Mathematical Modelling and Optimization of Engineering Problems (Nonlinear Systems and Complexity, 30) 3030370615, 9783030370619

Table of contents :
Preface
Contents
1 Heuristic Techniques for Real-Time Order Acceptance and Scheduling in Metal Additive Manufacturing
1.1 Introduction
1.2 Literature Review
1.3 Problem Statement
1.3.1 Assumptions
1.3.2 Notation
Decision Variables
Indicators
1.3.3 Basic Formulations
1.3.4 Objective Function
1.3.5 Constraints
1.4 Meta-heuristic Procedures
1.4.1 Generation of Feasible Solutions
1.4.1.1 Single Machine
1.4.1.2 Multiple Machines
1.4.2 Selection Rules
1.4.2.1 Stochastic Selection
1.4.2.2 Profit-Time Based Selection
1.4.2.3 Cost Benefit Based Selection
1.5 Computational Experiments
1.5.1 Data Generation
1.5.2 Experimental Results and Discussions
1.5.2.1 The Difference of Stochastic Results
1.5.2.2 Performance of Non-random Selection Rules
1.6 Conclusions and Future Research
References
2 Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection
2.1 Introduction
2.2 Optimization Models in Energy Economics
2.2.1 Economic Dispatch Model
2.2.2 Unit Commitment Model
2.2.3 Energy Storage System Modeling in UC
2.2.4 AC Optimal Power Flow Model
2.2.5 DC Optimal Power Flow Model
2.2.6 Optimal Energy Storage System Placement and Sizing Model
2.3 Optimization of the Nationwide Energy Storage System
2.3.1 The Maximum Sizing of Energy Storage Systems
2.3.2 Distribution of Energy Storage Systems Within the Network
2.3.3 The Proposed Bi-level Optimization for Sizing and Siting of Storage Units
2.4 Case Study: Power Systems in Turkey
2.5 Discussion on a Nationwide Energy Policy
2.6 Conclusion
Nomenclature
References
3 Pontryagin's Principle for a Class of Discrete Time Infinite Horizon Optimal Growth Problems
3.1 Introduction
3.2 One-Sector Optimal Growth Model
3.2.1 Social Planner's Problem of E
3.2.2 Necessary and Sufficient Conditions of Optimality
3.3 Optimal Growth Model with an Natural Exhaustible Resource
3.3.1 Management Problem of Eenr
3.3.2 Necessary and Sufficient Conditions of an Optimal Management of a Natural Resource
3.4 Optimal Growth Model of a Forest: An Optimal Management Model of Forestry
3.4.1 Planner's Management Problem
3.4.2 Necessary Conditions of Optimality
3.5 Conclusion
References
4 A Medical Modelling Using Multiple Linear Regression
4.1 Introduction
4.2 Materials and Methods
4.2.1 Study Samples
4.2.2 Multiple Linear Regression Analysis
4.2.3 Test for the Model
4.2.4 Residual Analysis
4.3 Building Regression Analysis Model
4.4 Discussion and Analysis
4.5 Conclusions and Recommendations
References
5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow Driven by Injection Through a Deformable Rectangular Channel with Porous Walls
5.1 Introduction
5.2 Mathematical Modelling of the Problem
5.2.1 Problem Statement
5.2.2 Flow Configuration
5.2.3 Forces Affecting the Dynamics of the Flow
5.2.3.1 Surface Force
5.2.3.2 Body Forces
5.2.4 Derivation of Governing Equations
5.2.4.1 Conservation of Mass
5.2.4.2 Conservation of Momentum
5.2.4.3 Conservation of Energy
5.2.4.4 Boundary Conditions
5.3 Mathematical Representation of Problem
5.3.1 Governing Equations and Boundary Conditions
5.4 Solution of the Problem
5.4.1 Lie Group Analysis
5.4.2 Semi-Analytical Solution
5.5 Results and Discussion
5.5.1 Effects of Wall Dilation
5.5.2 Effects of Reynolds number inside the Filtration Chamber
5.5.3 Effects of Porosity Variable Inside the Filtration Chamber
5.5.4 Effects of Stuart Number Inside the Filtration Chamber
5.5.5 Temperature Distribution Inside the Chamber
5.6 Concluding Remarks
References
6 Optimal Siting of Wind Turbines in a Wind Farm
6.1 Introduction
6.2 Numerical Methods of the Present Study
6.2.1 Wake Model
6.2.2 Power Model
6.3 Methodology
6.3.1 Problem Formulation
6.3.2 Initial Population Based on Elevation Values
6.3.3 Genetic Algorithm for Optimization
6.3.3.1 Population Formation
6.3.3.2 Selection
6.3.3.3 Crossover
6.3.3.4 Mutation
6.3.3.5 Genetic Algorithm Parameters
6.4 Results and Discussion
6.5 Conclusion
References
7 RSM-Based Optimization of Excitation Capacitance and Speed for a Self-Excited Induction Generator
7.1 Introduction
7.2 Modelling of SEIG
7.3 Voltage Build-up process
7.4 Analysis
7.5 Response Surface Method
7.6 Results and Discussions
7.7 Conclusion
References
8 Distance-Constrained Vehicle Routing Problems: A Case Study Using Artificial Bee Colony Algorithm
8.1 Introduction
8.2 Research Background
8.3 Artificial Bee Colony (ABC) Algorithm
8.3.1 Initialization of the Population
8.3.2 Initialization of the Bee Phase
8.3.3 Onlooker Bee Phase
8.3.4 Scout Bee Phase
8.3.5 Stopping Phase
8.4 Case Study
8.5 Results and Discussion
8.6 Conclusion
References
9 Fractional Model for Type 1 Diabetes
9.1 Introduction
9.1.1 Some Concepts of Fractional Calculus
9.2 Description of the Model
9.3 Model Analysis
9.4 Global Stability of the Disease-Free Equilibrium
9.5 Numerical Results
9.6 Conclusion
References
10 Mathematical Modelling and Additive Manufacturingof a Gyroid
10.1 Infinite Periodic Minimal Surfaces (IPMS) Without Self-intersections: Gyroid
10.2 Additive Manufacturing Technology
10.3 3D Printing Process of an IPMS Gyroid
10.3.1 Creating the 3D Mathematical Model of the IPMS Gyroid with K3DSurf Program
10.3.2 Converting the CAD Model Data to ``.obj'' or ``.stl'' File Format
10.3.3 Generating a Solid, Thickened Shell or Hollow CAD Model
10.3.4 Slice the Model into Layers, Generate the Travel Movements and Support Structure
10.3.5 3D Printing of the Model
10.3.5.1 Fused Deposition Modelling
10.3.5.2 3D Printing of the IPMS Gyroid
10.3.6 Removing the Support Material If Any and Apply Finishing Process
10.4 Conclusion
References
Index

Citation preview

Nonlinear Systems and Complexity Series Editor: Albert C. J. Luo

J. A. Tenreiro Machado Necati Özdemir Dumitru Baleanu   Editors

Mathematical Modelling and Optimization of Engineering Problems

Nonlinear Systems and Complexity Volume 30

Series editor Albert C. J. Luo Southern Illinois University Edwardsville, IL, USA

Nonlinear Systems and Complexity provides a place to systematically summarize recent developments, applications, and overall advance in all aspects of nonlinearity, chaos, and complexity as part of the established research literature, beyond the novel and recent findings published in primary journals. The aims of the book series are to publish theories and techniques in nonlinear systems and complexity; stimulate more research interest on nonlinearity, synchronization, and complexity in nonlinear science; and fast-scatter the new knowledge to scientists, engineers, and students in the corresponding fields. Books in this series will focus on the recent developments, findings and progress on theories, principles, methodology, computational techniques in nonlinear systems and mathematics with engineering applications. The Series establishes highly relevant monographs on wide ranging topics covering fundamental advances and new applications in the field. Topical areas include, but are not limited to: Nonlinear dynamics Complexity, nonlinearity, and chaos Computational methods for nonlinear systems Stability, bifurcation, chaos and fractals in engineering Nonlinear chemical and biological phenomena Fractional dynamics and applications Discontinuity, synchronization and control.

More information about this series at http://www.springer.com/series/11433

¨ J. A. Tenreiro Machado • Necati Ozdemir Dumitru Baleanu Editors

Mathematical Modelling and Optimization of Engineering Problems

Editors J. A. Tenreiro Machado Institute of Engineering, Polytechnic of Porto Porto, Portugal

¨ Necati Ozdemir Balıkesir University Department of Mathematics Balıkesir, Turkey

Dumitru Baleanu Cankaya University Department of Mathematics & Computer Science Ankara, Turkey

ISSN 2195-9994 ISSN 2196-0003 (electronic) Nonlinear Systems and Complexity ISBN 978-3-030-37061-9 ISBN 978-3-030-37062-6 (eBook) https://doi.org/10.1007/978-3-030-37062-6 © Springer Nature Switzerland AG 2020 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

Preface

The International Conference on Applied Mathematics in Engineering (ICAME’18) was successfully held in the period of 27–29 June 2018 in Burhaniye, Turkey. The conference provided an ideal academic platform for researchers to present the latest research and evolving findings of applied mathematics on engineering, physics, chemistry, biology and statistics. During the conference: • Three plenary lectures (by Prof. Dr. Albert Luo, Prof. Dr. J. A. Tenreiro Machado and Prof. Dr. Jordan Hristov under the chairship of Prof. Dr. Dumitru Baleanu) • Three invited talks (by Prof. Dr. Carla Pinto, Prof. Dr. Mehmet Kemal Leblebicioglu and Prof. Dr. Ekrem Savas) • A total of 224 oral presentations (in eight parallel sessions) have been successfully presented by participants from 15 different countries, i.e. Algeria, Argentina, Bulgaria, Libya, Germany, India, Morocco, Nigeria, Portugal, Saudi Arabia, South Africa, Turkey, United Arab Emirates, United Kingdom and United States of America. The members of the organizing committee were Ramazan Yaman (Turkey), J. A. Tenreiro Machado (Portugal), Necati Özdemir (Turkey) and Dumitru Baleanu (Romania, Turkey). We would like to thank all the members of the Scientific Committee for their valuable contribution forming the scientific face of the conference, as well as for their help in reviewing the contributed papers. We are also grateful to the staff involved in the local organization. This work organized in two volumes publishes a selection of extended papers presented at ICAME’18 after a rigorous peer-reviewing process. The second volume of the book Mathematical Modelling and Optimization of Engineering Problems contains ten high-quality contributions. This book presents recent developments in modelling and optimization of engineering systems, and the use of advanced mathematical methods for solving complex real-world problems. It provides recent theoretical developments and new techniques based on control, optimization theory, mathematical modelling and v

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Preface

fractional calculus that can be used to model and understand complex behaviour in natural phenomena including latest technologies such as additive manufacturing. Specific topics covered in detail include combinatorial optimization, flow and heat transfer, mathematical modelling, energy storage and management policy, artificial intelligence, optimal control, modelling and optimization of manufacturing systems. We thank all the referees and other colleagues who helped in preparing this book for publication. Our special thanks also go to Albert Luo and Michael Luby from Springer for their continuous help and work in connection with this book. Porto, Portugal Balıkesir, Turkey Ankara, Turkey

J. A. Tenreiro Machado Necati Özdemir Dumitru Baleanu

Contents

1

2

3

Heuristic Techniques for Real-Time Order Acceptance and Scheduling in Metal Additive Manufacturing. . . . . . . . . . . . . . . . . . . . . . Qiang Li, David Zhang, Ibrahim Kucukkoc, and Naihui He

1

Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gokturk Poyrazoglu

25

Pontryagin’s Principle for a Class of Discrete Time Infinite Horizon Optimal Growth Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ay¸segül Yıldız Ulus

51

4

A Medical Modelling Using Multiple Linear Regression . . . . . . . . . . . . . . Arshed A. Ahmad, Murat Sari, and Tahir Co¸sgun

5

Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow Driven by Injection Through a Deformable Rectangular Channel with Porous Walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gabriel Magalakwe, M. L. Lekoko, K. Modise, and Chaudry Masood Khalique

71

89

6

Optimal Siting of Wind Turbines in a Wind Farm . . . . . . . . . . . . . . . . . . . . . 115 Melike Sultan Karasu Asnaz, Bedri Yuksel, and Kadriye Ergun

7

RSM-Based Optimization of Excitation Capacitance and Speed for a Self-Excited Induction Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Haris Calgan, José Manuel Andrade, and Metin Demirtas

8

Distance-Constrained Vehicle Routing Problems: A Case Study Using Artificial Bee Colony Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Aslan Deniz Karaoglan, Ismail Atalay, and Ibrahim Kucukkoc

9

Fractional Model for Type 1 Diabetes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Ana R. M. Carvalho, Carla M.A. Pinto, and João M. de Carvalho

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Contents

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Mathematical Modelling and Additive Manufacturing of a Gyroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Yılmaz Gür

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Chapter 1

Heuristic Techniques for Real-Time Order Acceptance and Scheduling in Metal Additive Manufacturing Qiang Li, David Zhang, Ibrahim Kucukkoc, and Naihui He

1.1 Introduction Metal additive manufacturing (MAM), as an advanced direct digital manufacturing method with shortened lead time and increased performance, has been increasingly applied in industrial sectors, in particular, those characterized by small production batches but high level of demand customization [1, 2]. The two major MAM processes, selective laser melting (SLM) and electron beam melting (EBM), are powder-bed based where the metal powder is coated to the building platform and melted with high-energy laser or electron beam layer-by-layer to form solid metal parts [3]. A powder-bed based MAM machine is a kind of batch processing machine (BPM) in which a batch of identical or non-identical parts can be processed simultaneously according to its capacity. In MAM, producing a batch of parts is called a production job. Parts can form a job only when they are able to fit the MAM machine’s capacity which is generally limited by the cuboid space of the machine’s building chamber. For each job, a specific set-up/clean-up time is required, while the processing time and cost of each job are varied according to the total material

Q. Li · D. Zhang College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK College of Mechanical Engineering, Chongqing University, Chongqing, China e-mail: [email protected]; [email protected] I. Kucukkoc () Balikesir University, Industrial Engineering Department, Balikesir, Turkey e-mail: [email protected] N. He College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6_1

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volume and the maximum height of the parts included in this job, as well as the efficiency of the MAM machine to conduct this job [4]. This chapter considers a problem of scheduling randomly arriving part orders on multiple MAM machines with order acceptance and due date constraints to maximize the average profit obtained within per unit of time. The part orders with preferred due dates arrive randomly in a chronological order, waiting for the acceptance from the service provider. Such scene often happens in online 3D printing service where customers place orders if they are satisfied with the price given online and let the providers to confirm whether the orders can be delivered before requested/promised due dates. The arrived part orders will be grouped into a batch to form a production job and assigned to a specific MAM machine under the constraints of the machine’s capacity as well as the requested due dates of the parts. A part order will be accepted only if it can be processed within one of machines’ job and the complete time of the job is not later than its requested due date. Otherwise, the part order will be rejected and handed over to another department for further negotiation with customers. All the accepted part orders will be produced according to the scheduled start time of the jobs they were assigned to. The completion time as well the start time of a production job can only be determined when all part orders included in this job are confirmed. Therefore, any part order added into an unconfirmed production job of a MAM machine will occupy the capacity of the machine and affect the completion time of the job, which may cause the machine unable to fit further part orders due to the constraints of their due dates as well as the machine’s capacity. The difference of production cost per volume of material could be more than 40% when the part order was scheduled into different jobs [4]. It is vitally important to appropriately determine which part orders should be accepted and how they should be scheduled simultaneously so as to maximize the profit and minimize makespan. Thus, this chapter contributes to literature by proposing a decision-making model and simulation-based approach for the order acceptance and scheduling (OAS) problem in MAM. The rest of this chapter is organized as follows. Related works are reviewed in Sect. 1.2 and the problem of real-time OAS in MAM production environment is defined in Sect. 1.3. In Sect. 1.4, the meta-heuristic procedures are proposed for the generation of feasible production jobs on a single machine as well as multiple machines to form a feasible schedule result. Further, four heuristic selection rules are proposed based on the analysis of the selective behaviours, which may affect the schedule results, during the generation of feasible schedule. A comprehensive experimental study is designed and conducted in Sect. 1.5, followed by conclusions and future research directions in Sect. 1.6.

1.2 Literature Review The problem of OAS is defined as the joint decision of which orders to accept for processing and how to schedule them and has been widely studied over the past

1 Real-Time Order Acceptance and Scheduling in Metal Additive Manufacturing

3

decades [5]. Recently, the research interest is rising in the dynamic OAS problems by considering arrival and release of the orders and heuristics/meta-heuristics are widely used as the problem is hard to solve using traditional mathematical models. Rahman et al. [6] proposed a genetic algorithm based real-time OAS approach for permutation flow shop problems to maximize the revenue of a flow shop production business. Khalili et al. [7] and Noroozi et al. [8] studied the productiondistribution problems with order acceptance and batch delivery to maximize the total profit. Aouam et al. [9] introduced the demand uncertainty in production planning problems integrated with order acceptance, and proposed a relax and fix heuristic for the construction of feasible solutions which are then improved by a fix and optimist heuristic. Although various approaches have been developed according to different OAS problems, it is hard to adopt these approaches directly in MAM production environment due to the uniqueness of MAM production. Over the past 20 years, the topic of additive manufacturing (AM) has received considerable attention [2, 10, 11]. However, the research on production planning and scheduling in AM is just catching up. In their preliminary work, Kucukkoc et al. [12] addressed to the scheduling of additive manufacturing machines to maximize the resource utilization considering order delivery times. Li et al. [4] introduced the production planning problem of powder-bed based MAM facilities and modelled mathematically for the first time. Moreover, a mathematical model and two heuristic procedures (namely best-fit and adapted best-fit) were proposed by Li et al. [4] to minimize the average production cost per volume of material. Furthermore, the production scheduling problem in a multiple MAM machines environment considering the release and due dates of part orders was studied by Kucukkoc et al. [13] and a genetic algorithm approach was developed to minimize the maximum lateness. Kucukkoc [14] proposed the mixedinteger programming models to minimize makespan in MAM machines in single, parallel identical, and parallel non-identical machine environments. Aside from the additive manufacturing researches, Li and Zhang [15] studied the problem of minimizing the makespan for jobs on a single batch BPM with two-dimensional bin packing constraints which can be potentially used for the scheduling of MAM machines with further development. However, their current model considered “p-batch” scheduling problem where the processing time of a batch equals the longest processing time of the jobs while the processing time of a batch in MAM machine is a function of the properties of all parts in the batch. Rudolph and Emmelmann [16] proposed a cloud-based platform for automated order processing in additive manufacturing where the order acceptance determined according to the checking of manufacturing restriction and design guidelines, while the scheduling problem was not considered. Ransikarbum et al. [17] developed a multi-objective optimization model considering operating cost and load balance among printers, total tardiness, and the total number of unprinted parts to aid decision-making on part-to-printer assignment of a batch of parts to multiple fused deposition modelling (FDM) printers. Also, the authors analysed the conflicting objectives through a trade-off analysis to help to understand the conflicting aspect among different objectives. However, the order acceptance and the arrival time of parts were not considered in their work. Zhou et al. [18] discussed

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the problem of multi-task scheduling of distributed 3D printing services in cloud manufacturing and proposed a 3D printing service scheduling (3DPSS) method to reduce the delivery time of tasks from candidate services obtained through service matching. The authors treated the 3D printer as a service which only can process one job at a time and thus the batching problem was not considered. To the best of our knowledge, no research has been conducted to address OAS problem in MAM production environment except our recent work [19] which proposed a principle decision-making model and simulation-based approach for the OAS problem in MAM. According to the projection estimated by Jiang et al. [1], in 2030, “a significant amount of small and medium enterprises will share industryspecific additive manufacturing production resources to achieve higher machine utilization”, and “the AM will be used to efficiently enable customized products (mass customization) for every customer, moving from build-to-stock to build-toorder”. By then, the OAS problem in MAM production environment will be a critical problem which must be addressed for the hubs with hundreds of industry-specific AM facilities.

1.3 Problem Statement The problem addressed in this research can be formally described as follows: a set of part orders N = 1, 2, 3, . . . , n is placed by customers randomly in chronological order, and the manufacturer owns a set of MAM machines M = 1, 2, 3, . . . , m makes decisions on which part order should be accepted and how to schedule the accepted part orders simultaneously to maximize the net profit obtained during the makespan. The part orders have specific arrival time, requested due dates, material volumes, and boundary dimensions (height, length, and width). The MAM machines have specifications including operation cost, production efficiency, building capacity (represented as a cuboid space with maximum height, length, and width), and service price per unit material volume. Each MAM machine can handle one production job at a time and a batch of non-identical parts can be processed simultaneously in this job according to the machine’s capacity. The part order will be accepted only when it can be processed within one of machines’ job and the complete time of the job is not later than its requested due date. The scheduled production jobs will be started for processing according to their planned start time. The total net profit equals the total revenue for producing all the accepted part orders minus the total production cost of all scheduled jobs, whereas the makespan is the difference between the latest complete time and the earliest start time of all scheduled jobs.

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1.3.1 Assumptions The assumptions of the problem are as follows: • Customers know and accept the service price, thus the customers will accept the service if the part order can be processed on time. • The rejected part orders will be processed by the related department for the further negotiating with customers and are not considered in this chapter. • A part order consists of only one specific part which has already orientated according to the requirements of MAM process. That is, the part only can be rotated around the vertical axis but cannot be titled. • Both the machine’s building platform and the projection shape of a part on the machine’s building platform are represented by rectangles. • A batch of parts assigned to a machine’s job is feasible only when the parts can be placed in the machine without overlapping with each other. • All the parts will be made of same material which can be processed by the MAM machines configured with same/different efficiencies and operation costs. • All the parts assigned to a machine’s production job will be processed simultaneously. That is, once a production job started, no parts can be added to the job and the processed parts can only be removed when the job is finished. • All the MAM machines are available at the beginning and the MAM machine can only handle one job each time. That is, the jobs scheduled to a machine will be processed one by one in sequence.

1.3.2 Notation The following notations are used in the formulation and definition of the problem: i k j hi wi li vi ri di Hk Wk Lk V Tk H Tk T Ck H Ck STk MC Pk

The index used for the part orders, i ∈ N The index used for the MAM machines, k ∈ M The index used for the jobs on machine k, j ∈ N The height of part order i The boundary width of part order i The boundary length of part order i The material volume of part order i The arrival time of part order i The requested due date of part order i The maximum height of building space on machine k The maximum width of building space on machine k The maximum length of building space on machine k Time for forming per unit volume of material for machine k Time for coating per unit height of material for machine k The operation cost per unit time for machine k The cost of human work per unit time for machine k The time for setting up a new job on machine k The cost of per unit volume of material The service price of per unit volume of material for machine k

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Decision Variables Xi,k,j Yk,j

1, if part order i is accepted and assigned to the j th job on machine k; 0, otherwise. 1, if the j th job on machine k is assigned with any parts; 0, otherwise.

Indicators J P Pk,j J P Tk,j J P Ck,j J STk,j J CTk,j PT

The profit obtained from the j th job on machine k The production time of the j th job on machine k The production cost of the j th job on machine k The start time of the j th job on machine k The completion time of the j th job on machine k The average net profit per unit of time of the schedule

1.3.3 Basic Formulations The production cost of j th job scheduled to machine k, represented by J P C k,j , can be formulated as follows: J P Ck,j = (T Ck × V Tk + MC) ×

 

 vi × Xi,k,j

i∈N

+ T Ck × H Tk × max (hi × Xi,k,j ) + STk × H Ck × Yk,j . ∀i∈N

(1.1)

The production cost of a MAM production job is comprised of three sections: the cost of material melting, which depends on the total material volume of the parts assigned to the job; the cost of powder coating, which depends on the maximum height of parts within the same job; and the cost of setting up a new job. Accordingly, the production time of a MAM job J P Tk,j can be formulated as follows: J P Tk,j = V Tk ×

 

 vi × Xi,k,j + H Tk × max (hi × Xi,k,j ) + STk × Yk,j . ∀i∈N

i∈N

(1.2) Given the service price P Vk for machine k, the net profit obtained from j th job scheduled to machine k, represented by J P Pk,j , can be formulated as follows:

J P Pk,j = (P Vk − T Ck × V Tk − MC) ×

 

 vi × Xi,k,j

i∈N

− T Ck × H Tk × max (hi × Xi,k,j ) − STk × H Ck × Yk,j . ∀i∈N

(1.3)

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1.3.4 Objective Function The objective of OAS in MAM production is to maximize the total net profit within minimized makespan for the whole system including all jobs scheduled on all machines, which is termed as profit-per-unit-time in this chapter and represented by P T . The makespan of the whole system is defined as the difference between the latest complete time and the earliest start time of all scheduled production jobs. The objective function can be formulated as follows: 

 max P T =

k∈M

min

∀k∈M,j ∈N

j ∈N

(J CTk,j ) −

J P Pk,j min

∀k∈M,j ∈N

(J STk,j )

.

(1.4)

1.3.5 Constraints Capacity Constraints A part can be assigned to a job on one machine only when the part can be placed in the machine. Firstly, the height of the part must be smaller than the maximum height supported by the machine. Secondly, the part must be placed on the building platform without overlapping with other parts which already assigned to this job. In this chapter, both the machine’s building platform and the projection of part are presented by rectangles. A Python function implemented by Jacobs [20] is used to calculate whether a part could fit in a machine’s building platform by considering the parts already included in the job. Part and Job Occurrence Constraints A part can only be either assigned to an exact job on a machine or rejected. Considering that an extreme case is each job only process one part, the number of all scheduled jobs should be no more than the number of part orders. The MAM machine can only handle one job at a time, thus the jobs should be scheduled to the machine in sequence. That is, the second job cannot be scheduled if the first job on the machine has not been scheduled yet. Time Constraints A part can only be assigned to a job after its arrival and if the complete time of the job is no later than the requested due date. In other words, the start time of a job should be no earlier than the part’s arrival time and the complete time of the job should be no later than any part’s requested due date. For the jobs scheduled on a machine, a job can be started only when the previous job has completed.

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1.4 Meta-heuristic Procedures The problem of OAS in MAM production is a joint decision on order acceptance and BPM scheduling which both have been proved as strong NP-hard problems [5]. Additionally, the generation of a feasible solution, particular batching parts to form a production job, is an extremely complicated procedure when considering the constraints of machine’s capacity as well as the arrival time and due date. Therefore, we propose heuristic procedures for solving the problem efficiently. An illustration of the processing flow diagram for OAS in MAM production is shown in Fig. 1.1. The part orders from customers arrive at the system in a chronological order. At any time point when new orders are coming in, a procedure is then started to schedule the arrived parts to the MAM machines to form a production job. The part order will be accepted only when it can be processed in a machine’s job on time and the job will be scheduled when constraints of due date or machine’s capacity is reached. Otherwise, the part order will wait for the confirmation of a job which can be assigned to and be rejected if no machine can process it within the requested due date. The accepted part orders will be produced in the scheduled production jobs and then delivered to the customers. Whereas, the rejected order parts will leave the system and be handed over to related department for the further negotiation with customers. The main aim of this research is realtime scheduling of arrived part orders and making decisions on the acceptance of part orders according to the feasible scheduling results.

1.4.1 Generation of Feasible Solutions 1.4.1.1

Single Machine

To generate a feasible production job on a MAM machine, we need to well understand the production with MAM processes. A MAM production job can process a batch of non-identical parts simultaneously and the process time of the job, which can be calculated with Eq. (1.2), is a function of the properties of all parts assigned to this job as well as the specifications of the MAM machine to conduct this job. The combination of parts assigned to a job as well as the schedule of the job to a specific machine will cause different production time. At the same time, a production job must be completed no later than any requested due date of all parts

Fig. 1.1 OAS processing flow diagram

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Fig. 1.2 Available time slot for a job in scheduling

assigned to this job, and only can be started no earlier than the complete time of the previous job scheduled to the machine. As one possibility, the available time slot for a job which is in scheduling on a machine can be illustrated as Fig. 1.2. At the time moment t, the available time slot T Sk,j for a job in scheduling on a MMA machine can be formulated as follows: T Sk,j = [max(t, J CTk,j −1 ), min (di )]. ∀i∈Nk,j

(1.5)

A MAM production job is feasible only when both the start time and the complete time of the job are located within its available time slot. Therefore, for part order i arriving at or before the time moment t, it can be assigned to the j th job on machine k as long as the machine still has available capacity to accommodate it and the adding of this part will not cause the start time or the complete time of the job out of the job’s available time slot. However, a production job should be confirmed to schedule on the machine and move forward to schedule the next job when one of the following limits has reached: • Time Limits (measured by min (di ) − max(t, J CTk,j −1 ) ≤ J P Tk,j ), ∀i∈Nk,j

• Capacity Limits (no part orders can be fitted in the machine). Once a production job has been confirmed, the start time of the job should be adjusted to its earliest available start time, that is, J STk,j = max(t, J CTk,j −1 ) and the complete time of the job J CTk,j = J STk,j + J P Tk,j . Based on the analysis given above, the heuristic procedure to generate feasible production jobs on a single MAM machine is described as Algorithm 1. At any time moment, a production job is in scheduling on the MAM machine. All waiting part orders will be considered to assign to the job if any existing part order can be put into the machine and the production time is still no longer than its available time slot. Then, one of the feasible parts will be selected for the job and removed from the waiting list. After adding the selected part order, the part orders remained in the waiting list will be reconsidered for the selection of next feasible part order until the job has reached its time or capacity limits for confirmation. Once the current in scheduling job is confirmed, it will be added to the machine’s confirmed job list and renewed the in scheduling job by emptying the part list in the job and update

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Algorithm 1 Feasible production jobs generation on a single machine Input: Part order list P waiting for scheduling at time moment t. 1: get confirmed job list J on the machine 2: while P is not empty do 3: create an empty list of feasible parts F ← ∅ 4: get current in scheduling job j 5: for each part p in P do 6: if p feasible for j then add p to F . 7: end for 8: if F is not empty then 9: select one part and add to j . 10: remove the selected part from P 11: end if 12: if j reached time or capacity limits then 13: add j to J , and open a new scheduling job j 14: end if 15: end while

the available time slot of the job. For a new in scheduling job, the available time slot starts from the current time moment or the complete time of the last confirmed job on this machine whichever is later, and the ending of the time slot is far enough from current time moment.

1.4.1.2

Multiple Machines

Within a multiple MAM machines environment, the part orders within waiting list will be considered by all MAM machines. Each MAM machine generates confirmable feasible jobs according to its schedule and the waiting part orders. However, a part order can only be assigned to a particular production job on one of the MAM machines. The assigned part orders then will not be available for any other MAM machines. The heuristic procedure to generate feasible production jobs on multiple MAM machines is described as Algorithm 2. At any time moment, each MAM machine generates potential confirmable production jobs for the part orders waiting for scheduling with the heuristic procedure described in Algorithm 1. One of the potential confirmable production jobs from all MAM machines will be selected for the confirmation of schedule, and the part orders assigned to the selected job will then be removed from the waiting list. In addition, the selected job will be added to the confirmed job list of the MAM machine where the job comes from. However, the unselected potential jobs will be renewed during the next iteration. In other words, the production jobs will be scheduled one by one and the potential jobs from all MAM machines will be regenerated after updating the waiting list.

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Algorithm 2 Feasible production jobs generation on multiple machines Input: Part order list P waiting for scheduling at time moment t. 1: get machine list M and scheduled job list S 2: while P is not empty do 3: create an empty list of feasible job list J ← ∅ 4: for each machine m in M do 5: get the copy of P as P  6: generate confirmable feasible jobs C with Algorithm 1 (P  ) 7: if C is not empty then add C to J . 8: end for 9: if F is not empty then 10: select one job j from J and add to S 11: remove the part orders which assigned to j from P 12: end if 13: end while

1.4.2 Selection Rules There are two selective behaviours during the generation of a feasible schedule solution for multiple MAM machines. One is the selection of part orders from the waiting list to generate a feasible production job on a particular machine, and the other is the selection of potential jobs generated by all machines to schedule. Different choices of part orders as well as potential jobs lead to different schedule results. For better understanding, let us consider a situation where two production jobs Job1 and Job2 have been assigned to a part order with boundary dimensions (hi × wi × li ) of 24 × 17 × 11 and 2 × 18 × 13, respectively. The material volume of these two part orders is 2859 cm3 and 212 cm3 , respectively. A new part order with boundary dimensions of 5 × 9 × 6 and material volume of 176 cm3 can be assigned to either Job1 or Job2. For a machine (whose specifications will be listed in Table 1.1 in Sect. 1.5.1), the production cost and time of the two jobs can be calculated by Eqs. (1.1) and (1.2), respectively. If the new part order is assigned to Job1, the production time is 112.5 h and the cost is £12,758.3; if it is assigned to Job2 the production time and cost is 17.5 h and £1764.5. The production cost of

Table 1.1 Specifications of MAM machine used in this chapter Parameters Hk × Wk × Lk (cm3 ) VTk (h/cm3 ) HTk (h/cm) STk (h) TCk (GBP/cm3 ) HCk (GBP/h) MC (GBP/cm3 ) Pk (GBP/cm3 )

Reference value 32.5 × 25 × 25 0.030864 0.7 2 60 30 2 6

Random range – 0.03–0.06 0.7–1.0 1–3 50–80 25–50 – –

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Job1 and Job2 is £739.8 and £800.4, respectively. Also, the schedule of the same combination of part orders to machines with different specifications will generate different production time and cost. As a strongly NP-hard problem, it is hard to generate the results for all possible choices particular for the problems with a big number of part orders and MAM machines. Therefore, several selection rules are introduced for the generation of high-quality schedule results within a reasonable CPU time. The comparison of the schedule results obtained from different selection rules will be given in Sect. 1.5.

1.4.2.1

Stochastic Selection

The most obvious option to select a feasible part order or potential production job is stochastic selection, named RDM, described in Selection Rule 1. Although stochastic selection cannot guarantee the performance of the generated solution, it is a practical way to know about how different the bad result is from the good result through comparing a set of schedule results generated randomly. Theoretically, the optimized schedule result could be found as long as the number of iterations for the stochastic selection is big enough.

Selection Rule 1: Stochastic selection (RDM) – Randomly select feasible part orders into production job. – Randomly select potential confirmable production jobs to schedule.

1.4.2.2

Profit-Time Based Selection

The objective of the OAS problem in this chapter is to maximize the average profit per unit time of all the MAM machines during the whole makespan. It is reasonable to consider the effects of different selective behaviours to the total net profit and makespan. Considering a production job in scheduling on a machine with multiple feasible part orders at some time moment, the choice of the feasible part order will result in different production time and production profit for the job which can be calculated with Eqs. (1.2) and (1.3), respectively. The profit per unit time during a i , can be formulated machine’s makespan if a part is selected, represented as P MSk,j as follows: i i i i = (J P Pk,j )/(J STk,j + J P Tk,j ), P MSk,j

(1.6)

i , J ST i , and J P T i are, respectively, the production profit, start where J P Pk,j k,j k,j time, and production time if part i is assigned to the j th job on machine k. For the

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whole system, the profit per unit time during the whole makespan if a confirmable job is selected, represented as P MSk,j , can be formulated as follows:  P MSk,j =



Yk,j × J P Pk,j

 + J P Pk,j /MSj , 

(1.7)

k∈M,j ∈N

where Yk,j = 1 if a job has been confirmed for schedule; otherwise Yk,j = 0. MSj is the current makespan of the system if the job is selected and can be formulated as follows: 

MSj = max( max Yk,j × J CTk,j , J CTk,j ) − k∈M,j ∈N



min

k∈M,j ∈N

Yk,j × J STk,j ,

(1.8)



where J P Pk,j , J CTk,j are the profit and complete time of the job under consideration, respectively. The selection rule based on profit per unit time during makespan, named PMS, is described in Selection Rule 2.

Selection Rule 2: Maximum profit to makespan (PMS) i into the production – Select feasible part order results in maximum P MSk,j job in scheduling. – Select potential confirmable production job results in maximum P MSk,j to schedule.

Another consideration is that the ratio of production time to makespan may affect the schedule results. A MAM machine makes a contribution to the total profit only during the production time. However, in a makespan, the MAM may idle and thus does not contribute to the profit. The ratio of production time to makespan for a i , is formulated as follows: MAM machine if a part is selected, represented as P P Tk,j i i i i P P Tk,j = (J P Tk,j )/(J STk,j + J P Tk,j ).

(1.9)

Similarly, the ratio of production time to the total makespan for the system if a confirmable job is selected, represented as P P Tk,j , is formulated as follows:  P P Tk,j =



  Yk,j × J P Tk,j + J P Tk,j /(MSj ).

(1.10)

k∈M,j ∈N

The selection rule based on the ratio of production time to the makespan, named PPT, is described in Selection Rule 3.

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Selection Rule 3: Maximum production time to makespan (PPT) i into the production – Select feasible part order results in maximum P P Tk,j job in scheduling. – Select potential confirmable production job results in maximum P P Tk,j to schedule.

1.4.2.3

Cost Benefit Based Selection

In the case of multiple MAM machines with different specifications, the machines may need to spend different costs to obtain the same profit per unit time. The service providers always prefer to obtain more profit with less production cost during a specific time duration. Therefore, we consider the selection rule based on the cost benefit and makespan (termed as cost-time benefit in this chapter), named BEN, which is described in Selection Rule 4.

Selection Rule 4: Maximum cost-time benefit (BEN) i into the production – Select feasible part order results in maximum P P Ck,j job in scheduling. – Select potential confirmable production job results in maximum P P Ck,j to schedule.

The cost-time benefit of a production job when a part order is selected, i , is formulated as follows: represented as P P Ck,j i i i i i P P Ck,j = (J P Pk,j × J P Pk,j )/(J P Ck,j × J P Tk,j ).

(1.11)

The cost-time benefit of the whole system when a confirmable production job is selected, represented as P P Ck,j , is formulated as follows:  P P Ck,j = 





k∈M,j ∈N





Yk,j × J P Pk,j + J P Pk,j 

k∈M,j ∈N Yk,j × J P Ck,j + J P Ck,j × MSj

.

(1.12)

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1.5 Computational Experiments The computational experiments were conducted to compare the performance of different selection rules proposed in Sect. 1.4.2. The difference between potential bad and good schedule results was investigated first with the proposed RDM selection rule. Then, the performance of selection rules PMS, PPT, and BEN was evaluated by comparing the results obtained through non-random selection with the best and the worst results obtained through RDM selection. The heuristic algorithms proposed in Sect. 1.4.1 were implemented by using Python language. All experiments were performed on a computer with Intel Core i7-7700 CPU @3.60 GHz processors and 32 GB RAM. The CPU time consumed on different problem sizes was compared as well to evaluate the efficiency of the proposed heuristic algorithms.

1.5.1 Data Generation A serial of test problems was randomly generated and solved to evaluate the performance of the proposed heuristic algorithms. The test problems consist of different numbers of MAM machines with same/different specifications as well as different numbers of part orders randomly arrived during a specific time duration. The specification related to the capacity and efficiency of the MAM machine is referred to the direct metal laser sintering (DMLS) system produced by EOS—a global industrial 3D printing system supplier from Germany. Other parameters of the machine are given empirically. For multiple MAM machines, all machines have the same capacity, material cost, and service price, while other parameters were generated randomly within the given ranges. The reference and random range for the specifications of MAM machines used in this chapter are shown in Table 1.1. Four levels of the number of MAM machines (2, 5, 10, and 20 same/different machines) were considered for each size of problem based on the number of part orders. The part orders considered in the test problems were generated randomly with specific arrival time ri , boundary dimensions (hi × wi × li ), material volume vi , and requested due date di . The random ranges of parameters for the part orders used in this chapter are shown in Table 1.2. All part orders were assumed to arrive randomly within a specific duration (e.g., 30 days) and different part orders may arrival at the same time. The requested due date of a part order is an empirical duration (e.g., 14 days) after its arrival. It is also a case where the service providers will promise a due date within a reasonable time duration after receiving orders from customers. The size of problems according to different number of part orders (20, 50, 100, 200, 400, and 600) was considered for each level of the number of MAM machines. Therefore, in total, 48 different problems were tested for the evaluation of the proposed heuristic algorithms. An example data set of 20 part orders randomly arriving within 30 days is shown in Table 1.3.

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Table 1.2 Parameters of part orders used in this chapter Parameters ri (time moment in hours) di (time moment in hours) hi (cm) wi (cm) li (cm) vi (cm3 ) Table 1.3 Data set of 20 part orders randomly arrived within 30 days

Random range 0–720 ri + 336 2–32 2–25 2–25 hi × wi × li × (0.3–0.8) Index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ri 87 87 156 156 227 227 227 227 324 324 337 337 337 337 337 337 337 337 692 692

Example 439 775 20 9 10 1132 di 423 423 492 492 563 563 563 563 660 660 673 673 673 673 673 673 673 673 1028 1028

hi 30 31 28 28 27 8 20 8 21 27 28 17 4 2 15 27 20 14 18 9

wi 21 4 21 22 10 5 24 21 23 22 18 6 11 6 23 19 25 10 22 22

li 12 22 24 24 12 24 10 10 15 5 15 23 18 18 20 22 7 6 3 4

vi 4564 2083 6592 7615 2478 725 2543 952 5397 2326 4278 1372 577 103 3700 8872 1221 592 384 507

1.5.2 Experimental Results and Discussions The difference of the worst schedule from the best schedule generated with RDM selection rule was investigated through several experiments and discussed in this section. The performances of non-random selection rules were evaluated through comparing with the results obtained from RDM selection.

1.5.2.1

The Difference of Stochastic Results

To discover how different the bad schedule is from the good schedule, 12 problems which consists of 3 or 5 same/different MAM machines and 20, 50, or

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Table 1.4 Experimental results with RDM selection rule Iterations Problems Same M3N20 M3N50 M3N100 M5N20 M5N50 M5N100 Average Different M3N20 M3N50 M3N100 M5N20 M5N50 M5N100 Average Iterations Problems Same M3N20 M3N50 M3N100 M5N20 M5N50 M5N100 Average Different M3N20 M3N50 M3N100 M5N20 M5N50 M5N100 Average

50 P T best (£/h) 142.71 139.15 162.50 176.54 151.18 244.82 57.57 67.98 86.25 98.72 94.25 147.06 179.64 43.45 200 P T best (£/h) 145.08 139.15 167.05 187.47 151.18 248.58 82.12 68.32 91.35 99.65 94.35 151.40 183.49 58.08

P T worst (£/h) 81.41 85.21 120.16 98.50 104.42 165.87

DP T (%) 75.31 63.29 35.24 79.23 44.78 47.60

CPU (s) 3.048 32.264 105.096 5.563 37.329 149.450

41.64 53.29 79.16 65.60 102.64 144.97

63.25 61.86 24.71 43.66 43.28 23.92

1.838 19.761 61.634 3.250 24.522 64.716

P T worst (£/h) 69.30 82.67 108.30 73.18 101.65 159.43 36.58 47.70 75.79 62.63 97.79 137.69

100 P T best (£/h) 144.61 139.15 163.92 176.54 151.18 248.58 60.44 67.98 91.35 99.65 94.25 147.06 181.40 54.86

P T worst (£/h) 78.44 85.21 120.16 98.50 101.65 165.05

DP T (%) CPU (s) 84.36 6.319 63.29 63.155 36.42 213.397 79.23 11.092 48.73 75.992 50.61 307.722

36.58 49.18 78.67 62.63 98.72 138.02

85.86 3.660 85.75 36.589 26.66 120.513 50.49 6.599 48.97 49.241 31.43 130.229

DP T (%) 109.34 68.31 54.25 156.18 48.73 55.92

CPU (s) 12.540 123.894 429.796 21.879 168.054 597.206

86.78 91.49 31.48 50.66 54.82 33.26

7.283 76.750 236.643 13.194 96.755 259.092

100 part orders were tested with RDM selection rule through 50, 100, and 200 iterations, respectively. The results are listed in Table 1.4 where the profit-perunit-time (P Tbest , P Tworst ) of the best and the worst schedule, and the CPU time consumptions of each problem are provided for comparison. The CPU time was measured in the developed Python program through recording the difference of system clock time before and after the execution of the scheduling program. It can be seen that the CPU time consumption increases linearly as the

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number of iteration is increased. However, it increases, exponentially as the problem size increases along with the number of machines and part orders. For example, the CPU time consumptions increased approximately 34.5 times from 3.048 s to 105.096 s for the problems with three same machines, while the number of part orders increasing from 20 to 100, and the time consumptions for problems with same part orders increases approximately 1.5 times in average when the number of machines increased from 3 to 5. The difference indicator for profit-per-unittime, represented as DP T , is defined as DP T = (P Tbest − P Tworst )/P Tworst . The significant differences between the best and the worst schedule results obtained with RDM selection rule can be seen from the experimental results, and the difference increases as the number of iterations increases where better and worse results are discovered. For the problems with the same MAM machines, the average difference is 57.57%, 60.44%, and 82.12%, respectively, for 50, 100, and 200 iterations. The maximum difference of 156.18% appeared in the problem with 5 same machines and 20 part orders. However, the average difference is 43.45%, 54.86%, and 58.08%, respectively, for problems with different MAM machines. The most likely reason is that the profit as well as the production time is different when the same combination of part orders are scheduled to machines with different specifications, whereas they are the same if the machines with same specifications. It thus needs more iterations to discover better and worse results for the problems with different machines. To further understand the difference between the good and bad schedule results, the extended information of the best and the worst schedules generated with 100 iterations for problems with 3 same/different machines and 50 part orders is given in Table 1.5. In general, the best schedules present shorter makespan and more

Table 1.5 An example for scheduling results with RDM selection rule Problems Total profit (GBP) Makespan (h) Production time (h) Production cost (GBP) Accepted orders Scheduled jobs Profit-per-unittime (GBP/h) Cost-time benefit (GBP/h) Ratio of production time to makespan

3 same machines, 50 part orders Best schedule Worst schedule 129,060 82,147.8

3 different machines, 50 part orders Best schedule Worst schedule 77,599.6 48,904.7

927.52 2252.46

964.03 1433.47

849.52 2359.07

994.5 1647.93

265,992

169,186

222,472

151,191

47 14 139.15

37 10 85.21

38 12 91.35

30 9 49.18

67.51

41.37

31.86

15.91

2.43

1.49

2.78

1.66

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profit which result in higher profit-per-unit-time compared to the worst schedules. Meanwhile, more part orders are accepted in the best schedules which results in longer total production time compared to the worst schedules. The ratio of production time to the whole makespan and the cost-time benefit of the best schedules which is calculated with Eqs. (1.10) and (1.12), respectively, are also higher than the worst schedules.

1.5.2.2

Performance of Non-random Selection Rules

The performance of non-random selection rules is evaluated by comparing the schedule results obtained with non-random selection rule to those obtained through 100 iterations with RDM selection rule. Particular, the profit-per-unit-time and the total profit of each schedule result are considered as comparative items. There total 24 different problems (3 and 5 same/different machines with 20, 50, 100, 200, 400, or 600 part orders) are tested with selection rules proposed in Sect. 1.4.2 and the experiment results are listed in Table 1.6. The distribution of results obtained from non-random selection rules to the range between the best and the worst results obtained with RDM selection rule is considered as the performance indicator of a selection rule, represented as Prule which equals (valuerule − valueRDW )/(valueRDB − valueRDW ). The rule can be PMS, PPT, or BEN. The performances of non-random selection rules in total profit and profit-per-unit-time are listed in Table 1.7. It can be seen from Table 1.7 that on average the best and the worst performance in total profit and profit-per-unit-time is represented by PMS selection rule (105.57% and 95.44%) and BEN selection rule (52.79% and 54.85%), respectively. However, the PMS selection rule does not always lead to the best performance for different problems. For the total 24 test problems, there are 16 problems’ maximum total profit (11 from PMS, 4 from PPT, and 1 from BEN) and 14 problem’s maximum profit-per-unit-time (10 from PMS, 3 from PPT, and 1 from BEN) obtained from non-random selection rules. Interestingly, though, the three non-random selection rules likely do not present poor performance concurrently for the same problem. In other words, when a non-random selection rule presented poor performance, some other non-random selection rule may present good performance. As a result, the best performance of the three non-random selection rules in total profit and profit-perunit-time on average is 112.64% and 101.94%, respectively. Another phenomenon is that maximize of profit-per-unit-time does not guarantee maximum total profit which can be seen from the problems of M3N100, M5N200 with same machines and M3N20, M3N100 with different machines. In general, the non-random selection rules, except BEN, are able to present excellent performances for different problems. One of the benefits of using non-random selection rule in scheduling is saving time. With non-random selection rule, high-quality schedule result can be generated without iteration. The CPU time consumption of each test problem is shown in Fig. 1.3. On average, the CPU time consumption of 100 iterations with RDM selection rule (about 5.64 or 3.91 h) could be more than 800 or 477 times of non-

M3N20 M3N50 M3N100 M3N200 M3N400 M3N600 M5N20 M5N50 M5N100 M5N200 M5N400 M5N600 Different machines M3N20 M3N50 M3N100 M3N200 M3N400 M3N600 M5N20 M5N50 M5N100 M5N200 M5N400 M5N600

Same machines

Total profit (GBP) RDB RDW 93,466.6 50,698.4 129,060.4 82,147.8 152,664.6 110,967.0 167,796.8 125,496.2 169,389.4 123,993.4 164,292.4 133,197.6 114,104.2 63,663.8 139,274.8 93,644.8 214,307.6 156,660.0 261,826.2 169,367.2 268,494.2 187,037.6 274,218.8 213,804.2 51,327.2 31,488.2 77,599.6 48,904.7 91,765.6 73,295.7 95,833.8 77,327.6 96,423.9 83,974.2 96,337.9 80,920.7 94,017.8 62,473.0 131,462.4 94,029.5 178,587.0 137,298.7 197,844.6 151,569.3 198,972.3 157,794.6 201,997.3 163,654.1 PMS 84,994.0 106,098.8 148,317.6 185,065.4 182,476.4 190,582.4 84,994.0 124,417.6 201,189.2 270,922.8 285,879.0 295,495.6 47,953.1 72,015.0 88,426.1 98,861.2 104,311.1 101,877.8 93,963.4 133,994.5 172,810.1 200,415.9 207,602.5 211,619.9

Table 1.6 Experimental results with different selection rules PPT 84,994.0 119,938.8 152,462.4 177,257.8 172,828.8 178,085.8 84,994.0 123,280.0 202,234.2 269,667.0 295,367.8 293,520.6 52,022.9 66,572.5 92,080.5 100,708.7 100,052.1 99,798.6 91,816.6 125,512.1 176,455.9 200,135.1 204,564.9 208,190.0

BEN 86,876.6 109,584.6 125,843.8 141,944.0 147,806.8 162,319.6 98,512.4 119,233.6 160,997.0 169,274.6 198,302.4 221,570.0 50,512.7 79,622.2 81,400.8 79,036.0 99,652.4 86,548.0 92,466.0 132,726.6 160,094.0 169,298.6 184,340.2 151,771.0

Profit-per-unit-time (GBP/h) RDB RDW PMS PPT 144.61 78.44 131.50 131.50 139.15 85.21 114.39 129.31 163.92 120.16 160.40 159.89 166.53 121.67 179.48 172.12 168.75 123.82 175.00 164.61 169.75 130.09 184.66 171.43 176.54 98.50 131.50 131.50 151.18 101.65 134.14 132.91 248.58 165.05 214.70 215.79 261.18 181.18 266.29 266.31 271.16 193.34 277.96 281.52 274.85 207.02 285.54 282.55 67.98 36.58 59.22 64.57 91.35 49.18 78.41 68.08 99.65 78.67 94.97 94.49 96.20 73.71 94.62 97.10 97.98 81.71 102.17 97.38 99.53 78.17 101.86 95.78 94.25 62.63 94.19 92.04 147.06 98.72 147.46 136.29 181.40 138.02 178.97 180.45 194.35 145.20 202.94 195.15 194.89 154.04 203.25 196.23 200.59 157.80 202.27 198.36 BEN 150.48 115.79 141.98 140.54 161.56 159.60 170.63 128.55 182.13 167.60 220.32 213.41 66.90 80.47 88.55 78.33 97.88 87.24 92.69 135.13 169.19 165.92 181.27 147.10

CPU time (s) RDM PMS 7.05 0.04 68.41 0.34 210.93 1.15 898.97 1.66 4753.35 7.03 15,278.90 11.69 11.46 0.06 78.83 0.49 321.73 2.02 1213.79 2.93 6149.41 10.09 20,306.61 22.90 3.72 0.04 37.47 0.18 123.92 0.36 490.28 1.10 3345.34 3.94 10,542.25 10.01 6.65 0.04 50.05 0.27 133.64 0.59 744.07 1.55 4483.70 5.76 14,092.34 17.56 PPT 0.04 0.24 1.07 2.38 6.81 15.84 0.06 0.37 1.76 4.35 10.57 21.34 0.04 0.20 1.09 1.13 5.21 15.71 0.04 0.33 0.42 1.72 6.26 17.65

BEN 0.04 0.20 0.95 2.25 8.84 23.08 0.07 0.34 1.59 3.49 12.57 31.70 0.03 0.30 0.84 3.80 16.04 53.01 0.05 0.49 1.40 7.08 21.79 53.38

20 Q. Li et al.

1 Real-Time Order Acceptance and Scheduling in Metal Additive Manufacturing

21

Table 1.7 Experimental results with different selection rules

Same machines

Different machines

∗ High

M3N20 M3N50 M3N100 M3N200 M3N400 M3N600 M5N20 M5N50 M5N100 M5N200 M5N400 M5N600 M3N20 M3N50 M3N100 M3N200 M3N400 M3N600 M5N20 M5N50 M5N100 M5N200 M5N400 M5N600

Performance in total profit (%) P P MS P P P T P BEN BEST 80.19 80.19 84.59 84.59 51.05 80.56 58.48 80.56 89.57 99.52 35.68 99.52 140.82 122.37 38.88 140.82 128.83 107.58 52.46 128.83 184.55 144.36 93.66 184.55 42.29 42.29 69.09 69.09 67.44 64.95 56.08 67.44 77.24 79.06 7.52 79.06 109.84 108.48 −0.10 109.84 121.34 132.99 13.83 132.99 135.22 131.95 12.85 135.22 82.99 103.51 95.89 103.51 80.54 61.57 107.05 107.05 81.92 101.70 43.88 101.70 116.36 126.34 9.23 126.34 163.35 129.14 125.93 163.35 135.93 122.45 36.50 135.93 99.83 93.02 95.08 99.83 106.76 84.10 103.38 106.76 86.01 94.84 55.21 94.84 105.56 104.95 38.31 105.56 120.96 113.58 64.47 120.96 125.10 116.15 −30.99 125.10

Performance in profit-per-unit-time (%) P P MS P P P T P BEN 80.19 80.19 108.87 54.09 81.76 56.70 91.94 90.79 49.86 128.87 112.47 42.07 113.90 90.79 83.99 137.61 104.24 74.42 42.29 42.29 92.43 65.60 63.12 54.31 59.44 60.74 20.44 106.40 106.41 −16.98 108.73 113.31 34.67 115.77 111.36 9.41 72.10 89.13 96.56 69.32 44.82 74.22 77.70 75.40 47.08 92.96 104.00 20.55 125.73 96.28 99.37 110.90 82.45 42.48 99.83 93.02 95.08 100.82 77.72 75.31 94.41 97.82 71.86 117.48 101.63 42.17 120.48 103.27 66.66 103.91 94.78 −25.01

BEST 108.87 81.76 91.94 128.87 113.90 137.61 92.43 65.60 60.74 106.41 113.31 115.77 96.56 74.22 77.70 104.00 125.73 110.90 99.83 100.82 97.82 117.48 120.48 103.91

profits in bold

random selection rules (25.31 or 29.53 s) for the problem with 5 same/different machines and 600 part orders.

1.6 Conclusions and Future Research In this research, the problem of real-time OAS in MAM production environment with multiple machines and dynamic arriving part orders was introduced and mathematical formulations are presented for the first time. The part orders with specified boundary dimensions according to predefined orientation, material volume, and requested due date are placed by distributed customers in chronological order. The manufacturer makes decisions simultaneously on whether to accept the already

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Fig. 1.3 CPU time consumptions of the test problems

arrived part orders and how to schedule them on their multiple MAM machines specified with building capacity, production efficiency, operation cost, and service price under constraints of the machines capacity and the requested due date of part orders. The problem was described with the aim of maximizing the average profitper-unit-time within the whole planning horizon. Considering that this was shown to be a strongly NP-hard problem, we proposed meta-heuristic based procedures to generate feasible jobs on a single machine as well as on multiple machines to form

1 Real-Time Order Acceptance and Scheduling in Metal Additive Manufacturing

23

a feasible schedule solution. Moreover, four selection rules are proposed for the generation of feasible schedule solutions according to the analysis of the influence of selective behaviours on the schedule results. To evaluate the performance of our proposed selection rule based heuristic algorithms, a comprehensive experimental study was designed and conducted. The experimental results indicate that a significant difference (more than 2.56 times) may exist between the potential feasible schedule solutions, and the CPU time consumption of 100 iterations increases exponentially (increased about 3792 times from 3 different machines and 20 part orders to 5 different machines and 600 part orders) as the problem size increases along with the number of machines and part orders. Compared with the best and the worst schedules obtained through 100 iterations with RDM selection rule, the heuristic algorithms with non-random selection rules presented quite promising performances. For the 24 tested problems, there are 14 schedule results generated with one of the three non-random selection rules better than the best schedule results obtained through 100 iterations with RDM selection rule. The results are sufficiently convincing to state that the high-quality schedule results can be obtained without iteration based on properly designed selection rules, which is worthy for further researches. As the first attempt to address real-time OAS problem in MAM production environment, we aimed to open up opportunities to study the different production planning and scheduling problems in industrial additive manufacturing field. With appropriate modifications in capacity constraints, the mathematical expressions and meta-heuristic algorithms proposed in this research could be easily adapted to the OAS problems in the production environment with other additive manufacturing processes such as selective laser sintering or binder jetting [10]. The experimental evaluations in this research also show several applicable areas for future works. The first one might be the further investigation of the possible selective behaviours during the generation of a feasible schedule solution to design more practical selection rules. Secondly, developing advanced heuristic algorithms based on bio-inspired algorithms to improve flexibility and applicability of the methodologies for solving different OAS problems. Acknowledgements The third author (I.K.) acknowledges the financial support received from Balikesir University—Scientific Research Projects Department under grant number BAP-2018131.

References 1. R. Jiang, R. Kleer, F.T. Piller, Predicting the future of additive manufacturing: a Delphi study on economic and societal implications of 3D printing for 2030. Technol. Forecast. Soc. Change 117, 84–97 (2017). https://doi.org/10.1016/j.techfore.2017.01.006 2. S.A.M. Tofail, E.P. Koumoulos, A. Bandyopadhyay, S. Bose, L. O-Donoghue, C. Charitidis, Additive manufacturing: scientific and technological challenges, market uptake and opportunities. Mater. Today 21, 22–37 (2017). https://doi.org/10.1016/j.mattod.2017.07.001

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3. L.E. Murr, S.M. Gaytan, D.A. Ramirez, E. Martinez, J. Hernandez, K.N. Amato, P.W. Shindo, F.R. Medina, R.B. Wicker, Metal fabrication by additive manufacturing using laser and electron beam melting technologies. J. Mater. Sci. Technol. 28, 1–14 (2012). https://doi.org/10.1016/ S1005-0302(12)60016-4 4. Q. Li, I. Kucukkoc, D.Z. Zhang, Production planning in additive manufacturing and 3D printing. Comput. Oper. Res. 83, 1339–1351 (2017). https://doi.org/10.1016/j.cor.2017.01.013 5. S.A. Slotnick, Order acceptance and scheduling: a taxonomy and review. Eur. J. Oper. Res. 212, 1–11 (2011). https://doi.org/10.1016/j.ejor.2010.09.042 6. H.F. Rahman, R. Sarker, D. Essam, A real-time order acceptance and scheduling approach for permutation flow shop problems. Eur. J. Oper. Res. 247, 488–503 (2015). https://doi.org/10. 1016/j.ejor.2015.06.018 7. M. Khalili, M. Esmailpour, B. Naderi, The production-distribution problem with order acceptance and package delivery: models and algorithm. Manuf. Rev. 3, 18 (2016). https:// doi.org/10.1051/mfreview/2016018 8. A. Noroozi, M.M. Mazdeh, M. Heydari, M. Rasti-Barzoki, Coordinating order acceptance and integrated production-distribution scheduling with batch delivery considering Third Party Logistics distribution. J. Manuf. Syst. 46, 29–45 (2018). https://doi.org/10.1016/j.jmsy.2017. 11.001 9. T. Aouam, K. Geryl, K. Kumar, N. Brahimi, Production planning with order acceptance and demand uncertainty. Comput. Oper. Res. 91, 145–159 (2018). https://doi.org/10.1016/j.cor. 2017.11.013 10. F. Calignano, D. Manfredi, E. Ambrosio, S. Biamino, M. Lombbardi, E. Atzeni, A. Salmi, P. Minetola, L. Iuliano, P. Fino, Overview on additive manufacturing technologies. Proc. IEEE 105, 593–612 (2017). https://doi.org/10.1109/JPROC.2016.2625098 11. M. Khorram Niaki, F. Nonino, Additive manufacturing management: a review and future research agenda. Int. J. Prod. Res. 55, 1419–1439 (2017). https://doi.org/10.1080/00207543. 2016.1229064 12. I. Kucukkoc, Q. Li, D.Z. Zhang, Increasing the utilisation of additive manufacturing and 3D printing machines considering order delivery times, in Nineteenth International Working Seminar on Production Economics, Innsbruck, Austria, vol. 3 (2016), pp. 195–201 13. I. Kucukkoc, Q. Li, N. He, D. Zhang, Scheduling of multiple additive manufacturing and 3D printing machines to minimise maximum lateness, in: Twentieth International Working Seminar on Production Economics, Innsbruck, Austria, vol. 1 (2018), pp. 237–247 14. I. Kucukkoc, MILP models to minimise makespan in additive manufacturing machine scheduling problems. Comput. Oper. Res. 105, 58–67 (2019). https://doi.org/10.1016/j.cor.2019.01. 006 15. X. Li, K. Zhang, Single batch processing machine scheduling with two-dimensional bin packing constraints. Int. J. Prod. Econ. 196, 113–121 (2018). https://doi.org/10.1016/j.ijpe. 2017.11.015 16. J.P. Rudolph, C. Emmelmann, A cloud-based platform for automated order processing in additive manufacturing. Procedia CIRP 63, 412–417 (2017). https://doi.org/10.1016/j.procir. 2017.03.087 17. K. Ransikarbum, S. Ha, J. Ma, N. Kim, Multi-objective optimization analysis for part-to-Printer assignment in a network of 3D fused deposition modeling. J. Manuf. Syst. 43, 35–46 (2017). https://doi.org/10.1016/j.jmsy.2017.02.012 18. L. Zhou, L. Zhang, Y. Laili, C. Zhao, Y. Xiao, Multi-task scheduling of distributed 3D printing services in cloud manufacturing. Int. J. Adv. Manuf. Technol. (2018). https://doi.org/10.1007/ s00170-017-1543-z 19. Q. Li, I. Kucukkoc, N. He, D. Zhang, S. Wang, Order acceptance and scheduling in metal additive manufacturing: an optimal foraging approach, in Twentieth International Working Seminar on Production Enconomics, Innsbruck, Austria, vol. 1 (2018), pp. 225–235 20. P. Jacobs, 2D Rectangle bin packing in Python (2016). Online material. https://github.com/ pellejacobs/2d-rectangle-bin-packing (Accessed: 19.12.2019)

Chapter 2

Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection Gokturk Poyrazoglu

2.1 Introduction According to the International Energy Agency (IEA), by 2020 developing countries will double their power generation capacity in order to meet the growing demand. The global goal of decreasing consequences of carbon emission is enhancing renewable energy production. Furthermore, the effective integration and usage of renewable energies require energy storage to take place [1, 2]. In the USA, the total energy storage deployment has increased by 188% in terms of energy capacity and 243% in terms of power capacity between 2014 and 2015. This indicates how energy storage becomes an increasing need in developed countries. Moreover, the cost of energy storage systems is reducing rapidly in recent years [3, 4]. Energy storage decreases the dependency on the conventional energy generation during the peak load period [5]. One of the applications of a storage system is to store energy during the low demand periods where the energy cost is low and to inject the stored energy back to the grid within the peak demand period where the energy cost is high. It allows power systems optimization within multiple periods of time [6]. An evaluation of energy storage integration and its roles and benefits in the future smart grid is examined in the literature [7]. The storage systems play an important role in energy management such as power bridging, reliability, and power quality [8]. There are different energy storage technologies such as pump storage hydroelectricity (PHS), batteries, compressed air energy storage (CAES), superconducting magnetic energy storage (SMES), thermal storage, and many other technologies [9]. Currently more than 141 GW of energy storage installed in terms of power capacity worldwide [10], but pump storage hydro occupies around

G. Poyrazoglu () Ozyegin University, Department of Electrical and Electronics Engineering, Istanbul, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6_2

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90% of the global installed energy storage systems. The problem of allocating optimal hybrid-storage units only on the buses that include generation units is being discussed in the energy sector. The capacity of each storage unit and the economic dispatch decision variables are optimized together in order to achieve the energy generation cost minimized [11]. However, some research [12, 13] has been done on the optimal sizing and placements of energy storage units in power distribution systems. There is an interesting argument of using single phase storage units that are located at a customer house can help to solve the voltage problem with less energy than the three-phase units located outside [14–16]. Storage applications with renewable resources are quite popular in both academia and industry [17, 18]. The integration of storage technologies within the scope of energy markets is also an interesting research area that would help improve the development of storage economy [19–24]. This chapter is intended to provide a detailed explanation of the structure of energy economics by providing a step by step development of optimization models being used today in power system operations. After the introduction of optimization problems, a novel model for energy storage operations and the procedure to develop a nationwide energy storage policy is being discussed.

2.2 Optimization Models in Energy Economics 2.2.1 Economic Dispatch Model The economic dispatch (ED) problem aims to find the optimal hourly dispatch of generators by minimizing the total production cost. The general energy balance of the system and the generator limits are considered; however, the economic dispatch does not consider any network flows or network constraints. It assumes a copperplate network in which the electricity can flow from A to B in a lossless and limitless network. The ED problem is in the form of linear programming and the mathematical program in the standard form is given in Model 1. Model 1: Economic Dispatch Model

minimize



Bg Pg,t

(2.1)

g,t

subject to Pg ≤ Pg,t ≤Pg

∀g ∀t

(2.2)

2 Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection



Pg,t = Dt

∀t

27

(2.3)

g

The given program in Model 1 has a linear cost function; however, the cost function can also be defined in a quadratic form which converts the model into the standard form of a quadratic program (QP). However, the ED problem considers the physical limitations of a power plant in a simplistic manner. It only considers the generators’ limit, yet there are some other physical limitations that should have an impact on realistic electricity operations. Some limitations are based on the characteristics of the turbine used in the plant and some are related to the type of fuel used. The following are a couple of physical limitations: – – – –

Minimum up time Minimum downtime Ramp up capability Ramp down capability

2.2.2 Unit Commitment Model The ramp characteristic is generally a consequence of a turbine design of a power plant. The ramping capability of gas-fired plants is significantly high in comparison to the coal-fired power plants. However, the ramping capability in either down or up direction is extremely low for a nuclear power plant. On the other hand, minimum up/downtime characteristics of a power plant are generally based on the type of fuel. The nuclear power plants, for instance, have a high minimum up/downtime which may go up to 2–3 days. Gas-fired power plants, however, are famous for their low minimum up/downtime which may go down to 15 min. The minimum up time is the minimum number of hours that a unit must run for once it has been started. Minimum downtime is the minimum number of hours that a unit must be down after it has been shut down. The former is modeled as a start-up cost in the objective function and the latter is modeled as a shutdown cost. These characteristics require binary variables to be modeled properly. Therefore, the start-up variable s that takes on one when the unit is scheduled to start -up. Similarly, the shutdown variable h that takes on one when the unit is scheduled to shut down. In order to develop a commitment schedule between the unit’s start-ups and shutdowns, a unit commitment variable u is modeled that takes on one (zero) when the unit is on (off). These binary variables of s, h, and u are associated with the generators’ startup cost, shutdown cost, and no load cost, respectively. These costs are considered as a production cost; therefore, they are included in the objective function of the program. The last but not least constraint is about the reserve need of the power systems due to the uncertainty related to the demand. The reserve requirement is an extra fictitious load in the system that the total capacity of the committed units

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should satisfy if this fictitious load becomes reality in the real-time operations. The overall mathematical program of the unit commitment (UC) problem in the form of mixed integer linear programming (MILP) is given in Model 2. Model 2: Unit Commitment Model

minimize



(Bg Pg,t β + Cg sg,t + Eg hg,t + Ag ug,t )

(2.4)

t,g

subject to Constraint (2.3) ug,t Pg ≤ Pg,t ≤Pg ug,t

∀g ∀t

− RgD ≤Pg,t − Pg,(t−1) ≤ RgU 

sg,τ ≤ ug,t

∀g ∀t

∀g ∀t

(2.5) (2.6) (2.7)

τ =t−MgU +1



hg,τ ≤ 1 − ug,t

∀g ∀t

(2.8)

∀g ∀t

(2.9)

τ =t−MgD +1

sg,t − hg,t = ug,t − ug,t−1 

P g ug,t ≥ Dt + Rt

∀t

(2.10)

g

The constraints of (2.1), (2.2) in Model 1 are exchanged with the constraints of (2.4), (2.5) in Model 2, respectively. Moreover, a couple of extra constraints are now considered to create a more physically realistic model for the operation of power systems. The constraint (2.6) represents the ramp up/down capabilities for each generation unit between two consecutive time periods. The constraint (2.7) limits the unit commitment decisions based on the minimum uptime of each generator. The constraint (2.8) is the alias of (2.7) for the minimum downtime. The constraint (2.9) defines the relationship between start-up and shutdown decisions for each generation unit. Lastly, the constraint (2.10) considers an extra reserve requirement for the operational security purposes on top of the expected demand.

2 Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection

29

2.2.3 Energy Storage System Modeling in UC In the final part of Sect. 2.2.1, the addition of the energy storage system into the unit commitment model is discussed. The energy storage system cannot be classified as either a load or a generator for the power grid; it indeed can behave like both of them in different time intervals. However, an energy storage system is not a generator by itself, so it cannot operate as a generator forever due to its maximum energy capacity. Likewise, it is not actually a demand by itself, so it cannot create demand forever but up to its energy capacity. Therefore, taking the role of a generator or a demand by an energy storage system is directly affected by the maximum power delivery and the maximum energy capacity of those units. The modeling of an energy storage system in a mathematical program can be done in different forms. One way of these forms can be creating two more variables in the program: hourly charge and discharge amount by the energy storage system. This way of modeling requires an update on the constraints (2.3) and (2.10). However, this study proposes a novel model that can be more useful in energy economics. Instead of having two separate variables for charging and discharging modes of an energy storage system, the proposed model only has one variable, denoted as e. The variable e is a matrix variable in the size of T x T. The upper triangle of the matrix holds the charging amount and the lower triangle of the matrix holds the discharging amount. So the component ei,j where i > j holds the amount of stored energy at time i to be discharged at time j . Likewise, the component ei,j where i < j holds the amount of discharged energy at time i which was charged at time j . In that manner, the matrix e holds the total energy charged at time t as well as when each portion of it will be discharged in the future. This property of the proposed model provides a lot more detailed information about the operation of the energy storage system. In order to complete the unit commitment model with the integration of an energy storage system, the constraints (2.3) and (2.10) are updated with the constraints (2.11) and (2.12). 

Pg,t +

t 

et,τ ≥ Dt +

τ =1

g

 g

P g ug,t ≥ Dt + Rt +

T 

et,τ

∀t

(2.11)

τ =t T 

et,τ

∀t

(2.12)

τ =t

The constraint (2.11) maintains the power balance of the grid throughout the time period T. The second term on the right-hand side of the inequality represents the charging of energy storage and it is considered as an extra demand. While the second term on the left-hand side of the inequality represents the discharging of energy storage and considered as a generation. The constraint (2.12) considers also the demand related to the stored energy by the storage system as an extra demand that

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is required for operational security purposes. Finally, the unit commitment problem with the integration of an energy storage system is given in Model 3. Model 3: Unit Commitment Model with an Energy Storage System

minimize (2.4) subject to Constraints (2.5–2.9) Constraints (2.11–2.12) et,τ ≥ 0

∀t ∀τ

(2.13)

2.2.4 AC Optimal Power Flow Model Power systems, in general, are controlled by setting four individual quantities for each bus in a way that satisfies both Kirchhoff’s laws and a system’s operational limits. These four quantities are (1) real power generation, (2) reactive power generation, (3) voltage magnitude, and (4) voltage angle. Any values of these quantities that satisfy the energy balance equation and their physical limits is indeed a feasible, stable, and reliable operational state of the power system. However, the modern power system includes several generators that produce energy from various type of fuels. So a generator may use coal, gas, uranium, solar radiation, wind, or biofuel to produce electricity and the cost of generating electricity varies with the price of fuel used. Carpentier [1] first introduced an optimal power flow scheme in 1962 permitting the optimal operation of power systems. Carpentier’s focus was mostly on the optimization of the system operation yet not so much on the market at that time. However, the OPF problem becomes the common ground on the market operations in the USA, especially in California Independent System Operator (CAISO) that solves OPF every day for the day-ahead market, every hour and every 15 min for the intra-day market, and every 5 min for the balancing market. In a realistic form, the OPF is in the form of a non-linear nonconvex optimization problem. It minimizes the total production cost, such that the mathematical programming seeks the optimal operation state for the system, one that satisfies the energy balance equation and physical and operational limits such as: – – – –

Ohm’s Law Kirchhoff’s Law Generator Limits Transmission Limits

2 Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection

31

– Voltage Magnitude and Angle Limits – Others. The resultant mathematical problem, AC-OPF, can be formulated as a nonlinear program (NLP). For clarity, the AC-OPF problem is given in Model 4 with no consideration on time interval; however, the inclusion of time into the model should be straightforward by assuming that all constraints should be satisfied at each interval. Model 4: AC Optimal Power Flow 

fg (Pg )

(2.14)

|Vj |(Gij cos(θi − θj ) + Bij sin(θi − θj ))∀i

(2.15)

minimize

g

subject to P gi − P di = |Vi |

 j

Qgi − Qdi = |Vi |



|Vj |(Gij sin(θi − θj ) − Bij cos(θi − θj ))

∀i

(2.16)

j

|Vi | ≤ |Vi | ≤ |Vi |

∀i

(2.17)

P gi ≤ P gi ≤ P gi

∀i

(2.18)

Qgi ≤ Qgi ≤ Qgi

∀i

(2.19)

|Sij | ≤ Sijmax

∀(i, j ) and (j, i)

(2.20)

The main difference of an OPF problem with the UD or the ED problem is the consideration of power flows on the network. Each branch in the network represents a physical conductor that connects two buses (nodes). The resistance and reactance of the conductor cause the difference between the sending end of the branch and the receiving end. Moreover, there is a maximum amount of power that can be transferred via a conductor. This phenomenon is modeled as an inequality constraint (2.20) in Model 4. In general, the thermal limit exists on the apparent power flow, which is a quantity derived from the real and reactive (imaginary) power flow. Due to the bidirectional flow observed in power systems, this constraint must be modeled for both ends of the branch indicating the power flowing from Bus i to Bus j , and the power flowing from Bus j to Bus i.

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2.2.5 DC Optimal Power Flow Model The complexity of the realistic model as in Model 4 is challenging to be utilized in the real power system operations. The non-linearity of the energy balance equalities, (2.15) and (2.16), may be linearized and non-convexity of the constraints (2.15), (2.16), (2.17), and (2.20) may be relaxed by three assumptions. 1. The voltage angle difference in radian between Bus i and Bus j is so small so that sin(θi − θj ) = θi − θj . 2. |Vi | and |Vj | are constant and equal to 1 per unit (pu). 3. Given that r 0, given

3.2.2 Necessary and Sufficient Conditions of Optimality Our aim, in this section, is to set the optimal growth problem (P ) as an optimal

) and to show that the necessary conditions given by Theorem control problem (P 3.1 of [9] and sufficient conditions given by Theorem 5.1 of [9]2 are fulfilled for ). (P Set x = (k, c) ∈ ∞ × ∞ and g(kt , ct ) := f (kt ) − ct for all t = 0, 1, . . . where kt ∈ R+ is the scalar state variable and ct ∈ R+ is the scalar control variable. The  ): dynamic system is then governed by the following difference inequation (DI  ) kt+1 ≤ g(kt , ct ) for all t = 0, 1, . . . (DI

) will be Then (P

2 These

are also Theorem 3.3 and Theorem 3.8 of [8].

3 Pontryagin’s Principle for Optimal Growth Problems

55

 ⎧ t max J (x) = J (k, c) := ∞ ⎪ t=0 β u(ct ) ⎪ ⎨

) s.t. (P ⎪ ≤ g(kt , ct ) k ⎪ ⎩ t+1 k0 > 0 given, ct ≥ 0, kt ≥ 0

) and the multipliers 1 and Pontryagin’s Hamiltonian function associated with (P λ is defined by Ht : R × R × R × R → R such that Ht (kt , ct , 1, λ) := β t u(ct ) + λg(kt , ct ) Proposition 3.1 Let the following assumptions be satisfied:  (P rod) (Production Assumption)  (P ref ) (Preferences Assumption)

f : R → R is continuously differentiable, u : R → R is continuously differentiable.

∞ If the feasible accumulation sequence x ∗ = (k ∗ , c∗ ) in int∞ + × int+ is an optimal

), then it is a solution of the following system: solution of (P

u (ct ) = βu (ct+1 )f  (kt+1 ) for all t = 1, 2 . . .

(3.1)

f (kt ) = ct + kt+1 for all t = 0, 1, 2 . . .

(3.2)

 Conversely, under P rod and P ref , let the above Eqs. (3.1) and (3.2) be fulfilled ∞ ∗ 1 for a feasible allocation x ∗ = (k ∗ , c∗ ) in int∞ + ×int+ . Let there exist λ ∈ + such  that the Pontryagin’s Hamiltonian function, associated with (P ) and the multipliers 1 and λ, Ht (kt , ct , 1, λ) is concave with respect to (kt , ct ) for all t = 0, . . .. Then ). x ∗ = (k ∗ , c∗ ) is an optimal solution of (P Proof Since u is independent of kt and supposed to be continuously differentiable and since f is continuously differentiable then so is g : R2 → R. Under the  assumptions P rod and P ref , the assumptions3 of Theorem 3.1 in [9] are verified; therefore, we can directly use its conclusion. There exists then a sequence of multipliers λ∗ ∈ 1+ such that the following conditions, which are so-called Adjoint Equation (AE), Weak Maximum Principle (W MP ), and Complementary Slackness (CS), hold: (AE) λ∗t = ∇k Ht (kt∗ , ct∗ , 1, λ∗t+1 ) (W MP ) ∇c Ht (kt∗ , ct∗ , 1, λ∗t+1 ) = 0 ∗ )=0 (CS) λ∗t+1 (g(kt∗ , ct∗ ) − kt+1

3 Essentially

case since

the Assumption (H 1) in [9]. Note that Assumption (H 4) is always satisfied in our = −1 = 0 for all t = 0, 1 . . .

∂g ∂c (kt , ct )

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which imply, respectively, λ∗t = λ∗t+1 · λ∗t+1 ·

∂g ∗ ∗ (k , c ) + β t · 0 for all t = 1, 2, . . . ∂kt t t

∂g ∗ ∗ (k , c ) + β t u (ct∗ ) = 0 for all t = 0, 1, . . . ∂ct t t

∗ λ∗t+1 (g(kt∗ , ct∗ ) − kt+1 ) = 0 for all t = 0, 1, . . .

(3.3)

(3.4) (3.5)

that give us the following system: λ∗t = λ∗t+1 f  (kt∗ ) for all t = 1, 2, . . .

(3.6)

λ∗t+1 (−1) + β t u (ct∗ ) = 0 for all t = 0, 1, . . .

(3.7)

∗ ) = 0 for all t = 0, 1, . . . λ∗t+1 (f (kt∗ ) − ct∗ − kt+1

(3.8)

From Eqs. (3.6) and (3.7), the system reduces to ∗ ∗ )f  (kt+1 ) for all t = 0, 1, . . . u (ct∗ ) = βu (ct+1 ∗ ) = 0 for all t = 0, 1, . . . λ∗t+1 (f (kt∗ ) − ct∗ − kt+1

(3.9) (3.10)

Remark that the multipliers associated with this problem are defined by λ∗t+1 = β t u (ct∗ ) and satisfy (3.9) which is Euler equation together with (3.10). Conversely, if Eqs. (3.9) and (3.10) are satisfied, then setting λ∗t+1 = β t u (ct∗ ), the first three assumptions of Theorem 5.1 of [9] are fulfilled. Moreover, if the Pontryagin’s Hamiltonian function is supposed to be concave with respect to (kt , ct ), then optimality holds. That is, the following assumption of Theorem 5.1 of [9]: (Co) For all t ∈ N, (kt , ct ) → Ht (k, c, 1, λt+1 ) = β t u(ct )+λt+1 (f (kt )−ct ) is concave allows us to conclude the optimality.4

 

holds if the utility function u is supposed to be concave since we have f (kt ) = ct + kt+1 , we will have the following concave Pontryagin’s Hamiltonian function:

4 (Co)

Ht (k, c, 1, λt+1 ) = β t u(ct ) + λt+1 (f (kt ) − ct ) = β t u(ct ) + kt+1

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57

Remark 3.1 1. The utility function u is supposed to be non-decreasing. Hence, the output will ∗ . not be wasted and at the optimum we have ct∗ = f (kt∗ ) − kt+1 2. The result is useful as the assumptions are easy to check and one may avoid the concavity assumptions of u and f . However, the concavity of the Hamiltonian is needed for the sufficient conditions of the optimality. Example 3.1 By weak Pontryagin’s principles approach given above, a solution to the optimal growth problem in which u(ct ) = ln ct , f (kt ) = (kt )α with 0 < α < 1 is x ∗ = (k ∗ , c∗ ) which is the solution of the following system which holds by (3.9) and (3.10): 1 1 = βα (kt+1 )α−1 for all t = 0, 1, . . . ct ct+1 (kt )α − ct − kt+1 = 0 for all t = 0, 1, . . . ∗ = αβ(kt∗ )α for all t = 0, 1 . . . generating the optimal sequence: kt+1

For a comparison of methods, one can find the same example in [11] whose solution is also given by other methods existing in the literature.

3.3 Optimal Growth Model with an Natural Exhaustible Resource In this section, we present a model of optimal growth with an exhaustible natural resource. The difference of this model from the one of the previous section is the feature of the good. This model refers to the management of an exhaustible natural resource like oil, coal, gas, etc. We consider an economy not only as a problem of capital accumulation but also a problem involving an extraction and consumption process with conservation or sustainability requirements. The primitives of the model are initial natural resource stock, production function in terms of capital and extraction flow and the preferences. The original model is due to [12] but we use the model presented in [1] and give the optimality conditions with weak Pontryagin’s principles approach. We consider an economy of infinite periods from time t = 0 to ∞ where there exists a single planner who manages the decision of extraction and consumption of the single natural resource at each period. We denote this economy by Eenr with reference to exhaustible natural resource. st stands for the stock of resource at the beginning of the period [t, t + 1[. During the discrete unit of time, an extraction is realized where the extraction flow per unit of time is denoted by rt . The accumulated capital at time t is denoted by kt and at time t = 0 the amount of capital is supposed to be k0 > 0 units. The output yt ∈ R+ is produced from capital and extracted

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quantity of resource by a production function5 f : R+ × R+ → R+ where yt = f (kt , rt ). The output is either consumed as ct ≥ 0 or saved as capital to the next period as kt+1 satisfying the following process being repeated until infinity: ct + kt+1 ≤ f (kt , rt ) + (1 − δ)kt with kt ≥ 0 where δ stands for the depreciation rate of capital. The discrete time dynamics is given with the following difference inequation system (DI):  (DI )

st+1 = st − rt kt+1 ≤ (1 − δ)kt + f (kt , rt ) − ct

The consumption level is determined according to the unique planner’s preferences which is defined by a one period non-decreasing utility (reward) function v : R4 → R. The intertemporal utility is then defined as follows: ∞ 

β t v(st , kt , rt , ct )

t=0

where 0 < β < 1 is the discount factor.

3.3.1 Management Problem of Eenr The extraction, consumption, saving, and conservation decision with respect to the constraints of the model is the decision problem that the economy Eenr must make. The state variables of the problem are the natural resource stock st and the capital kt . The control variables are the consumption level ct and the extraction level rt of the natural resource. The feasibility constraints of the model are given by the following definitions: Definition 3.2 (Irreversibility of the Extraction) The extraction quantity per unit time is irreversible in the sense that rt ≥ 0 for all t = 0, 1, . . .. Definition 3.3 (Conservation of the Natural Resource) The natural resource is scarce in the sense that st ≥ 0 for all t = 0, 1, . . ..6

general, the production function is supposed to be of type Cobb-Douglas f (k, r) = Ak α r β where the exponents α and β represent the elasticity of production with respect to capital and the extracted natural resource, respectively. 6 Sometimes, a stronger conservation constraint is taken into account in the sense that s ≥ s¯ where t s¯ > 0 for all t = 0, 1, . . .. This refers to a strong conservation concern implying sustainability. In this chapter, we suppose only the less strong one in order to be able to obtain the analytical results. 5 In

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59

Definition 3.4 (Irreversibility of the Investment in Reproducible Capital kt ) The growth of the capital is guaranteed in the sense that 0 ≤ f (kt , rt ) − ct for all t = 0, 1, . . .. Definition 3.5 (Sustainability) The capital is non-negative: k0 > 0 and 0 ≤ kt for all t = 1 . . .. A non-negative consumption level is guaranteed in the sense that 0 ≤ ct for all t = 0, 1, . . .. The objective of the planner of the natural resource is to maximize the utility by choosing the feasible allocation (s, k, r, c), i.e., subject to the feasibility constraints with a given positive initial resource stock and capital. The problem can then be written as follows: ⎧  t max ∞ ⎪ t=0 β v(st , kt , rt , ct ) ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎨ st+1 = st − rt , ∀t ≥ 0 (Penr ) ct + kt+1 ≤ f (kt , rt ) + (1 − δ)kt , ∀t ≥ 0 ⎪ ⎪ ⎪ st ≥ 0, rt ≥ 0, ct ≥ 0, ∀t ≥ 0 ⎪ ⎪ ⎪ ⎪ kt ≥ 0, ∀t ≥ 1 ⎪ ⎪ ⎩ k0 > 0, given The objective function states that the planner of natural resource must decide the extraction and consumption level at each period in order to maximize the utility under constraints. The first constraint reflects that the stock of resource at time t + 1 is the difference of the stock at the beginning of time period [t, t + 1[ and the extracted amount at time t. The second constraint reflects that non-consumed amount of output will be added to the capital of the next period and hence will determine the future production levels. Furthermore, at optimum, output will not be wasted so that the consumption at t will be equal to the difference of output and resource stock conserved, that is, ct = f (kt , rt ) + (1 − δ)kt − kt+1 . This problem is then to explain how the technology effects to arrive an optimal extraction and consumption paths of the natural exhaustible paths.

3.3.2 Necessary and Sufficient Conditions of an Optimal Management of a Natural Resource In this section, like in the previous one, we apply the result of weak Pontryagin’s principle given in [9] (Theorem 3.1 and Theorem 5.1 of [9]) to a problem of optimal growth with an exhaustible natural resource whose setup is described in [1].7

7 In [1], they propose a bounded case with a difference equation system; here, with a slight change we give the result for the bounded case with a difference inequation system.

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Recall that (st , kt ) ∈ R2 is a vector state variable and (rt , ct ) ∈ R2 is a vector control variable such that (st , kt )t ∈ ∞ (N, R2 ) and (rt , ct )t ∈ ∞ (N, R2 ). Let g = (g1 , g2 ) be the function defined on R 4 → R2 as follows: 

g1 (st , kt , rt , ct ) := st − rt g2 (st , kt , rt , ct ) := (1 − δ)kt + f (kt , rt ) − ct

We first transform the optimal growth problem (Penr ) to an optimal control  problem (P enr ) by transforming the dynamical system (DI) to the following one: ) (DI



st+1 = g1 (st , kt , rt , ct ) = st − rt kt+1 ≤ g2 (st , kt , rt , ct ) = (1 − δ)kt + f (kt , rt ) − ct

We then show that the necessary conditions given by Theorem 3.2 of [8] are satisfied.  Then (P enr ) will be: ⎧  t max ∞ ⎪ t=0 β v(st , kt , rt , ct ) ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ st+1 = g1 (st , kt , rt , ct ), ∀t ≥ 0 ⎨ kt+1 ≤ g2 (st , kt , rt , ct ), ∀t ≥ 0 (P enr ) ⎪ s t ≥ 0, rt ≥ 0, ct ≥ 0, ∀t ≥ 0 ⎪ ⎪ ⎪ ⎪ k ⎪ t ≥ 0, ∀t ≥ 1 ⎪ ⎪ ⎪ k > 0, given ⎪ ⎪ ⎩ 0 (st )t , (kt )t , (rt )t and (ct )t bounded  Pontryagin’s Hamiltonian function associated with (P enr ) and the multipliers 1, 2 and λ is defined by Ht : R × R × R × R × R → R such that

λ1 ,

Ht (kt , ct , 1, λ1t , λ2t ) := β t v(st , kt , rt , ct ) + λ1t g1 (st , kt , rt , ct ) + λ2t g2 (st , kt , rt , ct ) Proposition 3.2 Let the following assumptions be satisfied: (P rodenr ) (Production Assumption)  (P refenr ) (Preferences Assumption)

f : R2 → R is continuously differentiable, v : R4 → R is continuously differentiable.

∞ If the feasible accumulation sequence x ∗ = (s ∗ , k ∗ , r ∗ , c∗ ) in int∞ + × int+ × ∞  int∞ + × int+ is an optimal solution of (Penr ), then it is a solution of the following system:

∂v ∗ ∗ ∗ ∗ ∂v ∗ ∗ ∗ ∗ (st , kt , rt , ct ) = β (s , k , r , c ) ∂ct ∂ct t t t t+1 × ((1 − δ) +

∂f ∗ ∗ (k , r )) for all t = 0, 1, . . . ∂kt t t

(3.11)

3 Pontryagin’s Principle for Optimal Growth Problems ∗ f (kt∗ , rt∗ ) = (1 − δ)kt∗ − ct∗ − kt+1 for all t = 0, 1, . . .

61

(3.12)

Conversely, under P rodenr and P refenr , let the above Eqs. (3.11) and (3.12) ∞ be fulfilled for a feasible allocation x ∗ = (s ∗ , k ∗ , r ∗ , c∗ ) in int∞ + × int+ × ∞ ∞ 1∗ 2∗ 1 2 int+ × int+ . Let there exist (λt , λt ) ∈  (N, R ) such that the Pontryagin’s 1 2 Hamiltonian function, associated with (P enr ) and the multipliers 1 and λt and λt 1 2 Ht (st , kt , rt , ct , 1, λt , λt ) be concave with respect to (st , kt , rt , ct ) for all t = 0, 1, . . .. Then x ∗ = (s ∗ , k ∗ , r ∗ , c∗ ) is an optimal solution of (P enr ). Proof Since v, f are continuously differentiable, then so is g1 and g2 . Under the assumptions P rodenr and P refenr , the assumptions of Theorem 3.1 in [9] are verified; therefore, we can directly use its conclusion. Essentially the Assumption (H 1) in [9] is satisfied since u, f are continuously differentiable then so is g1 and g2 . Note also that the Assumption (H 4) in [9] is satisfied since the partial derivative with respect to the vector control variable (rt , ct ) ∈ R2 is Dg(r,c) (s ∗ , k ∗ , r ∗ , c∗ ) and it is an isomorphism from R2 to R2 since det Dg(r,c) (s ∗ , k ∗ , r ∗ , c∗ ) = 0.8 There exists then a sequence of multipliers (λ1∗ , λ2∗ ) ∈ 1 (N, R 2 ) such that the following conditions, which are so-called Adjoint Equation (AE), Weak Maximum Principle (W MP ), and Complementary Slackness (CS), hold: ∗ ∗ ∗ ∗ 1∗ (AE1) λ1∗ t = ∇s Ht (st , kt , rt , ct , 1, λt+1 ) ∗ ∗ ∗ ∗ 2∗ (AE1) λ2∗ t = ∇k Ht (st , kt , rt , ct , 1, λt+1 )

(W MP 1) ∇r Ht (st∗ , kt∗ , rt∗ , 1, ct∗ , λ∗t+1 ) = 0 (W MP 2) ∇c Ht (st∗ , kt∗ , rt∗ 1, ct∗ , λ∗t+1 ) = 0 ∗ ∗ ∗ ∗ ∗ (CS) λ2∗ t+1 (g2 (st , kt , rt , ct ) − kt+1 ) = 0

which imply, respectively,

mapping (st , kt , rt , ct ) → Dg(r,c) (st , kt , rt , ct ) is continuous since g is continuously differentiable and the composition with the determinant is also continuous; therefore, there exists a neighborhood containing (s ∗ , k ∗ , r ∗ , c∗ ) such that the set {(st , kt , rt , ct )| det Dg(r,c) (st , kt , rt , ct ) = 0} is open. Here,      ∂g1 (s ∗ , k ∗ , r ∗ , c∗ ) ∂g1 (s ∗ , k ∗ , r ∗ , c∗ )   −1 0   ∂rt t t t t ∂ct t t t t    ∂g2 ∗ ∗ ∗ ∗ ∂g2 ∗ ∗ ∗ ∗  =  ∂f ∗ ∗  = 0  ∂r (st , kt , rt , ct ) ∂c (st , kt , rt , ct )   ∂rt (kt , rt ) 1  t t

8 The

Moreover, σ := Dg(r,c) (s ∗ , k ∗ , r ∗ , c∗ )−1 = f (kt , rt ).

supt∈N ||Dg(r,c) (s ∗ , k ∗ , r ∗ , c∗ )−1 || ∈ (0, +∞) since Dg(r,c) (s ∗ , k ∗ , r ∗ , c∗ ) with a convenient production function

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t λ1∗ t =β

∂v ∗ ∗ ∗ ∗ ∂g1 ∗ ∗ ∗ ∗ (s , k , r , c ) + λ1∗ (s , k , r , c ) t+1 · ∂st t t t t ∂st t t t t

+ λ2∗ t+1

t λ2∗ t =β

∂g2 ∗ ∗ ∗ ∗ (s , k , r , c ) for all t = 1, 2, . . . ∂kt t t t t

(3.14)

∂v ∗ ∗ ∗ ∗ ∂g1 ∗ ∗ ∗ ∗ (st , kt , rt , ct ) + λ1∗ (s , k , r , c ) t+1 · ∂rt ∂rt t t t t

+ λ2∗ t+1

βt

(3.13)

∂v ∗ ∗ ∗ ∗ ∂g1 ∗ ∗ ∗ ∗ (s , k , r , c ) + λ1∗ (s , k , r , c ) t+1 · ∂kt t t t t ∂kt t t t t

+ λ2∗ t+1

βt ·

∂g2 ∗ ∗ ∗ ∗ (s , k , r , c ) for all t = 1, 2, . . . ∂st t t t t

∂g2 ∗ ∗ ∗ ∗ (s , k , r , c ) = 0 for all t = 0, 1, . . . ∂rt t t t t

(3.15)

∂v ∗ ∗ ∗ ∗ ∂g1 ∗ ∗ ∗ ∗ (st , kt , rt , ct ) + λ1∗ (s , k , r , c ) t+1 · ∂ct ∂ct t t t t + λ2∗ t+1

∂g2 ∗ ∗ ∗ ∗ (s , k , r , c ) = 0 for all t = 0, 1, . . . ∂ct t t t t

∗ ∗ ∗ ∗ ∗ λ2∗ t+1 (g(st , kt , rt , ct ) − kt+1 ) = 0 for all t = 0, 1, . . .

(3.16)

(3.17)

which give us the following equations: t λ1∗ t =β

∂v ∗ ∗ ∗ ∗ (s , k , r , c ) + λ1∗ t+1 for all t = 1, 2, . . . ∂st t t t t

(3.18)

∂v ∗ ∗ ∗ ∗ ∂f ∗ ∗ (st , kt , rt , ct ) + λ2∗ (k , r )) for all t = 1, 2, . . . t+1 ((1 − δ) + ∂kt ∂kt t t (3.19) t ∂v ∗ ∗ ∗ ∗ 1∗ 2∗ ∂f ∗ ∗ (s , k , r , c ) + (−1)λt+1 + λt+1 (k , r ) = 0 for all t = 1, 2, . . . β ∂rt t t t t ∂rt t t (3.20) t ∂v ∗ ∗ ∗ ∗ 2∗ β (s , k , r , c ) + (−1)λt+1 = 0 for all t = 1, 2, . . . (3.21) ∂ct t t t t

t λ2∗ t =β

∗ ∗ ∗ ∗ λ2∗ t+1 ((1 − δ)kt + f (kt , rt ) − ct − kt+1 ) = 0 for all t = 0, 1, . . .

(3.22)

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63

Remark that since v is non-decreasing in the variable ct , we have λ2∗ t+1 > 0 for all t by (3.21) which will imply together with (3.22) that (1 − δ)kt∗ + f (kt∗ , rt∗ ) − ∗ ct − kt+1 = 0 for all t. Together with (3.18), (3.19), and (3.20), the multipliers associated with this problem satisfy   ∂v ∗ ∗ ∗ ∗ ∂v ∗ ∗ ∗ ∗ ∂f ∗ ∗ t λ1∗ = β (s , k , r , c ) + (s , k , r , c ) (k , r ) t+1 ∂rt t t t t ∂ct t t t t ∂rt t t t λ2∗ t+1 = β

∂v ∗ ∗ ∗ ∗ (s , k , r , c ) ∂ct t t t t

Thus, the system reduces to ∂v ∗ ∗ ∗ ∗ ∂v ∗ ∗ ∗ ∗ (s , k , r , c ) = β (s , k , r , c ) ∂ct t t t t ∂ct t t t t+1   ∂f ∗ ∗ (k , r ) for all t = 0, 1, . . . × (1 − δ) + ∂kt t t ∗ for all t = 0, 1, . . . f (kt∗ , rt∗ ) = (1 − δ)kt∗ − ct∗ − kt+1

(3.23) (3.24)

which is the adapted Euler equation together with (3.24) and coincide with the desired Eqs. (3.11) and (3.12). Conversely, if in addition to the first three assumptions of Theorem 5.1 of [9] if the Pontryagin’s Hamiltonian function is supposed to be concave with respect to (st , kt , rt , ct ), then optimality holds.  

3.4 Optimal Growth Model of a Forest: An Optimal Management Model of Forestry This section presents the model referring management (harvesting and replanting) of a forest which is presented in [1]. The primitives of the model are initial resource (forest), exploitation rules, and the performance function which are obtained from harvesting and replanting activity. In this economy, denoted by Ef , we suppose that there are infinite periods from time t = 0 to ∞. There exists a planner who manages the successive harvesting and replanting activity with respect to the exploitation rules (or assumptions). The decision of cutting trees will bring benefits whereas the decision of replanting trees will bring some costs. This cost–benefit mechanism will bring future income which are aggregated in a performance function. We consider a forest whose age-classified structure is given by the vector:

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⎡

⎤ Nn (t) ⎢ N (t) ⎥ ⎢ n−1 ⎥ ⎢ ⎥ N(t) = ⎢ . ⎥ ⎢ ⎥ ⎣ ⎦ . N1 (t) where Nj (t) (j = 1 . . . n − 1) represents the number of trees whose age is between j − 1 and j during the period [t, t + 1[ and Nn (t) represents the number of trees whose age is greater than n − 1. The natural evolution under no exploitation of the age vector N (t) is described by a linear system of difference equations (DE): N(t + 1) = AN(t)

(3.25)

where A is the following n × n matrix whose terms are non-negative which guarantees that N (t) remains positive at all times: ⎡ ⎤ 1 − mn 1 − mn−1 0 . . . 0 ⎢ 0 0 . . 0 ⎥ 0 1 − mn−2 ⎢ ⎥ ⎢ ⎥ . 0 1 − mn−3 0 . .0 ⎥ ⎢ 0 A=⎢ ⎥ ⎢ . . . 0 . . . ⎥ ⎢ ⎥ ⎣ 0 . . . . . 1 − m1 ⎦ νn νn−1 . . . . ν1 where mj s are mortality and νj s are recruitment parameters and mj , νj ∈ [0, 1] (of Leslie type as in [13]) in which mj represents the proportion of number of trees of age j − 1 die before reaching age j over all trees. νj represents the proportion of newborn trees generated by trees of age j − 1 over all trees. Given the matrix A, the difference equation (DE) in (3.25) can be given as: Nn (t + 1) = (1 − mn )Nn (t) + (1 − mn−1 )Nn−1 (t)

(3.26)

Nj (t + 1) = (1 − mj −1 )Nj −1 (t) j = 2 . . . n − 1

(3.27)

N1 (t + 1) = νn Nn (t) + . . . + ν1 N1 (t)

(3.28)

The following definition gives the rules of harvesting and replanting activity. Definition 3.6 (Assumptions of Exploitation) 1. Only the oldest trees may be cut. The minimum age at which it is possible to cut a tree is n − 1. 2. New trees of age 0 may be planted. N (t) is the state variable and the assumptions of exploitation describe the control of harvesting and replanting by a vector control variable w(t) = (h(t), i(t)) where

3 Pontryagin’s Principle for Optimal Growth Problems

65

h(t) represents the number of trees harvested at time t and i(t) represents the number of trees replanted at time t. The assumptions of exploitation with the defined control variable lead us to the following controlled evolution of age structure of the forest: (DE) N (t + 1) = AN(t) + Bh h(t) + Bi i(t)

(3.29)

where ⎤ −1 ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ Bh = ⎢ . ⎥ ⎥ ⎢ ⎣ . ⎦ 0 ⎡

and ⎡ ⎤ 0 ⎢.⎥ ⎢ ⎥ ⎢ ⎥ Bi = ⎢ . ⎥ ⎢ ⎥ ⎣.⎦ 1 We have then a bounded process with a difference equation (DE) given in (3.29) as 0 ≤ h(t) ≤ CAN (t) where C = (1, 0, . . . , 0). Since one cannot plan to harvest more than it will exist at the end of unit time, the non-negativity of the state variable N(t) is then ensured.

3.4.1 Planner’s Management Problem In each period, the planner chooses the number of trees that she will harvest and replant taking into account her future benefit as well as the sustainability of the forest. That is why, she maximizes the aggregated performance on the whole period. In the setup of the model under the assumptions of exploitation, the forestry management problem can be written as follows: ⎧  t ⎪ max J (N, h, i) := ∞ ⎪ t=0 β L(h(t), i(t)) ⎪ ⎪ ⎪ ⎨ s.t. (Pf ) N(t + 1) = AN(t) + Bh h(t) + Bi i(t), ∀t ≥ 0 ⎪ ⎪ ⎪ N(t) ≥ 0, h(t) ≥ 0, i(t) ≥ 0, ∀t ≥ 0 ⎪ ⎪ ⎩ (Nt )t bounded, (ht )t bounded, (it )t bounded

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3.4.2 Necessary Conditions of Optimality The aim of this section is to apply the result of weak Pontryagin’s principle (Theorem 3.2. in [8]) to a problem of harvesting and replanting of forestry whose setup is described in [1]. The method of weak Pontryagin’s principle allows to transform the optimal growth problem of forest (Pf ) into a dynamical system. Recall that Nt ∈ Rn is the vector state variable and ut = (ht , it ) ∈ R2 is a vector control variable such that N = (Nt )t ∈ ∞ (N, Rn ) and w = (wt )t = (ht , it )t ∈ ∞ (N, R2 ). Let g be the real-valued linear function defined on R n × R2 as follows: g(Nt , wt ) = g(Nt , (ht , it )) = ANt + Bh ht + Bi ht for all t = 0, 1, . . .. The

dynamic system is then governed by the following difference equation (DE):

Nt+1 = g(Nt , wt ) = g(Nt , (ht , it )) for all t = 0, 1, . . . (DE)  Then P f will be ⎧  t ⎪ max J (N, h, i) := ∞ ⎪ t=0 β L(h(t), i(t)) ⎪ ⎪ ⎪ ⎨ s.t.  (P f ) Nt+1 = g(Nt , wt ) = g(Nt , (ht , it )), ∀t ≥ 0 ⎪ ⎪ ⎪ N(t) ≥ 0, h(t) ≥ 0, i(t) ≥ 0, ∀t ≥ 0 ⎪ ⎪ ⎩ (N ) bounded, (h ) bounded, (i ) bounded t t t t t t Proposition 3.3 Let (N ∗ , w ∗ ) ∈ ∞ (N, Rn ) × ∞ (N, Rn ) be a solution of the problem (Pf ) and let the following assumptions be satisfied: 1. (P reff ) (Performance Assumption) The mapping L : R2 → R is continuously differentiable. 2. supt∈N ||A|| < 19 Then there exists λ∗ ∈ 1 (N, Rn∗ ) such that the following n + 3 equations hold:

(∗1)

(∗1)

(∗n)

λt

= λt+1 (1 − mn ) + νn λt+1

(∗2)

= λt+1 (1 − mn−1 ) + νn−1 λt+1

(∗n)

(3.31)

(∗2) λ(∗3) = λt+1 (1 − mn−2 ) + νn−2 λ(∗n) t t+1

(3.32)

λt

(∗1)

(3.30)

...

9 This

result.

assumption on mortality and the recruitment parameters mj , νj ∈ [0, 1] is essential for the

3 Pontryagin’s Principle for Optimal Growth Problems (∗n−1)

λt (∗n)

λt

(∗n−1)

= λt+1 (∗1)

λt+1 = β t (∗n)

λt+1 = β t

67

(∗n)

= ν2 λt+1

(3.33) (n)

(1 − m1 ) + ν1 λt+1

(3.34)

∂L ∗ ∗ ∗ (N , h , i ) ∂ht t t t

(3.35)

∂L (Nt ∗, ht ∗, it∗ ) ∂it

(3.36)

lim λ∗ t→+∞ t

=0

(3.37)

Proof We will show that the necessary conditions given by Theorem 3.2. in [8])  are fulfilled for the problem P f , under the assumptions above, that is, there exists a λ∗ ∈ 1 (N, Rn∗ ) satisfying the following equations together with (3.37): λ∗t = λt+1 ∗ ◦D1 g(N ∗ , w ∗ ) + β t D1 L(N ∗ , w ∗ ) for all t = 1, 2 . . .

(3.38)

λ∗t+1 ◦ D2 g(N ∗ , w ∗ ) + β t D2 L(N ∗ , w ∗ ) = 0 for all t = 0, 1, 2 . . .

(3.39)

Pontryagin’s Hamiltonian function associated with the multipliers 1 and λ are defined by Ht : Rn+ × R2 × R × Rn∗ → R such that Ht (N, w, λ, 1) = β t L(N, w)+ < λ, g(N, w) > The Adjoint Equation (AE) (3.38) and Weak Maximum Principle (WMP) (3.39) are given as follows10 : (AE) λ∗t = D1 Ht (Nt∗ , wt∗ , λ∗t+1 , 1) for all t = 1, 2 . . . (W MP ) D2 Ht (Nt∗ , wt∗ , λ∗t+1 , 1) = 0 for all t = 0, 1, 2 . . . which imply, respectively, (∗1) ]1×n = [λt+1 . . . λ(∗n) [λt(∗1) . . . λ(∗n) t t+1 ]1×n · [A]n×n for all t = 1, 2 . . .

(3.40)

λ∗t · D2 g(Nt∗ , wt∗ ) + β t D2 L(Nt∗ , wt∗ ) = 0 for all t = 0, 1 . . .

(3.41)

lim λ∗ t→+∞ t

=0

(3.42)

where (3.40) will be written as follows by putting the values of the matrix A: 10 Since

λ∗ ∈ 1 (N, Rn∗ ) the Transversality Condition (TC) holds, that is, limt→+∞ λ∗t = 0 (3.37)

68

A. Y. Ulus (1∗)

(1∗)

(∗n)

λt

= λt+1 (1 − mn ) + νn λt+1

(∗2)

= λt+1 (1 − mn−1 ) + νn−1 λt+1

(∗3)

= λt+1 (1 − mn−2 ) + νn−2 λt+1

λt λt

(3.43)

(∗1)

(∗n)

(3.44)

(∗2)

(∗n)

(3.45)

... (∗n−1)

λt (∗n)

λt

(∗n−1)

= λt+1

(∗n)

= ν2 λt+1

(3.46) (∗n)

(1 − m1 ) + ν1 λt+1

(3.47)

This gives us exactly the first n equations of (3.30)–(3.34). Also (3.41) will be written as follows by putting the values D2 (Nt∗ , wt∗ ), D2 L(Nt∗ , wt∗ )) together with wt∗ = (h∗t , it∗ ): λ∗t+1 ·

∂L ∗ ∗ ∗ ∂g (Nt∗ , h∗t , it∗ ) + β t (N , h , i ) = 0 ∂ht ∂ht t t t

λ∗t+1 ·

∂L ∗ ∗ ∗ ∂g ∗ ∗ ∗ (Nt , ht , it ) + β t (N , h , i ) = 0 ∂it ∂it t t t

(3.48)

(3.49)

where (3.48) and (3.49) can be written as follows: λ∗t+1 ·

∂L ∗ ∗ ∗ ∂g (Nt∗ , h∗t , it∗ ) + β t (N , h , i ) = 0 ∂ht ∂ht t t t

λ∗t+1 ·

∂L ∗ ∗ ∗ ∂g ∗ ∗ ∗ (Nt , ht , it ) + β t (N , h , i ) = 0 ∂it ∂it t t t

(3.50)

(3.51)

By recalling the linear map g(Nt∗ , (h∗t , it∗ )) = ANt∗ + Bh h∗t + Bi h∗t , (3.50) and (3.51) can be written still in a more explicit form as in the following two equations which are in fact equivalent to (3.35) and (3.36): (∗1)

∂L ∗ ∗ ∗ (N , h , i ) ∂ht t t t

(3.52)

(∗n)

∂L ∗ ∗ ∗ (N , h , i ) ∂it t t t

(3.53)

λt+1 = β t λt+1 = β t

 

3 Pontryagin’s Principle for Optimal Growth Problems

69

3.5 Conclusion Optimal growth theory in economics has always been one of the major application fields of the discrete time infinite horizon optimal control problems. Besides, recent developments in optimal management theory of natural resources such as oil, coal, gas reserves, forests, and fisheries and sustainable development provide other motivations for the study of these problems. Blot and Hayek [8] introduces weak Pontryagin’s principles approach for a solution of discrete time infinite horizon optimal control problems. This approach is given for bounded process which are governed by difference equations or inequations which proves to be useful. This chapter applies the results of [8] to three specific problems: macroeconomic optimal growth, optimal growth of an exhaustible natural resource, and optimal growth of a forest. Acknowledgements The author gratefully acknowledges the conference team ICAME18 where a preliminary version of this chapter was presented. The author thanks the chairman and the public of control theory session of ICAME18 for all their remarks and comments.

References 1. M. De Lara, L. Doyen, Sustainable Management of Natural Resources: Mathematical Models and Methods (Springer Science and Business Media, Berlin, 2008) 2. P. Dasgupta, G. Heal, The optimal depletion of exhaustible resources. Rev. Econ. Stud. 41, 1–28 (1974) 3. R.M. Solow, Intergenerational equity and exhaustible resources. Rev. Econ. Stud. 41, 29–45 (1974) 4. C.W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation (Wiley, Hoboken, 2010) 5. N. Stokey, R.E. Lucas Jr., E.C. Prescott, Recursive Methods in Economic Dynamics (Harvard University Press, New York, 1989) 6. R.A. Dana, C. Le Van, Dynamic Programming in Economics (Springer Science & Business Media, Dordrecht, 2003) 7. C. Le Van, C. Saglam, Optimal growth models and the Lagrange multiplier. J. Math. Econ. 40(3), 393–410 (2004) 8. J. Blot, N. Hayek, Infinite-horizon Optimal Control in the Discrete-time Framework. Springer Briefs in Optimization (Springer, New York, 2014) 9. J. Blot, N. Hayek, F. Pekergin, N. Pekergin, Pontryagin principles for bounded discrete-time processes. Optimization 64(3), 505–520 (2015) 10. J. Blot, N. Hayek, F. Pekergin, N. Pekergin, The competition between Internet service qualities from a difference game viewpoint. Int. Game Theory Rev. 14(01), 1250001 (2012) 11. A.Y. Ulus, On discrete time infinite horizon optimal growth problem. Int. J. Optim. Control 8(1), 102–116 (2017) 12. H. Hotelling, The economics of exhaustible resources. J. Polit. Econ. 39(2), 137–175 (1931) 13. H. Caswell, Matrix Population Models, 2nd edn. (Sinauer Associates, Sunderland, 2001), pp. 91–92

Chapter 4

A Medical Modelling Using Multiple Linear Regression Arshed A. Ahmad, Murat Sari, and Tahir Co¸sgun

4.1 Introduction A mathematical model is a relationship that includes all variables of a problem. Therefore, it is a description of a system that uses mathematical concepts and language. How well any particular objective is achieved depends on both the state of knowledge about a system and how well the modelling is done. Therefore, it can be used for various reasons. The scientific problems faced in nature are usually modelled mathematically. Mathematical modelling has been proven to be an essential tool for analyzing pathological characteristics as well [1–7]. Data availability is one of the key determinants of the potential of a particular model to assist in such measures to parameterize the model. To assess the conditions we see in hospitals, it is important to understand the types of necessary data for the model. Therefore, most results in real situations are affected by multiple input variables. To understand these relationships, we take models that use more than one input to produce one output; it is important to consider that a single disease (output) has several effects. As pointed out in the literature [8–12], the anemia of chronic disease was initially thought to be associated primarily with the infectious, inflammatory diseases. The aim of this study is to predict different types of pathological subjects from a population through the physical observational variables (eight blood variables,

A. A. Ahmad · M. Sari () Yildiz Technical University, Department of Mathematics, Istanbul, Turkey e-mail: [email protected] T. Co¸sgun Yildiz Technical University, Department of Mathematics, Istanbul, Turkey Amasya University, Department of Mathematics, Amasya, Turkey © Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6_4

71

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sex, and age) and output (the type of disease). It is important to predict the type of anemia because there has been an increase in the incidence of anemia among different segments of the society. To make the best biomedical decisions, medical predictions play a very important role for health providers. There have been many studies carried out in the literature [13–18]; the methods used in the corresponding studies have produced pretty accurate results by using relatively less number of input variables to predict the type of anemia. As can be realized, the more number of input variables makes the derived model more realistic. The literature relatively used limited number of blood variables and they have not considered such a large number of input variables in the model. To the best of authors’ knowledge, for the first time, for such a realistic model, for such a large number of input variables, a study has been accomplished here. Therefore, the present study focuses on the determination of the type of anemia through a very large number of input variables because the literature used models with relatively less number of variables. In the literature, different strategies of mathematical methods have been applied in biomedical problems. Since many researchers have effectively and successfully used regression analysis to deal with their own problems [19–29], we take into account the linear regression analysis in modelling our problem. Multiple linear regression analysis is to determine a relationship between a set of independent variables, and a dependent variable using observations. Thus, to make predictions about the value of a dependent variable, you must know only values of the independent variables. From the advantages of using the multiple regression analysis of the current model, the multiple analysis has the ability to determine the relative effectiveness of one or more variables. In the multiple regression, the data is used to describe a relationship between the state variables. This article is structured as follows. Section 4.2 illustrates the study samples and explains the regression analysis procedure for the model. Building the model of the data is given in Sect. 4.3. The analysis of the model is carried out and discussed in Sect. 4.4. Some final remarks are reported in Sect. 4.5.

4.2 Materials and Methods A mathematical model is a platform for understanding behavior of a dynamical system. Therefore, a model can help us to explain a real system. Understanding the nature of the system can be improved by using iterations repeatedly while generating a modelling process. However, the mathematical model depends on the data model. The model figures out the type of required data and how it should be organized instead of the processes that will be performed on the data. A data model being independent of some constraints concentrates on the representation of the data by preserving physical meaning of environment. It is usual to divide the model process

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into the following main categories: building, studying, testing, and use. Notice that if any changes are made to a model, then it is necessary to repeat the studying and test steps [30, 31].

4.2.1 Study Samples Anemia is a lower blood hemoglobin level below normal limits of less than 130 g/l for adult males and less than 120 g/l for non-pregnant adult females. Hemoglobin thresholds depend on the age of children. This decrease in the hemoglobin level leads to decreased oxygen delivery to the organs of the body and therefore appear in the symptoms of headache, fatigue, and inability to focus and attention [32]. As pointed out by the corresponding researchers, anemia is one of the most common blood diseases worldwide. The diagnosis of anemia depends on the concentration of hemoglobin less than the normal limits followed by the World Health Organization (WHO), and it is worth noting that the concentration of hemoglobin varies by age and sex as seen in Table 4.1 [33]. Anemia is classified into several types and those types differ in terms of their causes. Some types of anemia are hereditary. These types may affect children and may cause health problems for a lifetime. Women after adulthood may experience iron deficiency anemia, blood loss during the menstrual cycle, the most common type, may occur during pregnancy due to excessive need of minerals in the blood by the fetus during pregnancy, older people may be exposed to anemia due to malnutrition and other medical conditions [34]. Anemia is also classified according to the red cell form; this classification helps to identify some types of it based on the change in the size of red blood cells and the concentration of hemoglobin in the red blood cells. Therefore, it is a sign of recognition of the type of anemia when a change is taken place in one of the mean corpuscular volume (MCV) and the mean corpuscular hemoglobin concentration (MCHC) or both [35]. The data used here were collected from observations of blood variables in order to identify the types of anemia and included 539 subjects provided from blood laboratories in Iraq. We have taken observations of the ages of individuals between

Table 4.1 Hemoglobin thresholds used to define anemia [33] Age or gender group Children (0.50–4.99 years) Children (5.00–11.99 years) Children (12.00–14.99 years) Non-pregnant women (≥15.00 years) Pregnant women Men (≥15.00 years)

Hemoglobin threshold (g/l) 110 115 120 120 110 130

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(6–56) years and 211 healthy subjects and 328 sick subjects to build our model. For each case, we have samples for the people and for each person readings of the blood variables are hemoglobin, red blood cell, mean corpuscular hemoglobin, white blood cell, mean corpuscular volume, hematocrit, mean corpuscular hemoglobin concentration, platelets, and sex (male (1) and female (2)), and age. For the same subject, some blood diseases are iron deficiency anemia (1), deficiency of vitamin B12 (2), thalassemia (3), sickle cell (4), and spherocytosis (5). The types of anemia and blood variables for our data are shown in Fig. 4.1. The number of variables studied for building the model consists of ten independent variables and a dependent variable. The dependent variable consists of six different types of output (healthy subject: 0 and blood diseases: 1–5).

4.2.2 Multiple Linear Regression Analysis The regression analysis is the engine behind a large number of data analysis applications used in many estimations. Consider a multiple linear regression (MLR) model with k predictor variables y = b0 + b1 x1 + b2 x2 + . . . + bk xk + = b0 +

k 

bi xi + .

(4.1)

i=1

The observations recorded for each of these n levels can be expressed in the following way: y1 = b0 + b1 x11 + b2 x12 + . . . + bk x1k + 1 y2 = b0 + b1 x21 + b2 x22 + . . . + bk x2k + 2 .. . yi = b0 + b1 xi1 + b2 xi2 + . . . + bk xik + i

(4.2)

.. . yn = b0 + b1 xn1 + b2 xn2 + . . . + bk xnk + n . The dependent observations y1 , y2 , . . . , yn , and the independent observations x1 , x2 , . . . , xk , have n levels. Then xij represents the ith level of the jth predictor variable, xj . System (4.2) can be represented as follows: y = bX + ,

(4.3)

4 A Medical Modelling Using Multiple Linear Regression

a

b

6 5 4 3 2 1 0 0

c

5

10 HB

15

d

2 0

e

40

60 MCH

80

2 0 0

f

50

100

100

0

150

h

i

60

1.5

1 Sex

2

2.5

500

1000 PLT

1500

2000

Type of anemia and Age Type of anemia

Type of anemia

0

j

6 5 4 3 2 1 0 0.5

6 5 4 3 2 1 0

80

Type of anemia and Sex

0

60

Type of anemia and PLT Type of anemia

Type of anemia

Type of anemia and MCHC

40 MCHC

40

20 HCT

6 5 4 3 2 1 0 20

200

6 5 4 3 2 1 0

MCV

0

150

Type of anemia and HCT Type of anemia

50

4

WBC

Type of anemia and MVC

g

60

6

100

6 5 4 3 2 1 0 0

40

Type of anemia and WBC

4

20

20 RBC

Type of anemia

Type of anemia

0

6

0

6 5 4 3 2 1 0

20

Type of anemia and MCH

Type of anemia

Type of anemia and RBC Type of anemia

Type of anemia

Type of anemia and HB

75

6 5 4 3 2 1 0 0

20

40

60

Age

Fig. 4.1 Types of anemia and blood variables: (a) Types of anemia and HB; (b) Types of anemia and RBC; (c) Types of anemia and MCH; (d) Types of anemia and WBC; (e) Types of anemia and MCV; (f) Types of anemia and HCT; (g) Types of anemia and MCHC; (h) Types of anemia and PLT; (i) Types of anemia and sex; (j) Types of anemia and age

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with ⎡ ⎡ ⎤ 1 y1 ⎢1 ⎢y2 ⎥ ⎢ ⎢ ⎥ y = ⎢ . ⎥,X = ⎢ . ⎣ .. ⎣ .. ⎦ yn

x11 x21 .. .

x12 . . . x22 . . . .. .

⎡ ⎤ ⎤ b0 x1k ⎢ ⎥ ⎥ x2k ⎥ ⎢b1 ⎥ .. ⎥ , b = ⎢ .. ⎥ ⎣.⎦ . ⎦

1 xn1 xn2 . . . xnk

bk

and

⎡ ⎤

1 ⎢ 2 ⎥ ⎢ ⎥

= ⎢ . ⎥, ⎣ .. ⎦

(4.4)

n

where y, X, b, and stand for the observations, the regression coefficients, and an unobserved random variable that adds noise to the linear relationship between the dependent variable and regressors, respectively. To obtain the regression model, b should be known. Therefore, b is estimated by using the least square estimates as follows: bˆ = (XT X)−1 XT y,

(4.5)

where XT represents the transpose of the matrix X while (XT X)−1 represents ˆ the MLR model can now inverse of the matrix (XT X). Knowing the estimate b, be expressed as [36, 37] ˆ yˆ = bX,

(4.6)

where yˆ is the estimated value for y from the regression.

4.2.3 Test for the Model The coefficient of the determination, usually referred to as R 2 , is presented. This goodness-of-fit measure calculates the extent of the regression line in the estimation of the dependent variable using the independent variable and how effective the regression equation is used for predicting the values of the dependent variable. If the percentage explained by the coefficient of the determination is small, compatibility may not be very appropriate. Here, some initial considerations are presented. Consider the variance of the observations y by analyzing the total sum of squares of y around its mean as given by SST =

n  (yj − y) ¯ 2.

(4.7)

j =1

The total sum of squares, denoted by SST, can be decomposed into the sum of squares explained by the regression, denoted by SSR, and the sum of squared errors, denoted by SSE. That is, SST = SSR + SSE,

(4.8)

4 A Medical Modelling Using Multiple Linear Regression

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with SSR =

n  (yˆj − y) ¯ 2,

(4.9)

j =1

and n n   2 SSE = (yj − yˆj ) = ej2 . j =1

(4.10)

j =1

The SSR is that part of the total sum of squares that is explained by the regression. The SSE is the part of the sum of squares of the errors. Now, the coefficient of determination is defined by R2 =

SSE SSR =1− , SST SST

(4.11)

where R 2 takes on values in the interval [0,1]. All of the variations in the dependent variable is explained by the variation in the independent variable. More details on this topic can be found in references [38–43].

4.2.4 Residual Analysis In the regression analysis, the difference between the observations yi and the fitted values yˆi from the model is the residual, ei . The vector of residuals, e, is thus given by e = y − y. ˆ

(4.12)

The MSE stands for the average of the mean squared error, and is used to determine the extent to which the model fits the data around the regression model. It is given by 1 2 ej . n n

MSE =

(4.13)

j =1

Root mean squared error (RMSE) is a quadratic scoring base that measures the average error size. Residual is a measure of how far away from the regression line data points are. In other words, it tells you how the data is centered around the most appropriate line. Root mean square error is commonly used in the prediction. Then it is given by RMSE =

√

MSE.

(4.14)

For details of the topic, interested readers are referred to the references [38–43].

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4.3 Building Regression Analysis Model Model building is one of those skills in science that are difficult to estimate. It is difficult to determine the steps, because at every step, one has to evaluate the situation and make decisions on the next step. Building multiple regression models of a data is one of the most challenging regression problems. We now pay our attention to the process of constructing the model in the sense that we are trying to find the best relationship between the independent variables and the dependent variable y so that the complete final model is investigated in the regression model. Building a linear model is an iterative process. Given the problem and data but without a model, the process of building linear models can often be aided by graphs that help to visualize the relationship between different variables in the data. Also, multiple linear regression enables us to include additional variables to improve the predictive power of the regression equation [42, 43]. Main steps in building a linear model of a dataset are given by conducting the regression analysis in Fig. 4.2.

Data collection

Transform data into a stationary series

Building a linear regression model for data

Parameters estimation

No

Is the linear model adequate?

Yes The linear model

Fig. 4.2 Main steps in the linear regression analysis procedure

Examine the linear model

4 A Medical Modelling Using Multiple Linear Regression

79

Following the aforementioned manner, we have produced the following MLR model for our data as follows: y = b0 + b1 HB + b2 RBC + b3 MCH + b4 WBC + b5 MCV + b6 HCT + b7 MCHC + b8 PLT + b9 Sex + b10 Age + ,

(4.15)

where y: type of the anemia, bi : 0 ≤ i ≤ 10 are the parameters to be determined. Here HB, RBC, MCH, WBC, MCV, HCT, MCHC indicate hemoglobin, red blood cell, mean corpuscular hemoglobin, white blood cell, mean corpuscular volume, hematocrit, mean corpuscular hemoglobin concentration, platelets, respectively. The regression model is then estimated to be yˆ = 6.377 − 0.224HB − 0.224RBC − 0.029MCH + 0.001WBC + 0.0005MCV − 0.016HCT + 0.007MCHC + 0.001PLT

(4.16)

− 0.311Sex − 0.009Age. As previously mentioned, the model can be represented in matrix form as follows: ˆ yˆ = bX,

(4.17)

where ⎡ ⎤ ⎤ 1 HB11 RBC12 . . . Age110 b0 ⎢ 1 HB21 RBC22 . . . Age ⎢ b1 ⎥ ⎢ ⎥ ⎥ 210 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ yˆ = ⎢ ⎥,X = ⎢ . ⎥ , bˆ = ⎢ . ⎥. .. .. .. ⎣ .. ⎣ .. ⎦ ⎣ ⎦ ⎦ . . . 1 HB539,1 RBC539,2 . . . Age539,10 y539 b10 ⎡

y1 y2 .. .

⎤

⎡

(4.18)

Here yˆ and X represent the estimates for output (the types of anemia) and the independent observations matrix, respectively. The vector bˆ is thus obtained as ⎤ ⎡ 6.377 ⎢ −0.224 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ −0.224 ⎥ ⎥ ⎢ ⎢ −0.029 ⎥ ⎥ ⎢ ⎢ 0.001 ⎥ ⎥ ⎢ ⎥ (4.19) bˆ = ⎢ ⎢ 0.0005⎥. ⎢ −0.016 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0.007 ⎥ ⎥ ⎢ ⎢ 0.001 ⎥ ⎥ ⎢ ⎣ −0.311 ⎦ −0.009

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4.4 Discussion and Analysis Different strategies of mathematical methods have been implemented to deal with blood variables, as seen in the literature [13, 14, 17, 18]. However, as far as the authors know, they used relatively limited number of blood variables and they did not study the prediction of the types of anemia. Therefore, for the sake of increasing realisticity, the current study concentrates on the investigation of the relationship between a very large number of blood variables and the types of anemia. We produce various versions of models based on the variables (see Tables 4.2, 4.3, 4.4, and 4.5). The models produced in terms of larger number of blood variables show better correlation than the models produced in terms of less number blood variables for predicting the types of anemia in Eq. (4.16). The MLR model has been derived in predicting the types of anemia in Eq. (4.16). The parameter values used are all the latest values in producing results, to provide an approach that is more realistic. The variables have been included for the MLR analysis. Those variables consist of regression coefficients b, the blood variables (HB, RBC, MCH, WBC, MCV, HCT, MCHC, PLT), sex, and age. Here, the MLR model shows a strong effect in predicting the types of anemia better than the ones used fewer blood variables. The diagnosis of anemia depends on hemoglobin thresholds used to define anemia followed by the WHO for age, and it is worth noting that the concentration of hemoglobin varies by age. Here we classify the data into three categories as age (6–11) years old, (12–14) years old, and (≥15) years old as seen in Table 4.1 [33]. We have compared the results for the age group (6–56) with other classified age groups (6–11), (12–14), and (15–56). It has been found out that the results produced for the age group (6–56) are better than all other classified groups (see Tables 4.2, 4.3, 4.4, and 4.5). This difference is believed to stem from the decreasing the data as seen in Table 4.3. In the outcome of the current analysis, it has been found that there is more significant relation of the MLR model for the data (6–56) comparison to the other cases (6–11), (12–14), and (15–56). It explains 69.90% of the change in the relationship between all blood variables, sex, age, and the types of anemia as seen in Table 4.5. It is the best comparison to the results 48.2%, 83.8%, and 68.6% for the three categories (6–11), (12–14), and (15–56), respectively, as seen in Tables 4.2, 4.3, and 4.4. Thus, it is concluded that the regression model with the blood variables, sex, and age is seen to be significant (p < 0.000). Consideration of the blood variables, sex, and age simultaneously has a significant effect on the relationship on the determination of the types of anemia as shown in Table 4.6. The standardized coefficient (Beta) tells about the strength of particular parameters in the regression equation, and therefore it shows the effect of changing each of the blood variables, sex, and age to the types of anemia. It is thus given by Standardized Betaj = bj

SD(Xj ) . SD(Y )

(4.20)

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81

Table 4.2 Various forms of linear regression models: Blood variables, sex, and age (6–11) Models Model 1 for (HB, sex, and age) Model 2 for (RBC, sex, and age) Model 3 for (WBC, sex, and age) Model 4 for (PLT, sex, and age) Model 5 for (HB, RBC, sex, and age) Model 6 for (MCH, WBC, sex, and age) Model 7 for (MCV, HCT, sex, and age) Model 8 for (MCHC, PLT, sex, and age) Model 9 for (HB, RBC, MCH, sex, and age) Model 10 for (WBC, MCV, HCT, sex, and age) Model 11 for (HB, MCHC, PLT, sex, and age) Model 12 for (WBC, MCV, HCT, MCHC, sex, and age) Model 13 for (HB, RBC, MCH, PLT, sex, and age) Model 14 for (HB, RBC, MCH, WBC, MCV, HCT, MCHC, PLT, sex, and age)

R 0.561 0.497 0.185 0.212 0.577 0.198 0.620 0.304 0.612 0.646 0.626 0.648 0.656 0.694

R-square 0.315 0.247 0.034 0.045 0.333 0.039 0.385 0.092 0.375 0.417 0.392 0.420 0.431 0.482

RMSE 0.82956 0.86971 0.98506 0.97960 0.82221 0.98658 0.78956 0.95889 0.79909 0.77142 0.78824 0.77270 0.76579 0.74322

Table 4.3 Various forms of linear regression models: Blood variables, sex, and age (12–14) Models Model 1 for (HB, sex, and age) Model 2 for (RBC, sex, and age) Model 3 for (WBC, sex, and age) Model 4 for (PLT, sex, and age) Model 5 for (HB, RBC, sex, and age) Model 6 for (MCH, WBC, sex, and age) Model 7 for (MCV, HCT, sex, and age) Model 8 for (MCHC, PLT, sex, and age) Model 9 for (HB, RBC, MCH, sex, and age) Model 10 for (WBC, MCV, HCT, sex, and age) Model 11 for (HB, MCHC, PLT, sex, and age) Model 12 for (WBC, MCV, HCT, MCHC, sex, and age) Model 13 for (HB, RBC, MCH, PLT, sex, and age) Model 14 for (HB, RBC, MCH, WBC, MCV, HCT, MCHC, PLT, sex, and age)

R 0.816 0.753 0.448 0.511 0.823 0.619 0.620 0.732 0.869 0.630 0.893 0.762 0.870 0.916

R-square 0.666 0.567 0.201 0.261 0.678 0.383 0.385 0.536 0.755 0.397 0.798 0.580 0.757 0.838

RMSE 0.64123 0.73044 0.99263 0.95426 0.64979 0.89857 0.89766 0.77954 0.58501 0.91793 0.53143 0.79267 0.60328 0.58174

The HB absolute value of the Beta coefficient is (−0.663) which has the strongest relationship with the types of the disease comparison to the other variables RBC (−0.345), Sex (−0.106), HCT (−0.100), MCH (−0.090), PLT (0.080), Age (−0.065), WBC (0.016), MCHC (0.016), and MCV (−0.001). The interpretation of the Beta value for HB signifies that for every change in the HB, the dependent variable will be changed by the Beta coefficient value (see Table 4.7).

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Table 4.4 Various forms of linear regression models: Blood variables, sex, and age (15–56) Models Model 1 for (HB, sex, and age) Model 2 for (RBC, sex, and age) Model 3 for (WBC, sex, and age) Model 4 for (PLT, sex, and age) Model 5 for (HB, RBC, sex, and age) Model 6 for (MCH, WBC, sex, and age) Model 7 for (MCV, HCT, sex, and age) Model 8 for (MCHC, PLT, sex, and age) Model 9 for (HB, RBC, MCH, sex, and age) Model 10 for (WBC, MCV, HCT, sex, and age) Model 11 for (HB, MCHC, PLT, sex, and age) Model 12 for (WBC, MCV, HCT, MCHC, sex, and age) Model 13 for (HB, RBC, MCH, PLT, sex, and age) Model 14 for (HB, RBC, MCH, WBC, MCV, HCT, MCHC, PLT, sex, and age)

R 0.750 0.184 0.365 0.439 0.823 0.521 0.799 0.543 0.825 0.801 0.761 0.816 0.828 0.828

R-square 0.562 0.034 0.134 0.193 0.677 0.272 0.638 0.295 0.680 0.642 0.580 0.665 0.685 0.686

RMSE 0.95026 1.41107 1.33618 1.28994 0.81736 1.22649 0.86461 1.20694 0.81427 0.86102 0.93290 0.83370 0.80897 0.81159

Table 4.5 Various forms of linear regression models: Blood variables, sex, and age (6–56) Models Model 1 for (HB, sex, and age) Model 2 for (RBC, sex, and age) Model 3 for (WBC, sex, and age) Model 4 for (PLT, sex, and age) Model 5 for (HB, RBC, sex, and age) Model 6 for (MCH, WBC, sex, and age) Model 7 for (MCV, HCT, sex, and age) Model 8 for (MCHC, PLT, sex, and age) Model 9 for (HB, RBC, MCH, sex, and age) Model 10 for (WBC, MCV, HCT, sex, and age) Model 11 for (HB, MCHC, PLT, sex, and age) Model 12 for (WBC, MCV, HCT, MCHC, sex, and age) Model 13 for (HB, RBC, MCH, PLT, sex, and age) Model 14 for (HB, RBC, MCH, WBC, MCV, HCT, MCHC, PLT, sex, and age)

R 0.754 0.417 0.479 0.520 0.828 0.551 0.806 0.561 0.832 0.810 0.763 0.817 0.836 0.836

R-square 0.568 0.174 0.229 0.271 0.686 0.304 0.649 0.314 0.692 0.656 0.582 0.668 0.698 0.699

RMSE 0.96685 1.33677 1.29106 1.25593 0.82466 1.22844 0.87201 1.21903 0.81825 0.86483 0.95278 0.85008 0.80985 0.81171

We used the t-test to measure the effect of the variables HB, RBC, MCH, WBC, MCV, HCT, MCHC, PLT, sex, and age on the types of anemia, one by one. Notice that these variables have been seen to affect quite much the types of anemia but at various levels (see Table 4.7). Figure 4.3 shows the histogram of the residuals which confirm that the data are distributed according to a normal distribution with a mean of zero and a standard deviation.

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Table 4.6 Analysis of variance for the correlation in Eq. (4.16) Regression Residual Total

Sum of squares 809.354 347.889 1157.243

Degrees of freedom 10 528 538

Mean square 80.935 0.659

F -stat. 122.838

p-Value 0.000

t-Stat. 11.563 −3.581 −3.392 −1.931 0.549 −0.015 −0.575 0.464 2.637 −4.191 −2.303

p-Value 0.000 0.000 0.001 0.054 0.583 0.988 0.565 0.643 0.009 0.000 0.022

Table 4.7 Analysis of the regression coefficients given in Eq. (4.16)

(Const.) HB RBC MCH WBC MCV HCT MCHC PLT Sex Age

Unstandardized coefficients b Std. error 6.377 0.552 −0.224 0.062 −0.224 0.066 −0.029 0.015 0.001 0.003 0.0005 0.008 −0.016 0.028 0.007 0.016 0.001 0.000 −0.311 0.074 −0.009 0.004

Standardized coefficients Beta −0.663 −0.345 −0.090 0.016 −0.001 −0.100 0.016 0.080 −0.106 −0.065

The present investigation is to create the most appropriate line for observations. The observations vary and therefore will never fit exactly on the line. However, the best-fitted line for the data leaves the least amount of unexplained variation, such as the dispersion of observed points around the line. To find out the extent of dispersal of the random error around the regression model, the MLR uses the mean square residuals. In Tables 4.2, 4.3, 4.4, and 4.5, small values of the RMSE indicate the concentration of data around the produced model and thus the smaller RMSE, whenever the results are more accurate as shown in Fig. 4.4. Therefore, we see the convergence of the data from the regression line for the model of the age group (6–56); it is the best comparison to the other classified age groups (6–11), (12–14), and (15–56). Therefore, the present study provides an accurate model for prediction of the types of anemia.

4.5 Conclusions and Recommendations This study has discovered types of anemia through biomedical information under the consideration of eight different blood variables, sex, and age of individuals. The observational blood variables are HB, RBC, MCH, WBC, MCV, HCT, MCHC, and PLT. The linear regression model, for the first time, has been derived in predicting

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Fig. 4.3 Histogram plot of the residuals for classification of the data according to age: (a) (6–56); (b) (6–11); (c) (12–14); (d) (15–56)

the types of anemia. The parameter values used are all the latest values obtained results, to secure the approach that is more realistic. The results revealed that the regression model is very promising and is capable of making the prediction. In addition, the considered model has measured the effect of the blood variables, sex, and age on the blood diseases. It has been concluded that the model is expected to be very helpful for investigation of the types of anemia for health providers and designing appropriate treatment programs for their patients. For future research, these mathematical models may be improved under the consideration of various computational methods.

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Fig. 4.4 Normal PP Plot of Regression Standardized Residual for classification of the data according to age: (a) (6–56); (b) (6–11); (c) (12–14); (d) (15–56)

Acknowledgements The first author is thankful to the Diyala University and the Iraqi government for their financial support during his PhD studies.

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Chapter 5

Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow Driven by Injection Through a Deformable Rectangular Channel with Porous Walls Gabriel Magalakwe, M. L. Lekoko, K. Modise, and Chaudry Masood Khalique

5.1 Introduction It is vitally important to have mathematical representations (models) of various industrial phenomena that represent what take place in real life. Most industrial applications lead to three-dimensional mathematical models for a model to be more realistic, hence yield complex models. On the other hand, a simpler model can be obtained by careful consideration of the system configuration (design) that might lead to lower dimension without loss of generality. The design of the system is one of the most important steps during modelling process that prioritizes realistic models and appropriate boundary conditions. Finding solutions of such mathematical representations of industrial phenomena provides better understanding of the complicated processes and dynamics of these phenomena. Due to this, many researchers have studied various models in the past and recently to gain better insight of the dynamics of such case studies. Various solution processes became prominent part of research over the years. These two arms of research provide engineers, physicists, etc. with theoretical insight information of what to expect from the final product. These also provide financial benefits to various industries. G. Magalakwe () · M. L. Lekoko · K. Modise School of Mathematical and Statistical Sciences, North-West University, Potchefstroom, Republic of South Africa e-mail: [email protected] C. M. Khalique International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mmabatho, Republic of South Africa e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6_5

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Various integration techniques addressing different case studies gained lot of attention in the past. Some of the works found in literature are the study of two-dimensional laminar flow in a porous channel with permeable surface which expands or contracts. Such studies are closely linked to the current case study and other various applications because of their wide application and relevance in the fields of engineering, science and biological sciences. Applications in biological sciences largely revolve around the transport of biological fluids through contracting or expanding vessels with some degree of permeability. Transport of biological fluids includes blood flow in arteries, veins and capillaries, blood flow in artificial kidneys, air circulation in the respiratory system and osmosis. It should be noted that the application of porous surface goes beyond biological sciences. The study of laminar flow in a deformable porous channel has been undertaken under various assumptions for various values of the permeation Reynolds number Re and wall dilation rate α. The values of the permeation Reynolds number Re and wall dilation rate α include small, intermediary, large and arbitrary values, respectively. In the pioneering study by Berman [1], Navier–Stokes equations were investigated to study the effects of walls permeability on a steady-state laminar flow in a porous channel. Dauenhauer et al. [2] studied an unsteady state regime of the flow which was studied by Berman [1]. The author found numerical solutions by using Runge–Kutta along with Jacobian method to find unknown initial guesses for the numerical scheme. Similarity transformation was used to reduce systems of partial differential equations governing the flow to a single fourth order ordinary differential equation. Numerical solution of the reduced equation obtained was used to study effects of wall dilation, injection and suction on flow field. Majdalani et al. [3] investigated the flow field by authors [2] analytically by using perturbation method. Co-relation between numerical solutions [2] and the analytical solutions [3] was investigated. Boutros et al. [4] analytically solved the flow studied by Majdalani et al. [3], varying Re and α for small ranges −5 < Re < 5 and −1 < α < 1. They performed similarity transformation by using Lie-symmetry. Thereafter double perturbations method was used to solve the problem analytically and the results were compared with numerical solutions obtained by shooting method coupled with the fourth order Runge–Kutta method. Mahmood et al. [5] extended the model in [3] by taking into consideration the effect of surface force on the flow field. The study investigated effect of wall dilation, injection or suction and porosity which is due the porous nature of the medium. B. T. Matebese et al. [6] have studied a two-dimensional flow of an incompressible, viscous and magneto-hydrodynamic (MHD) fluid in a porous rectangular domain and the effect of body force due to the presence of a variable magnetic field. The author of [7] investigated thermal-diffusion and diffusion-thermo effects in a viscous flow between deformable permeable walls. The study found analytical solutions for velocity field, temperature and concentration. Recently, authors [8] presented analytical solutions of laminar flow in a flat mini-channel. Authors took into account dissipation effect of the fluid related to shear viscosity.

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Many researchers studied the flow between permeable surface which expands or contracts. Also the effects of surface and body forces were investigated. A literature review revealed that no attempt was made to use permeable surface, porous medium and wall dilation to study the dynamics of filtration process. Also mathematical representations of various system studied in the literature do not show modelling process of those studies in details. Authors applied various methods to solve standard problems which were posed in standard form without proficient modelling process. Motivated by the above-mentioned works, the current case study extends the flow studied by Berman [1], Dauenhauer and Majdalani [2], Majdalani et al. [3], and Boutros et al. [4] in three directions by taking into consideration various forces affecting flow field during filtration process and their importance in finding optimal outflow, thus, more filtrates. The first direction is concerned with the influence of a constant magnetic field on the fluid taking into consideration that the fluid is electrically conducting while the second investigates the effect of porous medium on the axial velocity for optimal outflow (filtrates). Lastly, effects of heat are taken into consideration. The main aim of this work is to explore a practical problem by studying flow dynamics during filtration process. For this purpose we consider a two-dimensional laminar flow driven by fluid injection through porous surface which represents an incompressible fluid inside a filtration chamber during extraction of particles from the fluid. Also we construct a mathematical model for internal flow field during filtration process proficiently by using basic conservation laws of mass, momentum and energy. Solutions are obtained which leads to stable filtration process and hence provide an insight of the dynamics that leads to an optimal filtration process.

5.2 Mathematical Modelling of the Problem In this section we derive a set of equations representing the flow and heat transfer inside a filtration chamber taking into consideration various forces which affect the dynamics of the flow. The mathematical representation of the filtration system under investigation will be derived from basic conservation laws of mass, momentum and energy. Physical arguments that lead to the final system of equations from the abovementioned conservation laws will be presented.

5.2.1 Problem Statement Lubricant cleanliness is highly important and lubrication practitioners are provided with various options for filtering and controlling contamination. Due to this reason, the study of filtration process gained great deal of attention over the years. One of the most persisting challenges during filtration is flow restriction due to fine pore size of the filter medium. To optimize the outflow, the configuration of the current

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study is such that the magnetic field is integrated in the system to produce load zone that collect iron, steel particles, etc. by placing magnets of equal strength parallel to top and bottom. The filter chamber is placed between the magnets with an equal distance at the top and bottom to allow the fluid flow between the chamber surface and the magnets. Thus, the permeable surface will filter less particles and pores will take more time to become plugged with particles. The system is such that magnet attracts iron particles, from there onwards the remaining particles will be restricted by permeable surface and porous medium to pass through the chamber. With time the plugged particles will pile up forming a filter cake which reduces the efficiency of fluid flow through the surface and the chamber. Thus, every now and then the lubricant practitioners need to stop the operation to flush out the filter cake. Hence, the current study is important to reflect more on dynamical operation of filtration process. To provide lubrication practitioners with more information about the dynamics of filtration process, it is therefore desirable to formulate a mathematical model which can be used to predict the effects of magnetic field and heat transfer on the flow field during this process. Also effects of permeability and porosity are important due to the permeability of the surface and the porous nature of the medium. These important parameters will be derived clearly in the next subsection.

5.2.2 Flow Configuration Consider an unsteady two-dimensional flow of a viscous, incompressible and magneto-hydrodynamic (MHD) fluid in a porous medium bounded by two deformable walls of equal permeability parallel to the axial direction. The length of the walls is assumed to be infinitely greater than the distance between them. The parallel plates are located at a distance y = h(t) and y = −h(t) from the centre of the chamber. The chamber is closed on the left side and open on the right side to allow outflow. Due to the nature of the walls being permeable, fluid mass is able to flow in at the surface and flow out of the open face on the right while filtering particles. The inflow of the fluid through the permeable walls is minimal since the walls are weakly permeable. Furthermore, the permeable nature of the walls and the porous nature of the medium allow the fluid into the system through the walls and pores of medium, this is called injection (Fig. 5.1).

5.2.3 Forces Affecting the Dynamics of the Flow 5.2.3.1

Surface Force

Flow in a porous medium can be understood by examining Darcy’s law. Darcy’s law is a generalized relationship for flow in porous medium and it governs fluid

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Fig. 5.1 Coordinate system, bulk motion and velocity profile. Note: For this configuration, symmetric nature of flow is taken into account at y = 0

flow through porous medium. The law was formulated by Henry Darcy as a result of his experiments on the flow of water through beds of sand. Darcy discovered that the volumetric flow rate of the fluid is dependent on the number of variables, namely the flow area through which the fluid is flowing, elevation, fluid pressure/hydraulic head and a proportionality constant. Thus, Darcy’s law is mathematically formulated as follows: Q = AK

Δh , ΔL

(5.1)

where Q is volumetric flow rate, A is the flow area perpendicular to the flow path L, K is the hydraulic conductivity (proportionality constant) and h is the hydraulic head. The hydraulic head h at a specific point is the sum of the pressure head and the elevation z given by the following equation: h=

P + z, ρg

(5.2)

where P is the fluid pressure, ρ is the fluid density, g is the gravitational acceleration and z is the elevation. For the problem at hand there is no change in elevation between any two points within the surface and it follows that z = 0. Taking into account that the flow is incompressible, Darcy’s law in differential form is given by Q = −AK

  d P . dL ρg

(5.3)

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Let it be noted that the negative accounts for the known fact that a fluid flows from a region of high to low pressure. We also note that the velocity and flux equations are given by u=

q , φ

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q=

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where φ is the porosity of the medium and q is the Darcy flux. Taking Eqs. (5.4) and (5.5) into Eq. (5.3) and making pressure gradient the subject of the formula yields ∂P φρg =− u. ∂L K

(5.6)

The conductivity K can be replaced by the permeability of the media. The two properties are related by the following equation: K=

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where k is the permeability, ν is the fluid dynamic viscosity and u is the velocity vector. Since L is the flow path and the problem under investigation is twodimensional, the flow is in the x–y directions. Hence (5.8) yields two surface forces in both directions as ∂P νφ =− u ∂x k

and

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(5.9)

Since the flow under investigation is a pressure driven flow, number of deductions can be made from Eq. (5.9) based on work and energy principle. The pressure difference in the y-direction is constant since the work done on the surface by fluid is constant due to constant injection. The symmetric nature of the configuration results in no driving force along the y-direction. Also the pressure gradient along the axial direction yields a driving force towards the right, since the work done by the fluid on the left side (closed) is more than the work done on right (open). Since the pressure is force per area the system has a closed area on the left only, the pressure on the left is more than the atmospheric pressure on the right (open). Thus, the filtrates outflow is towards the right.

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

5.2.3.2

95

Body Forces

Buoyancy Flow between the walls has net force (the difference between the buoyancy and gravity) acting on a unit volume of the fluid inside the boundary layers and it is always in the vertical direction [9]. Thus, the net force vector is given by Fn = g(ρw − ρ),

(5.10)

where g is gravity, ρw is density of the fluid close to the surface (at the boundary layer) and ρ is the density of the fluid inside the boundary layer. For the problem under investigation, the density is a function of temperature, hence the variation of density of the fluid with temperature at constant pressure can be expressed in terms of the volumetric expansion coefficient β as follows:   1 ∂ρ , ρ ∂T P

(5.11)

  1 ρ . ρ T

(5.12)

β=− ≈−

The current flow is incompressible which means the fluid density is constant except in the gravitational term due to Boussinesq approximation which states that for any incompressible fluid that has density variation due to temperature difference, density variation can be neglected except along the direction of gravity. Equation (5.12) can be expressed by replacing differential quantities by differences such that β≈−

1ρ 1 ρ − ρw =− , ρT ρ T − Tw

(5.13)

which can be rewritten as ρw − ρ = ρβ(T − Tw ),

(5.14)

where ρw and Tw are density and temperature at the boundary layer, respectively. Substituting (5.14) into (5.10) yields the force in the y-direction as Fn = gρβ(T − Tw ).

(5.15)

Magnetic Field The fluid is electrically conducting in the presence of a transverse uniform magnetic field. This flow can be understood by a careful consideration of Lorenz’s law of

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Fig. 5.2 Flow of an electrically conducting fluid through magnetic load zone

electromagnetism. Lorenz’s law describes the electromagnetic force Fe acting on a moving charged particle. For more information the reader is referred to the following books [10–12]. To illustrate the basic concept describing magneto-hydrodynamics phenomena, we consider an electrically conducting fluid moving with a velocity vector u. A magnetic field Bapp is applied perpendicular to the velocity vector as shown in Fig. 5.2. A number of deductions can be made from the interaction of the two vector fields. Firstly, an electric field is induced perpendicular to both u and Bapp . This electric field, denoted by Eind , is formulated as Ohm’s law given by Eind = u × Bapp .

(5.16)

Secondly, a current is induced in the fluid. That is, the positive and the negative charges comprising the fluid are each accelerated in such a way that their average motion gives rise to an electric current, the current density is denoted by J and is given by J = σ (u × B).

(5.17)

Here σ is the electrical conductivity of the fluid and B is the total magnetic field. The magnetic field (Bind ) is induced as a result of the induced current (J). That is, intrinsic to a current flowing through a medium (a fluid in our case) is a magnetic field. Thus the total magnetic field in the system is given by the sum of the applied magnetic field (Bapp ) and the induced magnetic field (Bind ) given by B = (Bapp + Bind ).

(5.18)

Using the result in (5.18), Eq. (5.17) can be written as J = σ [u × (Bapp + Bind )].

(5.19)

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

97

Simultaneously occurring with the induced current J is the induced electromagnetic force or the so-called Lorentz force denoted by Fe , given by Fe = J × B.

(5.20)

The Lorentz force occurs because the conducting fluid gives rise to a current perpendicular to the magnetic field. The vector Fe is the cross product of vectors J and B and is perpendicular to the plane of both J and B. In the current study the induced magnetic field is assumed to be small in comparison to the applied magnetic field, i.e., (Bapp  Bind ), thus its effect is negligible. Hence, Eq. (5.19) reduces to the equation J = σ (u × Bapp ).

(5.21)

For this problem the MHD fluid is moving with a velocity u along the x-axis, thus u = ui

(5.22)

and the magnetic field acts perpendicular to the direction of fluid flow, thus B = B0 j.

(5.23)

Substituting Eq. (5.22) and (5.23) into (5.21) gives a current density along the z-axis J = σ uB0 k.

(5.24)

Also, substituting (5.23) and (5.24) into (5.20) yields the Lorentz force acting along the negative x-axis given by Fe = −σ uB02 i,

(5.25)

the magnetic induction B0 is given by B0 = μH0 ,

(5.26)

with μ being the magnetic permeability and H0 is the magnetic strength which is allowed to vary with time. Substituting Eq. (5.26) into Eq. (5.25) yields Fe = −σ μ2 H02 ui.

(5.27)

Therefore, the Lorentz force induced by the interaction of the axial velocity field u and the transverse magnetic field acts opposite to the axial velocity. The electromagnetic force acts as a drag force. We note that a change in the direction of the magnetic field or the velocity field changes the Lorenz’s force (5.27) into a flow propelling force.

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5.2.4 Derivation of Governing Equations The mathematical model of the above-specified flow is represented by the governing equations derived below. The flow is considered to be fully developed, thus, the fluid saturates the space between the pores of the medium. The permeable walls are allowed to expand or contract uniformly at a dilation rate (α). Due to the configuration, symmetric nature of flow is taken into consideration at y = 0. The porous walls are considered to be stationary along the x-axis.

5.2.4.1

Conservation of Mass

The principle of mass conservation states that the rate of change of the fluid mass in a closed system is constant [9]. In other words the amount of fluid flowing into the system must equal the amount of fluid flowing out of the system. The equation of mass conservation is given by the following equation in Einstein tensor notation: ∂ρ ∂(ρ.uk ) = 0, + ∂t ∂xk

(5.28)

where k = 1, 2 and 3 represents the x, y and z directions of flow, respectively. Since the flow is two-dimensional, along the x and y axis and the fluid is incompressible, the above Eq. (5.28) reduces to ∂u ∂v + = 0. ∂x ∂y 5.2.4.2

(5.29)

Conservation of Momentum

The law of conservation of momentum states that the rate of change of the momentum of a fluid mass in a control volume is equal to the net external force acting on the fluid [9]. The external forces that act on a fluid mass are classed as either surface forces or body forces. Surface forces act across the surface of the fluid mass. Examples of such forces are pressure and viscosity forces. Body forces act throughout the body of the fluid. Examples of such forces are gravitational, centrifugal and electromagnetic force. The equation of conservation of momentum is given by the following Naiver–Stokes equation in Einstein tensor notation: ∂uj ∂ 2 uj Fj ∂uj 1 ∂P + uk , =− +s + 2 ∂t ∂xk ρ ∂xj ρ ∂xi

(5.30)

where Fj is both surface and body forces. Similarly k = j = i = 1, 2 represent the x and y directions. Substituting (5.9), (5.15), and (5.27) into (5.30) yields

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

99

  2 σ μ2 H02 ∂ u ∂ 2u ∂u sφ ∂u ∂u 1 ∂P + − +u +v =− +s u − u, (5.31) ∂t ∂x ∂y ρ ∂x k ρ ∂x 2 ∂y 2   2 sφ ∂ v ∂v ∂v 1 ∂P ∂ 2v ∂v − +u +v =− +s v + gβ(T − Tw ). + 2 2 ∂t ∂x ∂y ρ ∂y k ∂x ∂y (5.32)

5.2.4.3

Conservation of Energy

Law of conservation of energy states that the time rate of change of the internal and kinetic energy of a body is equal to the rate at which heat is transmitted to the body plus the rate at which work is done on the body and the rate at which electromagnetic energy is liberated within the body [13]. The governing differential equation for energy transport is given by  ρcp

∂T ∂T ∂T +u +v ∂t ∂x ∂y





∂ 2T ∂ 2T =k + ∂x 2 ∂y 2

 + Φ,

(5.33)

where Φ is the source term. For convective heat transfer processes of practical interest the reversible work and the viscous dissipation are small enough to be neglected and since the flow is unsteady two-dimensional, the above equation reduces to   ∂T ∂T ∂T k ∂ 2T ∂ 2T . (5.34) +u +v = + ∂t ∂x ∂y ρcp ∂x 2 ∂y 2 5.2.4.4

Boundary Conditions

The appropriate boundary conditions of the flow are defined by the following physical conditions. The configuration of the system leads to a symmetric nature of the flow at y = 0. At the wall there is no axial velocity u¯ since the wall is stationary along the axial direction. This is due to no slip condition at the boundary. During filtration the fluid passes through the permeable surface into the chamber in the direction normal to the surface. Also the surface is kept at constant temperature to allow an efficient flow injection. Thus, we have u¯ = 0,

v¯ = −Vw ,

T = Tw

at y¯ = h(t).

(5.35)

At the centre of the chamber, the axial velocity is constant, due to the symmetric nature of flow which yields a parabolic velocity profile in the axial direction. The normal velocity is zero since the particle paths inside the chamber are symmetric.

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This condition is due to the impact of symmetric injection. Also the temperature profile is parabolic with minimum temperature at the centre. Thus, ∂ u¯ = 0, ∂ y¯

∂T = 0 at y¯ = 0. ∂ y¯

v¯ = 0,

(5.36)

Along the y-axis, the chamber is closed. Thus the axial velocity is zero along x = 0 and hence u¯ = 0

at

x¯ = 0.

(5.37)

5.3 Mathematical Representation of Problem The two-dimensional flow of the viscous, incompressible and magnetohydrodynamic fluid inside a filtration chamber is governed by the following equations and boundary conditions:

5.3.1 Governing Equations and Boundary Conditions The governing equations and boundary conditions for the case study are: ∂ v¯ ∂ u¯ + = 0, ∂ x¯ ∂ y¯

(5.38)

  2 σ μ2 H02 ∂ u¯ sφ ∂ u¯ ∂ u¯ ∂ u¯ 1 ∂ P¯ ∂ 2 u¯ − + u¯ + v¯ =− +s u¯ − u, ¯ (5.39) + 2 2 ∂t ∂ x¯ ∂ y¯ ρ ∂ x¯ k ρ ∂ x¯ ∂ y¯   2 ∂ v¯ sφ ∂ v¯ ∂ v¯ ∂ v¯ 1 ∂ P¯ ∂ 2 v¯ − + u¯ + v¯ =− +s v¯ + gβ(T − Tw ), + ∂t ∂ x¯ ∂ y¯ ρ ∂ y¯ k ∂ x¯ 2 ∂ y¯ 2 (5.40)   2 ∂T ∂ T ∂T ∂T ∂ 2T . (5.41) + u¯ + v¯ =α + ∂t ∂ x¯ ∂ y¯ ∂ x¯ 2 ∂ y¯ 2

Here u¯ is axial velocity, v¯ is normal velocity, T is temperature, ρ is density, s is kinematic viscosity, g is gravitational acceleration, β is thermal expansion, α is thermal diffusivity, P¯ is pressure, t is time, φ and k are the porosity and permeability of porous medium, respectively. The appropriate boundary conditions are (i) u¯ = 0,

v¯ = −Vw ,

T = Tw

at y¯ = h(t),

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

∂ u¯ = 0, ∂ y¯

(ii)

101

∂T = 0 at y¯ = 0, ∂ y¯

v¯ = 0,

(iii) u¯ = 0 at x¯ = 0.

(5.42)

The stream function Ψ¯ (x, ¯ y, ¯ t) satisfies continuity Eq. (5.38) such that u¯ =

∂ Ψ¯ , ∂ y¯

v¯ = −

∂ Ψ¯ . ∂ x¯

(5.43)

Invoking dimensionless normal coordinate y = y/ ¯ h(t) into (5.43) we obtain u¯ =

1 ∂ Ψ¯ , h ∂y

v¯ = −

∂ Ψ¯ . ∂ x¯

(5.44)

Substituting (5.44) into momentum equations (5.39), (5.40) and energy equation (5.41) together with the non-dimensional quantities given below u=

u¯ , Vw

t=

t¯ , hVw

v=

u¯ , Vw

α=

hh˙ , s

x=

x¯ , h(t)

N=

y=

y¯ , h(t)

σ hμ2 H 2 , ρVw

Ψ =

1 sφh , = R kVw

Ψ¯ , hVw θ=

P =

P¯ , ρVw2

T − Th , Tw − Th (5.45)

yield the following dimensionless system: Ψy t¯ + Ψy Ψxy − Ψx Ψyy + Px −

 1 1  αΨy + αyΨyy + Ψxxy + Ψyyy + Ψy Re R

+N Ψy = 0,

(5.46)

Ψx t¯ + Ψy Ψxx − Ψx Ψxy − Py − +Gr θ = 0,



 1 1  αyΨxy + Ψxyy + Ψxxx + Ψx Re R ∂ 2θ

(5.47)

 ∂ 2θ

1 ∂θ ∂θ ∂θ + Ψy − Ψx = + 2 . ∂ t¯ ∂x ∂y Pr Re ∂x 2 ∂y

(5.48)

Similarly boundary conditions (5.42) becomes (i) Ψy = 0, (ii) Ψyy = 0,

Ψx = 1, Ψx = 0,

(iii) Ψy = 0 at x = 0.

θ = 1, θy = 0,

at y = 1, at y = 0, (5.49)

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Also, the relation between the stream function and the velocity (5.44) yields the following non-dimensional relation: u=

∂Ψ , ∂y

v=−

∂Ψ . ∂x

(5.50)

5.4 Solution of the Problem In this section, Lie group analysis [14–19] is used to transform (5.46)–(5.49) to system of ordinary differential equations. Thereafter double perturbation is used to find semi-analytical solutions of the problem under investigation.

5.4.1 Lie Group Analysis To obtain a balanced filtration system (stable), the system must be in such a way that fluid temperature is constant throughout the chamber, i.e., T ≈ Tw . This equilibrium effect of temperature results in a small Gr . The dimensionless system (5.46)–(5.48) admits the following six symmetries: X1 =

∂ , ∂ t¯

X6 = F1 (t¯)

X2 = θ

∂ , ∂ t¯

X3 =

∂ , ∂θ

X4 =

∂ , ∂Ψ

X5 = F2 (t¯)

∂ , ∂y

∂ . ∂x

Lie symmetries X1 and X2 are the only generators that leave the system and the boundary conditions invariant. It follows from these two symmetries that solutions Ψ = Ψ (x, y, t¯), P = P (x, y, t¯) and θ = θ (x, y, t¯) are invariant under X1 and X2 , if ΦΨ = X(Ψ − Ψ (x, y, t¯)) = 0,

ΦP = X(P − P (x, y, t¯)) = 0,

Φθ = X(θ − θ (x, y, t¯)) = 0, where X is Lie symmetry operator. The general solutions of invariant surface conditions are given by Ψ = h(y)H (x, y),

P = Γ (x, y),

Substituting (5.51) into (5.46) and multiplying by

θ = τ (x, y).

(5.51)

1 1 , we have , letting K = H Re

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

103

 2  d h d 3h −K 3 + − αKy − hK1 − 3KK2 dy dy 2   dh 1 + − αK − 2αKyK2 − hK3 + hK4 − KK5 − 3KK6 + + N R dy  2   dh 1 K1 + − αKK2 + K2 + NK2 − αKK6 y − KK9 − KK10 h dy R   1 dΓ + K7 − K8 h2 + , (5.52) H dx where K1 = Hx , K6 =

Hyy , H

Hy Hx Hy Hxx , K3 = , K4 = Hxy , K5 = , H H H Hy Hxy Hx Hyy Hxxy Hyyy K7 = , K8 = , K9 = , K10 = . H H H H (5.53)

K2 =

Solving K1 = Hx for H (x, y) from (5.53) gives H (x, y) = K1 x + K11 (y).

(5.54)

Using the above Eq. (5.54) into Ψ = h(y)H (x, y) from (5.51) we get Ψ = (K1 x + K11 (y))h(y).

(5.55)

The first derivative of (5.55) with respect to y together with (5.49) (iii) gives K11 (y)h(y) = K12 ,

(5.56)

where K12 is an arbitrary constant. Thus Eq. (5.55) becomes Ψ = xG(y) + K12

(5.57)

by letting G(y) = K1 h(y). Using P = Γ (x, y) from (5.51), and (5.54) into (5.52) yields K11 = 0,

(5.58)

which implies that K12 = 0 from (5.56). Thus Ψ = xG(y).

(5.59)

Substituting the above stream function (5.59) into (5.50) yields the velocity components as

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dG u = , x dy

v = −G(y).

(5.60)

The above velocity components confirm that normal flow is injected into the filter chamber and axial velocity changes along the normal direction per length of the chamber. This information confirms the parabolic flow field inside the chamber. Differentiating equation obtained from substituting (5.59) in (5.47) with respect to x yields Pxy =

1 Gr θx . h2

(5.61)

Similarly by substituting (5.59) into (5.46) and differentiating with respect to y and then using the result in (5.61) yields 

  3 d 4G d G d 2G d 3 G Re d 2 G d 2G + Re G 3 − + α y + 2 − Re 2 N 4 3 2 2 R dy dy dy dy dy dy  1 dG d 2 G x + 2 Gr θx = 0. (5.62) −Re dy dy 2 h

Using (5.59) and θ = τ (x, y) from (5.51) in (5.48), we obtain x

 2  ∂ τ ∂τ 1 ∂ 2τ dG ∂τ = 0, −G − + dy ∂x ∂y Pr Re ∂x 2 ∂y 2

(5.63)

and the boundary conditions (5.49) become (i)

dG(1) = 0, dy

(iv) G(0) = 0,

(ii) G(1) = 1, (v) τ (x, 1) = 0,

(iii)

d 2 G(0) = 0, dy 2

(5.64)

(vi) τ (x, 0) = 1.

Equating powers of h from (5.62) and thereafter using the value of θ from Eq. (5.51) yields Gr τx = 0. This result implies that τ = E(y) and Gr = 0 which satisfy dynamics of the stable filtration system. Since Gr = 0, Eq. (5.61) yields Pxy = 0 which confirms that the flow is laminar. Since the driving force is only in the axial direction at a constant rate, using the fact that Gr = 0, for a stable system and temperature τ = E(y), (5.62)–(5.64) yields   3 d 4G d G d 2G d 3 G Re d 2 G d 2G + R + α y + 2 G − − Re N e R dy 2 dy 4 dy 3 dy 2 dy 3 dy 2 −Re

dG d 2 G = 0, dy dy 2

(5.65)

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

 2  ∂ E 1 ∂E = 0, + G(y) ∂y Pr Re ∂y 2

105

(5.66)

and the boundary conditions (i)

dG(1) = 0, dy

(iv) G(0) = 0,

(ii) G(1) = 1, (v)

E(1) = 1,

(iii)

d 2 G(0) = 0, dy 2

(5.67)

(vi) E  (0) = 0.

5.4.2 Semi-Analytical Solution In this subsection we find solutions of Eqs. (5.65)–(5.66) with the corresponding boundary conditions (5.67) using double perturbation method. For more information the reader is referred to the studies [3, 4]. The solutions of axial velocity and temperature can be represented by decreasing perturbation series as G = G1 + Re G2 + O(Re2 ),

(5.68)

where G1 = G10 + αG11 + O(α 2 ) and

G2 = G20 + αG21 + O(α 2 ). (5.69)

Similarly E = E1 + Re E2 + O(Re2 ),

(5.70)

where E1 = E10 + αE11 + O(α 2 )

and

E2 = E20 + αE21 + O(α 2 ).

(5.71)

Substituting (5.68) and (5.69) into (5.65) and (5.66) gives momentum and energy equations in terms of Re which yields four equations from equating like powers of Re . Thereafter substituting (5.70) and (5.71) into those resulting equations yields four equations in terms of the second perturbation parameter α which yields system of twelve ordinary differential equations by equating powers of α. Solving the resulting system yields G10 (y) =

  1 3y − y 3 , 2

(4)

G11 (y) − 9y = 0,

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 1 G21 (y) = 480NRy 7 − 1512NRy 5 + 1584N Ry 3 100800R − 552NRy − 65Ry 9 − 324Ry 7



+162Ry + 908Ry − 681Ry + 480y − 1512y + 1584y − 552y , 5

1 G22 (y) = 232848000R

3

7

5

3

 − 136675NRy 9 + 540540N Ry 7

− 760914NRy 5 + 446908NRy 3 − 89859NRy + 12600Ry 11 + 204435Ry 9 − 345708Ry 7 − 314622Ry 5 + 785628Ry 3 − 342333Ry − 136675y 9 + 540540y 7 − 760914y 5  + 446908y 3 − 89859y , E11 = 1,

E12 = 0,

E21 = 0,

E22 = 0.

Thus   1 y{α 2 (13 − 25y 2 )(y 2 − 1)2 + 210α(y 2 − 1)2 − 1400(y 2 − 3)} , 2800 (5.72)   1 y(y 2 − 1)2 831600{R(y 2 − 7N + 2) − 7} G2 (y) = 232848000R  − 2310α − 2y 2 {(240N − 227)R + 240} + (552N + 681)R  + 65Ry 4 + 552 + α 2 [−35y 4 {(3905N − 6561)R + 3905}] G1 =

+ 2y 2 [{133595N + 50481}R + 133595] − 3[{29953N + 114111}R  6 (5.73) + 29953] + 12600Ry and E1 = 1,

E2 = 0.

(5.74)

Finally, we obtain the solution of equation axial velocity and temperature distribution as

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107

  1 2 2 2 2 2 2 2 G= y{α (13 − 25y )(y − 1) + 210α(y − 1) − 1400(y − 3)} 2800    2  1 2 y y −1 831600{R(y 2 − 7N + 2) − 7} + Re 232848000R   − 2310α − 2y 2 {(240N − 227)R + 240} + (552N + 681)R + 65Ry 4 + 552 + α 2 [−35y 4 {(3905N − 6561)R + 3905}] + 2y 2 [{133595N + 50481}R  + 133595] − 3[{29953N + 114111}R + 29953] + 12600Ry 6 and E(y) = 1. It is noted that the solution G(y) in [4] is recovered when the pore size of the medium becomes extremely large and magnetic field becomes extremely small. This verifies the correctness of the present case study, since in the absence of a magnetic field and porous medium the present work reduces to the problem in [4].

5.5 Results and Discussion In this section the graphical representations of the self-axial velocity u/x are shown to investigate optimal filtrates. An analysis is carried out to interpret the behaviour of the self-axial velocity subject to some physical parameters such as the wall dilation α, permeation Reynolds number Re , the porosity variable R, the Stuart number N . Temperature distribution is also presented. The analysis gives more insight information about the dynamics of the filtration process that yields optimal outflow (area under the graph) of filtrates. The area as an average flow applies to all figures.

5.5.1 Effects of Wall Dilation Results from Figs. 5.3 and 5.4 illustrate that when the chamber (wall dilation) increases, the axial velocity for both small and average pore size increases. Thus, more fluid outflow. Area between the graphs u/x for different values of α and axis u/x = 0 depicts that average outflow for both flow fields (filtrates) is relatively the same for small and average pore size.

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1.4 1.2 1.0 u 0.8 x 0.6 0.4 0.2 0.0

-1.0

-0.5

0.0 y

0.5

1.0

Fig. 5.3 Self-axial velocity profiles over a range of α where blue = 1, pink = 0.5, black = 0, green = −0.5 and red = −1 at R = 0.5, Re = 1 and N = 2.5

1.4 1.2 1.0 u 0.8 x 0.6 0.4 0.2 0.0

-1.0

-0.5

0.0 y

0.5

1.0

Fig. 5.4 Self-axial velocity profiles over a range of α where blue = 1, pink = 0.5, black = 0, green = −0.5 and red = −1 at R = 0.25, Re = 1 and N = 2.5

5.5.2 Effects of Reynolds number inside the Filtration Chamber Figures 5.5 and 5.6 show that the self-axial velocity increases when more fluid is injected inside the chamber. Also, the increase in Stuart number leads to a reverse

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

109

2.0

1.5 u x 1.0

0.5

0.0 -1.0

-0.5

0.0

0.5

1.0

y

Fig. 5.5 Self-axial velocity profiles over a range of Re where blue = 5, pink = 4, black = 3, green = 2 and red = 1 at R = 0.5, α = 1 and N = 2.5 3.0 2.5 2.0 u x

1.5 1.0 0.5 0.0 -0.5

-1.0

-0.5

0.0

0.5

1.0

y

Fig. 5.6 Self-axial velocity profiles over a range of Re where blue = 5, pink = 4, black = 3, green = 2 and red = 1 at R = 0.5, α = 1 and N = 5

flow close to the walls which is not ideal during filtration process (to obtain optimal filtrates). Figure 5.6 shows two regions above the line u/x = 0 and below which indicates the effect of propulsion force and drag force, respectively. Hence less permeates propel fast.

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1.4 1.2 1.0 u 0.8 x 0.6 0.4 0.2 0.0

-1.0

-0.5

0.0 y

0.5

1.0

Fig. 5.7 Self-axial velocity profiles over a range of R where blue = 1, pink = 0.75 and black = 0.5 at Re = 1, α = 1 and N = 5

5.5.3 Effects of Porosity Variable Inside the Filtration Chamber Figure 5.7 illustrates that when the chamber volume increases while fluid permits at an extremely lower speed with an average magnetic strength, variation of porosity has no effects during filtration process. Thus, the small pore size should be taken under this situation. Figure 5.8 on the other hand indicates that with more injection, the effect of magnetic field yields reverse flow. The fluid flow is pressure driven, thus, the increase in chamber size decreases internal pressure, as a result decrease outflow.

5.5.4 Effects of Stuart Number Inside the Filtration Chamber Figures 5.9 and 5.10 illustrate that expansion and contraction of the chamber yield the increase and the decrease of drag force, respectively. Expansion leads to more effects of magnetic field on the fluid bulk compared to contraction since the surface walls move closer to the magnets during expansion.

5.5.5 Temperature Distribution Inside the Chamber Figure 5.11 indicates constant temperature for filtration process (when the system is stable).

5 Lie Group Method Solution for Two-Dimensional Heat and Viscous Flow

111

3.0 2.5 2.0 u 1.5 x 1.0 0.5 0.0 -0.5

-1.0

-0.5

0.0 y

0.5

1.0

Fig. 5.8 Self-axial velocity profiles over a range of R where blue = 1, pink = 0.75 and black = 0.5 at Re = 5, α = 1 and N = 5 3.0 2.5 2.0 u 1.5 x

1.0 0.5 0.0 -0.5

-1.0

-0.5

0.0 y

0.5

1.0

Fig. 5.9 Self-axial velocity profiles over a range of N where blue = 5, pink = 3.5, black = 2.5, green = 1.5 and red = 0.5 at R = 0.5, α = 1 and Re = 5

5.6 Concluding Remarks In this paper, Lie group analysis along with double perturbation method is used to study the internal flow. The configuration and design of the filter revealed the important parameters which affect the internal flow during filtration process.

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Fig. 5.10 Self-axial velocity profiles over a range of N where blue = 5, pink = 3.5, black = 2.5, green = 1.5 and red = 0.5 at R = 0.5, α = −1 and Re = 5

2.0

1.5 u x 1.0

0.5

0.0 -1.0

-0.5

0.0 y

0.5

1.0

2.0

1.5

T 1.0

0.5

0.0

-1.0

-0.5

0.0

0.5

1.0

y Fig. 5.11 Temperature distribution profile

Also proficient mathematical formation of the case study is obtained from basic conservation laws of mass, momentum and energy. The findings from the study indicate that to have an optimal outflow the following dynamics are critical: 1. Wall dilation rate: The increase in wall dilation rate creates more space inside the chamber. Thus the pressure difference between the internal and external pressure increases in such a way that the system allows more injection of fluid inside chamber, which results in more flow out of the chamber. 2. Reynolds: The flow injection into the filter chamber has a positive effect during filtration process. The more fluid mass into the chamber leads to more permeates out of the system. 3. Porosity: The variation of pore size of the medium does not affect fluid flow for weak injection along with average magnetic field and expansion. The average outflow is more when the fluid injection is minimal.

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4. Magnetic field: Increasing magnetic load zone leads to a reverse flow away from the centre of the chamber and hence decreases the average outflow. To counter this negative effect, more injection is needed to result in a net force towards the chamber since both magnetic field and the injection act normal to the surface. 5. Temperature: Temperature is constant throughout the chamber, hence its effect during filtration plays an important role based on the following reasons. Firstly, it increases the internal pressure, thus, increases the driving force which is needed to have more filtrates. Secondly, it decreases the drag force by decreasing the fluid density, thus, yields more flow output. Lastly effects of buoyancy force which leads to turbulent flow become minimal due to the temperature effects. Hence, variation of temperature is not ideal for a stable filtration process. Acknowledgements Authors would like to thank CSIR and North-West University South Africa for their financial support.

References 1. A.S. Berman, Laminar flow in channels with porous walls. J. Appl. Phys. 24, 1232–1235 (1953) 2. E.C. Dauenhauer, J. Majdalani, Exact self-similarity solution of the Navier–Stokes equations for a deformable channel with wall suction or injection. AIAA 3588, 1–11 (2001) 3. J. Majdalani, C. Zhou, C.A. Dawson, Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. J. Biomech. 35, 1399–1403 (2002) 4. Z. Boutros, B. Minas, A. Badran, S. Hassan, Lie group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. Appl. Math, Model. 31, 1092–1108 (2007) 5. M. Mahmood, M.A. Hossain, S. Asghar, T. Hayat, Application of homotopy perturbation method to deformable channel with wall suction and injection in a porous medium. Int. J. Nonlinear Sci. Numer. Simul. 9, 195–206 (2008) 6. B. Matebese, A. Adem, C.M.Khalique, T. Hayat, Two-dimensional flow in a deformable channel with porous medium and variable magnetic field. Math. Comput. Appl. 31, 674–684 (2010) 7. S. Srinivas, A. Subramanyam Reddy, T.R. Ramamohan, A study on thermal-diffusion and diffusion-thermo effects in a two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. Int. J. Heat Mass Transf. 55, 3008–3020 (2012) 8. C. Bao, Z. Jiang, X. Zhang, T.S. Irvine, Analytical solution to heat transfer in compressible laminar flow in a flat minichannel. Int. J. Heat Mass Transf. 127, 975–988 (2018) 9. I.G. Currie, Fundamental Mechanics of Fluids, 4th edn. (CRC Press, New York, 2016) 10. B. Lehnert, in Dynamics of Charged Particles, 4th edn. Proceeding of the Third international Conference on the Physics of Electronic and Atomic Collisions, London, 1964 11. P.H. Roberts, An Introduction to Magnetohydrodynamics, 4th edn. (The Whitefriars Press, Great Britain) 12. D.J. Griffiths, Introduction to Electrodynamics, 3th edn. Library of Congress Cataloging, United States of America 13. S. Whitaker, Fundamental Principles of Heat Transfer (Pergamon Press, New York, 1977) 14. L.V. Ovsiannikov, Group Analysis of Differential Equations (Academic Press, New York, 1982) 15. P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993) 16. G.W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer, New York, 1989)

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17. H. Stephani, Differential Equations. Their Solutions Using Symmetries (Cambridge University Press, Cambridge, 1989) 18. N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations (Wiley, Chichester, 1999) 19. F.M. Mahomed, Recent trends in symmetry analysis of differential equations. Notices SAMS 33, 11–40 (2002)

Chapter 6

Optimal Siting of Wind Turbines in a Wind Farm Melike Sultan Karasu Asnaz, Bedri Yuksel, and Kadriye Ergun

6.1 Introduction 2015 is the year that the international community worked to reach a global climate change agreement. In December 2015, 195 countries adopted a universal global climate deal with Paris Agreement. All these governments agreed to limit global warming to well below 2 ◦ C, and outlined their national post-2020 mitigation commitments throughout the year. In this context, renewables must take center stage in achieving 2 ◦ C Scenario for climate goals. According to the 2015 report of Global Wind Energy Council (GWEC), the mainstream source of renewable energy supply will be wind power, and it will play a major role in decarbonization [1]. However, becoming mainstream means to function the overall energy system cost-effectively. Thus, the factors that affect energy production adversely have to be considered. One of the most important factors is the placement of wind turbines in a wind farm. Upwind turbines create wind wakes that impact the natural wind flow to adjacent downwind turbines, causing the downwind turbines to produce less energy production, and less overall lifetime of because of increased mechanical loads [2]. So, the wind energy industry has to use technical and financial innovation to drive costs down, and keep sustain the improvement of wind farm reliability. The wind farm layout optimization (WFLO) problem consists of finding the turbine positions that maximizes the expected power production. In the literature, there are several researchers who addressed this problem. In 1994, Mosetti et al. [3]

M. S. K. Asnaz () · K. Ergun Balikesir University, Balikesir, Turkey e-mail: [email protected]; [email protected] B. Yuksel Istanbul Gelisim University, Istanbul, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6_6

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attempted to optimize the placement of wind turbines in a wind farm by utilizing a genetic algorithm. He discretized the terrain in a matrix, used Jensen’s wake effect model, and he obtained results for three different wind regime scenarios considering cost and power production. Mosetti’s problem was examined by many other researchers. Grady et al. [4] used same approach as Mosetti, and proofed that Mosetti et al.’s results were not showing the optimal placement. Emami et al. [5] proposed a different objective function for a better layout for the same three cases. On the other hand, Marmidis et al. [6] investigated the same problem by using Monte Carlo simulation for the first scenario. Bilbao and Alba [7] used a simulated annealing in their study, while in their second study Bilbao and Alba [8] utilized CHC which is a non-traditional genetic algorithm that combines a conservative selection strategy, and geometric particle swarm optimization in order to maximize the profit per year. Kusiak and Song [9] presented optimizing a multi-objective function that uses evolutionary strategy algorithms. Unlike other studies, a circular plot of wind farm terrain was considered instead of a rectangular shape. Eroglu and Seckiner [10] proposed an ant colony optimization algorithm to optimize the same wind farm model as proposed by Kusiak and Song [9]. Also, in 2017 Bansal et al. [11] studied Kusiak and Song’s [9] model, and presented a new evolutionary population-based optimization technique called bio-geography based optimization which was inspired by migration of species from one island to another island. Migration and mutation operators are the key parameters since they are responsible to evolve new candidate solutions. Minimizing velocity deficits, and so maximizing the energy production was the objective function in the study. Chowdhury et al. [12] proposed a particle swarm optimization for optimum design of wind farm, and optimum turbine selection in order to maximize the net power production. Identical turbines and different rotor sized turbines were evaluated in two scenarios. They found that installing wind turbines with different rotor diameters improved the efficiency of the wind farm. Frandsen wake model was used for turbine interaction calculations. Chen et al. [13] proposed to install wind turbines that have multiple hub heights. Three-dimensional greedy algorithm was utilized on both linear and particle wake model over flat terrain and complex terrain, respectively. Simulations revealed that in case of using different hub height wind turbines on a complex terrain increases the power production, and decreases the cost per unit power production. Saavedra et al. [14] considered orography and shape of the wind farm, and carried out Monte Carlo simulations of several years of wind speeds. The authors optimized the wind farm model by offering an initial solution obtained by a greedy algorithm, and then proposed a final layout by using an evolutionary algorithm. Rasuo et al. [15] tried a different type of genetic algorithm, called differential evolution for WFLO problem. Instead of placing the turbines at the center of each cell, the locations of turbines were adjusted freely. By this way, they could manage to reduce wake effect, and produce more energy from a given wind farm. Simulation results showed suitability of the proposed algorithm. A bio-inspired algorithm called Coral Reefs Optimization (CRO) was presented for off-shore wind farm design in Salcedo-

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Sanz et al.’s [16] paper. The proposed algorithm’s performance was compared to three different approaches: evolutionary algorithm, differential evolution, and harmony search algorithm. It was proven that CRO approach produces the layout with the highest power production. Kallioras et al. [17] proposed a music-inspired meta-heuristic algorithm called harmony searched for WFLO problem. Two different objective functions were presented: profit maximization for a specific number of wind turbines and the profit maximization for a given energy per year. Jensen wake model was utilized for turbine interactions, while the wind characteristics of the terrain were modeled stochastically. Kwong et al. [18] presented a continuous location model, and included noise minimization with energy maximization as objective functions. They formulated previous test cases of Mosetti’s [3]; Jensen’s approach for the wake model and multi-objective non-domination sorting genetic algorithm II (NSGAII) for the optimization were used in this paper. Pareto frontiers were identified regarding the relative importance of the energy production and noise objectives. Mittal et al. [19] also studied energy-noise trade-off problem, proposed a hybrid method of a multi-objective evolutionary algorithm and a single-objective gradient approach. Khan et al. [20] outlined the iterative non-deterministic algorithms in WFLO literature including design issues, different constraints, single and multi-objective aspects of the problem. Serrano Gonzales et al. [21] also presented a review of the optimal placement of wind turbines discussing the main features concerning objective function, application of several algorithms, and wake effect models. The main topic of this paper is to provide the optimum number of wind turbines, and their optimum locations in a given site by applying both heuristic and metaheuristic algorithms. In this case, a 350×1000 m rectangle shaped area is considered for the wind farm. The majority of previous approaches consider square wind farms, and divided into grids where turbines could be placed [14]. Instead of discretizing the wind farm area by meshing, a continuous layout model is used, since using real location variables can avoid choosing optimal grid size [22]. To do so, latitude, longitude (angles in radians), and elevation (in meters) values of the wind farm are generated by scanning a digital map from Google Earth. Then, these three values are turned into a dataset which introduced to algorithm as candidate locations for wind turbines. By doing so, the algorithm gives the opportunity to explore every potential location on a wind farm to reduce wake losses. Also, this approach provides the opportunity to exploit any irregular shape wind farm micrositing problem. Since the latitude, longitude values are in angles in radians, the distances between each point needed to be calculated by a geodesic approach which is a novelty in a WFLO problem. After turning all the data into metrics, a three-dimensional Cartesian coordinate is generated to create a digital elevation model (DEM) and a contour by Surfer 3D surface mapping software program. Basically, the main novelties of the paper are to introduce a DEM to model the terrain, and assign the locations of wind turbines based on elevation values which are obtained from a digital map. The annual hourly wind data at 10 m is provided from data portal of national meteorological station [23]. According to this data, wind characteristics of the

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terrain are evaluated. Average wind speed is 6.2 m/s at the height of a 10 m mast. Three prevailing wind directions are considered for the calculations, since the total frequency of them is 92.06% of all times. A single type, identical wind turbine is considered. Herein, the objective function is to minimize the velocity deficits while maximizing the total output. A combination of a heuristic and a meta-heuristic approach is offered for the optimal placement. Heuristic method is set based on elevation values that ensures the minimum distance between the turbines, and this approach is used for the formation of the initial population. Then, a genetic algorithm is employed for optimal positioning of wind turbines. The rest of the paper is organized as follows. Section 6.2 discusses wake and power model, Sect. 6.3 presents the proposed methodology, Sect. 6.4 focuses on results and discussions, and the conclusion is given in Sect. 6.5.

6.2 Numerical Methods of the Present Study 6.2.1 Wake Model The term “wake effect” originates from the wake behind a ship. Like ships, wind turbines also create wakes. For wind turbines, wake effect relates to the velocity deficit of the wind and decreased energy content after leaving a wind turbine. By extracting energy from the wind, a wind turbine formed an imaginary cone (wake) that creates slower and more turbulent air behind it [2]. When a uniform incoming wind encounters a wind turbine, a linearly expanding wake behind the turbine occurs. A portion of the free stream wind’s speed will be reduced from its original speed u0 to u. According to this model the wake is turbulent, it expands linearly with downstream distance as shown in Fig. 6.1. The velocity deficit is defined as the fractional reduction of freestream wind speed in the wake of the turbine. Based on the momentum conservation assumption in the wake, the velocity deficit at turbine i(vel_ defij ) which has the distance of xi,j from turbine j can be calculated by Eq. (6.1). vel_ def ij = 1 −

u 2a = 2 x u0 (1+ ∝ ri,jr )

(6.1)

where u0 (m/s) is the wind speed perpendicular to the rotor plane, xi,j (m) is the downstream distance of the wind turbine, u is the downstream wind speed after xi,j distance, rr (m) is the rotor radius, a is the axial induction factor which is calculated from the thrust coefficient (CT ) of the wind turbine. According to IEC 61400-1 standard [25], CT is the characteristic wind turbine thrust coefficient for the corresponding hub height wind velocity which is shown as uhub in Eq. (6.2). CT and a calculations are given in Eq. (6.2) [26] and Eq. (6.3):

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Fig. 6.1 Wake model [24]



2uhub − 3.5 CT = 3.5 × uhub 2  a = 0.5(1 − 1 − CT )

 (6.2) (6.3)

and a is the wake spreading or entrainment constant, and shows how fast the wake expands. It can be calculated from Eq. (6.4). 0.5 ∝ = ln



zH z0

 (6.4)

Here, z0 (m) represents the surface roughness height of the site, and zH (m) represents the hub height of the wind turbine. xi,j is the distance between the turbine i and j , and it is calculated based on the given wind direction θ (degree). Details of Eq. (6.5) can be found in Kusiak and Song’s [9] paper.     xi,j = | xi − xj cos θ + yi − yj sin θ |

(6.5)

In order to calculate the produced power from a wind turbine, it should be ensured whether the wind turbine is located in the wake of other wind turbine(s) or not. To illustrate the wake area behind a wind turbine, an imaginary cone can be

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Fig. 6.2 Imaginary cone of a wind turbine [24]

drawn, see in Fig. 6.2 [24]. Wind blows from left with a given wind direction θ . Any two turbines i and j positioned perpendicular to the wind direction and locate at (xi , yi ) and (xj , yj ), respectively. The point A is the imaginary vertex, the angle α (0 ≤ α ≤ π /2) is calculated as arctan(α), and the distance between A and the hub is rr /α. β (0 ≤ β ≤ π ) is the angle used to determine if turbine i is in the cone of turbine j given the wind direction θ . For example, in Fig. 6.2, if the angle between the vectors AT2 and AT1 which is β is greater than the angle α, then T2 will not be inside the cone, which means T2 will not be under the wake of T1 . On the contrary, if β is smaller than the angle α, then T2 will face velocity deficit due to wake effect of T1 . The calculation of the angle β is shown in Eq. (6.6). ⎧ ⎨

⎫     rr ⎬ xi − xj cos θ + yi − yj sin θ + ∝ βij = cos−1 2  2 ⎭ ⎩  xi − xj + r∝r cos θ + yi − y + r∝r sin θ

(6.6)

Large wind farms experience a cumulative effect of multiple wakes. When many turbines are located in a wind farm, the direction of the wind changes regularly, that causes certain turbines to be in the wake of other turbines. In this case, multiple wakes have to be considered. ui (m/s) which is the downstream wind speed of the turbine i can be calculated by Eq. (6.7). ui = u0 (1 − vel_ def ij )

(6.7)

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And the calculation of multiple wake deficits on turbine i can be seen in Eq. (6.8). $ % N %  % vel_ def i = & vel_def 2i,j (6.8) j =1, j =i, β ij C2 > C1 wr3>wr 2>wr 1

Im (A) Exciting current

Fig. 7.3 Magnetizing curve of induction generator

The function of capacitor bank is to provide necessary reactive power to induction machine. During the occurrence of reactive power balance which is noticed as horizontal dashing line in Fig. 7.3, nominal voltage is generated under constant speed and load. In case of insufficient excitation capacitance value, the induction machine does not manage to generate nominal voltage. When the capacitance over the nominal value is connected to the terminal, loads in the system can be damaged because of high voltage. Higher speeds cause larger voltage magnitudes at the same capacitance value. Therefore, the point O on the figure is the equilibrium point for both speed and excitation capacitance.

7.4 Analysis The per phase equivalent circuit can be simplified as representation of three series of impedance as shown in Fig. 7.4. The steady-state performance of an induction generator may be determined from this simplified circuit. The impedances are given by

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Fig. 7.4 Simplified equivalent circuit

Z1

Z2

Vg F

I1 Vt F

Z¯ 1 = (R1 /F + j X1 ) −1  1 1 1 ¯ + Z2 = + Rc /F j/Xm R2 /(F − ω) + j X2 −1  1 1 ¯ + Z3 = RL /F −j Xc /F 2

Z3

(7.2)

According to Kirchhoff voltage law, the loop equation in Fig. 7.4 is   I¯1 Z¯ 1 + Z¯ 2 + Z¯ 3 = 0

(7.3)

where the stator current I¯1 is not zero. Thus sum of the three impedances equals to zero as follows: Z¯ 1 + Z¯ 2 + Z¯ 3 = 0

(7.4)

Above equation has to cover all operation conditions. Real and imaginary parts of Eq. (7.4) have to be zero, separately and thus   g1 = real Z¯ 1 + Z¯ 2 + Z¯ 3   g2 = imag Z¯ 1 + Z¯ 2 + Z¯ 3

(7.5) (7.6)

The two unknown parameters which are frequency (F) and excitation capacitance (Xc ) can be calculated from Eqs. (7.5) and (7.6) if the values of resistive load (RL ), magnetizing reactance (Xm ), rotor speed (ω) and the rest of the machine parameters are known. The relationship between the terminal voltage V¯t and air gap voltage (V¯g ) can be written as

V¯t = V¯g ∗

Z¯ 3 Z¯ 1 + Z¯ 3

(7.7)

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Here, the terminal voltage V¯t also represents the load voltage. If V¯t and required machine parameters are known for the given F, V¯g can be found from Eq. (7.7). Herewith, Xm can be calculated from Eq. (7.1). As a conclusion, required rotor speed can be determined using Xm value from Eqs. (7.5) and (7.6) to maintain desired frequency and terminal voltage. On the other hand, Xc value can be calculated to have desired V¯t . The load current can be written as V¯t I¯L = RL

(7.8)

Thus, the power consumed by the load is as follows: PL = 3 ∗ I¯L2 ∗ RL

(7.9)

The reactive power supplied by the excitation capacitance Xc becomes [24] Qc =

3 ∗ V¯t2 Xc /F

(7.10)

7.5 Response Surface Method Response surface is one of the most useful ways to optimize and develop processes in advance using requisite statistical methods in concurs with mathematical techniques. It is generally used in industry in order to characterize and evaluate the factors (input variants) that affect the performance of processes or products. RSM has advantage of having less number of experiments that ensures it more profitable and time-saving method [25]. An optimization process of using RSM consists of three stages. The determination of factors and their levels composes the preliminary work and first stage. The next is selection of the experimental design. The last stage is formed by obtaining response surface model as a function of the factors and defining the optimum points according to this model. The model is composed when the significant factors are defined. In this study, capacitor value and shaft speed are handled as independent variable factors. Besides, narrow range resistive load is included as factor to show the impact to the output voltage and frequency. Face central composite (FCC) design is utilized as response surface type. Second order equations are provided with this design when mathematical models are insufficient [26]. Hereby, second order terms are involved to the necessary model to find the optimum conditions. The most important reason of high order terms not to be included in the system is difficulty of calculating the coefficients. The major drawback of RSM emerges at this point. It is hard to fit the data to second order polynomial for highly non-linear systems. To overcome this problem, independent parameters can be chosen in a smaller range [27]. The second order centre composite designed response surface model is obtained as follows: [28]

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ˆ G(X) =a+

n  i=1

bi xi +

n 

ci xi2

(7.11)

i=1

ˆ G(X) is the second order polynomial that shows the output of the system, whereas a, bi , ci are regression coefficients needed to be calculated and xi are independent variables. The total amount of experiment number can be found using Eq. (7.12) [29]. N = 2k + 2k + nc

(7.12)

where k corresponds quantity of the factors, nc is number of repetitions of central point and N is the total experiment number. In face centred system, the points are chosen without exiting from surface of the composite. The system equations given in (7.11) are solved to obtain regression coefficients using the least squares method (LS) which is multiple regression technique [30]. After the determination of coefficients, one can calculate the response of the model for various input factors. Therefore, optimum input parameters can be easily determined by using this model.

7.6 Results and Discussions In this study, Minitab program has been firstly used to determine experiment design as shown in Table 7.1. The experiments shown in table have been carried out on Matlab/Simulink environment. The proposed method of selecting capacitance and speed values is simulated on a three phase, 4 kW, 400 V, 50 Hz, Y-connected induction machine as shown in Fig. 7.5. The per-phase parameters of the generator are R1 = 1.405 , R2 = 1.395 , X1 = X2 = 1.8344 . The nominal speed of the machine is 1465 rpm. The input parameters from Table 7.1 have been implemented to the simulation (Fig. 7.5) and output voltage and frequency of the system which has induction generator have been obtained and recorded into the same table. Secondly, the correlation has to be analysed between inputs and outputs of the system. To determine this correlation, data in Table 7.1 has been implemented to the least squares algorithm using Minitab program as mentioned in Sect. 7.5. As a result, response surface model of the outputs is obtained separately as follows: Output Frequency = −1.77 + 0.0791 ∗ C + 0.03558 ∗ ω − 0.00038 ∗ RL − 0.001667 ∗ C 2 + 0.000001 ∗ RL2 − 0.000047 ∗ C ∗ ω − 0.000001 ∗ ω ∗ RL

(7.13)

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Table 7.1 Experimental design Experiment number 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Capacitor (μF) 5 15 5 15 5 15 5 15 5 15 10 10 10 10

Speed (rpm) 1450 1450 1600 1600 1450 1450 1600 1600 1525 1525 1450 1600 1525 1525

Fig. 7.5 Simulink model of SEIG

Resistive load () 900 900 900 900 1000 1000 1000 1000 950 950 950 950 900 1000

Output frequency (Hz) 48.23 48.00 53.20 52.92 48.24 48.03 53.21 52.91 50.72 50.47 48.15 53.12 50.63 50.65

Output voltage (V) 174.57 299.31 217.32 350.37 171.96 303.60 210.96 339.95 193.50 321.05 257.94 320.58 286.33 287.90

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Fig. 7.6 Response surface optimization results

Output Voltage = −665 + 32.9 ∗ C − 0.08 ∗ ω + 1.19 ∗ RL − 1.209 ∗ C 2 + 0.000314 ∗ ω2 − 0.00015 ∗ RL2 + 0.00189 ∗ C ∗ ω + 0.00142 ∗ C ∗ RL − 0.000615 ∗ ω ∗ RL

(7.14)

The values between minimum and maximum limit of the inputs have been substituted in Eqs. (7.13) and (7.14). The response of the model has been recorded as shown in Fig. 7.6. The outputs of the system (V and F) have to be optimum at the same time which makes it multi-objective optimization problem [31]. Vertical straight red lines show level of the C, ω and RL , respectively. Rest of the curves illustrates the response of the outputs against various levels of inputs. The points where the vertical red lines crossed the horizontal curves are optimal points. The d value in Fig. 7.6 shows the approximation of the model to the real system. The maximum value of d is one. The desired value of output voltage and frequency are 230 V and 50 Hz, respectively. According to Fig. 7.6, the optimum values for capacitor and speed are found as 6.8974 μF, 1504.09 rpm, respectively, for the given 950.85  resistive load. The load

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234 Output Voltage Desired Output Voltage (230V)

Output voltage magnitude (Vrms)

232

230

230.8

228

230.6 226

230.4 230.2

224

230 5

4.5 0

1

2

3

4

5

5.5 6

7

8

9

Time (s)

Fig. 7.7 Variation of output voltage

can be changed on the graph and new optimum values can be found for different load conditions. RSM also gives an option of estimating the output of the system when the optimum values are used. On the left-hand side of Fig. 7.6, blue lined y values show the estimated outputs of the system as 50 Hz and 229.43 V. Moreover, regression value (R 2 ) of the model has been found as 98.85% which shows that the obtained model demonstrates the system accurately. Additionally, variation of outputs can be estimated by changing the input parameters using Fig. 7.6. The simulation has been executed for the obtained optimum values to have desired outputs. Figure 7.7 shows the output voltage of the system for a 6.897 μF capacitor value, 1504 rpm shaft speed under 950  resistive load. As shown from the zoomed part of Fig. 7.7 between 4 and 6 s, the average of the output voltage is obtained around as 230.4 V per phase with 0.004% variations. It can be made as inference that the maximum output error is between acceptable range with 0.5 V (0.21%) overshoot. Figure 7.8 shows the variation of output frequency and it clearly indicates that the frequency always maintained within acceptable range. It was mentioned in Sect. 7.4 that the all reactances depend on frequency and its stability is one of the most important points during self-excitation operation.

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Fig. 7.8 Variation of output frequency

It can be shown clearly from Figs. 7.7 and 7.8 that the desired output voltage magnitude and frequency are achieved using response surface optimization results without employing complex calculations. Besides, this method does not require any information of machine parameters. According to response surface model, following contour plots have been obtained to show the variation of frequency and voltage magnitude against speed, capacitor and resistive load. Figure 7.9 demonstrates that the value of resistive load and capacitor value have a very small effect on frequency as like speed. Figure 7.9a, b shows directly the effect of the speed to the output frequency. The speed value has to be between 1490– 1510 rpm to have 50 Hz frequency. It is seen from Fig. 7.9c that frequency does not vary that much in case of change in load and capacitor value. The effect of resistive load to the output voltage is not too much while the speed is under 1525 rpm and capacitor is under 13 μF. It is shown from Fig. 7.10a, b that above these values, the more resistive load causes a reduction on output voltage magnitude. Capacitor and speed both have respectable effects on output voltage. Figure 7.10c shows that the desired 230 Vr ms can be obtained when the capacitor value is between 8–11 μF. On the other hand, the more speed causes an increment in output voltage, whereas the capacitor is constant.

7 RSM-Based Optimization of Excitation Capacitance and Speed for SEIG Contour Plot of Frequency vs Speed; Capacitor 1600

Frequency < 48,8 49,6 50,4 51,2 52,0 52,8 > 52,8

48,8 49,6 50,4 51,2 52,0

1575

Speed

1550

Hold Values Resistive Load 950

1525

1500 1475 1450 5,0

7,5

10,0 Capacitor

12,5

15,0

(a) Contour Plot of Frequency vs Resistive Load; Speed 1000

Frequency < 49 50 51 52 53 53

49 50 51 52

980 Resistive Load


310

270 280 290 300

Resistive Load

980

Hold Values Capacitor 10

960

940

920

900 1450 1475 1500 1525 1550 1575 1600 Speed

(b) Contour Plot of V_rms vs Speed; Capacitor 1600

V_rms < 200 200 240 240 280 280 320 320

1575


limit Then Do Scout bee: achieve a new food source. End If Save essential information End While

8.3.1 Initialization of the Population The initial population of food sources is generated with N P number of food sources randomly. The vector Xi = (xij ) represents the ith food source in the population of solutions. The interval of search boundaries is given in Eq. (8.1):

8 Distance-Constrained Vehicle Routing Problems: A Case Study Using ABC. . .

xij = xj min + r(xj max − xj min ),

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(8.1)

where i = 1, . . . , N P (number of food sources or problem solutions); j = 1, . . . , n (number of decision variables or length of the food source vector) and r is a uniform random number between [0 − 1]. xj max and xj min are the upper and lower bounds for the dimension j . A sample food source will be presented in Sect. 8.4.

8.3.2 Initialization of the Bee Phase In this phase, each employed bee determines a new neighborhood source using the food source that is associated with them: vij = xij + φ(Xij − Xkj )

(8.2)

where j and k are randomly selected integers between [1−n] and [1−N P ] (must be different from the food source xi ), respectively. φ is a uniform real random number between [−1, 1], vij is the generated new solution.

8.3.3 Onlooker Bee Phase The nectar amount that is taken from all employed bees is calculated and a food source is selected based on its probability value by onlooker bee: P (i) = F V (i)/

NP 

 F V (k) ,

(8.3)

k=1

where FV is the fitness value for food source i (vi ). The FV for a minimization problem is given in Eq. (8.4):  fitness value (F V ) =

1/(1 + fi ) : when fi ≥ 0 1 + abs(fi ) : when fi < 0,

where fi is the objective function value of the solution i (food source vi ). Then the xi and vi are compared. After each solution is generated and then evaluated by artificial bee, it is compared with the old one and the better one is selected depending on FV. If the new food source has an equal or bigger nectar amount food source, it is replaced with the old one. Otherwise, the old one is retained. In other words, a greedy selection mechanism is applied between these solutions. This process is reiterated for whole employed bees in the population. In this study, FV is determined as the total length of the route. The solutions that are calculated with an adequate

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goodness values are called employed bees. Solutions which do not have adequate quality are classified as onlooker bees. In the onlooker bee phase, roulette wheel selection is employed. In this phase each food source is assigned a probability as given in Eq. (8.3).

8.3.4 Scout Bee Phase The corresponding employed bee becomes a scout when a food source cannot improve through a number of trials limit. In this phase, the food source is abandoned and scout produces a food source randomly. Then the new food source is replaced with xij : ( ' xij = xjmin + r xjmax − xjmin .

(8.4)

8.3.5 Stopping Phase There are two basic stopping criteria in this algorithm. One is the maximum number of iterations and the other is maximum elapsed time. In this study, both criteria are considered. In other words, the stopping criterion is the maximum number of iterations or maximum elapsed time. According to the equations above and the given pseudocode, the main steps of the proposed ABC algorithm for this case study are presented in Fig. 8.1.

8.4 Case Study In this study, we consider a single city center with a single vehicle. It is aimed to visit all the towns of Balikesir (Turkey) with the vehicle used by the controllers of Balikesir Directorate of Science, Industry and Technology under the given constraints. We studied the distance-constrained VRP (also known as DVRP). The problem calls for the determination of a set of minimum pathway routes to be performed by a vehicle to serve a group of controllers, where each route originates and terminates at a single point (city center). The aim is to minimize the pathway of the vehicle and to visit all the towns by the routes with a length of 550 km at maximum. In other words, the vehicle will return to the point where it starts from Balikesir city center on every turn and starts again without exceeding the 550 km range limit. It is aimed to determine the optimum routes so that the total length of all routes is minimized and each of the 19 towns is visited at least once. The map for the towns those have to be visited by the controllers and the distance matrix for the towns are given in Figs. 8.2 and 8.3, respectively.

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Fig. 8.1 Main steps of ABC algorithm for the presented case study [28]

In the ABC algorithm, each food source (possible solution) is transformed into a route. To clarify, N = 19 points are to be visited by a vehicle. In this solution, 0 represents the city center that has to be returned back; hence, the vehicle leaves from the center and delivers to points 2, 6, and 8; then, the vehicle must return to the city center (assuming that vehicle capacity is exhausted) to load. After that, it visits

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Fig. 8.2 The map for the towns those have to be visited by the controllers

points 1 and 3 and goes back to the city center. Later, the vehicle visits points 4, 5, 7, 9, and 10 in that respective order and this scheduling process continues until all the points those have to be visited are scheduled. The representation of the sample solution is given in Fig. 8.4 [17]. The sample schedule that is given in Fig. 8.4 can be a candidate solution to the problem, which is to minimize total traveled distance, assuming that capacity constraint is not violated. An optimization problem is seen as a search problem, where x represents a possible solution to the problem, which is within the search space. When it comes to the VRP, the search space is restricted if only it is considered the feasible region when trying to find the best solution via heuristics. Hence, it is necessary to allow the search in either the feasible or infeasible region (at least a restriction is violated). In this case, a solution is infeasible if the distance constraint is violated [17]. Routes used by the controllers in the current practice are: [City Center (1)— Savastepe (2)—Bigadic (3)—Sindirgi (4)— City Center (1)], [City Center (1)— Kepsut (5)—Dursunbey (6)—City Center (1)], [City Center (1)—Susurluk (7)— Manyas (8)—Gonen (9)—City Center (1)], [City Center (1)—Bandirma (10)— Erdek (11)—Marmara Island (12)—City Center], [City Center (1)—Balya (13)—

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Fig. 8.3 Distance matrix

Fig. 8.4 Representation of a sample solution

Ivrindi (14)—Havran (15)—Edremit (16)—City Center (1)], [City Center (1)— Burhaniye (17)—Gomec (18)—Ayvalik (19)—City Center (19)] and the total length of the routes is 1452 km. The details of the routes and lengths are summarized in Table 8.1 and Fig. 8.5, respectively. In order to solve the problem some assumptions are used. For example, only one vehicle is used. Also, the effect of road traffic is ignored. The locations of the facilities those have to be visited by the controllers are assumed to be at the center of the town so the distance matrix that is given in Fig. 8.3 can be used. The vehicle traveling speed is constant and they must return to the original starting point after finishing their daily task. The objective is to minimize the total traveling distance of the vehicle.

8.5 Results and Discussion In order to solve the problem, 10 onlooker bees and 10 employed bees are used. The parameters of the algorithm are determined as: the number of food is 100, the

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Table 8.1 Routes used by the controllers in current practice Route number Steps of route City Center (1)—Savastepe (2)—Bigadic (3)—Sindirgi (4)—City Center (1) 1 2 City Center (1)—Kepsut (5)—Dursunbey (6)—City Center (1) City Center (1)—Susurluk (7)—Manyas (8)—Gonen (9)—City Center (1) 3 City Center (1)—Bandirma (10)—Erdek (11)—Marmara Island (12)—City Center 4 City Center (1)—Balya (13)—Ivrindi (14)—Havran (15)—Edremit (16)—City Center (1) 5 City Center (1)—Burhaniye (17)—Gomec (18)—Ayvalik (19)—City Center (1) 6 Total length of the route

Length (km) 214 213 275 274 218 258 1452

Fig. 8.5 Routes used by provincial directorate in current practice

number of limit is 100, the distance limit for each route is 550 km, and the number of towns is 19. The algorithm is coded in MATLAB and run for 300 thousand iterations. The algorithm is run on a personal computer with Intel Core i7-3500 M 2.5 GHz processor. The algorithm coded in Matlab is run for 5 times and the best

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Table 8.2 The best result determined by the ABC algorithm Route number Steps of route City Center (1)—Edremit (16)—Gomec (18)—Ayvalik (19)—Burhaniye (17)—Havran (15)—City Center (1) 1 2 City Center (1)—Savastepe (2)—City Center (1) City Center (1)—Ivrindi (14) —Balya (13)—Gonen (9)—Marmara Island (12)—Erdek (11)—Bandirma (10)—Manyas (8)—Susurluk (7)—City Center (1) 3 City Center (1)—Kepsut (5)—Dursunbey (6)—Bigadic (3)—Sindirgi (4)—City Center 4 Total length of the route

Length (km) 264 94

346 327 1031

Fig. 8.6 Routes determined by the ABC algorithm

solution is recorded. Table 8.2 reports the details of the routes obtained by the ABC algorithm. The routes are also shown in Fig. 8.6. The total length of the routes is calculated as 1031 km. This total length is composed of 4 routes each of which is less than 550 km, the range limit for each route. These routes are: [City Center (1)—Edremit (16)—Gomec (18)— Ayvalik (19)— Burhaniye (17)—Havran (15)—City Center (1)], [City Center

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Table 8.3 Solutions obtained under different range limits Range limit Run 1 Run 2 Run 3 Run 4 Run 5

500 1169 1106 1124 1096 1123

550 1031 1091 1074 1031 1091

600 1026 1029 1021 1023 997

650 998 997 997 974 964

Best in bold

Boxplot of Total Length

1150

Total Length

1100

1050

1000

950 500

550

600

650

Range Limit

Fig. 8.7 The boxplot of total length for the solutions obtained under different range limits

(1)—Savastepe (2)—City Center (1)], [City Center (1)— Ivrindi (14)—Balya (13)—Gonen (9)—Marmara Island (12)—Erdek (11)—Bandirma (10) —Manyas (8)—Susurluk (7)— City Center (1)], [City Center (1)—Kepsut (5)—Dursunbey (6)—Bigadic (3)—Sindirgi (4)—City Center (1)]. The results indicate that using the ABC algorithm, the total length of the vehicle is reduced by 421 km which corresponds to a 29% improvement. Results indicate that the ABC algorithm can be used effectively for DVRP. For more scenarios, the range limit (550 km) is varied and the problem is solved again for five times. For this aim, the range limit is assumed to be 500, 600, and 650 km. The solutions are reported in Table 8.3. The boxplots of the solutions are also provided in Fig. 8.7. As seen from Table 8.3, the total length of the routes tends to reduce when the range limit is increased. For example, the lengths of the routes reduce to 997 and 964

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when the range limit is increased to 600 and 650 km, respectively. On the contrary, it increases to 1123 km when the length of each route is limited to 500 km.

8.6 Conclusion Vehicle routing literature consists of a broad range of problem variants and researchers pay even more attention to VRP variants that include real-life characteristics and assumptions, thereby making their models more realistic and their solution approaches more applicable in practice. However, real-life characteristics are often considered either individually or with a limited number of other characteristics [2]. In this paper, a case study for ABC algorithm is presented to cope with the distance-constrained vehicle routing problem which has been known as an NP-hard optimization problem. The aim is to minimize the pathway of the vehicle to visit 19 towns by the routes with a max length of 550 km, called range limit. The problem is also solved under different range limits and the best solutions are reported. The computational results show that the ABC algorithm can be used to solve the distance restricted vehicle routing problems. For the given case study under 550 km limit for each route, the total length of the route is reduced by 29%. Also the number of the routes is reduced from 6 to 4. Future research could consider multiple real-life characteristics. The problem solution can be expanded in such a way that the towns with population over 100 thousand will be visited at least 3 times. Also performance comparisons of different heuristics—those are used to solve this problem—can be presented. Furthermore, since the case study considered here includes 20 locations, exact solvers may have a potential to find the optimum solution in a reasonable time. Therefore, the result of the ABC can be compared with an exact solver solution. Acknowledgement The authors would gratefully like to thank Balikesir Directorate of Science, Industry and Technology whose valuable supports lead to reveal this paper.

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Chapter 9

Fractional Model for Type 1 Diabetes Ana R. M. Carvalho, Carla M. A. Pinto, and João M. de Carvalho

9.1 Introduction Diabetes Mellitus is a disease that affects the levels of glucose in our body. It is a result of the insufficient amount of insulin that is produced by the body or of the inability of cells to respond appropriately to insulin. Insulin is a natural hormone produced in the pancreas, by the β-cells. It controls the level of the sugar glucose in the blood and is the “key” which unlocks the cell to allow sugar to enter the cell and be used for energy. There are several types of diabetes. The most important are type 1 (T1D) and type 2 diabetes. T1D diabetes is caused by the destruction of pancreas insulin-producing cells, by the immune system, due to an autoimmune reaction. Type 2 diabetes is caused by an imbalance in insulin metabolism. There is a shortage of insulin and insulin resistance, that is, a greater amount of insulin is needed for the same amount of glucose in the blood. Common symptoms of diabetes are the urge to urinate and the increased need of drinking more water. Having more appetite is also a common symptom. Although diabetes is a disease that can be controlled, when it is not, it can lead to cardiovascular problems, blindness or even lead to death. By 2014, 422 million adults had diabetes, 314 million more than in 1980. Today 8.5% of adults have diabetes, which is the eighth-largest cause of death. The area

A. R. M. Carvalho · J. M. de Carvalho Faculty of Sciences, University of Porto, Porto, Portugal e-mail: [email protected] C. M. A. Pinto () School of Engineering, Polytechnic of Porto, Porto, Portugal Center for Mathematics of the University of Porto, Porto, Portugal e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6_9

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most affected by diabetes is the Eastern Mediterranean [1]. In 2015, 415 million people were estimated to live with diabetes. In 2040, this number is expected to grow up to 642 million. There are 542 thousand children and teenagers leaving with diabetes. Every 6 s, a person dies with diabetes, and the total amount of deaths caused by diabetes in 2015 was 5 million people. A striking perturbing fact is that 192 million people are unaware of diabetes. Still in 2015, diabetes was responsible for 4406 deaths in Portugal, which represents 4% of deaths. This is a worrying fact given that in 2001 the number of deaths due to diabetes was 3138 [18]. In the last few decades, some mathematical models have been proposed in the literature to understand the dynamics of diabetes. In 2007, Mahaffy et al. [7] present a mathematical model for the immune response that suggests a possible explanation for the cyclic pattern of disease behavior. The authors explain T-cell activation as an increasing function of the autoantigen level, whereas decreasing levels lead to the production of memory cells. Moreover, high β-cells death rates increase autoantigen levels, which turn off memory cells production, leading to less activated T-cells. After clearance of the peptide, the production of memory cells is recovered and the cycle repeats. In 2010, Magombedze et al. [6] formulate a mathematical model that incorporates the role of cytotoxic T-cells and regulatory T-cells in T1D diabetes. The authors show that diabetes is a complex disease, resulting from a sequence of events. Numerical results indicate that high levels of regulatory T-cells reduce the activity of auto-reactive T-cells, permitting β-cells to replenish and allow insulin production. In 2012, Marinkovic et al. [10] develop a model for the dynamics of diabetes that includes metabolism and the immune system at early stages of the disease. The model fits well with clinical and non-clinical data and suggests that amplitude and duration of autoimmune response may explain β-cell loss. In 2013, Nielsen et al. [11] provide a bifurcation analysis for a known model for early stages of the development of T1D. Several innovative treatment strategies are proposed, such as increasing the phagocytic abilities of resting or activated macrophages.

9.1.1 Some Concepts of Fractional Calculus The calculation of non-integer order, known as fractional calculus, generalizes the integral and differential calculus. In a simple and objective way, we think of fractional order (FO) operators as operators that represent memory functions. There is more than one possible formulation for the FO derivative. The most common definitions are those of Euler, Abel, Riemann–Liouville, Grünwald– Letnikov, and Caputo [13, 17]. Models of fractional order have been proposed to deepen the understanding of epidemiological phenomena [2, 14, 15]. In 2015, Goharimanesh et al. [4] propose an optimized control policy on T1D. The fractional order is used as a control method. They concluded that by using the fractional order, not only did the controller performance improve considerably, but also, unlike the traditional method, the concentration of glucose in the blood is maintained at the required range. In

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2017, Sakulrang et al. [16] present a model for continuous glucose monitoring of individuals with T1D. The results showed that the FO models may produce better agreement with data than models of integer order. Fostering the aforementioned ideas, we generalize an integer-order mathematical model for T1D to include a FO derivative. We study the role of macrophages from non-obese diabetic (NOD) mice and from control (Balb/c) mice in triggering autoimmune T1D. The outline of the paper is as follows. In Sect. 9.2, we describe the model. In Sects. 9.3 and 9.4, we study the local and the global stability of the disease-free equilibrium. We present and discuss the results of the simulations of the model, for biological relevant parameters and distinct values of the order of the fractional derivative, in Sect. 9.5. In the last section, we conclude our work.

9.2 Description of the Model Five populations are considered in the model: the resting macrophages, M, activated macrophages, Ma , apoptotic β-cells, Ba , necrotic β-cells, Bn and cytokines, C. The resting macrophages’ tissue entrance rate and mean residence time are a α and 1/cα , respectively. Their activation rate, through contact with apoptotic cells, is f1α . The mean activation duration time is 1/k α . Macrophages’ recruitment to tissue by activated macrophages is done at rate bα . The parameter eα represents crowding effects, that is, at high densities there is a reduced entry and/or increased efflux of macrophages from tissue. Apoptotic β-cells have a neonatal wave, W (t). β-cells’ cytokine-induced apoptosis is a Michäelis–Menten saturated function of C, where Aαmax is the maximal rate and kc the half-max cytokine concentration. Removal of apoptotic β-cells by resting/activated macrophages is done at rates f1α and f2α , respectively. Other non-specific processes contribute at rate d α . The cytokines, C, are produced by the necrotic cells, Ba and the activated macrophages, Ma , at rate αCα and are removed at a linear rate δ α . The nonlinear system of FO equations is given by dαM = a α + (k α + bα )Ma − cα M − f1α MBa − eα M(M + Ma ) dt α d α Ma = f1α MBa − k α Ma − eα Ma (M + Ma ) dt α d α Ba Aαmax C − f1α MBa − f2α Ma Ba − d α Ba = W (t) + dt α kc + C d α Bn = d α Ba − f1α MBn − f2α Ma Bn dt α dαC = αCα Bn Ma − δ α C, dt α

(9.1)

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where α ∈ (0, 1] is the order of the fractional derivative. We consider the definition of a FO derivative proposed by Caputo d α y(t) = I p−α y (p) (t), t > 0, dt α in which p = [α] is the integer part of α, y (p) is the p-th derivative of y(r), and I p1 is the Riemann–Liouville fractional integral 1 I z(t) = Γ (p1 ) p1

) t' (  p1 −1   t −t z(t )dt . 0

9.3 Model Analysis We now show that the solutions of system (9.1) are positive. 5 = {x ∈ R 5 | x ≥ 0} and x(t) = (M, M , B , B , C)T . We start by Let R+ a a n stating the following generalized mean value theorem [12] and corollary. Lemma 9.1 ([12]) Let f (x) ∈ C[a, b] and Daα f (x) ∈ C(a, b], and 0 < α ≤ 1, then f (x) = f (a) +

1 (D α f )(ξ )(x − a)α , Γ (α) a

(9.2)

where a ≤ ξ ≤ x, ∀x ∈ (a, b] and Γ (·) is the gamma function. Corollary 9.1 Consider f (x) ∈ C[a, b] and Daα f (x) ∈ C(a, b], for 0 < α ≤ 1. If Daα f (x) ≥ 0, ∀x ∈ (a, b), then f (x) is non-decreasing for each x ∈ [a, b]. If Daα ≤ 0, ∀x ∈ (a, b), then f (x) is non-increasing for each x ∈ [a, b]. It follows the proof of the main theorem. Theorem 9.1 There is a unique solution x(t) = (M, Ma , Ba , Bn , C)T to sys5. tem (9.1) for t ≥ 0. Moreover, the solution is in R+ Proof The solution on R0+ of the initial value problem exists and is unique, by Theorem 3.1 and Remark 3.2 of [5]. It is then sufficient to prove that the non5 is positively invariant. To do this, we show that the vector negative orthant R+ 5 field points to R+ on each hyperplane, limiting the non-negative orthant. From system (9.1), we get: D α M |M=0 = a α + (k a lpha + bα )Ma ≥ 0, D α Ba |Ba =0 v = W (t) +

Aαmax C ≥ 0, kc + C

D α C |C=0 = αCα Bn Ma ≥ 0.

D α Ma |Ma =0 = f1α MBa ≥ 0,

D α Bn |Bn =0 = d α Ba ≥ 0,

(9.3)

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5. Thus, we conclude that by Corollary 1, the solution stays in R+

Lemma 9.2 ([8]) The disease-free equilibrium P1 of the system (9.1) is locally asymptotically stable iff all eigenvalues λi of the linearization matrix of system (9.1), satisfy |arg(λi )| > α π2 . Proof The disease-free equilibrium is  P0 = (M

0

, Ma0 , Ba0 , Bn0 , C 0 )

=

−cα +

 c2α + 4eα a α , 0, 0, 0, 0 . 2eα

√

The matrix of the linearization, M1 , of system (9.1), around P0 , is ⎛

−cα − 2eα M 0 k α + bα − eα M 0 −f1α M 0 0 0 ⎜ α α 0 f1α M 0 0 0 0 −k − e M ⎜ ⎜ Aαmax M1 = ⎜ 0 0 0 −f1α M 0 − d α k ⎜ ⎝ 0 0 dα −f1α M 0 0 0 0 0 0 −δ α

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

Expanding, det (λp I5 − M) = 0, where I5 is the identity matrix of order 5, we obtain the characteristic equation given as '

(' ( ( ' λp + cα + 2eα M 0 λp + f1α M 0 λp + δ α λp + k α + eα M 0 ( ' × λp + f1α M 0 + d α = 0.

(9.4)

Now, the arguments of the roots of the equations λp + cα + 2eα M 0 = 0, λp + f1α M 0 = 0, λp + δ α = 0, λp + k α + eα M 0 = 0 and λp + f1α M 0 + d α = 0, satisfy: arg(λk ) =

2π π π π +k > > , p p M 2M

where k = 0, 1, . . . , (p − 1). Thus, the disease-free equilibrium P0 is locally asymptotically stable.  

9.4 Global Stability of the Disease-Free Equilibrium In this section, we derive the global stability of the equilibrium P0 of system (9.1). System (9.1) is rewritten as follows:

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dαX = F (X, Z) dt α dαZ = G(X, Z), dt α

(9.5) G(X, 0) = 0,

where X = M and Z = (Ma , Ba , Bn , C), with X ∈ R+ being the number of resting macrophages and Z ∈ R4+ denoting the number of activated macrophages, apoptotic and necrotic β-cells, and cytokines. F (X, Z) = a α + (k α + bα )Ma − cα M − f1α MBa − eα M(M + Ma ) and G(X, Z) is the subsystem composed by the last four equations of system (9.1). √ c +4e a . The equilibrium P0 is written as U = (X , 0), where X = −c + 2e α The global asymptotic stability of P0 is valid when conditions (H1 ) and (H2 ) are satisfied: α

(H1 ) : For

dαX dt α

2α

α α

= F (X, 0), X is globally asymptotically stable

ˆ ˆ ≥ 0 for (X, Z) ∈ Υ1 , (H2 ) : G(X, Z) = AZ − G(X, Z), G

(9.6)

where A = DZ G(X , 0) can be written in the form A = M − D, where M ≥ 0 (mij ≥ 0) and D > 0 is a diagonal matrix. Υ1 is the region where the model makes biological sense. If system (9.5) satisfies the conditions in (9.6) the following theorem holds [3]. Theorem 9.2 The fixed point U = (X , 0) is a globally asymptotically stable equilibrium of system (9.5) provided that R0 < 1 and that the conditions in (9.6) are satisfied. Proof Let 0 1 F (X, 0) = a α − cα M − eα M 2

(9.7)

and ⎛

−k α − eα M 0 f1α M 0 0 0 ⎜ Aαmax α 0 α 0 0 −f1 M − d ⎜ kc A=⎜ ⎝ 0 dα −f1α M 0 0 0 0 0 −δ α and

⎞ ⎟ ⎟ ⎟ ⎠

(9.8)

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⎞ ⎛ ˆ G1 (X, Z) ⎜ Gˆ2 (X, Z) ⎟ ⎟ ˆ G(X, Z) = ⎜ ⎝ Gˆ3 (X, Z) ⎠ Gˆ4 (X, Z) ' ( ⎛ ⎞   M f1α Ba M 0 1 − M + e α Ma M + Ma − M 0 0 ( ' ' ( ⎜ ⎟ ⎜−W (t) + Aα C 1 − 1 ⎟ M α + f1α M 0 Ba M ⎜ 0 − 1 + f2 Ma Ba ⎟ max k k +C c c =⎜ ' ( ⎟. ⎜ ⎟ M α f1α M 0 Bn M ⎝ ⎠ 0 − 1 + f2 Ma Bn −αCα Bn Ma (9.9) Condition H2 in (9.6) is not satisfied since Gˆ4 < 0. Consequently, we cannot say that U0 is generically globally asymptotically stable.  

9.5 Numerical Results In this section we simulate system (9.1). Table 9.1 contains the values of the parameters used in the simulations. The initial conditions are set to M(0) = 4.77 × 105 and Ma (0) = Ba (0) = Bn (0) = C(0) = 0 [9]. In Figs. 9.1, 9.2, and 9.3, we consider W (t) = 4 × 107 exp(−((t − 9)/3)2 ) and two different cases: non-obese diabetic (NOD) and control (Balb/c) mice. For α = 1, in the case of NODs, the apoptotic cell wave causes an increase in active macrophages and macrophages at rest, soon after the activation of the resting macrophages. The amount of apoptotic β-cells decreases but remains high because

Table 9.1 Values of the parameters used in the numerical simulations of system (9.1) Parameter a b c d k f1 f2 e Amax kc αC δ

Value 5 × 104 0.09 0.1 0.5 0.4 1 × 10−5 1 × 10−5 1 × 10−8 2 × 107 1 5 × 10−9 25

Units ml−α day−α day−α day−α day−α day−α ml day−α ml day−α day−α ml−α day−α day−α day−α

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of the damage induced by inflammation. Instead, in the case of the Balb/c, the macrophages can eliminate the apoptotic wave quite rapidly, which is sufficient for chronic inflammation to be avoided. In the case of Balb/c, activated macrophages, apoptotic β-cells, and other cells (C, Bn ), go to zero and resting macrophages’ number returns to normal, 20 days after the end of the apoptotic wave. When α < 1, chronic inflammation is observed in both NOD and control mice. Concentrations of cytokines are higher for smaller α. Moreover, it is observed a faster decrease of cytokines for smaller α. Now, we consider that β-cells are induced by Ma at a rate l, rather than being induced by cytokines. The equation that reflects this is d α Ba dt α

= W (t) + lMa − f1α MBa − f2α Ma Ba − d α Ba ,

(9.10)

where l = 0.41. In Figs. 9.4, 9.5, and 9.6, in both NOD and Balb/c cases, chronic inflammation occurs for all α. However, the result is similar to the previous figures,

9 Fractional Model for Type 1 Diabetes α=0.7

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Fig. 9.4 Log cells density of the system (9.1) and the apoptosis of β-cells are induced by Ma , for α = 1. Left—NOD case. Right—Balb/c case (f1 = 2 × 10−5 and f2 = 5 × 10−5 )

the maximum value of the cytokines is reached for smaller values of the order of the fractional derivative α.

9.6 Conclusion We generalize a mathematical model for T1D to include a FO derivative, α. We study the role of macrophages from non-obese diabetic (NOD) mice and from control (Balb/c) mice in triggering autoimmune T1D. For α = 1, we find that an apoptotic wave can trigger T1D in NOD but not in Balb/c mice. The apoptotic wave is cleared efficiently in Balb/c mice preventing the onset of T1D. For smaller values of α, the inflammation persists for NOD and control mice. This suggests that α may be used as a parameter to distinguish between distinct disease progressions (characterized by different immune systems’ response) in several patients.

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Acknowledgements The authors were partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT— Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/ UI0144/2013. The research of AC was partially supported by a FCT grant with reference SFRH/BD/96816/2013.

References 1. BBC, http://www.bbc.com/portuguese/noticias/2016/04/160406_diabetes_aumento_lab. Accessed May 26 2019. 2. A.R.M. Carvalho, C.M.A. Pinto, Within-host and synaptic transmissions: contributions to the spread of HIV infection. Math. Methods Appl. Sci. 40, 1231–1264 (2016) 3. C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of RO and its role in global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An 388, ed. by S. Tennenbaum, T.G. Kassem, S. Roudenko, C. Castillo-Chavez

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4. M. Goharimanesh, A. Lashkaripour, A.A. Mehrizi, Fractional order PID controller for diabetes patients. J. Comput. Appl. Mech. 46(1), 69–76 (2015) 5. W. Lin, Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007) 6. G. Magombedze, P. Nduru, C.P. Bhunu, S. Mushayabasa, Mathematical modelling of immune regulation of type 1 diabetes. BioSystems 102, 88–98 (2010) 7. J.M. Mahaffy, L. Edelstein-Keshet, Modeling cyclic waves of circulating T cells in autoimmune diabetes. SIAM J. Appl. Math. 67(4), 915–937 (2007) 8. D. Matignon, Stability results for fractional differential equations with applications to control processing, in Computational Engineering in Systems Applications, vol. 2 (Lille, France, 1996), p. 963 9. A.F.M. Maree, R. Kublik, D.T. Finegood, L. Edelstein-Keshet, Modelling the onset of type 1 diabetes: can impaired macrophage phagocytosis make the difference between health and disease? Philos. Trans. R. Soc. Lond. A 364, 1267–1282 (2006) 10. T. Marinkovic, M. Sysi-Aho, M. Oresic, Integrated model of metabolism and autoimmune response in β-cell death and progression to type 1 diabetes. PLoS One 7(12), e51909 (2012) 11. K.H.M. Nielsen, F.M. Pociot, J.T. Ottesen, Bifurcation analysis of an existing mathematical model reveals novel treatment strategies and suggests potential cure for type 1 diabetes. Math. Med. Biol. 31(3), 205–225 (2014). https://doi.org/0.1093/imammb/dqt006 12. Z.M. Odibat, N.T. Shawagfeh, Generalized Taylor’s formula. Appl. Math. Comput. 186, 286– 293 (2007) 13. K. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974) 14. C.M.A. Pinto, A.R.M. Carvalho, The role of synaptic transmission in a HIV model with memory. Appl. Math. Comput. 292, 76–95 (2017) 15. C.M.A. Pinto, A.R.M. Carvalho, Persistence of low levels of plasma viremia and of the latent reservoir in patients under ART: a fractional-order approach. Commun. Nonlinear Sci. Numer. Simul. 43, 251–260 (2017) 16. S. Sakulrang, E.J. Moore, S. Sungnul, A. Gaetano, A fractional differential equation model for continuous glucose monitoring data. Adv. Difference Equ. (2017), 150 (2017). https://doi.org/ 10.1186/s13662-017-1207-1 17. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, London, 1993) 18. Sociedade Portuguesa de Diabetologia: Diabetes: Factos e Números—O Ano de 2015— Relatório Anual do Observatório Nacional da Diabetes (2016)

Chapter 10

Mathematical Modelling and Additive Manufacturing of a Gyroid Yılmaz Gür

10.1 Infinite Periodic Minimal Surfaces (IPMS) Without Self-intersections: Gyroid The minimal surfaces research is around for over 200 years. The study began in 1760 when Lagrange asked the question “what does a surface bounded by a given curve look like, when it has smallest surface area?” [1]. Minimising a surface leads us to a partial differential equation of the surface and a tool to study such equation is not available yet. The first example of an infinite periodic minimal surface (IPMS) was published by H. A. Schwarz in 1865. The Schwarz P surface which is a triply periodic minimal surface was described by Hermann Schwarz in 1890 [2]. Gyroid is discovered by Alan Schoen, who is a physicist and computer scientist born in NY, in 1970 while he was studying super-strong, super light structures. This surface is known that the only example of an intersection-free IPMS which contains neither straight lines nor mirror reflections, and the axes of rotational symmetry do not lie in the surface. The Gyroid belongs to the cubic crystal system and the Gyroid G has a body-centred cubic Bravais lattice. The mathematical equation of the Gyroid is very complicated because it consists of elliptic integrals [3]. The parameterisation of the Gyroid is given by the following equations: ) x = Re

exp(iθG )F (τ )(1 − τ 2 ) dτ,

(10.1)

i. exp(iθG )F (τ )(1 + τ 2 ) dτ,

(10.2)

) y = Re

Y. Gür () Balikesir University, Balikesir, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6_10

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) z = Re

2. exp(iθG )τ F (τ ) dτ,

(10.3)

where i 2 = −1,

(10.4)

τ = τ1 + iτ2 ,

(10.5)

F (τ ) = (1 − 14τ 4 + τ 8 )−1/2 .

(10.6)

cos x · sin y + cos y · sin z + cos z · sin x = 0

(10.7)

However,

equation gives an approximation to the Gyroid surface looks like the actual Gyroid [4]. Because of this, the above equation is considered to create a mathematical model of the Gyroid which is the subject of this study.

10.2 Additive Manufacturing Technology It is perceived that additive manufacturing (AM) is a new and emerging technology of the twenty-first century. In fact, the roots of the technology date back to 1850s. Hideo Kodama who was the first describing the layer by layer approach for manufacturing in 1980. Later on, an additive manufacturing device and material were invented by Charles Hull in the 1980s. The device named as “stereolithography apparatus” and the material used was photopolymer. The invention was patented in 1986 as “Apparatus for production of three dimensional objects by stereolithography” [5]. Scott Crump has patented another printing technique named “Fused Deposition Modelling” (FDM) in 1989 for creating three-dimensional objects [6]. It is the most commonly used 3D printing technique today. The fastly growing 3D technologies have benefited from the expired 3D printing patents for FDM. Fused deposition modelling technique is a 3D printing process in which heated thermo-plastic material is extruded from a nozzle through a computer controlled printing head to build up the object layer by layer. In this study, in order to print the IPMS Gyroid FDM technology is used. The use of low-cost 3D desktop printers is fast growing and they have huge potential for digital fabrication. Various types of 3D objects can be digitally fabricated by 3D desktop printers. The objects can be simple geometrical shapes, complex mathematical models, archaeological artifacts, and medical prosthesis. They hold a promising future for science, technology, manufacturing industry, and education. The use of additive manufacturing or 3D printing technology makes the production of mathematical models much easier even though they are very complex.

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Fabrication of complex mathematical models is nearly impossible by using traditional engineering methods such as subtractive chip removal methodology whose material is removed from a solid work piece block by using milling, drilling, lathing, and forming method [7]. The industry of 3D printing is considered a part of 4th Industrial Revolution and it is the latest piece in a chain of visualisation techniques [8]. It allows everyone to create custom parts, will localise manufacturing and create less waste. 3D printers allow for creating models of what are traditionally abstract concept to math teachers and students. Printed models will provide to express mathematical ideas easily and more clearly in mathematical fields such as calculus, topology, or geometry [9]. 3D printing is a highly flexible digital fabrication process.

10.3 3D Printing Process of an IPMS Gyroid A basic 3D printing process has six stages to print a 3D mathematical model of an IPMS Gyroid; 1. 2. 3. 4.

Creating a 3D mathematical model based on the surface equation, Converting the CAD model data to .OBJ or .STL file format, Generating a solid, thickened shell, or hollow CAD model, Slicing the model into layers and generating the travel movements and support structures if necessary, 5. 3D printing of the model, 6. Removing the support material if any and apply finishing process [10, 11].

10.3.1 Creating the 3D Mathematical Model of the IPMS Gyroid with K3DSurf Program In order to create the mathematical model of the IPMS Gyroid is required a mathematical modelling software to calculate x, y, and z coordinate values of the vertex points which lay on the surface of the Gyroid. There are many types of 3D mathematical modelling software available on the market, some of them are free licence and some are commercial. For the generation of the mathematical model, K3DSurf v0.6.2 software is used [12]. K3DSurf software lets you to visualise and manipulate multi-dimensional surfaces by using mathematical equations. It supports parametric equations and isosurfaces as well. K3DSurf requires the right-hand side of the considered equation to be zero. So the IPMS Gyroid’s trigonometric approximation equation should be turned into cos x · sin y + cos y · sin z + cos z · sin x = 0,

(10.8)

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Fig. 10.1 List of evaluated surface points and the 3D mathematical model

where x(−4, 4),

y(−4, 4),

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(10.9)

Equation (10.8) defines the F (x, y, z) = 0 function in Cartesian coordinate system. To evaluate the equation in the range of boundary conditions (10.9) Eq. (10.8) is entered into the K3DSurf software. The list of the evaluated coordinate points and the 3D model generated based on these points is given in Fig. 10.1. Then the generated surface data is imported to “.obj” data file format.

10.3.2 Converting the CAD Model Data to “.obj” or “.stl” File Format After successful creation of the Cartesian coordinate points of the mathematical model, K3DSurf program allows the conversion of the data into wavefront’s “.obj” R data file format. MakerBot MakerWareTM (MW) software that prepares 3D models for being made on a 3D FDM printer support “.obj”, “.stl” and “thing” file formats and can load them without any problem [13]. Some CAD software require “.stl” files rather than “.obj” file format to modify and edit the model such as giving thickness. “.stl” data file format consists of the coordinates of the vertices and the direction of the outward normal of each triangle. “.stl” data format cannot represent

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the curved surfaces precisely because it uses triangular shaped planar elements. This inadequacy can be mitigated by increasing the number of triangles. But this time file size and processing time increased [10].

10.3.3 Generating a Solid, Thickened Shell or Hollow CAD Model In order to 3D print the IPMS Gyroid mathematical model it should be thickened, otherwise it would not be possible to fabricate it. To do that Autodesk’s MeshMixer 3D sculpting based computer aided design (CAD) program is used. Meshmixer software not only allows to modify the 3D models and optimise them for 3D printing but also analyse different properties of the models such as thickness for successful 3D printing [14]. The thickened and 3D printable version of the IPMS Gyroid can be seen in Fig. 10.2.

Fig. 10.2 Thickened shell CAD model of the IPMS Gyroid

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10.3.4 Slice the Model into Layers, Generate the Travel Movements and Support Structure In this step, pre-processing and slicing program MakerWareTM is used. Initially, the model is located on the build platform, resized, and orientated. In order to shorten the print time minimising the number of layers in z-axis is important. MakerWareTM slices the 3D mathematical model into finite number of layers. The thinner the layer thickness is the higher the number of layers. FDM process starts with slicing of a 3D model of IPMS Gyroid, which is in “.stl” file format, into layers. This stage not only consists of slicing, calculation of printing time, and material necessary to be used but also generates travel movements of extrusion nozzle and model support structure that holds the model standstill and prevent the leaky connections, overhangs, bridges, and internal cavities. Slicing information is then exported to “.gcod” or “.x3” file format that 3D FDM printer can understand to print the model. For the IPMS Gyroid example, 459 horizontal R layers are created by the MakerBot MakerWareTM software. Because the IPMS Gyroid has some overhangs and bridges, it requires support structures in order to be printed correctly. These support structures are also generated during this slicing process. Once the part is completed these support scaffoldings can be removed. As a printing time, it is calculated that it is going to take 17.5 h (see Fig. 10.3).

Fig. 10.3 Slicing of the mathematical model by MakerWare software

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10.3.5 3D Printing of the Model The new 3D printing technology makes the realisation of mathematical models more accessible than ever. Fused deposition modelling (FDM) technique that is one of the many additive manufacturing technologies is used to fabricate a tangible sample of the IPMS Gyroid model.

10.3.5.1

Fused Deposition Modelling

Fused deposition modelling (FDM) technology is a layered additive manufacturing process which uses thermo-plastic material such as ABS (acrylonitrile butadiene styrene) and PLA (polylactic acid) to produce concept models, functional prototypes, manufacturing aids, and low volume end-use parts. The thermo-plastic material is uncoiled slowly and extruded through heated extrusion nozzle. The material is precisely laid down upon the precedent layers. Following each sequence the building platform is lowered down by the thickness of one layer while the extrusion nozzle continues to move on a horizontal X-Y plane. The process is repeated, adding layer upon layer, until the object is finished (see Fig. 10.4).

Fig. 10.4 Fused deposition modelling (FDM) technology

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FlashForge Creator Dual Extruder 3D printer, which uses fused deposition modelling principle and available in the University of Balikesir’s mechanical engineering department is used. Its dimensions are 467 × 320 × 38 mm and small enough to use on a desktop in an office room. Building volume of the printer is 225×145×150 mm. Layer thickness can be changed between 250 μm and 100 μm. As a consumable material either Acrylonitrile Butadiene Styrene (ABS) ((C8 H8 )x · (C4 H6 )y · (C3 H3 N)z ) or biodegradable Poly Lactic Acid (PLA) (C3 H4 O2 )n thermo plastics can be used. It uses open filament system and filament diameter it accepts is R 1.75 mm. As a slicing software, both open source ReplicatorG 0040 or MakerBot TM MakerWare can be used. Layer thickness for the printing was 150 μm. As a consumable material acrylonitrile butadiene styrene is used (ABS) ((C8 H8 )x · (C4 H6 )y · (C3 H3 N)z ). The filament diameter it accepts is 1.75 mm. The building platform is heated up around 110 ◦ C prior to printing otherwise part does not stick on to the build platform well even though it is covered with kapton tape. Meanwhile, the extrusion nozzle is also heated up to 226 ◦ C in order to provide fluent flow for the ABS plastic. During the FDM printing process, extrusion nozzle moves along the X-Y axis and the building platform moves in Z-axis. Printing of the Gyroid took approximately 17.5 h.

10.3.6 Removing the Support Material If Any and Apply Finishing Process After the 3D printing of the IPMS Gyroid, support scaffolds on the object have been broken off by hand carefully, and finally, the Gyroid is polished in a heated acetone vapour bath (see Fig. 10.5).

10.4 Conclusion The aim of the concrete cases described in this study was to create a real tangible object from a mathematical model of an IPMS Gyroid that would not be possible to produce with traditional engineering methods. First, a mathematical model from the IPMS Gyroid approximation equation is generated with K3DSurf mathematical software. Then, Cartesian coordinate point values are obtained and then the data is converted to “.OBJ” file format that 3D R FDM printer software MakerBot TM MakerWare can slice the 3D model into layers. During the slicing process, travel movements and speed of the extrusion nozzle on X-Y plane of the building platform are defined, inlay filling rate and type is specified, number of layers

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Fig. 10.5 3D printed and polished IPMS Gyroid object

are determined according to the thickness of the layer, places, and thickness of supporting scaffolding structures, which support the model while printing and broken off by hand after printing, are appointed. 3D FDM printing of the Gyroid, layer by layer, took 17.5 h. The printed IPMS Gyorid object would serve as an educational tool and can be used for the presentation purposes. Nowadays, lowcost desktop FDM 3-D printers give great opportunity to the users for creating such complex mathematical models [15]. Today, layer thickness for the FDM printers can go down as low as 50 μm and this gives good surface quality and smoothness for the objects printed.

References 1. E. Denme, Visions in math. https://mathvis.academic.wlu.edu/2016/02/29/schwarz-p-surfacethe-math. Accessed 4 Oct 2018 2. H.A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 1 (Julius Springer, Berlin, 1890) 3. A.H. Schoen, Infinite periodic minimal surfaces without self-intersections. NASA Technical Note TN D-5541, 1970 4. A. Weyhaupt, 3-D printing software [online]. https://plus.maths.org/content/-meet-gyroid. Accessed 11 Oct 2018 5. C.W. Hull, US Patent 4575330. Apparatus for production of three-dimensional objects by stereolithography (Date of patent: 11 Mar 1986)

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6. S.S. Crump, US. Patent 5121329. Apparatus and method for creating three-dimensional objects. U.S. Class: 364/468, ASSIGNEES: Stratasys, Minneapolis, MN (filed: 30 Oct 1989. Date of patent: 9 June 1992) 7. H. Segerman, 3D printing for mathematical visualisation. Math. Intell. 34(4), 54-62 (2012) 8. O. Knill, E. Slavkovsky, Illustrating mathematics using 3D printers (2013). arXiv:1306.5599 (math.HO) https://arxiv.org/pdf/1306.5599.pdf. Accessed 12 Mar 2018 9. E.A. Slavkovsky, Feasibility Study For Teaching Geometry and Other Topics Using ThreeDimensional Printers. M.L.A thesis. Harvard University, 2012 10. V.M. Chapela, M.J. Percino, F.D. Calvo, F. Calvo, L. Trinidad, in Proceedings of the World Congress on Engineering, Manufacture of 3D Möbius–Listing Models with a 3D Printer, vol. I (London, 2013) 11. J.N. Marcincin, L.N. Marcincinova, J. Barna, M. Janak, Tehniˇoki Vjesnik Vjesnik—Technical Gazette. Appl. FDM Rapid Prototyping Tech. Exp. Gearbox Dev. Process 19(3), 689–694 2012 12. A. Taha, K3DSurf software package. http://k3dsurf.sourceforge.net (2014). Accessed 12 Mar 2018 TM R 13. MakerBot MakerWare v2.4.1.24. 3-D printing software (2013). Accessed 30 Sept 2014 14. Autodesk Meshmixer 3.5. http://www.meshmixer.com (2014). Accessed 12 June 2018 15. Y. Gür, Digital fabrication of mathematical models via low-cost 3D FDM desktop printer. Acta Phys. Pol. A 127(2-B), B-100-B-102 (2015)

Index

A AC optimal power flow (AC-OPF), 30–31 Additive manufacturing (AM) description, 188 FDM (see Fused deposition modelling (FDM)) invention, 188 layer by layer approach, 188 low-cost 3D desktop printers, 188 MAM (see Metal additive manufacturing (MAM)) “stereolithography apparatus,” 188 3D printing, 188–189 Anemia classification, 73 description, 73 diagnosis, 73, 80 hemoglobin thresholds, 72 inflammatory diseases, 71 medical medical, 72 MLR model, 80 regression analysis, 79 standardized coefficient (Beta), 80 types and blood variables, 74, 75 Artificial bee colony (ABC) algorithm case study, 164–167 CVRP, 160 description, 161 distance-constrained problem, 159–160 EVRP, 160 food sources, bees, 162 groups of bees, 161–162 initialization, bee phase, 163 initial population, food sources, 162–163

onlooker bee phase, 163–164 parameters, 167–168 scout bee phase, 164 steps, 164, 165 stopping phase, 164 vehicle routing problem, 160 Artificial fish swarm algorithm (AFSA), 161

B Batch processing machines (BPM), 1, 3, 8 Batteries, 25 Bio-geography, 116 Biological fluids, 90 Biological sciences, 90 Boundary conditions of the flow, 99–100 Building multiple regression models, 78 Buoyancy, 95

C Capacitated VRP (CVRP), 158 Carbon emission, 25 Compressed air energy storage (CAES), 25 Conservation laws, 91 energy, 99 mass, 99 momentum, 98–99 Conservation of energy, 99 Conservation of mass, 99 Conservation of momentum, 98–99 Coral Reefs Optimization (CRO), 116–117 Cost–benefit mechanism, 52, 63

© Springer Nature Switzerland AG 2020 J. A. T. Machado et al. (eds.), Mathematical Modelling and Optimization of Engineering Problems, Nonlinear Systems and Complexity 30, https://doi.org/10.1007/978-3-030-37062-6

197

198 D Darcy’s law, 93–94 Data availability, 71 DC optimal power flow (DC-OPF), 32, 33 Decarbonization, 115 Deformable channel, 90, 92 Diabetes cause of death, 175–176 description, 175 dynamics, 176 FO (see Fractional order (FO)) fractional calculus, 176–177 symptoms, 175 T1D, 175, 183 T2D, 175 types, 175 Digital elevation model (DEM), 117, 122, 133, 135 Dimensionless system, 101, 102 Direct metal laser sintering (DMLS) system, 15 Distance and capacity constrained VRP (DCVRP), 158, 159 Distance-constrained VRPs (DVRP), 158, 159, 164, 170 Double perturbations method, 90, 102, 105–107 Downwind turbines, 115 Dynamical system, 52, 60, 66

E Economic dispatch (ED) problem, 26–27 Electricity, 34–35 Electronic load controller (ELC), 140–141 Energy economics AC-OPF model, 30–31 DC-OPF model, 32 ED model, 26–27 optimal placement and sizing, 33–34 physical limitations, 27 power systems, 30 UC model (see Unit commitment (UC)) Energy storage conventional energy generation, 25 deployment, 25 nationwide policy, 44–46 optimal placement and sizing, 33 storage system, 25 system modeling, UC, 29–30 technologies, 25 Turkish transmission grid, 42–44 UC problem, 29–30

Index See also Nationwide energy storage system Environmental vehicle routing problem (EVRP), 160 Excitation capacitance, 139, 140, 143

F Face central composite (FCC) design, 145 Filtration process, 91, 92, 107, 109, 110 Fluid injection, 91 Forestry dynamical systems and control theory, 52 natural resources, 51 optimal management cost–benefit mechanism, 63 harvesting and replanting, 63, 66 management problem, 65 natural evolution, 64 Pontryagin’s principle, 66–68 Fractional calculus, 176–177 Fractional order (FO) as control method, 176 global stability, disease-free equilibrium, 179–180 integer-order mathematical model, T1D, 177 model analysis, 178–179 nonlinear system, FO equations, 177–178 numerical simulations of system, 181 operators, 176 Frandsen wake model, 116 Fuels, 27, 30, 38–42 Fused deposition modelling (FDM) description, 193 IPMS Gyroid FDM technology, 188 multi-objective optimization model, 3 printing technique, 188 thermo-plastic material, 193 3D model, IPMS Gyroid, 192

G Gas-fired power plants, 27 Genetic mutation, 130

H Hemoglobin, 73, 74, 79, 80 Horizon optimal growth problems, 51

I Immune system, 175, 176, 183

Index Incompressible fluid, 91, 92, 95, 98, 100 Infinite periodic minimal surface (IPMS) AM technology, 188–189 Gyroid, 187 Schwarz P surface, 187 3D printing process CAD model, 190–191 FDM technology, 193 K3DSurf program, 189–190 polishing, 194, 195 printing, IPMS Gyroid, 189–190 thickened shell CAD model, 191 travel movements and support structure, 192 Injection, 90, 92, 110, 112 Insulin, 175, 176 Intertemporal utility, 53, 54, 58 Irregular shape wind farm micrositing, 117, 136

J Jensen’s wake effect model, 116

K Kirchhoff’s law, 30, 144

L Lie group method, 102–105 Linear regression model, 72, 78, 81, 83 Lorentz force, 97 Lorentz’s law, 96 Lubricant cleanliness, 91

M Macroeconomic optimal growth problem, 51, 52, 69 Magnetic field, 95–97, 113 Magnetization curve, 139 Magneto-hydrodynamic (MHD) fluid, 90, 92, 97 Mass conservation, 98 Mathematical modelling fabrication, 189 IPMS Gyroid (see Infinite periodic minimal surface (IPMS)) low-cost 3D desktop printers, 188 Mathematical pathology, 71–72 Matlab/Simulink model, 141 Mean squared error (MSE), 77 Medical modelling, 71, 83

199 See also Multiple linear regression (MLR) Metaheuristics, 159, 160 Metal additive manufacturing (MAM) computational experiments CPU time consumption, 17–18 DMLS system, 15 non-random selection rules, 19–21 part orders, 15, 16 scheduling results with RDM selection rule, 16–19 size of problems, 15 test problems, 15 meta-heuristic procedures cost benefit based selection, 14 multiple machine, feasible production job, 10–11 profit-time based selection, 12–14 single machine, feasible production job, 8–10 stochastic selection, 12 online 3D printing service, 2, 4 “p-batch” scheduling problem, 3 problem of real-time OAS (see Order acceptance and scheduling (OAS)) production job, 1 production scheduling problem, 3 SLM and EBM, 1 Micrositing, 117, 136 Minitab program, 146 Mixed integer linear programming (MILP), 28 Modelling FDM, 3, 188, 192, 193 SEIG, 141–142 Multiple linear regression (MLR) anemia and blood variables, 74, 75 building multiple regression models, 78–80 coefficient of the determination, 76 diagnosis of anemia, 80–82 hemoglobin thresholds, anemia, 73–74 mathematical model, 72–73 multiple analysis, 72 regression analysis, 74, 76 RMSE, 78 Multiple MAM machines, 2, 3, 10–11 Multiple traveling repairmen problem with distance constraints (MTRPD), 159 Multiple traveling salesman problem (MTSP), 159

N Nationwide energy policy, 44–47 Nationwide energy storage system AC-OPF problem, 31

200 Nationwide energy storage system (cont.) bi-level optimization methodology, 37–38 daily energy demand, 35 electricity, 34–35 grid reliability and economics, 34 maximum power and energy capacity, 35–36 optimal placement and sizing, 36–37 optimization models, 35 Natural exhaustible resources difference inequation system (DI), 58 intertemporal utility, 58 management problem of economy, 58–59 primitives of model, 57 Natural resources, 51 Navier–Stokes equations, 90 Non-domination sorting genetic algorithm II (NSGA-II), 117 Non-integer order system, 176

O Optimal control theory, 51 Optimal growth cost–benefit mechanism, 52 dynamical systems and control theory, 52 forest, 63–68 models, 52 natural exhaustible resources, 57–63 one-sector optimal growth, 53–57 Pontryagin’s principles approach, 52 Optimal management exhaustible natural resources, 51, 52, 59–63 forestry, 63–68 Optimal power flow (OPF) AC-OPF problem, 30–31 constraints, 33 DC-OPF problem, 32 Optimization ABC algorithm (see Artificial bee colony (ABC) algorithm) bio-geography, 116 CRO, 116–117 in energy economics AC-OPF model, 30–31 DC-OPF model, 32 ED model, 26–27 energy storage system modeling, 29–30 optimal placement and sizing, 33–34 UC model, 27–28 MAM machine, 3 nationwide energy storage system, 34–38 RSM (see Response surface methodology (RSM))

Index VRPSTW, 160 WFLO (see Wind farm layout optimization (WFLO)) wind farm, 116 Order acceptance and scheduling (OAS) objective function, 7 problem of OAS, 2–3 processing flow diagram, 8 real-time scheduling assumptions, 4 basic formulations, 6 constraints, 7 formulation and definition, 5–6 objective function, 7 production cost, 6 specifications, 4, 11

P Perturbation method, 90, 102, 105, 111 Pontryagin’s principle Hamiltonian function, 55, 56, 60, 61 harvesting and replanting, forestry, 66 natural exhaustible resources, 58 optimal growth problems, 52 Porosity, 90, 92, 94, 110, 112 Powder-bed based MAM machine, 1, 3 Power production, 121 Power systems AC-OPF problem, 30–31 DC-OPF problem, 32 in Turkey, 38–44 Profit-per-unit-time, 6, 7, 12–14, 17–19 Pump storage hydroelectricity (PHS), 25

Q Quadratic program (QP), 27

R Reactive power generation, 30 Real power generation, 30 Regression model, 74–79 Renewable energy, 25, 115 Response surface methodology (RSM) advantage, 145 description, 145 drawback, 145 FCC design, 145 Minitab program, 146 optimization process, 145 response of the model, 148

Index Reynolds number, 90, 107–109, 112 Root mean squared error (RMSE), 77, 81–83

S Self-excited induction generator (SEIG) excitation curve, 139–140 external reactive source, 139 per-phase equivalent circuit, 142 simplified equivalent circuit, 143, 144 Simulink model, 146, 147 steady-state analysis, 140 utilization, 139 voltage and delivers, 141 voltage build-up process, 142–143 Self-induction generator, see Self-excited induction generator (SEIG) Stable filtration process, 91, 102, 104 Static compensator (STATCOM), 140–141 Stochastic selection (RDM), 12, 15–19, 23 Superconducting magnetic energy storage (SMES), 25 Sustainability, 51, 52, 59, 65 Swarm intelligence (SI), 161 Switchable capacitor bank, 141 System configuration, 89

T Temperature, 113 Terminal voltage, 144–145 Thermal storage, 25 3D printing additive manufacturing, 188–189 3DPSS method, 4 FDM, 188 IPMS Gyroid (see Infinite periodic minimal surface (IPMS)) MAM machine, 15 3D printing service scheduling (3DPSS), 4 Traveling salesman problem (TSP), 158 “Truck Dispatching Problem,” 157 Type 1 diabeties (T1D), 175, 183 Type 2 diabeties (T2D), 175

U Unit commitment (UC) constraints, 28 energy storage system modeling, 29–30 gas-fired power plants, 27

201 MILP, 28 ramping capability, gas-fired plants, 27 Upwind turbines, 115

V Vehicle routing problem (VRP) applications, 157 characteristics, 157–158 definition, 157 driver fatigue, 158 DVRP, 158, 159, 164, 170 logistics distribution practice, 158 real-life characteristics and assumptions, 171 Velocity deficit, 118 Viscous laminar flow conservation of energy, 99 conservation of momentum, 99 equations and boundary conditions, 100–102 MHD fluid, 90 two-dimensional flow, 92 Voltage magnitude, 30, 31, 140, 143, 152 VRP with soft time windows (VRPSTW), 160

W Wake effect, 118–121, 135, 136 Wall dilation, 90, 107–108, 112 Wind farm layout optimization (WFLO) DEM, 133 differential evolution, 116 genetic algorithm (GA) crossover, 129–130 description, 126 genetic mutation, 130 operators, 126 parameters, 130–132 population formation, 126–128 selection, 128–129 genetic mutation, 130 geodesic approach, 117 harmony searched, WFLO problem, 117 iterative non-deterministic algorithms, 117 wake effects, 135 Wind speed multipliers, 124 Wind turbines (WTs) coordinates, 126 DEM, 122 features, 124 identical and rotor sized turbines, 116 maximization, 117

202 Wind turbines (WTs) (cont.) placement model, 135 power production, 121 3D Cartesian coordinates, 128 three-dimensional surface model, 124 2D contour map, 124, 125 wake effect, 118–121

Index wind speed multipliers, 124 wind speeds, 123 yaw alignments, 126, 127 Y Yaw alignments, WTs, 126, 127