and the other is infeasible. 3. Both problems are infeasible. We now state and prove an important result of duality theory, commonly named as complementary slackness. Theorem 2.6 (Complementary slackness). Let x∗ and w∗ be any feasible solutions to primal linear programming problem and its corresponding dual problem. Then, x∗ and w∗ are respectively optimal if and only if (w∗ A − c)x∗ = 0, w∗ (b − Ax∗ ) = 0

(2.34)

Remark 2.7 The complimentary slackness conditions (2.34) is equivalent to (below, refers to the i-th row and refers to the j-th column of A) w∗ a j < c j ⇒ x j∗ = 0, or x j∗ > 0 ⇒ w∗ a j = c j , j = 1, . . . , n, a¯ i x∗ < bi ⇒ wi∗ = 0, or wi∗ > 0 ⇒ a¯ i x∗ = bi , i = 1, . . . , m. We introduce here a new dual method to solve the linear programming problem directly, working on the primal problem and its associated simplex tableau. Consider the following linear programming problem, max z = cx s.t. Ax ≤ b x ≥0

(2.35)

2.2 Linear Programming

53

where c T = (c1 , . . . , cn )T ∈ Rn , b = (b1 , . . . , bm )T ∈ Rm , x = (x1 , . . . , xn )T ∈ Rn , A = ai j m×n ∈ Rm×n . Now, we may rewrite (2.35) as follows: max z = cx + 0y s.t. Ax + y = b x ≥ 0, y ≥ 0

(2.36)

where y = (y1 , . . . , ym )T ∈ Rm . Define x¯ ∈ R(n+m) and c¯ ∈ Rn+m as x¯ j =

j = 1, . . . , n, xj, c¯ = y j−n , j = n + 1, . . . , n + m, j

j = 1, . . . , n, cj, 0, j = n + 1, . . . , n + m.

(2.37)

Suppose that a basic solution for (2.37) is given by x B = B −1 b, with the basis matrix B. Now, let z j = c¯ B B −1 a¯ j , y0 = B −1 b, where c¯ B = c¯ B1 , . . . , c¯ Bm , and a¯ j is the jth column of the coefficient matrix [A I ]. Let (x¯ B )r is the r th basic variable and y j = B −1 a¯ j . Suppose that for j = n + 1, . . . , n + m, we have z j − c¯ j ≥ 0

(2.38)

Define w = c¯ B B −1 , where w = (w1 , . . . , wm ). For j = 1, . . . , n, we have y0 j = z j − c¯ j = c¯ B B −1 a j − c j = wa j − c j

(2.39)

Therefore, for j = 1, . . . , n from z j − c j ≥ 0, it follows that wa j − c j ≥ 0. Then, wA ≥ c

(2.40)

On the other hand, using (2.38), we have, 0 ≤ z n+i − c¯n+i = c B B −1 ei − 0 = wei = wi , i = 1, ..., m

(2.41)

and hence, w≥0

(2.42)

yielding the dual feasibility. If yr 0 ≥ 0, for all r = 1, ..., m, then a fuzzy feasible solution for the linear programming problem is at hand. Moreover, we will have, c¯ x¯ = c¯ B y0 = c¯ B B −1 b = wb

(2.43)

and thus, by Corollary 2.1, establishing the optimality of x and w for the LP and DLP problems, respectively. Therefore, we have the following result. Corollary 2.3 The optimality criteria z j − c¯ j ≥ 0 for all j, for the LP problem is equivalent to the feasibility condition for the DLP problem. If, in addition, x¯

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corresponding to a basis B is primal feasible then x¯ is optimal for the LP problem and w = c¯ B B −1 is optimal to the DLP problem. Now, lets assume that the DLP problem is feasible and x, corresponding to a basis B, is dual feasible but primal infeasible (and hence not optimal). That is, we have, y0 j = z j − c¯ j ≥ 0, j = 1, ..., n + m and there exists at least one r such that yr 0 < 0. We can immediately deduce that the primal problem must be finite. Thus, the LP problem can be either infeasible (in which case, the DLP problem is unbounded), or it has an optimal solution. In what follows we will show how to work on row r of the simplex tableau corresponding to B, as the pivoting row, and either (1) detect infeasibility of the LP problem (or unboundedness of the DLP problem), or (2) find a column l, as the pivoting column, to pivot on yrl and obtain a new dual feasible tableau with a nondecreasing primal objective value. We explain these cases as Lemmas 2.2 and 2.3. Lemma 2.2 If in a dual feasible simplex tableau an r exists such that yr 0 < 0 and yr j ≥ 0, for all j, then the LP problem is infeasible. Lemma 2.3 If in a dual feasible simplex tableau, an r exists such that yr 0 < 0 and there exists a nonbasic index j such that yr j < 0, then a pivoting column l can be found so that pivoting on yrl < 0 will yield a dual feasible tableau with a corresponding nondecreasing objective value. Now, using the above results, we introduce a dual algorithm to solve the LP problem directly, making use of the dual feasible simplex tableau. Thus, we refer this algorithm as a dual simplex method. Algorithm 2.2 A dual simplex method Step 1: (Dual feasibility) Given a basis B for the LP problem such that y0 j = z j − c¯ j ≥ 0, for all j. Compute the simplex tableau. Step 2: If y0 ≥ 0, then Stop (the current solution is optimal) else select the pivot row r with yr 0 < 0. Step 3: If yr j ≥ 0 for all j, then Stop (the LP problem is infeasible) else select the pivot column l by the following minimum ratio test: y0l = min j= Bi yrl

yoj |yr j < 0 . yr j

Step 4: Pivot on yrl and go to step (2). Remark 2.8 One suggestion for choice of r in step (2) may be r so that yr 0 = min {yi0 } 1≤i≤m

2.2 Linear Programming

55

Remark 2.9 Pivoting on yrl in step (4) is the usual Gaussian elimination process that, using yrl , converts column l to the unit vector er yielding the new simplex tableau corresponding to the new basis (the one that is obtained by replacing column r of B with al ). In this subsection, we introduced the dual of linear programming problem and hence developed some duality results for the primal and dual problems. Using these results and the dual feasible primal simplex tableau, we introduced a dual simplex algorithm for solving the primal and the dual problems directly. We emphasize that the capabilities offered here will be useful for sensitivity (or post optimality) analysis using the primal simplex tableau too.

2.3 Fuzzy Linear Programming Linear programming (LP) is a mathematical technique for the optimal allocation of scarce resources to several competing activities on the basis of given criteria of optimality. In conventional LP problems, it is assumed that the data have precise values. However, the observed values of the data in real-life problems are often imprecise because of incomplete or non-obtainable information. In such situations, fuzzy sets theory is an appropriate approach to handle imprecise data in LP by generalizing the notion of membership in a set and this leads to the concept of fuzzy linear programming problems. Fuzzy Linear Programming (FLP) problems allow working with imprecise data and constraints, leading to more realistic models. They have often been used for solving a wide variety of problems in sciences and engineering [2– 8]. Fuzzy mathematical programming has been researched by a number of authors. One of the earliest works on fuzzy mathematical programming problems was first presented by Tanaka et al. [9] based on the fuzzy decision framework of Bellman and Zadeh [10]. Since Tanaka et al. [11] there have been a number of FLP models. In the literature, FLP has been classified into different categories, depending on how imprecise parameters are modeled by subjective preference-based membership functions or possibility distributions. In this section, we briefly introduce fuzzy linear programming problems and present the various views to the classification of FLP problems. There are generally five FLP classifications in the literature: • Zimmermann (1987)[12] has classified FLP problems into two categories: symmetrical and non-symmetrical models. In a symmetrical fuzzy decision there is no difference between the weight of the objectives and constraints while in the asymmetrical fuzzy decision, the objectives and constraints are not equally important and have different weights [13]. • Leung (1988)[14] has classified FLP problems into four categories: – a precise objective and fuzzy constraints – a fuzzy objective and precise constraints

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– a fuzzy objective and fuzzy constraints – robust programming. • Luhandjula (1989)[15] has classified FLP problems into three categories: – flexible programming – mathematical programming with fuzzy parameters – fuzzy stochastic programming. • Inuiguchi et al. (1990)[16] have classified FLP problems into six categories: – – – – – –

flexible programming possibilistic programming possibilistic LP using fuzzy max robust programming possibilistic programming with fuzzy preference relations possibilistic LP with fuzzy goals.

• Nasseri et al. [17–19] have divided FLP problems into three categories: – LP problems with fuzzy numbers in coefficients of constraints and objective function – LP problems with fuzzy variables and resources – LP problems with fuzzy coefficients in objective function, resources and fuzzy variables Some authors [20–22] have proposed different methods for solving FLP problems with inequality constraints where initially the FLP problem is converted into crisp LP problem and then the obtained crisp LP problem is solved to find the fuzzy optimal solution of the FLP problems. Other authors [23, 24] have proposed methods for solving FLP problems with equality constraints. However, the solutions to FLP problems with equality constraints are generally approximate [24]. In the past decade, researchers have discussed various properties of FLP problems and proposed an assortment of models. Zhang et al. [25] proposed an FLP with fuzzy numbers for the coefficients of objective functions. They introduced a number of optimal solutions for the FLP problems and developed a number of theorems for converting the FLP problems into multi-objective optimization problems. Stanciulescu et al. [26] proposed an FLP model with fuzzy coefficients for the objectives and the constraints. He used fuzzy decision variables with a joint membership function instead of crisp decision variables and linked the decision variables together to sum them up to a constant. He considered lower-bounded fuzzy decision variables that set up the lower bounds of the decision variables. He then generalized the method to lowerupper-bounded fuzzy decision variables that set up also the upper bounds of the decision variables. Ganesan and Veeramani [27] proposed an FLP model with symmetric trapezoidal fuzzy numbers. They proved fuzzy analogues of some important LP theorems and obtained some interesting results which in turn led to the solution for FLP problems without converting them into crisp LP problems. Ebrahimnejad [28] showed that the method proposed by Ganesan and Veeramani [27] stops in a

2.3 Fuzzy Linear Programming

57

finite number of iterations and proposed a revised version of their method that was more efficient and robust in practice. He also proved the absence of degeneracy and showed that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it also has an optimal fuzzy basic solution. Mahdavi-Amiri and Nasseri [29] developed some methods for solving FLP problems by introducing and solving certain auxiliary problems. They applied a linear ranking function to order trapezoidal fuzzy numbers and deduced some duality results by establishing the dual problem of the LP problem with trapezoidal fuzzy variables. Rommelfanger [30] showed that both the probability distributions and fuzzy sets should be used in parallel or in combination, to model imprecise data dependent on the real situation. Van Hop [31] presented a model to measure attainment values of fuzzy numbers/fuzzy stochastic variables and used these new measures to convert the FLP problem or the fuzzy stochastic linear programming problem into the corresponding deterministic LP problem. Mahdavi-Amiri and Nasseri [32] proposed an FLP model where a linear ranking function was used to rank order trapezoidal fuzzy numbers. They established the dual problem of the LP problem with trapezoidal fuzzy variables and deduced some duality results to solve the FLP problem directly with the primal simplex tableau. Ebrahimnejad [33] introduced a primal-dual algorithm for solving FLP problems by using the duality results proposed by Mahdavi-Amiri and Nasseri [29]. Ebrahimnejad [34] has also generalized the concept of sensitivity analysis in FLP problems by applying fuzzy simplex algorithms and using the general linear ranking functions on fuzzy numbers. Ghodousian and Khorram [35] studied the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max-min composition in which fuzzy inequality replaces ordinary inequality in the constraints. They showed that their method attains the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints. Tan et al. [36] developed an FLP extension of the general life cycle model using a concise and consistent linear model that makes identification of the optimal solution straightforward. Wu [37] derived the optimality conditions for FLP problems by proposing two solution concepts based on similar solution concept, called the non-dominated solution, in the multi-objective programming problem. Hosseinzadeh Lotfi et al. [24] considered fully fuzzy linear programming problems where all variable and parameters were triangular fuzzy numbers. They showed that there is no method in the literature for finding the fuzzy optimal solution of fully fuzzy linear programming problems and introduced a new method for solving fully fuzzy linear programming problems with equality constraints. They used the concept of the symmetric triangular fuzzy numbers and proposed an approach to defuzzify a general fuzzy quantity. They first approximated the fuzzy triangular numbers to its nearest symmetric triangular numbers with the assumption that all decision variables were symmetric triangular. They then converted every FLP model into two crisp complex LP models and used a special ranking for fuzzy numbers to transform their fully fuzzy linear programming problems model into a multi-objective LP where all variables and parameters were crisp. Kumar et al. [38] further studied the fully FLP problems

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with equality introduced by Hosseinzadeh Lotfi et al. [24] and proposed a new method for finding the fuzzy optimal solution in these problems. Gupta and Mehlawat [39] studied a pair of fuzzy primal-dual LP problems and calculated duality results using an aspiration level approach. Their approach is particularly important for FLP where the primal and dual objective values may not be bounded. Peidro et al. [40] used fuzzy sets and developed an FLP to model the supply chain uncertainties. Chen and Ko [41], Inuiguchi and Ramk [42] and Peidro et al. [40] have developed a number of FLP models to solve problems ranging from supply chain management to product development. Saati et al. [43] based on the fuzzy linear programming models which are appeared in the literature have been classified into the following seven groups: • Group 1: the FLP problems in this group involve fuzzy numbers for the coefficients of the decision variables in the objective function (e.g., [29, 37, 44]). max z˜ ≈ cx ˜ s.t. Ax ≤ b, x ≥ 0.

(2.44)

• Group 2: the FLP problems in this group involve fuzzy numbers for the coefficients of the decision variables in the constraints and the right-hand-side of the constraints (e.g., [45]). max z = cx ˜ b, ˜ (2.45) s.t. Ax x ≥ 0. • Group 3: the FLP problems in this group involve fuzzy numbers for the coefficients of the decision variables in the objective function, the coefficients of the decision variables in the constraints and the right-hand-side of the constraints (e.g., [32, 46, 47]). max z˜ ≈ cx ˜ ˜ ˜ (2.46) s.t. Ax b, x ≥ 0. • Group 4: the FLP problems in this group involve fuzzy numbers for the decision variables and the right-hand-side of the constraints (e.g., [29]). max z˜ ≈ c x˜ ˜ s.t. A x˜ b, ˜ x˜ 0.

(2.47)

• Group 5: the FLP problems in this group involve fuzzy numbers for the decision variables, the coefficients of the decision variables in the objective function and the right-hand-side of the constraints (e.g., [33]).

2.3 Fuzzy Linear Programming

59

max z˜ ≈ c˜ x˜ ˜ s.t. A x˜ b, ˜ x˜ 0.

(2.48)

• Group 6: the FLP problems in this group involve fuzzy numbers in the decision variables, the coefficients of the decision variables in the constraints and the righthand-side of the constraints (e.g., [48]). max z˜ ≈ c x˜ ˜ s.t. A˜ x˜ b, ˜ x˜ 0.

(2.49)

• Group 7: the FLP problems in this group, so-called fully fuzzy linear programming (FFLP) problems, involve fuzzy numbers in the decision variables, the coefficients of the decision variables in the objective function, the coefficients of the decision variables in the constraints and the right-hand-side of the constraints (e.g., [24]). max z˜ ≈ c˜ x˜ ˜ s.t. A˜ x˜ b, ˜ x˜ 0.

(2.50)

Remark 2.10 Although the FFLP group is the general case of the FLP models, it may not be suitable for all FLP problems with different assumptions and sources of fuzziness. In the other hand, for the problems in Group 6 similar to the problems in Group 7 there is not suitable approaches which are accepted by a wide range of the researchers, so we omit them in our discussion and hope to add the new version of the solving approaches which will be appeared in the literature in the future versions of this book. In the next chapters we will focus on the models and solving methods of the FLP problems in three main categories: 1) Fuzzy Numbers Linear Programming (FNLP), 2) Linear Programming with Fuzzy Variables (FVLP) and 3) Semi-Fully Fuzzy Linear Programming (SFFLP). In fact, all five first groups of FLP problems which are mentioned in the above discussion will be include in these three categories (Chaps. 3–5).

References 1. Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear Programming and Network Flows. WileyInterscience, Wiley, Hoboken (2005) 2. Ebrahimnejad, A.: A new link between output-oriented BCC model with fuzzy data in the present of undesirable outputs and MOLP. Fuzzy Inf. Eng. 3(2), 113–125 (2011) 3. Ebrahimnejad, A., Karimnejad, K., Alrezaamiri, H.: Particle swarm optimisation algorithm for solving shortest path problems with mixed fuzzy arc weights. Int. J. Appl. Decis. Sci. 8(2), 203–222 (2015)

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4. Ebrahimnejad, A., Tavana, M., Alrezaamiri, A.: A novel artificial bee colony algorithm for shortest path problems with fuzzy arc weights. Measurement 93, 48–56 (2016) 5. Hatami-Marbini, A., Ebrahimnejad, A., Lozano, S.: Fuzzy efficiency measures in data envelopment analysis using lexicographic multiobjective approach. Comput. Ind. Eng. 105, 362–376 (2017) 6. Melián, B., Verdegay, J.L.: Fuzzy optimization models for the design of WDM networks. IEEE Trans. Fuzzy Syst. 16(2), 466–476 (2008) 7. Melián, B., Verdegay, J.L.: Using fuzzy numbers in network design optimization problems. IEEE Trans. Fuzzy Syst. 19(5), 797–806 (2011) 8. Nasseri, S.H., Bavandi, S.: Amelioration of Verdegay’s approach for fuzzy linear programs with stochastic parameters. Iran. J. Manag. Stud. 11(1), 71–89 (2018) 9. Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming. J. Cybern. 3(4), 37–46 (1974) 10. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manage. Sci. 17(4), 141–164 (1970) 11. Tanaka, H., Ichihashi, H., Asai, K.: A formulation of fuzzy linear programming problems based on comparison of fuzzy numbers. Control Cybern. 13, 186–194 (1984) 12. Zimmermann, H.J.: Fuzzy Sets. Decision Making and Expert Systems. Kluwer Academic Publishers, Boston (1987) 13. Amid, A., Ghodsypour, S.H., O’Brien, C.: Fuzzy multiobjective linear model for supplier selection in a supply chain. Int. J. Prod. Econ. 104, 394–407 (2006) 14. Leung, Y.: Spatial Analysis and Planning under Imprecision. North-Holland, Amsterdam (1988) 15. Luhandjula, M.K.: Fuzzy optimization: an appraisal. Fuzzy Sets Syst. 30, 257–282 (1989) 16. Inuiguchi, M., Ichihashi, H., Tanaka, H.: Fuzzy programming: a survey of recent developments. In: Slowinski, R., Teghem, J. (eds.) Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainty, pp. 45–68. Kluwer Academic Publishers, Dordrecht (1990) 17. Nasseri, S.H., Ebrahimnejad, A.: A fuzzy primal simplex algorithm and its application for solving flexible linear programming problems. Eur. J. Ind. Eng. 4(3), 372–389 (2010) 18. Nasseri, S.H., Mahdavi-Amiri, N.: Some duality results on linear programming problems with symmetric fuzzy numbers. Fuzzy Inf. Eng. 1, 59–66 (2009) 19. Nasseri, S.H., Attari, H., Ebrahimnejad, A.: Revised simplex method and its application for solving fuzzy linear programming problems. Eur. J. Ind. Eng. 6(3), 259–280 (2012) 20. Allahviranloo, T., Hosseinzadeh Lotfi, F., Kiasary, M.K., Kiani, N.A., Alizadeh, L.: Solving full fuzzy linear programming problem by the ranking function. Appl. Math. Sci. 2(1), 19–32 (2008) 21. Buckley, J.J., Feuring, T.: Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming. Fuzzy Sets Syst. 109(1), 35–53 (2000) 22. Hashemi, S.M., Modarres, M., Nasrabadi, E., Nasrabadi, M.M.: Fully fuzzified linear programming, solution and duality. J. Intell. Fuzzy Syst. 17(3), 253–261 (2006) 23. Dehghan, M., Hashemi, B., Ghatee, M.: Computational methods for solving fully fuzzylinear systems. Appl. Math. Comput. 179(1), 328–343 (2006) 24. Hosseinzadeh Lotfi, F., Allahviranloo, T., Alimardani Jondabeh, M., Alizadeh, L.: Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Appl. Math. Model. 33(7), 3151–3156 (2009) 25. Zhang, G., Wu, Y.H., Remias, M., Lu, J.: Formulation of fuzzy linear programming problems as four-objective constrained optimization problems. Appl. Math. Comput. 139, 383–399 (2003) 26. Stanciulescu, C., Fortemps, Ph, Installá, M., Wertz, V.: Multiobjective fuzzy linear programming problems with fuzzy decision variables. Eur. J. Oper. Res. 149(3), 654–675 (2003) 27. Ganesan, K., Veeramani, P.: Fuzzy linear programs with trapezoidal fuzzy numbers. Ann. Oper. Res. 143(1), 305–315 (2006) 28. Ebrahimnejad, A.: Some new results in linear programs with trapezoidal fuzzy numbers: finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method. Appl. Math. Model. 35(9), 4526–4540 (2011a)

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29. Mahdavi-Amiri, N., Nasseri, S.H.: Duality results and a dual simplex method for linear programming with trapezoidal fuzzy variables. Fuzzy Sets Syst. 158, 1961–1978 (2007) 30. Rommelfanger, H.: A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values. Fuzzy Sets Syst. 158(17), 1892–1904 (2007) 31. Van Hop, N.: Solving linear programming problems under fuzziness and randomness environment using attainment values. Inf. Sci. 177(14), 2971–2984 (2007) 32. Mahdavi-Amiri, N., Nasseri, S.H.: Duality in fuzzy number linear programming by use of a certain linear ranking function. Appl. Math. Comput. 180(1), 206–216 (2006) 33. Ebrahimnejad, A., Nasseri, S.H., Hosseinzadeh Lotfi, F., Soltanifar, M.: A primal-dual method for linear programming problems with fuzzy variables. Eur. J. Ind. Eng. 4(2), 189–209 (2010) 34. Ebrahimnejad, A.: Sensitivity analysis in fuzzy number linear programming problems. Math. Comput. Model. 53, 1878–1888 (2011b) 35. Ghodousian, A., Khorram, E.: Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition. Inf. Sci. 178(2), 501–519 (2008) 36. Tan, R.R., Culaba, A.B., Aviso, K.B.: A fuzzy linear programming extension of the general matrix-based life cycle model. J. Clean. Prod. 16(13), 1358–1367 (2008) 37. Wu, H.C.: Optimality conditions for linear programming problems with fuzzy coefficients. Comput. Math. Appl. 55(12), 2807–2822 (2008) 38. Kumar, A., Kaur, J., Singh, P.: A new method for solving fully fuzzy linear programming problems. Appl. Math. Model. 35(2), 817–823 (2011) 39. Gupta, P., Mehlawat, M.K.: Bector-Chandra type duality in fuzzy linear programming with exponential membership functions. Fuzzy Sets Syst. 160(22), 3290–3308 (2009) 40. Peidro, D., Mula, J., Jiménez, M., Botella, M.: A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environment. Eur. J. Oper. Res. 205(1), 65–80 (2010) 41. Chen, L.H., Ko, W.C.: Fuzzy linear programming models for NPD using a four-phase QFD activity process based on the means-end chain concept. Eur. J. Oper. Res. 201(2), 619–632 (2010) 42. Inuiguchi, M., Ramík, J.: Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst. 111(1), 3–28 (2000) 43. Saati, S., Tavana, M., Hatami-Marbini, A., Hajiakhondi, E.: A fuzzy linear programming model with fuzzy parameters and decision variables. Int. J. Inf. Decis. Sci. 7(4), 312–333 (2015) 44. Maleki, H.R., Tata, M.: Mashinchi, M: Linear programming with fuzzy variable. Fuzzy Sets Syst. 109, 21–33 (2000) 45. Xinwang, L.: Measuring the satisfaction of constraints in fuzzy linear programming. Fuzzy Sets Syst. 122(2), 263–275 (2001) 46. Hatami-Marbini, A., Agrell, P., Tavana, M., Emrouznejad, A.: A stepwise fuzzy linear programming model with possibility and necessity relation’. J. Intell. Fuzzy Syst. 25(1), 81–93 (2013) 47. Hatami-Marbini, A., Tavana, M.: An extension of the linear programming method with fuzzy parameters. Int. J. Math. Oper. Res. 3(1), 44–55 (2011) 48. Saati, S., Hatami-Marbini, A., Tavana, M., Hajiahkondi, E.: A two-fold linear programming model with fuzzy data. Int. J. Fuzzy Syst. Appl. 2(3), 1–12 (2012)

Chapter 3

Fuzzy Number Linear Programming

3.1 Introduction The aim of this chapter is to study linear programming problem with fuzzy coefficients as one of the convenient model of fuzzy linear programs. In this way, we first introduce a general form of Fuzzy Number Linear programming (FNLP) problems and then give the fundamental concepts which are useful in throughout of the chapter. Then, some other standard methods and results such as primal, dual and primal-dual simplex algorithms, duality results, and etc. will be discussed. Sections 3.8 and 3.9 is assigned to some applications of FNLP problems in the real world such as Data Envelopment Analysis (DEA), Network Flows (NF) and Diet Problem (DP). Finally, the last of part this chapter is allocated to the sensitivity analysis.

3.2 Formulation of a Fuzzy Number Linear Programming A general form of Fuzzy Number Linear Programming (FNLP) problem is given in the below. Definition 3.1 An FNLP Problem is defined as follows: max z˜ ≈ cx ˜ ˜ b, ˜ s.t. Ax x ≥ 0,

(3.1)

where ≈ and mean equality and inequality with respect to a given linear ranking function R , A˜ = (a˜i j )m×n , c˜ = (c˜1 , . . . , c˜n ), b˜ = (b˜1 , . . . , b˜m )T and a˜ i j , b˜i , c˜ j ∈ F(R) for i = 1, . . . , m, j = 1, 2, . . . , n. © Springer Nature Switzerland AG 2019 S. H. Nasseri et al., Fuzzy Linear Programming: Solution Techniques and Applications, Studies in Fuzziness and Soft Computing 379, https://doi.org/10.1007/978-3-030-17421-7_3

63

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Definition 3.2 Any x which satisfies the set of constraints of FNLP is called a feasible solution. Let Q be the set of all feasible solutions of FNLP. We say that ˜ cx ˜ 0 for all x ∈ Q. x 0 ∈ Q is an optimal feasible solution for FNLP if cx Definition 3.3 We say that the real number a corresponds to the fuzzy number a, ˜ with respect to a given linear ranking function R, if a = R(a). ˜ The following theorem shows that any FNLP can be reduced to a linear programming problem (see [18, 19]). Theorem 3.1 The following Linear Programming (LP) problem and the FNLP problem in (3.1) are equivalent: max z = cx Ax ≤ b s.t. x ≥0

max z = or

s.t.

n

n

cjxj

j=1

ai j x j ≤ bi , i = 1, . . . , m

j=1

x j ≥ 0,

(3.2)

j = 1, . . . , n

where ai j , bi , c j are real numbers corresponding to the fuzzy numbers a˜ i j , b˜i , c˜ j with respect to a given linear ranking function R, respectively. Remark 3.1 The above theorem shows that the sets of all feasible solutions of FNLP and LP problems are the same. Also if x¯ is an optimal feasible solution for FNLP, then x¯ is an optimal feasible solution for LP problem. Corollary 3.1 If LP problem does not have a solution then FNLP problem does not have a solution either. For an illustration of Theorem 3.1, consider following example used by Maleki [19]. Example 3.1 Consider the following FNLP programming: max z˜ ≈ (2, 3, 1, 1)x1 + (3, 4, 1, 2)x2 s.t. (1, 2, 1, 1)x1 + (2, 3, 1, 2)x2 (5, 6, 2, 2) (2, 3, 1, 3)x1 + (1, 2, 1, 1)x2 (4, 6, 2, 1) x1 , x2 ≥ 0 where (a L , a U , a α , a β ) is a trapezoidal fuzzy number. We apply the ranking function which is defined in (1.19) to solve the above FNLP. The problem reduces to: max z = 5x1 + 7.5x2 s.t. 3x1 + 5.5x2 ≤ 11 6x1 + 3x2 ≤ 9.5 x1 , x2 ≥ 0

3.2 Formulation of a Fuzzy Number Linear Programming

65

We write the above linear programming problem in the standard form: max z = 5x1 + 7.5x2 s.t. 3x1 + 5.5x2 + x3 = 11 6x1 + 3x2 + x4 = 9.5 x1 , x2 , x3 , x4 ≤ 0 where new variables x3 and x4 are called slack variables. The optimal solution and the optimal value of the objective function, rounded to three decimal places, are respectively as: x1∗ = 0.802, x2∗ = 1.562 and z˜ ∗ = (6.292, 2.365, 8.656, 3.927).

3.3 A Fuzzy Primal Simplex Method for FNLP Problem Consider the FNLP problem as the standard form, max z˜ ≈ cx ˜ ˜ ≈ b, ˜ s.t. Ax x ≥ 0,

(3.3)

where the parameters of the problem are as defined in (3.1). ˜ where R is a linear ranking function. Let A = (ai j )m×n = (R(a˜ i j )) = R( A), Assume rank(A) = m. Partition A as [B N ] where B, m × m, is nonsingular. ˜ where B˜ is the fuzzy matrix in A˜ It is obvious that rank(B) = m and B = R( B), corresponding to B. Let y j be the solution to By = a j . It is apparent that the basic solution (3.4) x B = (x B1 , ..., x Bm )T = B −1 b, x N = 0 ˜ If x B 0, then the basic solution is is a solution of Ax = b, where b = R(b). feasible and the corresponding fuzzy objective value is: z˜ ≈ c˜ B x B , where c˜ B = (c˜ B1 , ..., c˜ Bm ). Now, corresponding to every nonbasic variable x j , ( j = 1, ..., n) , j = Bi , i = 1, ..., m, define the fuzzy variable z˜ j as z˜ j ≈ c˜ B y j ≈ c˜ B B −1 a j .

(3.5)

It is important to note that for the future discussions, we need to analyze a special kind of FNLP problem, having only fuzzy coefficients for the objective function, as follows: max z˜ ≈ cx ˜ s.t. Ax = b, (3.6) x ≥ 0,

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We name this problem as Reduced Fuzzy Numbers Linear Programming (RFNLP) problem. Maleki et al. [19] proved fuzzy analogous of some important theorems of linear programming that lead to a new method for solving RFNLP problems. Consider the RFNLP problem (3.6) in its standard form. Let A = ai j m×n and assume rank (A) = m. Partition A as [B, N ] where B, m × m, is nonsingular. It is obvious that rank (B) = m. Let y j be the solution to By = a j . It is apparent that the T basic solution x B = x B1 , ..., x Bm = B −1 b, x N = 0 is a solution of Ax = b. In fact, T T T x = x B , x N . If x B ≥ 0, then the basic solutionis feasible and the corresponding fuzzy objective value is: z˜ ≈ c˜ B x B , where c˜ B = c˜ B1 , ..., c˜ Bm . Here, B and N are called the basis matrix and nonbasis matrix, respectively. The components of x B and x N are called the basic variables and nonbaisc variables, respectively. If x B > 0, then x is called a nondegenerate basic feasible solution, and if at least one component of x B is zero, then x is called a degenerate basic feasible solution. The RFNLP problem (3.6) is said to be totally nondegenerate if every basis for (3.6) is nondegenerate. Now, dene the fuzzy variable z˜ j as z˜ j ≈ c˜ B y j ≈ c˜ B B −1 a j , for all j = 1, ..., n and suppose a basic feasible solution for problem (3.6) is at hand. Below, we state some important results concerning to the improving a basic feasible solution, unbounded criteria and optimality conditions (taken from [15, 19]). The result corresponds to the so-called nondegenerate problems, where all basic variables corresponding to every basis B are nonzero and hence positive. Theorem 3.2 If we have a basic feasible solution with fuzzy objective value z˜ such that z˜ k ≺ c˜k for some nonbasic variable xk , and meanwhile yk 0, then it is possible to obtain a new basic feasible solution with new fuzzy objective value z˜ new , that satisfies z˜ new z¯ . Theorem 3.3 If we have a basic feasible solution with z˜ k ≺ c˜k for some nonbasic variable xk , and meanwhile yk ≤ 0, then the problem (3.6) has an unbounded optimal solution. Theorem 3.4 If a basic solution x B = B −1 b, x N = 0 is feasible to (3.6) and z˜ j c˜ j for all j, 1 ≤ j ≤ n, then the basic solution is an optimal solution to (3.6). Theorem 3.5 Assume the FNLP is nondegenerate. A basic feasible solution x B = B −1 b, x N = 0 with basis matrix B is optimal to (3.6) if and only if z˜ j c˜ j for all j, ( j = 1, ..., n). Proof The “if” part has been stated and proved in [18]. Here, we prove the“only if” part. Suppose that x = (x B T x N T )T is an optimal feasible solution to (3.6), where x B = B −1 b, x N = 0, corresponding to an optimal basis B. Then, the corresponding optimal objective value is: ˜ ≈ c˜ B x B ≈ c˜ B B −1 b. z˜ ∗ ≈ cx Now, since x is feasible, we have x 0, and

(3.7)

3.3 A Fuzzy Primal Simplex Method for FNLP Problem

67

b = Ax = Bx B + N x N

(3.8)

Hence, we can rewrite (3.8) as follows: x B = B −1 b − B −1 N x N

(3.9)

Substituting (3.9) in (3.7), we obtain z˜ ≈ cx ˜ ≈ c˜ B x B + c˜ N x N ≈ c˜ B B −1 b − (c˜ B B −1 N − c˜ N )x N ≈ c˜ B B −1 b − (c˜ B B −1 a j − c˜ j )x j j= Bi

Thus, z˜ ≈ z˜ ∗ −

(˜z j − c˜ j )x j

(3.10)

j= Bi

Now, from (3.10) it is obvious that if for any nonbasic variable x j we have z˜ j ≺ c˜ j , then we can enter x j into the basis and obtain z˜ ∗ ≺ z˜ (because the problem is nondegenerate and x j > 0 in the new basis). This is contradiction to z˜ ∗ being optimal and hence we must have z˜ j c˜ j . Algorithm 3.1 A primal simplex algorithm for the RFNLP problem Initialization Step: Suppose a basic feasible solution with basis matrix B is at hand. Form the following initial tableau (Table 3.1). Main Step:

Step 1: Calculate y0 j = R y˜0 j = R c˜ B B −1 a j − c˜ j = R z˜ j − c˜ j for all nonbasic variables. Let

y0k = R ( y˜0k ) = R (˜z k − c˜k ) = min R z˜ j − c˜ j = min R y˜0 j j= Bi

j= Bi

where j = Bi means the index of nonbasic variable. If R (˜z k − c˜k ) ≥ 0, then stop; the current solution is optimal. Otherwise, go to step(2).

Table 3.1 The initial RFNLP simplex tableau Basis xB xN z z˜ xB

0 0˜ I

R (˜z N − c˜ N ) = R (c˜ B Y N − c˜ N ) z˜ N − c˜ N = c˜ B Y N − c˜ N YN

R.H.S.

y00 = R c˜ B B −1 b y˜00 = c˜ B B −1 b y0 = B −1 b

68

3 Fuzzy Number Linear Programming

Step 2: Let yk = B −1 ak . If yk ≤ 0, then stop; the problem is unbounded. Otherwise, determine the index of the variable x Br leaving the basis as follows: yr 0 = min 1≤i≤m yr k

yi0 |yik > 0 . yik

Step 3: Update the tableau by pivoting at yr k . Update the basic and nonbasic variables where xk enters the basis and x Br leaves the basis, and go to step (1). Remark 3.2 While solving the RFNLP problem (3.6) by the Algorithm 3.1, a degenerate pivot step may occur. The next pivot step in the algorithm also turns out to be a degenerate pivot step. Actually there might be a sequence of consecutive degenerate pivot steps. In all these steps of basic feasible solution of the problem and the objective value do not change at all. After several of these degenerate pivot steps, the algorithm might return to the basis that started this sequence of degenerate pivot steps, completing a cycle. The algorithm can now go through the same cycle again and again indefinitely without ever reaching a basis satisfying the optimality condition. This phenomenon is known as cycling under degeneracy. Throughout this book, we assume that cycling does not occur.

3.4 Duality in Fuzzy Number Linear Programming 3.4.1 Formulation of the Dual Problem For the following FNLP,

max cx ˜ ˜ b, ˜ s.t. Ax x ≥ 0,

(3.11)

define the dual fuzzy number linear programming problem as: min w b˜ s.t. w A˜ c, ˜ w ≥ 0.

(3.12)

Note that there is exactly one dual variable (of the form 0) for each FNLP constraint of the form and exactly one dual constraint (of the form ) for each variable of the form in FNLP. Example 3.2 Consider the given FNLP in Example 3.1 The dual to this problem follows:

3.4 Duality in Fuzzy Number Linear Programming

69

min u˜ ≈ (5, 6, 2, 2)w1 + (4, 6, 2, 1)w2 s.t. (1, 2, 1, 1)w1 + (2, 3, 1, 3)w2 (2, 3, 1, 1) (2, 3, 1, 2)w1 + (1, 2, 1, 1)w2 (3, 4, 1, 2) w1 , w 2 ≥ 0 Now if we apply the Yager linear ranking function which is defined in (1.19), we have min u = 11w1 + 9.5w2 s.t. 3w1 + 6w2 ≥ 5 5.5w1 + 3w2 ≥ 7.5 w1 , w2 ≥ 0 The optimal solution, rounded to three decimal places, is w1∗ = 1.250, w2∗ = 0.208 and u˜ ∗ = (7.083, 8.750, 2.917, 2.708). Applying the Yager ranking function in (1.19), we then obtain R(u˜ ∗ ) = 15.729. On the other hand, from Example 3.1, we have R(˜z ∗ ) = 15.729. Remark 3.3 It is important to note that the FNLP may have alternative optimal solutions, but the value of the ranking function for the fuzzy value of the objective function corresponding to all optimal solutions is unique.

3.4.2 The Relationships Between FNLP and Dual FNLP We shall discuss here the relationships between the fuzzy number linear programming problem and its corresponding dual. Lemma 3.1 Dual of dual FNLP is FNLP. Proof Use Lemma 1.1 and the definition of dual FNLP. Remark 3.4 Lemma 3.1 indicates that the duality results can be applied to any of the primal or dual problems posed as the primal problem. Theorem 3.6 (The Weak Duality Property.) If x0 and w0 are feasible solutions to ˜ FNLP and dual FNLP, respectively, then cx ˜ 0 w0 b. ˜ 0 b˜ on the left by w0 ≥ 0 and w0 A˜ c˜ on the right by Proof Multiplying Ax ˜ 0 w0 b. ˜ ˜ 0 w0 Ax x0 ≥ 0, we get cx Remark 3.5 The value of the ranking function for the fuzzy value of the objective function at any feasible solution to FNLP is always lower than or equal to the value of the ranking function for the fuzzy value of the objective function for any feasible solution to dual FNLP ©.

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3 Fuzzy Number Linear Programming

As an illustration of the application of this theorem, suppose that in Examples 3.1 and 3.2, we select the feasible FNLP and dual FNLP solutions x0 = (1, 1)T and w0 = (2, 1). Then cx ˜ 0 = (5, 7, 2, 3) and w0 b˜ = (14, 18, 6, 5). Now if we apply ˜ = 31.5. Thus at the the ranking function (R), we obtain R(cx ˜ 0 ) = 12.5 < R(w0 b) optimal solution, the value of the ranking function for the fuzzy value of the objective function lies between 12.5 and 31.5. The following corollaries are immediate consequences of Theorem 3.6. Corollary 3.2 If x0 and w0 are feasible solutions to FNLP and dual FNLP, respec˜ then x0 and w0 are optimal solutions to their respective tively, and cx ˜ 0 ≈ w0 b, problems. The following corollary relates unboundeness of one problem to infeasibility of the other. We use the definition below. Definition 3.4 We say FNLP (or dual FNLP) is unbounded if feasible solutions exist with arbitrary large (or small) ranking function for the fuzzy objective value. Corollary 3.3 If either problems is unbounded, then the other problem has no feasible solution. We now state and prove the main duality result. Theorem 3.7 (Strong Duality.) If any one of the FNLP or dual FNLP has an optimal solution, then both problems have optimal solutions and the two optimal values of ranking functions for the fuzzy objective values are equal. Proof Because of Lemma 3.1, it suffices to show the result assuming the existence of optimal solution for the FNLP. Assume that the FNLP has an optimal solution, ˜ b. ˜ The and rank(A) = m. Let y 0 be the slack variables for the constraints Ax new equivalent problem to the FNLP is: max z˜ ≈ cx ˜ +0y ˜ + y ≈ b, ˜ s.t. Ax x ≥ 0, y ≥ 0.

(3.13)

Assume B is the basis matrix and x∗ = (x B T 0)T is the basic optimal solution corresponding to the FNLP. From Theorem 3.5 we have, c B B −1 a j − c j 0,

j = 1, ..., n, n + 1, ..., n + m,

or equivalently, c B B −1 a j c j ,

j = 1, ..., n

3.4 Duality in Fuzzy Number Linear Programming

c B B −1 ei 0,

71

i = 1, ..., m.

Hence we must have: c B B −1 A c c B B −1 0. Thus, ˜ R(c) ˜ c B B −1 R( A) c B B −1 0 Let w∗ = c B B −1 . Using the above inequality, we can write, w∗ A˜ c˜ w∗ 0 Thus, w∗ is a feasible solution to the dual FNLP and ˜ = w∗ b = c B B −1 b = c B x B = cx∗ = R(c)x ˜ ∗, w∗ R(b) and hence ˜ ∗. w∗ b˜ ≈ cx Therefore, the result follows immediately from Corollary 3.3. For an illustration of the above theorem consider the FNLP and dual FNLP given in Examples 2.1 and 2.2, respectively. We see that the basic optimal solution

11for 3 2 T T . the FNLP is x B = (x1 , x2 ) = (0.802, 1.562) , with the basis matrix B = 6 3 Hence, if we let w = c B B −1 , we obtain w = (1.250, 0.208), which is equal to the optimal solution of the dual FNLP. Using the results of the lemmas, corollaries, and remarks above we obtain the following important duality result. Theorem 3.8 (Fundamental Theorem of Duality.) For any FNLP and its corresponding dual FNLP, exactly one of the following statements is true. ˜ ˜ ∗ ≈ w∗ b. 1. Both have optimal solutions x∗ and w∗ with cx 2. One problem is unbounded and the other is infeasible. 3. Both problems are infeasible. Proof Using the above results, (1) and (2) are obviously correct. We show (3) by an example.

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3 Fuzzy Number Linear Programming

Example 3.3 Consider FNLP and its corresponding dual FNLP as follows: max z˜ ≈ (1, 2, 5, 1)x1 + (−1, 0, 2, 6)x2 s.t. (1, 2, 5, 1)x1 + (−1, 1, 3, 1)x2 (−1, 1, 4, 2) (0, 1, 6, 2)x1 − (−1, 1, 3, 1)x2 (−2, −1, 3, 7) x1 , x2 ≥ 0 and min u˜ ≈ (−1, 1, 4, 2)w1 + (−2, −1, 3, 7)w2 s.t. (1, 2, 5, 1)w1 + (0, 1, 6, 2)w2 (1, 2, 5, 1) (−1, 1, 3, 1)w1 − (−1, 1, 3, 1)w2 (−1, 0, 2, 6) w1 , w 2 ≥ 0 We clearly see that both problems are infeasible.

3.4.3 Complementary Slackness Property in FNLP An important corollary of the strong duality theorem is known as the complimentary slackness theorem. When problems in mathematical economics are formulated as LPs, this theorem can be interpreted as indicating an economic stability when optimality conditions are satisfied. The complementary slackness theorem does not say anything the values of unrestricted variables (corresponding to equality constraints in the other problems) in a pair of optimal feasible solution of the primal and dual problems, respectively. It is concerned only with nonnegative variables of one problem and the slack variables corresponding to the associated inequality constraints in the other problems of the primal and dual linear programming problems [9]. Theorem 3.9 (Complementary Slackness.) Let x∗ and w∗ be any feasible solutions to FNLP problem and its corresponding dual problem. Then x∗ and w∗ are respectively optimal if and only if ˜ ∗ − b) ˜ ≈ 0. ˜ ∗ ≈ 0, ˜ w∗ ( Ax ˜ (c˜ − w∗ A)x

(3.14)

Proof Suppose that x∗ and w∗ are feasible solutions to FNLP and dual FNLP prob˜ ∗≤ ˜ ∗ b˜ and w∗ A˜ c˜ which imply gives R( A)x lems, respectively. Therefore, Ax ˜ ˜ ˜ ˜ ˜ respectively. Multiplying R( A)x∗ ≤ R(b) on the left by R(b) and w∗ R( A) ≤ R(c), ˜ ∗ ≤ w∗ R(b). ˜ Also, Multiplying w∗ R( A) ˜ ≤ R(c) ˜ on the right w∗ ≥ 0 yields w∗ R( A)x ˜ ∗ ≤ R(c)x ˜ ∗ . Therefore, we will have by x ∗ ≥ 0 yields w∗ R( A)x ˜ ≤ w∗ R( A)x ˜ ∗ ≤ R(c)x ˜ ∗. w∗ R(b)

(3.15)

3.4 Duality in Fuzzy Number Linear Programming

73

On the other hand, since x∗ and w∗ are optimal solutions to the primal and dual ˜ = R(c)x ˜ ∗ , and (3.15) problems respectively, then by Theorem 3.7, we have w∗ R(b) is written as ˜ = w∗ R( A)x ˜ ∗ = R(c)x ˜ ∗. (3.16) w∗ R(b) ˜ ∗ ≈ 0˜ and w∗ (R( A)x ˜ ∗− From (3.16) we immediately have (R(c) ˜ − w∗ R( A))x ˜ ∗ − b) ˜ ≈ 0, ˜ ∗ ≈ 0˜ and w∗ ( Ax ˜ ≈ 0˜ which imply (c˜ − w∗ A)x ˜ respectively. R(b)) ˜ − R(c))x ˜ and ˜ ∗ ≈ 0, The converse of the theorem follows from the fact that (w∗ R( A) ˜ − R( A)x ˜ ∗ ) ≈ 0˜ imply that R(c)x ˜ or c x˜∗ ≈ w∗ b. ˜ ˜ ∗ = w∗ R(b) w∗ (R(b) Remark 3.6 The conditions (3.14) is equivalent to (below, a i refers to the ith row and a j refers to the jth column of A): w∗ a˜ j ≺ c˜ j ⇒ x j∗ = 0, or x j∗ > 0 ⇒ w∗ a˜ j ≺ c˜ j ,

j = 1, ..., n,

a˜ i x∗ b˜i ⇒ w∗i = 0, or w∗i > 0 ⇒ a˜ i x∗ ≈ b˜i ,

i = 1, ..., m,

or equivalently w∗ R(a˜ j ) < R(c˜ j ) ⇒ x j∗ = 0, or x j∗ > 0 ⇒ w∗ R(a˜ j ) < R(c˜ j ),

j = 1, ..., n,

R(a˜ i )x∗ > R(b˜i ) ⇒ w∗i = 0, or w∗i > 0 ⇒ R(a˜ i )x∗ = R(b˜i ),

i = 1, ..., m.

˜ ≥ 0, i = 1, 2, ..., m be the m slack variables in Suppose that u i = R(a˜ i )x − R(b) the FNLP primal problem and let v j = R(c˜ j ) − w R(a˜ j ) ≥ 0, j = 1, 2, ..., n be the n slack variables in the its dual problem. Then we rewrite the complementary slackness condition as follows: j = 1, 2, ..., n, v ∗j x ∗j = 0, wi∗ u i∗ = 0,

i = 1, 2, ..., m.

For an illustration of the above theorem we consider the following example. Note that here we use from Yager linear ranking function which is defined in (1.19) as a pattern of linear ranking function on F(R).

74

3 Fuzzy Number Linear Programming

Example 3.4 Consider the following FNLP problem: min z˜ ≈ (1, 3, 1, 1)x1 + (2, 4, 1, 1)x2 + (3, 5, 1, 1) ⎧ ⎨ (0, 2, 21 , 21 )x1 + (1, 3, 1, 1)x2 + (0, 2, 21 , 21 )x3 (2, 4, 1, 1) s.t. (1, 3, 1, 1)x1 − (0, 2, 21 , 21 )x2 + (2, 4, 1, 1)x3 (3, 5, 1, 1) ⎩ x1 , x2 , x3 ≥0. The dual problem is given as follows: min z˜ ≈ (2, 4, 1, 1)w1 + (3, 5, 1, 1)w2 ⎧ (0, 2, 21 , 21 )w1 + (1, 3, 1, 1)w2 (1, 3, 1, 1) ⎪ ⎪ ⎨ (1, 3, 1, 1)w1 − (0, 2, 21 , 21 )w2 (2, 4, 1, 1) s.t. (0, 2, 21 , 21 )w1 + (2, 4, 1, 1)w2 (3, 5, 1, 1) ⎪ ⎪ ⎩ w1 , w2 ≥0. We apply the Yager ranking function to the fuzzy coefficient matrix A˜ and the fuzzy right hand side vector b˜ in dual problem. Thus, we have: min z˜ ≈ (2, 4, 1, 1)w1 + (3, 5, 1, 1)w2 ⎧ w1 + 2w2 ≤2 ⎪ ⎪ ⎨ 2w1 − w2 ≤3 s.t. w1 + 3w2 ≤4 ⎪ ⎪ ⎩ w1 , w2 ≥0. Now by introducing the slack variables v3 , v4 and v5 the problem reduces to the standard form: min z˜ ≈ (2, 4, 1, 1)w1 + (3, 5, 1, 1)w2 ⎧ w1 + 2w2 + v3 = 2 ⎪ ⎪ ⎨ 2w1 − w2 + v4 = 3 s.t. w1 + 3w2 + v5 = 4 ⎪ ⎪ ⎩ w1 , w2 , v3 , v4 , v5 ≥0. The dual problem is an FNLP problem and can be solved by Maleki’s method [18]. The optimal to the dual problem is w1∗ = 85 , w2∗ = 15 with optimum fuzzy value W˜ ∗ ≈ ( 19 , 37 , 11 , 11 ) and R(W˜ ∗ ) = 28 . Substituting w1∗ = 85 , w2∗ = 15 in each dual 5 5 5 5 5 constraints, we find that the third dual constraint are not tight and hence the corresponding fuzzy variable in the FNLP primal problem must be zero. Thus, we get x3∗ = 0. Also, since w1∗ > 0, w2∗ > 0, then the slack variables in each primal constraint must be zero. Therefore, from the FNLP constraints we have

3.4 Duality in Fuzzy Number Linear Programming

75

1 1 (0, 2, , )x1∗ + (1, 3, 1, 1)x2∗ ≈ (2, 4, 1, 1) 2 2 1 1 (1, 3, 1, 1)x1∗ − (0, 2, , )x2∗ ≈ (3, 5, 1, 1) 2 2 Applying the Yager ranking function to the fuzzy coefficient matrix and the fuzzy right hand side vector, this system reduce to: x1∗ + 2x2∗ = 3 2x1∗ − x2∗ = 4 , x2∗ = So, we obtain x1∗ = 11 5 28 R(˜z ) = 5 to FNLP problem.

2 5

and objective value z˜ ≈ ( 15 , 41 , 13 , 13 ) with 5 5 5 5

3.5 A Fuzzy Dual Simplex Method for FNLP Problem 3.5.1 Methodology Consider the following FNLP problem, min z˜ ≈ cx ˜ ˜ b˜ s.t. Ax x ≥ 0.

(3.17)

We first write this problem in the following form: min z˜ ≈ cx ˜ s.t. Ax ≥ b x ≥ 0,

(3.18)

˜ = [R(a˜ i j )] and b = R(b) ˜ = (R(b˜i )), for all i = 1, ..., m and j = where A = R( A) 1, ..., n. Now, with introducing the nonnegative slack variables si , i = 1, ..., m for the ith constraint, we may rewrite (3.18) as follows: ˜ min z˜ ≈ cx ˜ + 0s s.t. Ax − I s = b x ≥ 0, s ≥ 0, where s = (s1 , .., sm )T .

(3.19)

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3 Fuzzy Number Linear Programming

Table 3.2 A fuzzy dual feasible simplex tableau Basis x¯1 ... x¯k ... ˜ ˜ z R(˜z 1 − c¯1 ) . . . R(˜z k − c¯k ) . . . z˜ z˜ 1 − c˜¯1 ... z˜ k − c˜¯k ... (x¯ B )1 .. .

y11 .. .

...

y1k .. .

...

(x¯ B )r .. .

yr 1 .. .

...

yr k .. .

...

(x¯ B )m

ym1

...

ymk

...

x¯n+m

R(R.H.S.) ˜ R(˜z n+m − c¯n+m ) R(c˜¯ B b) c˜¯ B b z˜ n+m − c˜¯n+m y1,n+m R(b˜¯1 ) .. .. . . yr,n+m R(b˜¯r ) .. .. . . ym,n+m R(b˜¯m )

Define x¯ ∈ Rn+m and c˜¯ T ∈ (F(R))n+m as: c˜j , j = 1, ..., n x j , j = 1, ..., n ˜ x¯ j = , c¯ j = ˜ s j−n , j = n + 1, ..., n + m 0 , j = n + 1, ..., n + m.

(3.20)

Suppose that a basic solution for (3.19) is given by x¯ B = B −1 b, with the basis matrix B. Now, let z j = c¯ B B −1 a j , where ˜¯ B = ((R(c)) ˜¯ B1 , ..., (R(c) ˜¯ Bm )) c¯ B = (R(c)) ˜ I ]. Conand a j is the jth column of the coefficient matrix [A I ] = [R( A) −1 sider Table 3.2, where (x¯ B )r is the r th basic variable, y j = B a j and R(˜z j − c˜ j ) = R(c˜ B )B −1 a j − R(c˜ j ) = R(c˜ B )y j − R(c˜ j ) is the real number corresponding to z˜ j − c˜ j ≈ c˜ B y j − c˜ j obtained by a linear ranking function. Suppose that for j = 1, ..., n + m, we have z˜ j − c˜¯ j 0˜ or equivalently z j − c j ≤ 0. Define ˜¯ B . For j = 1, ..., n, we have w = c¯ B B −1 , where c¯ B = (R(c)) z j − c¯ j = c¯ B B −1 a j − c j = w a j − c j . Therefore, z j − c¯ j ≤ 0, for j = 1, ..., n, follows that wa j − c j ≤ 0 or w R(a˜ j ) ≤ R(c˜ j ). Then, w A˜ c. ˜ In addition, for i = 1, ..., m, we have, z n+i − c¯n+i = c¯ B B −1 ˜ for ei − 0 = wei = wi and hence, w ≥ 0. We have just shown that z˜ j − c˜¯ j 0, −1 j = 1, ..., n + m implies that w A˜ c˜ and w ≥ 0, where w = c¯ B B . In other words, dual feasibility is precisely the optimality condition for primal FNLP problem. If b¯ = ˜ ≥ 0, then a feasible solution for the FNLP problem is at hand. Moreover, B −1 R(b) we will have, ˜ ˜¯ x¯ = c¯ x¯ = c¯ B b¯ = c¯ B B −1 b = wb = w R(b), R(c)

3.5 A Fuzzy Dual Simplex Method for FNLP Problem

77

and thus, by Corollary 3.2, establishing the optimality of x¯ and w for the FNLP and dual FNLP problems, respectively. Therefore, we have the following result. ˜ for all j, for the FNLP problem Corollary 3.4 The optimality criteria z˜ j − c˜ j 0, is equivalent to the feasibility condition for the dual FNLP problem. If, in addition, x¯ corresponding to a basis B is primal feasible then x¯ is optimal for the FNLP and w = c¯ B B −1 is optimal to the dual FNLP problem. Now, let’s assume that the dual FNLP problem is feasible and x, ¯ corresponding to a basis B, is dual feasible but primal infeasible (and hence not optimal). That is, we have, z˜ j − c˜¯ j 0,

j = 1, ..., n + m.

Thus, R(˜z j − c˜¯ j ) ≤ 0,

j = 1, ..., n + m,

and there exists at least one r such that b˜¯r ≺ 0˜ or R(b˜¯r ) < 0. We can immediately deduce that the primal problem must be finite. Thus, the dual FNLP problem can be either infeasible (in which case, the dual FNLP problem is unbounded), or it has an optimal solution. In what follows we will show how to work on row r of the simplex tableau corresponding to B, as the pivoting row, and either (1) detect infeasibility of the FNLP problem (or unboundedness of the dual FNLP problem), or (2) find a column k, as the pivoting column, to pivot on yr k and obtain a new fuzzy dual feasible tableau with a nondecreasing primal objective value. We explain these cases as Lemmas 3.2 and 3.3. Lemma 3.2 If in a fuzzy dual feasible simplex tableau an r exists such that R(b˜¯ ) < 0 and yr j 0, for all j, then the FNLP problem is infeasible.

r

Proof Suppose that a fuzzy dual simplex tableau is feasible, and an r exists such that R(b˜¯r ) < 0 and yr j 0, for all j. Corresponding to the r th row of the tableau, we yr j x¯ j = R(b˜¯r ). Since yr j 0 for all j and x¯ j is required to be nonnegative, have j

then

yr j x¯ j ≥ 0 for any basic feasible solution. However, R(b˜¯r ) < 0. This shows

j

that FNLP problem is infeasible. Lemma 3.3 If in a fuzzy dual feasible simplex tableau, an r exists such that R(b˜¯r ) < 0 and there exists a nonbasic index j such that yr j < 0, then a pivoting column k can be found so that pivoting on yr k will yield a fuzzy dual feasible tableau with a corresponding nondecreasing objective value. Proof We need a criterion for choosing a nonbasic variable to enter the basis so that the new fuzzy simplex tableau will remain dual feasible and the new objective value

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3 Fuzzy Number Linear Programming

is nondecreasing. Assume column k is the pivot column. Pivoting on the pivot yr k will result in the new zero row as follows: (˜z j − c˜¯ j )new = z˜ j − c˜¯ j −

z˜ k − c˜¯k yr j , j = 1, ..., n + m, j = Bi . yr k

(3.21)

For the new tableau to be dual feasible we need to have (˜z j − c˜¯ j )new 0˜ , j = Bi

(3.22)

which, using (3.21), results in z˜ k − c˜¯k z˜ j − c˜¯ j , ∀ j = Bi . yr k yr j

(3.23)

To satisfy (3.22), it is sufficient to let R(˜z k − c˜¯k ) = min j= Bi yr k

R(˜z j − c˜¯ j ) | yr j < 0 . yr j

(3.24)

We note that the new objective value is nondecreasing, since ˜ ˜ − R(˜z k − c¯k ) R(b˜r ) ≥ w R(b), ˜ ˜ new = w R(b) (w R(b)) yr k using the fact that R(˜z k − c˜¯k ) ˜ R(br ) ≤ 0. yr k Now, using the above results, we introduce a new fuzzy dual algorithm to solve the FNLP problem directly, making use of the fuzzy dual feasible simplex tableau. Thus, we refer to the new algorithm as a fuzzy dual simplex method [25]. Algorithm 3.2 A fuzzy dual simplex method: Step 1: (dual feasibility) Given a basis B for the FNLP problem such that z˜ j − c˜¯ j 0˜ for all j. Compute the fuzzy simplex tableau. ˜ then Stop (the current solution is optimal) Step 2: If b˜¯ 0, else select the pivot row r with b˜¯r ≺ 0˜ (that is, r so that R(b˜¯r ) < 0) Step 3: If yr j ≥ 0 for all j, then Stop (the FNLP problem is infeasible) else select the pivot column k by the following minimum ratio test:

3.5 A Fuzzy Dual Simplex Method for FNLP Problem

R(˜z k − c˜¯k ) = min j= Bi yr k Step 4:

R(˜z j − c˜¯ j ) | yr j < 0 yr j

79

Pivot on yr k and go to step (2).

Remark 3.7 One suggestion for choice of r in step (2) may be r so that R(b˜¯r ) = min R(b˜¯i ) . 1im

3.5.2 Numerical Example For an illustration of the above method we consider the following example that can be solved by the Maleki’s method (see [16, 18]). Note that here we use from Yager linear ranking function as a pattern of linear ranking function on F(R). Example 3.5 Consider the following FNLP problem: min z˜ ≈ (1, 3, 1, 1)x1 + (2, 4, 1, 1)x2 + (3, 5, 1, 1)x3 ⎧ ⎨ (0, 2, 21 , 21 )x1 + (1, 3, 1, 1)x2 + (0, 2, 21 , 21 )x3 (2, 4, 1, 1) s.t. (1, 3, 1, 1)x1 − (0, 2, 21 , 21 )x2 + (2, 4, 1, 1)x3 (3, 5, 1, 1) ⎩ x1 , x2 , x3 ≥0. We apply the Yager ranking function to the fuzzy coefficient matrix A˜ and the ˜ Thus, with introducing the slack variables x4 and x5 , fuzzy right hand side vector b. the problem reduces to: min z˜ ≈ (1, 3, 1, 1)x1 + (2, 4, 1, 1)x2 + (3, 5, 1, 1)x3 ⎧ ⎨ x1 + 2x2 + x3 − x4 = 3 s.t. 2x1 − x2 + 3x3 − x5 = 4 ⎩ x1 , x2 , x3 , x4 , x5 ≥0. We may write the first fuzzy dual feasible simplex tableau as follows (Tables 3.3). x1 is an entering variable and x5 is a leaving variable. The new tableau is (Table 3.4). x2 is an entering variable and x4 is a leaving variable. The new tableau is Table 3.5. Therefore the optimal solution obtained by the fuzzy dual simplex is x1 = 11 17 17 , x2 = 25 and the fuzzy optimal value of the objective function is z˜ ≈ ( 11 5 5 , 9, 5 , 5 ).

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3 Fuzzy Number Linear Programming

Table 3.3 The first tableau Basis x1 x2 z z˜ x4 x5

−2 (−3, −1, 1, 1) −1 −2

−3 (−4, −2, 1, 1) −2 1

Table 3.4 The second tableau Basis x1 x2 z z˜ x4 x1

−4 (−4, −4, 23 , 23 ) − 25 − 21

0 0˜ 0 1

0 0˜

0 0˜

0 1

1 0

− 95 ( − 31 5 , − 15 7 5

x4

x5

R.H.S.

−4 (−5, −3, 1, 1) −1 −3

0 0˜ 1 0

0 0˜ 0 1

0 (0,0,0,0) −3 −4

x3

x4

x5

R.H.S.

−1 (−3, 1, 2, 2)

0 0˜

1 2 3 2

1 0

−1 (−1, −1, 0, 0) − 21 − 21

4 (0,0,0,0) −1 2

Table 3.5 The third tableau Basis x1 x2 x3 z z˜ x2 x1

x3

13 18 18 5 , 5 , 5 )

x4

x5

R.H.S.

− 85 (− 85 , − 85 , 0, 0) − 25 − 15

− 15 (− 15 , − 15 , 0, 0)

28 5 17 17 ( 11 5 , 9, 5 , 5 ) 2 5 11 5

1 5

− 25

3.6 Bounded Fuzzy Number Linear Programming We recall that the linear programming problem with fuzzy cost coefficients denoted by RFNLP, is a kind of FNLP problem in which its coefficient matrix, the right-handside vector and its variables are crisp and the cost coefficients are trapezoidal fuzzy numbers. We denote a RFNLP problem with bounded variables by BRFNLP and propose a new method for solving this problem [11]. The proposed method specially will be useful for situations in which the auxiliary problem associated with an FVLP problem is bounded and so the RFNLP primal simplex method (Algorithm 3.1) is not efficient for such situation. Definition 3.5 A BRFNLP problem in its standard from is defined as follows: max z˜ ≈ cx ˜ s.t. Ax = b l≤x ≤u

(3.25)

where b ∈ Rm , c˜ ∈ (F(R))n , l, u ∈ Rn and A ∈ Rm×n (rank(A) = m) are given and x ∈ Rn is to be determined.

3.6 Bounded Fuzzy Number Linear Programming

81

We note that any lower bound vector can be transformed into the zero by the change of variables x´ = x − l. In this case, a method for handling the constraints l ≤ x ≤ u is to introduce the slack vector x1 , leading to x + x1 = u, which increase both the constraints and variables in problem (3.25) by n. The most straightforward (and the least efficient) method of handling the constraints l ≤ x ≤ u is to introduce the slack vectors x1 and x2 , leading to x + x1 = u and x − x2 = l. This increases the number of equality constraints in problem (3.25) from m to m + 2n and the number of variables from n to 3n. Thus, the problem size and computational effort would increase significantly if the constraints l ≤ x ≤ u are treated in the RFNLP simplex algorithm by introducing slack vectors. Here, we generalize the Algorithm 3.1 such that our method handles these constraints implicitly in a fashion similar to that used by the Algorithm 3.1. Definition 3.6 A feasible solution x¯ to the equations Ax = b is a basic feasible solution

(3.25) if the set {a j : j ∈ J } is linear independent, where (BFS) of problem J = j : l j < x¯ j < u j . It needs to point out that corresponding to every basic feasible solution of problem (3.25), matrix A can be partitioned into [B, N1 , N2 ], where the matrix B has rank m such that with x partitioned accordingly as (x B , x N1 , x N2 ), we have x¯ N1 = l N1 , x¯ N2 = u N2 and accordingly x¯ B = B −1 b − B −1 N1 l N1 − B −1 N2 u N2 . The matrix B is called the (working) basis, x B are the basic variables, and x N1 and x N2 are the nonbasic variables at their lower and upper limits, respectively. In a similar way, the vectors x and c˜ are decomposed into (x B , x N1 , x N2 ) and (c˜ B , c˜ N1 , c˜ N2 ), respectively. The basic variables can be represented in terms of the nonbasic variables x N1 and x N2 as follows: Ax = b ⇔ Bx B + N1 x N1 + N2 x N2 = b. Thus, we have:

x B = B −1 b − B −1 N1 x N1 − B −1 N2 x N2 .

(3.26)

Also, the objective function can be represented in terms of the nonbasic variables x N1 and x N2 as follows: z˜ ≈ cx ˜ ≈ c˜ B x B + c˜ N1 x N1 + c˜ N2 x N2 , or

z˜ ≈ c˜ B (B −1 b − B −1 N1 x N1 − B −1 N2 x N2 ) + c˜ N1 x N1 + c˜ N2 x N2 .

(3.27)

(3.28)

Therefore, we have: z˜ ≈ c˜ B B −1 b + (c˜ N1 − c˜ B B −1 N1 )x N1 + (c˜ N2 − c˜ B B −1 N2 )x N2 . Now, suppose that a current basic solution to problem (3.25) is given by

(3.29)

82

3 Fuzzy Number Linear Programming

Table 3.6 The initial BRFNLP simplex tableau Basis l u xB x N1 x N2 z

R(c˜ B

0 0˜ I

z˜ xB

B −1 N

1

− c˜ N1 )

R(c˜ B

c˜ B B −1 N1 − c˜ N1 B −1 N1

B −1 N

2

− c˜ N2 )

c˜ B B −1 N2 − c˜ N2 B −1 N2

R.H.S. R(z˜ˆ )

z˜ˆ bˆ

⎛

⎞ ⎛ ⎞ xB bˆ = B −1 b − B −1 N1l N1 − B −1 N2 u N2 ⎝ x N1 ⎠ = ⎝ ⎠. l N1 x N2 u N2

(3.30)

The fuzzy value of the objective function for this solution is z˜ˆ ≈

m

c˜ Bi bˆi +

i=1

c˜ j l j +

j∈R1

c˜ j u j ,

(3.31)

j∈R2

where R1 and R2 are the set of indices of nonbasic variables at their lower and upper bounds respectively. Now form Table 3.6 as the initial tableau of BFNLP method. It will be helpful to distinguish between nonbasic variables at their lower and upper bounds during the BRFNLP algorithm iterations. This is done by flagging the corresponding columns by l and u respectively. Since B is a basis matrix corresponding to A, thus every nonbasic column a j can be written as a linear combination of columns of B, that is, a j = B y j or y j = B −1 a j . Define the fuzzy variable z˜ j as follows: m (c BLi , cUBi , α Bi , β Bi )yi j z˜ j ≈ c˜ B B −1 a j ≈ c˜ B y j ≈ i=1 ≈ (c BLi yi j , cUBi yi j , α Bi yi j , β Bi yi j ) + (cUBi yi j , c BLi yi j , −β Bi yi j , −α Bi yi j ) {i:yi j ≥0}

{i:yi j