Metaheuristics for Resource Deployment under Uncertainty in Complex Systems [1 ed.] 1032065206, 9781032065205

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Acknowledgments
Author Bios
CHAPTER 1: Introduction
1.1. APPLICATIONS OF NODE DEPLOYMENT PROBLEM
1.1.1. Unmanned Systems
1.1.2. Wireless Sensor Networks
1.1.3. Healthcare
1.1.4. Public Sectors
1.1.5. Railway Network Design
1.1.6. Distributed Simulation Systems
1.2. FUNDAMENTAL ISSUES OF NODE DEPLOYMENT PROBLEM
1.2.1. Task
1.2.2. Node
1.2.3. Environment
1.3. RESEARCH PROGRESS OF NODE DEPLOYMENT MODELING
1.3.1. Deployment Space
1.3.1.1. Candidate Locations
1.3.1.2. Deployment Formation
1.3.2. Constraints
1.3.3. Objective Functions
1.3.3.1. Node Deployment in Wireless Sensor Networks
1.3.3.2. Node Deployment in Air Defense
1.3.3.3. Other Types of Optimization Objective
1.4. RESEARCH PROGRESS OF NODE DEPLOYMENT METHODS
1.4.1. Encoding
1.4.2. Constraints Handling
1.4.3. Multi-Objective Handling
1.4.4. Algorithms
1.4.4.1. Exact Algorithm
1.4.4.2. Metaheuristic Algorithm
1.5. MAIN ISSUES AND CHALLENGES
1.6. BOOK OUTLINE
CHAPTER 2: Stochastic Node Deployment for Area Coverage Problem
2.1. INTRODUCTION
2.2. PROBLEM FORMULATION
2.2.1. Detection Models
2.2.1.1. Binary Detection Model
2.2.1.2. Probabilistic Detection Model
2.2.2. Network Model
2.2.3. Problem Statement
2.2.4. NP-Hardness Proof
2.3. SOLUTION ALGORITHMS
2.3.1. D-VFCPSO
2.3.2. Other PSO-Based Algorithm for Area Coverage Problem
2.3.3. Complexity Analysis
2.4. EXPERIMENTS AND DISCUSSION
2.4.1. Test Instances
2.4.2. Parameter Setting
2.4.3. Analysis of Results
2.5. CONCLUSION
CHAPTER 3: Stochastic Dynamic Node Deployment for Target Coverage Problem
3.1. INTRODUCTION
3.2. PROBLEM FORMULATION
3.2.1. Mathematical Model
3.2.2. Scenario-Based Model Reformulation
3.3. SOLUTION ALGORITHMS
3.3.1. NSGA-II
3.3.2. MOPSO
3.3.2.1. Personal Best Selection
3.3.2.2. Non-Dominated Solutions Maintaining and Global Best Selection
3.3.2.3. Diversity Maintaining
3.3.3. Complexity Analysis
3.4. EXPERIMENTS AND DISCUSSION
3.4.1. Test Instances
3.4.2. Performance Metrics
3.4.3. Parameter Turning
3.4.4. Analysis of Results
3.5. CONCLUSION
CHAPTER 4: Robust Node Deployment for Cooperative Coverage Problem
4.1. INTRODUCTION
4.2. PROBLEM FORMULATION
4.2.1. The Deterministic and Uncertain Two-Level Cooperative Set Covering Problem
4.2.1.1. Two-Level Cooperative Set Covering Problem
4.2.1.2. Generalized Uncertain Two-Level Cooperative Set Covering Problem
4.2.2. Modeling the Robust Uncertain Two-Level Cooperative Set Covering Problem
4.2.2.1. Compact Formulation of the RUTLCSCP
4.3. SOLUTION ALGORITHMS
4.3.1. Dealing with Subproblem
4.3.2. Rule-Based Heuristic for RUTLCSCP
4.3.2.1. Processing Procedure
4.3.2.2. Complexity Analysis of MRBCH-k
4.3.3. Proposed SaDE for RUTLCSCP
4.3.3.1. Encoding
4.3.3.2. Constraints Handling
4.3.3.3. Complexity Analysis of SaDE
4.4. EXPERIMENTS AND DISCUSSION
4.4.1. Test Instances
4.4.2. Analysis of Results
4.4.2.1. Solving RUTLCSCP-LA-RC through CPLEX
4.4.2.2. Comparisons of MRBCH-k with Different k
4.4.2.3. Comparisons of SaDE and Its Variants
4.4.2.4. Comparisons on RUTLCSCP
4.5. CONCLUSION
CHAPTER 5: Fuzzy Node Deployment for Cooperative Coverage Problem
5.1. INTRODUCTION
5.2. PROBLEM FORMULATION
5.2.1. Fuzzy Conditional Value-at-Risk
5.2.2. Mathematical Model
5.2.3. Some Properties on CVaR-FTLCNDP
5.2.4. Linear Approximation of CVaR-FTLCNDP
5.3. SOLUTION ALGORITHMS
5.3.1. Fuzzy Simulation
5.3.2. Improved Decomposition-Based Multi-Objective Evolutionary Algorithms
5.3.2.1. Encoding
5.3.2.2. Updating of Individuals
5.3.2.3. Complexity Analysis
5.4. EXPERIMENTS AND DISCUSSION
5.4.1. Performance Metrics
5.4.2. Analysis of Results
5.4.2.1. Case Study 1
5.4.2.2. Case Study 2
5.5. CONCLUSION
CHAPTER 6: Simulation-Based Evaluation Analysis of Node Deployment under Risk Preference
6.1. INTRODUCTION
6.2. SIMULATION-BASED EVALUATION ANALYSIS OF WORST-CASE CVAR NODE DEPLOYMENT
6.2.1. Uncertain Initial Position of Penetration Paths
6.2.2. Penetration Paths under Uncertainty
6.2.3. Scenario-Based Simulation
6.2.4. Evaluation Model with Decision Makers’ Risk Preference
6.3. EXPERIMENTS AND DISCUSSION
6.3.1. Case Study 1: Deployment of Sensor Nodes
6.3.2. Case Study 2: Deployment of Weapon Nodes
6.3.3. Case Study 3: Cooperative Deployment of Sensor and Weapon Nodes
6.4. CONCLUSION
CHAPTER 7: Overview and Future Directions
Bibliography
Index

Citation preview

Metaheuristics for Resource Deployment under Uncertainty in Complex Systems

Metaheuristics for Resource Deployment under Uncertainty in Complex Systems

Shuxin Ding Chen Chen Qi Zhang Bin Xin Panos M. Pardalos

First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Shuxin Ding, Chen Chen, Qi Zhang, Bin Xin, Panos M. Pardalos CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf. co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Ding, Shuxin, 1991- author. | Chen, Chen, 1982- author. | Zhang, Qi, 1968- author. | Xin, Bin, 1982- author. | Pardalos, Panos M., 1954author. Title: Metaheuristics for resource deployment under uncertainty in complex systems / Shuxin Ding, Chen Chen, Qi Zhang, Bin Xin, Panos M. Pardalos. Description: First edition. | Boca Raton, FL : CRC Press, 2022. | Includes bibliographical references and index. | Summary: “This book analyzes how to set locations for the deployment of resources to incur the best performance at the lowest cost. Resources can be static nodes and moving nodes while services for a specific area or for customers can be provided. Theories of modeling and solution techniques are used with uncertainty taken into account and real-world applications used”-Provided by publisher. Identifiers: LCCN 2021010674 (print) | LCCN 2021010675 (ebook) | ISBN 9781032065205 (hardcover) | ISBN 9781032065243 (pbk.) | ISBN 9781003202653 (ebook) Subjects: LCSH: Business logistics. | Metaheuristics. Classification: LCC HD38.5 .D548 2022 (print) | LCC HD38.5 (ebook) | DDC 658.5/1015196--dc23 LC record available at https://lccn.loc.gov/2021010674 LC ebook record available at https://lccn.loc.gov/2021010675 ISBN: 978-1-032-06520-5 (hbk) ISBN: 978-1-032-06524-3 (pbk) ISBN: 978-1-003-20265-3 (ebk) DOI: 10.1201/9781003202653 Typeset in LatinModern by KnowledgeWorks Global Ltd.

Contents Preface

xi

Acknowledgments

xv

Author Bios Chapter 1.1

1.2

1.3

xvii

1  Introduction

1

APPLICATIONS OF NODE DEPLOYMENT PROBLEM

1

1.1.1

Unmanned Systems

1

1.1.2

Wireless Sensor Networks

3

1.1.3

Healthcare

4

1.1.4

Public Sectors

6

1.1.5

Railway Network Design

6

1.1.6

Distributed Simulation Systems

8

FUNDAMENTAL ISSUES OF NODE DEPLOYMENT PROBLEM

9

1.2.1

Task

10

1.2.2

Node

11

1.2.3

Environment

11

RESEARCH PROGRESS OF NODE DEPLOYMENT MODELING

12

1.3.1

Deployment Space

12

1.3.1.1

Candidate Locations

12

1.3.1.2

Deployment Formation

14

1.3.2

Constraints

15

1.3.3

Objective Functions

16 v

vi  Contents

1.4

1.3.3.1

Node Deployment in Wireless Sensor Networks

16

1.3.3.2

Node Deployment in Air Defense

17

1.3.3.3

Other Types of Optimization Objective 20

RESEARCH PROGRESS OF NODE DEPLOYMENT METHODS

20

1.4.1

Encoding

21

1.4.2

Constraints Handling

21

1.4.3

Multi-Objective Handling

21

1.4.4

Algorithms

22

1.4.4.1

Exact Algorithm

22

1.4.4.2

Metaheuristic Algorithm

23

1.5

MAIN ISSUES AND CHALLENGES

25

1.6

BOOK OUTLINE

27

Chapter

2  Stochastic Node Deployment for Area Coverage Problem 29

2.1

INTRODUCTION

29

2.2

PROBLEM FORMULATION

31

2.2.1

Detection Models

31

2.2.1.1

Binary Detection Model

31

2.2.1.2

Probabilistic Detection Model

32

2.3

2.4

2.2.2

Network Model

32

2.2.3

Problem Statement

33

2.2.4

NP-Hardness Proof

34

SOLUTION ALGORITHMS

35

2.3.1

D-VFCPSO

35

2.3.2

Other PSO-Based Algorithm for Area Coverage Problem

38

2.3.3

Complexity Analysis

39

EXPERIMENTS AND DISCUSSION

39

2.4.1

40

Test Instances

Contents  vii

2.5

Chapter

2.4.2

Parameter Setting

40

2.4.3

Analysis of Results

41

CONCLUSION

41

3  Stochastic Dynamic Node Deployment for Target Coverage Problem 45

3.1

INTRODUCTION

46

3.2

PROBLEM FORMULATION

47

3.2.1

Mathematical Model

49

3.2.2

Scenario-Based Model Reformulation

49

3.3

SOLUTION ALGORITHMS

50

3.3.1

NSGA-II

50

3.3.2

MOPSO

52

3.3.3 3.4

3.5

Chapter

3.3.2.1

Personal Best Selection

52

3.3.2.2

Non-Dominated Solutions Maintaining and Global Best Selection

53

3.3.2.3

Diversity Maintaining

53

Complexity Analysis

55

EXPERIMENTS AND DISCUSSION

56

3.4.1

Test Instances

56

3.4.2

Performance Metrics

57

3.4.3

Parameter Turning

59

3.4.4

Analysis of Results

60

CONCLUSION

4  Robust Node Deployment for Cooperative Coverage Problem

74

75

4.1

INTRODUCTION

76

4.2

PROBLEM FORMULATION

78

4.2.1

The Deterministic and Uncertain Two-Level Cooperative Set Covering Problem

78

viii  Contents

4.2.2

4.2.1.1

Two-Level Cooperative Set Covering Problem

78

4.2.1.2

Generalized Uncertain Two-Level Cooperative Set Covering Problem

79

Modeling the Robust Uncertain Two-Level Cooperative Set Covering Problem 4.2.2.1

4.3

4.5

Chapter

87

SOLUTION ALGORITHMS

88

4.3.1

Dealing with Subproblem

89

4.3.2

Rule-Based Heuristic for RUTLCSCP

92

4.3.2.1

Processing Procedure

94

4.3.2.2

Complexity Analysis of MRBCH-k

95

4.3.3

4.4

Compact Formulation of the RUTLCSCP

84

Proposed SaDE for RUTLCSCP

96

4.3.3.1

Encoding

97

4.3.3.2

Constraints Handling

97

4.3.3.3

Complexity Analysis of SaDE

100

EXPERIMENTS AND DISCUSSION

101

4.4.1

Test Instances

102

4.4.2

Analysis of Results

102

4.4.2.1

Solving RUTLCSCP-LA-RC through CPLEX

102

4.4.2.2

Comparisons of MRBCH-k with Different k

105

4.4.2.3

Comparisons of SaDE and Its Variants 109

4.4.2.4

Comparisons on RUTLCSCP

CONCLUSION

5  Fuzzy Node Deployment for Cooperative Coverage Problem

109 113

115

5.1

INTRODUCTION

116

5.2

PROBLEM FORMULATION

117

5.2.1

118

Fuzzy Conditional Value-at-Risk

Contents  ix

5.3

5.4

5.5

Chapter

5.2.2

Mathematical Model

118

5.2.3

Some Properties on CVaR-FTLCNDP

121

5.2.4

Linear Approximation of CVaR-FTLCNDP

122

SOLUTION ALGORITHMS

123

5.3.1

Fuzzy Simulation

124

5.3.2

Improved Decomposition-Based Multi-Objective Evolutionary Algorithms 126 5.3.2.1

Encoding

126

5.3.2.2

Updating of Individuals

127

5.3.2.3

Complexity Analysis

130

EXPERIMENTS AND DISCUSSION

130

5.4.1

Performance Metrics

131

5.4.2

Analysis of Results

131

5.4.2.1

Case Study 1

131

5.4.2.2

Case Study 2

134

CONCLUSION

136

6  Simulation-Based Evaluation Analysis of Node Deployment under Risk Preference 139

6.1

INTRODUCTION

139

6.2

SIMULATION-BASED EVALUATION ANALYSIS OF WORST-CASE CVAR NODE DEPLOYMENT

141

6.2.1

Uncertain Initial Position of Penetration Paths

142

6.2.2

Penetration Paths under Uncertainty

143

6.2.3

Scenario-Based Simulation

144

6.2.4

Evaluation Model with Decision Makers’ Risk Preference

148

6.3

EXPERIMENTS AND DISCUSSION

148

6.3.1

Case Study 1: Deployment of Sensor Nodes

148

6.3.2

Case Study 2: Deployment of Weapon Nodes

153

6.3.3

Case Study 3: Cooperative Deployment of Sensor and Weapon Nodes

155

x  Contents

6.4

Chapter

CONCLUSION

7  Overview and Future Directions

164

167

Bibliography

171

Index

189

Preface Effective use of the resources at the best location is becoming more important, bringing better services and lower costs. Optimizing the resource deployment problem is a critical issue in complex systems, which involves how to substantially apply the limited resources to maximize the systems’ ability. These resources can be unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) in unmanned systems which provide continuous surveillance, reconnaissance, command, control, etc. They can also be static nodes, e.g., supermarkets, surveillance cameras, etc. The resources are denoted as nodes in this book. In realworld applications, the data obtained is uncertain. These data may be stochastic, time-varying, unknown distribution, or fuzzy. These factors should be considered when analyzing the resource deployment problem. This book mainly focuses on the modeling and methods for solving the resource deployment problem under uncertainty. The models are related to stochastic programming, robust optimization, fuzzy programming, risk management, and single/multi-objective optimization. Heuristic and metaheuristic algorithms will be applied to solve the proposed problems. The resources are heterogeneous, which can be sensors and actuators providing different tasks, e.g., sensing and attacking. As a result, both separate coverage and cooperative coverage of the resources are analyzed. The content of this book can be divided into six chapters and summarized as follows: Chapter 1: The applications, fundamental issues, literature review of the modeling and methods, main issues, and challenges of the book are presented. Chapter 2: The stochastic node deployment problem for area coverage is considered. The detection probabilities of the sensors are under uncertainty with known distribution. A perturbation mechanism is used in the particle’s velocity updating equation, which prevents the particle from local convergence. Comparative experiments show the advantage of disturbance-virtual force co-evolutionary particle swarm optimization (d-VFCPSO) in finding a good sensor deployment scheme. xi

xii  Preface

Chapter 3: The stochastic dynamic node deployment problem for target coverage is investigated. The positions, the threat of the target, and the kill probabilities of the nodes are under uncertainty with known distribution. It is a bi-objective problem, which minimizes the threat of the targets and redeployment distance. Elitist learning strategy (ELS) is used for mutation of the global best particles to keep diversity. The Taguchi method with a novel response value is utilized to tune the parameters of the proposed algorithms. Comparative experiments with non-dominated sorting genetic algorithm with an elitist strategy (NSGAII) and other multi-objective particle swarm optimization (MOPSO) algorithm show the effectiveness of the ELS. Chapter 4: The robust node deployment problem for cooperative coverage is discussed. The covering probabilities of the nodes are under uncertainty with unknown distribution but a known range. Deployment cost is used as the objective, while cooperative covering probability is used as a constraint. It is assumed that there are two types of facilities to be located for covering the demand node cooperatively. We proposed a novel Two-Level Cooperative Set Covering Problem and its related uncertain problems. A linear approximation method is proposed to deal with the robust nonlinear constraints. The problem is solved by CPLEX to obtain the linear approximation solution under relaxed constraints. However, these solutions may violate the nonlinear constraints. Therefore, a rule-based method is proposed to deal with the subproblem from the robust nonlinear constraints. Besides, two marginal return values are constructed by exploiting the objective and nonlinear constraints. Then, a marginal-return-based constructive heuristic (MRBCH) method is developed by exploiting the two marginal return values. Meanwhile, an improved self-adaptive differential evolution (SaDE) algorithm is also developed with four constraint-handling methods. Finally, a hybrid algorithm is obtained by combining the constructive heuristic method MRBCH-1 with ordered repair operator SaDE-OR. Chapter 5: We investigate the fuzzy node deployment problem for cooperative coverage. To deal with the fuzzy uncertainty in the target threat, we first proposed a method to calculate fuzzy conditional valueat-risk (CVaR). What is more, we develop a fuzzy two-level cooperative node deployment problem (CVaR-FTLCNDP), minimizing fuzzy CVaR of target threat and deployment cost. Two decomposition-based multiobjective evolutionary algorithms, MOEA/D and DMOEA-εC, are used. Since the number of nodes is unknown a priori, the individual size varies during the evolutionary process. Therefore, a variable individual size

Preface  xiii

(VIS) encoding method is adopted, and MOEA/D-VIS and DMOEAεC-VIS are obtained. Experiments show the effectiveness of these two algorithms, and MOEA/D-VIS shows better performance in diversity and convergence. As a result, effective deployment schemes are obtained at different costs. Chapter 6: We study the simulation-based analysis of the node deployment problem. The decision maker’s risk preference, the uncertainty of the starting position, and the target’s penetration route are considered. To deal with the starting position’s uncertainty, a systematic sampling method is adopted to obtain the scenarios of different starting positions. As for the uncertainty of the target’s penetration route, the Dijkstra algorithm is used for the target optimal penetration route planning to avoid detection by sensors or being shot by weapons. Worst-case CVaR is used to describe the risk of the deployment scheme, and its confidence level is used to describe the decision maker’s risk preference. Experiments show that the proposed evaluation method has successfully demonstrated the incoming target’s uncertainty and evaluated deployment schemes under different risk preferences. Chapter 7: The overview and future directions are presented. The models and approaches proposed in this book can be applied in complex systems, e.g., unmanned systems, sensor networks, etc. Besides, they can also be used in some facility location-related problems, e.g., railway network design, hospital layout optimization, locations problems in emergency medical services (EMS), etc. Shuxin Ding Chen Chen Qi Zhang Bin Xin Panos M. Pardalos

Acknowledgments We wish to thank all the researchers of the School of Automation, Beijing Institute of Technology, the State Key Laboratory of Intelligent Control and Decision of Complex Systems, and the Signal and Communication Research Institute, China Academy of Railway Sciences Corporation Limited for reading and commenting on draft versions of the book. We wish to acknowledge the support of the National Natural Science Foundation of China under Grants U1834211, U1934220, 62022015, 62088101, 61773066, 61822304, 61673058, and 61790575 and the National Key R&D Program of China (2018YFB1308000). Panos M. Pardalos was supported by a Humboldt Research Award (Germany). Most of all, Shuxin Ding would like to express his greatest gratitude to his parents for their support, without which this book would not have been finished. We would appreciate any comments, questions, criticisms, or corrections of this book. Readers may kindly provide to Dr. Shuxin Ding at [email protected].

xv

Author Bios Shuxin Ding received the B.E. degree in automation and the Ph.D. degree in control science and engineering from the Beijing Institute of Technology, Beijing, China, in 2012 and 2019, respectively. He is currently an assistant researcher with the Signal and Communication Research Institute, China Academy of Railway Sciences Corporation Limited. His current research interests include railway scheduling, evolutionary computation, multi-objective optimization, and optimization under uncertainty. Chen Chen received the B.S. degree in automation and the Ph.D. degree in control science and engineering from the Beijing Institute of Technology, Beijing, China, in 2004 and 2009, respectively. She is currently a professor with the School of Automation, Beijing Institute of Technology. Her current research interests include complicated systems, multi-objective optimization, and distributed simulation. Qi Zhang received the B.S. degree in electronic engineering from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1991, the M.S degrees in computer applications, and the Ph.D. degree in traffic information engineering and control from China Academy of Railway Sciences, Beijing, in 1993 and 1998, respectively. He is currently a chief researcher of China Academy of Railway Sciences Corporation Limited and a leader in railway technical expertise. His research interests include railway signal and communication, automatic train operation, train operation control, intelligent dispatching, and cooperative control of multiple trains. Dr. Zhang was a recipient of the Youth Award, the Contributional Award, and the Achievement Award of Zhan Tianyou Railway Science and Technology. Bin Xin received the B.S. degree in Information Engineering and Ph.D. degree in Control Science and engineering, both from the Beijing Institute of Technology, Beijing, China, in 2004 and 2012, respectively. He xvii

xviii  Author Bios

is currently a professor with the School of Automation, Beijing Institute of Technology. His current research interests include search and optimization, evolutionary computation, unmanned systems, and multiagent systems.

Panos M. Pardalos received the B.S. degree in mathematics from Athens University, Athens, Greece, in 1977, the M.S. degree in mathematics and computer science from Clarkson University, Potsdam, NY, USA, in 1978, and the Ph.D. degree in computer and information sciences from the University of Minnesota, Minneapolis, MN, USA, in 1985. Dr. Pardalos is a Distinguished Professor in the Department of Industrial and Systems Engineering at the University of Florida, Florida, USA, and an affiliated faculty of the Biomedical Engineering and Computer Science & Information & Engineering departments. In addition, he is the director of the Center for Applied Optimization. Dr. Pardalos is a worldrenowned leader in Global Optimization, Mathematical Modeling, Energy Systems, and Data Sciences. He is a Fellow of AAAS, AIMBE, and INFORMS and was awarded the 2013 Constantin Caratheodory Prize of the International Society of Global Optimization. In addition, Dr. Pardalos has been awarded the 2013 EURO Gold Medal prize bestowed by the Association for European Operational Research Societies. This medal is the preeminent European award given to Operations Research (OR) professionals for “scientific contributions that stand the test of time.” Dr. Pardalos has been awarded a prestigious Humboldt Research Award (2018–2019). The Humboldt Research Award is granted in recognition of a researcher’s entire achievements to date—fundamental discoveries, new theories, and insights that have had significant impact on their discipline. Dr. Pardalos is also a Member of several Academies of Sciences, and he holds several honorary Ph.D. degrees and affiliations. He is the Founding Editor of Optimization Letters and Energy Systems, and CoFounder of the International Journal of Global Optimization, Computational Management Science, and Springer Nature Operations Research Forum. He has published over 500 journal papers and edited/authored over 200 books. He is one of the most cited authors and has graduated 65 Ph.D. students so far.

CHAPTER

1

Introduction

Resource deployment problem analyzes how to set the locations for deploying resources with the best performance and lowest cost. The resources can be static nodes and moving nodes. These resources can provide services for a specific area or some customers. This chapter introduces the resource deployment problem. It provides some real-world applications and fundamental issues of node deployment problem, and research progress of node deployment modeling and methods. This chapter provides a basic foundation for the whole work and gives the main issue and challenges that lead to the following chapters.

1.1 APPLICATIONS OF NODE DEPLOYMENT PROBLEM In this section, we outline a number of different applications related to the node deployment problem. They can be complex systems, e.g., unmanned systems, sensor networks, etc. Besides, there are many applications in facility location problems, e.g., healthcare, public sectors, railway network design, etc. There also exists resource deployment problems in distributed simulation systems. 1.1.1 Unmanned Systems

Unmanned systems are man-made and can be operated or managed through advanced technologies. They are complex systems created by the fusion of various technologies related to mechanics, control, computer, communication, and materials [1]. They can be controlled by humans or perform tasks autonomously. Various types of unmanned systems are emerging include unmanned aerial (UAV), ground (UGV), and underwater (UUV) vehicles, etc. They may be applied in applications from DOI: 10.1201/9781003202653-1

1

2  Metaheuristics for Resource Deployment

:DWHU

Figure 1.1

UAVs, UGVs, and UUVs in an unmanned system.

the civil domain to the military domain, such as logistics, surveillance, building and environment monitoring, search and rescue, intruder detection and attacking, etc. [2]. Thus, one objective might be to minimize the number of vehicles for completing the tasks, and the other objective might be to maximize the payoff of the tasks [3]. Figure 1.1 shows the UAVs, UGVs, and UUVs in an unmanned system. A brief introduction and example applications of unmanned systems are presented as follows. • Video surveillance. Camera-mounted UAVs can provide coverage of multiple-oriented targets. The positions of these UAVs are decided by the ground control station with a master camera. Except for surveillance [4] and crowd monitoring [5], they can be used for infrastructure inspections [6], cinematography [7], etc. • Networks. The UAVs can help formulate coverage when there are disturbances and disruptions in the cellular networks caused by concerts, natural disasters, etc. [8]. Problems such as minimizing the number of UAVs required for continuous coverage, maximizing the area coverage, and preserving network connectivity require an optimized deployment strategy [2]. The vehicles may also perform different functional roles. Four main functional roles are defined as sensors, actuators, decision makers (DMs), and auxiliary facilities [9, 10]. For example, in UAV-UGV coordination systems, the characteristics of UAVs and UGVs are strongly complementary. The combination of different functional roles makes cooperative

Introduction  3

UAV-UGV coordination systems promising. UGVs can act as actuators limited by their speed and environmental occlusion, while UAVs can act as sensors quickly deployed for finding targets. UAVs can also help in formulating communication links for UGVs, which may be blocked by obstacles. Besides, small-scale UAVs are restricted by their short voyage due to the energy limitation, while UGVs can act as carriers providing auxiliary facilities. This also suggests that we would like to make better use of the systems by optimizing the deployment of the vehicles. 1.1.2 Wireless Sensor Networks

Wireless sensor networks (WSNs) are formed by small, inexpensive, lowpowered sensors. WSNs have recently become a popular research area for many applications in military, environmental, industrial, home, medical, etc. [11]. The objective is to monitor the environment and communicate information with each sensor. Figure 1.2 shows a target detection scenario in the three-dimensional (3D) space of WSNs. Decision makers need to decide the numbers and locations for the WSNs. Some performance metrics to be optimized in WSNs are introduced in Section 1.3.3.1. A brief introduction and example applications of wireless sensor networks are presented as follows. • Military applications. In command, control, communications, computers, intelligence, surveillance, and reconnaissance (C4ISR) systems, WSNs can be rapidly scattered in critical terrains, routes to provide battlefield intelligence. They can be used to detect and track enemy targets. • Environmental applications. Environmental problem is a critical issue for human on the earth. WSNs can be used for wildlife monitoring, fire detection, measuring CO2 level, flood detection, air pollution detection, etc. [12]. • Industrial applications. WSNs can be used in manufacturing process management, monitoring the gas, water, and electric, lighting control, etc. They can also be used for monitoring the structural health of buildings, bridges, roads, physical condition of water and gas pipes, smart railway stations, etc. • Home applications. WSNs in home environments connect everyday objects and devices at home through networks and create an

4  Metaheuristics for Resource Deployment

Figure 1.2

Wireless sensor networks in 3D space for target detection.

Internet of Things (IoT)-based smart environment [13]. It is an environment that learns from our daily activity. People can conduct remote control of the home devices through IoT-based smart home systems [14]. • Medical applications. The Body Area Sensor Network (BASN) consists of multiple interconnected nodes for sensing, data processing, and wireless communication. These sensor nodes are placed on, near, or within the human body. The BASN sensor nodes constantly monitor and analyze different physiological signals, e.g., the electrical activities of the heart, muscles, and the brain; body temperature, blood glucose, blood pressure, blood oxygen saturation, etc. [15].

1.1.3 Healthcare

Healthcare is defined as the prevention and treatment for illness or injury through professional medical services. The facility location problems related to healthcare covers from locating healthcare facilities to layout problems in hospitals [16]. Figure 1.3 shows the locations of historical cardiac arrests and candidate sites for AED deployment. A brief introduction and example applications of healthcare are presented as follows.

Introduction  5

+LVWRULFDOFDUGLDFDUUHVWV &DQGLGDWHVLWHV

Figure 1.3

Locations of historical cardiac arrests and candidate sites for AED deployment. • Healthcare facility location. The facility location problems for healthcare are mainly related to healthcare facilities, e.g., community health clinics, public and private hospitals, etc. The optimization criteria for healthcare facilities are minimizing access cost for healthcare consumers, maximizing population with access, etc. • AED location. Optimizing the deployment of public automated external defibrillators (AEDs) can help to increase the probability of survival when sudden cardiac arrest occurs [17, 18]. • Ambulance location. Ambulance location belongs to the emergency vehicles sitting problems. The goal is to find the locations for the ambulances (or ambulance bases) with a minimal number and provide a certain level of service. Meanwhile, relocation decisions for ambulances should be periodically made to avoid areas unprotected [19]. • Hospital layout planning. The layout planning problems for hospitals aim at minimizing in-house travel distances or costs inside the building. It is classified as a resource capacity planning problem, which directly influences the quality and efficiency of healthcare, and patient satisfaction.

6  Metaheuristics for Resource Deployment

1.1.4 Public Sectors

For public sectors, the main feature is the optimization criteria of the decision makers compared with private sectors. Public sectors like government organizations aim at maximizing the provided services to people with given resources, while private sectors aim at maximizing their profit as well as minimizing their cost [20]. Possible applications include the location of schools, bike sharing systems, hospitals, and bus stops [21]. Figure 1.4 shows the locations of sharing bakes in a bike sharing system. A brief introduction and example applications of public sectors are presented as follows. • Locating schools. Schools are located to meet the students’ demand. Some optimization criteria include student travel distance, fixed and overhead school costs, the average number of students enrolled per school, etc. • Bike sharing systems. Bike sharing systems are designed to increase the use of bicycles, to decrease congestion, and to provide a service to people available to other means of transportation [22]. It consists of bike stations with several bike slots. A bike station should have empty slots for arrivals and full slots for departures. The bike sharing systems company would like to maximize the total profit related to the revenue from bike renting, cost of moving empty bikes, cost of maintenance, etc. The number and location of bicycle stations and the capacity of each station, and the number of bikes in the system should be optimized. • Locating electric vehicle charging stations. Electric vehicles (EVs) are eco-friendly transport modes to reduce carbon emissions. The need for EV charging or refueling stations increases with the developments in the EV industry. Some optimization criteria are maximizing the coverage, maximizing the traffic flow, and minimizing the costs when optimizing the number, location of stations, and the scale of each station.

1.1.5 Railway Network Design

Decisions involved in railway planning belong to different stages: strategic, tactical, operational, and real-time. The first step in the railway planning process is the network design followed by line planning, timetabling,

Introduction  7

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rolling-stock circulation and maintenance, crew scheduling, and real-time management [23]. These problems can be analyzed separately or together with two or more by iterative/integrated approach. Figure 1.5 shows a railway network with lines and high-speed railway stations. A brief introduction and example applications of the railway are presented as follows. • Railway rapid transit network design. This problem consists of the selection of nodes and links to formulate a potential or underlying railway network with stations and links between them. Some optimization criteria include maximizing the expected trip coverage and the total population covered of the planned network. The existing network may also be extended by adding new lines or extending existing ones. The robustness of the network is also an important factor in case of a disturbance or disruption. Complex network theory may be applied in the assessment of transportation networks [24]. • Location of Stations. Locating stations on a network from scratch is influenced by the attraction to large volume of passengers. There are also some problems dealing with locating new stations on existing railway lines. Besides, when extreme weather events cause damage to the trains, proper locations of the railway emergency rescue stations may help in saving rescue time [25]. • Line planning. This problem is the next task in the railway planning process after the network design. The origin, itinerary, and

8  Metaheuristics for Resource Deployment

   









 



























 















  

  



  



 

 



  



 



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An illustration of a high-speed railway network.

destination of each line, with the frequency of the train, are to be decided with a given railway network. Cost related to the operating company and travel time related to the users are concerned for line planning [23].

1.1.6 Distributed Simulation Systems

With the advantages of high utilization ratio, high scalability, and parallelism, distributed systems can be run parallel to handle computationally intensive problems. Besides, several difficult problems are derived from the distributed structure, such as the communication problems among the nodes, the mapping and assignment problems of system resources and computing nodes. High level architecture (HLA) is widely used for distributed simulation development. Although HLA supports large-scale distributed simulation, the load balancing mechanism in the simulation process is not standardized. When the simulation system runs under non-optimized allocation, load imbalance problems will occur due to the shortage of computing resources, which will seriously affect the simulation efficiency and confidence. The main purpose of load balancing is to even out the computation load distribution among the nodes and minimize the communication

Introduction  9

Figure 1.6

Simulation process in a distributed simulation system.

load among them. The results of load balancing are shown in the following aspects: shortening the average response time of tasks effectively, maximizing the use of simulation resources of the whole system, and reducing the unnecessary waste of resources [26]. During the process of simulation, additional federates or components will be added to complete specific simulation tasks. As shown in Figure 1.6, when the simulation process is advanced to step k, a certain number of federates or components must be added. The key to solving this dynamic load balancing problem is transferring dynamic states to static states, converting the problems into static load balancing problems.

1.2 FUNDAMENTAL ISSUES OF NODE DEPLOYMENT PROBLEM With the development of computer technology, communication technology, network technology, and other emerging technologies, there has been a significant impact on optimizing resource deployment. Optimizing the resource deployment problem is a critical issue in complex systems, which involves applying the limited resource to maximize the systems’ ability. These resources can be UAVs and UGVs in unmanned systems,

10  Metaheuristics for Resource Deployment

which provide continuous surveillance, reconnaissance, command, control, etc. [9,10,27]. They can also be static nodes, e.g., surveillance cameras, supermarkets, etc. The resources are denoted as nodes in this book. Traditional decision-making methods for resource deployment are not suitable for the large-scale, dynamic, and uncertain characteristics of the system. The network connects the resources in an area as a united and efficient system, which shares the condition of the resources and the environment [28]. As a result, it decreases the total time consumed for decision-making and increases the command speed. Node deployment is affected and restricted by many factors in a complex environment. Traditional deployment methods mainly rely on experience; thus, the nodes’ characteristics and environmental conditions are not precisely described. As a result, the result is not ideal. The node deployment problem in this book mainly studies how to determine the position and number of nodes under the environment’s constraints, the total number of nodes, etc., to obtain the deployment’s optimal performance. Compared with the traditional experience deployment, it provides a reasonable and effective deployment scheme. Optimizing the resource deployment can be regarded as a resource-space matching problem. In the following, the key elements of node deployment problem are introduced, which are task, node, and environment. 1.2.1 Task

Optimizing the resource deployment problem is a critical issue in complex systems. Heterogeneous resources, e.g., sensors and actuators, can work cooperatively through the network, improving the system’s performance. It has been applied in the network-centric systems with sensors and weapon systems, which significantly improve the performance in air defense systems [29]. In unmanned systems, both UAVs and UGVs can act as sensor nodes or actuator nodes. Sensor nodes usually perform monitoring and surveillance tasks. Actuator nodes perform some control tasks, e.g., weapon nodes perform attacking towards the enemy targets. For example, by deploying UAVs and UGVs, surveillance tasks on a specific area and attacking tasks on the targets within the area are conducted. The sensor nodes’ environmental information and warning information can be transmitted to the actuator (weapon) nodes through the network. Under the guidance of sensor nodes, weapon nodes could perform effective attacking.

Introduction  11

Many research works and reviews focused on deploying sensor nodes, especially the wireless sensor networks (WSNs) [30–37]. Most of these studies analyze the maximum sensor coverage satisfying the connectivity and energy consumption. For weapon nodes, characteristics of the nodes and the tasks should be considered [38]. Since there are fewer works on actuator nodes, most of the studies will concentrate on deploying actuator (weapon) nodes in this section. Some research only focuses on deciding the number of nodes to be deployed in some regions but neglects the specific position of the nodes [39]. Therefore, this kind of problem can be regarded as an assignment problem of nodes, assigning different numbers of nodes for regions with different demands. Scholars have widely discussed assignment problems in recent decades. For example, generalized assignment problem (GAP) [40], resource assignment problem (RAP) [41], weapon target assignment (WTA) [42], etc. Other research focuses on deciding the nodes’ positions, which are more common and widely studied [43–47]. 1.2.2 Node

In real-world applications, there are usually different kinds of nodes deployed in a complex system. These nodes are deployed to complement each other’s advantages. However, these nodes are homogeneous resources, which are belong to the same type, i.e., sensor node or actuator node. Few studies consider the joint deployment of heterogeneous resources, e.g., sensor and actuator nodes. Therefore, the cooperation between heterogeneous resources and rational utilization of them needs further investigation. 1.2.3 Environment

The node deployment problem can also be divided into static deployment and dynamic deployment. Most of the previous studies analyze the static deployment problem. Nodes are deployed before any possible related tasks. It is an off-line node deployment problem without any time limit. However, when the states of the environment and nodes change during the service, the deployment’s performance may not meet the future requirements. Therefore, the nodes’ location should be adjusted by redeployment to meet the future requirement. Besides, most of the studies are deterministic, which means the parameters of the environment, resources, and targets are determined val-

12  Metaheuristics for Resource Deployment

ues. However, uncertainties always exist in real-world applications. For example, some nodes are not reliable after deployment [44], the targets may have uncertain route [48,49], etc. Therefore, it is necessary to model and effectively deal with uncertainty. As a result, previous research only deals with one resource type and considers static and deterministic problems. Therefore, cooperative coverage with heterogeneous resources and the resources’ uncertain and dynamic properties should be considered. It is of great significance to study this kind of problems both in theory and in practice.

1.3

RESEARCH PROGRESS OF NODE DEPLOYMENT MODELING

Node deployment problem brings a number of questions: 1. How many nodes should be deployed? 2. Where should the nodes be deployed? 3. How to select the type of the nodes? The answers to this question depend on how the model is formulated and the method is developed. To solve the node deployment problem, we need to analyze the problem by mathematical modeling first. Three factors are related to studying the model: deployment space, constraints, and objective functions. The research progress on modeling is stated as follows. 1.3.1 Deployment Space

Deployment space is the area for locating nodes with different amounts and positions. Figure 1.7 shows the classification of deployment space based on the candidate locations and deployment formation. 1.3.1.1

Candidate Locations

In order to obtain the candidate locations, the deployment space can be classified into two types: discrete space-based and continuous spacebased. In discrete space-based deployment, different discrete points in an area are regarded as candidate locations. A common way to obtain the discrete points is by dividing the area by grids. As a result, grid points are generated and regarded as both candidate locations and sampling points for evaluating the coverage. The grid size should be selected

Introduction  13

Figure 1.7

Classification of deployment space.

(a) Square-shaped grid. Figure 1.8

(b) Ring-shaped grid.

(c) Sector-shaped grid.

Methods for obtaining grid points.

appropriately based on the demand. More locations represent more calculation, but the result will be more precise. There are several methods to obtain grid points in the discrete space-based deployment. Most are divided by squares and sectors shown in Figure 1.8. Except for grid points, predefined positions can also be regarded as candidate locations. Ref. [50] considers squared grids for node deployment in a square area shown in Figure 1.8(a). The whole area is uniformly divided by grids and covered by nodes. Refs. [39,47] adopt sectors to divide a circle area shown in Figure 1.8(b). Ref. [51] considers a sector area for deployment shown in Figure 1.8(c). For the circle area and sector area, they are divided by sectors from the original point. The original point is also regarded as a protected object. Therefore, more gird points are close to the protected object for selecting locations. This non-uniform discretization

14  Metaheuristics for Resource Deployment

method can effectively reduce the unnecessary search space and make the search more efficient. Besides squared grids and sectors, other discretization methods such as the triangular grid and hexagonal grid are usually adopted in coverage problems in WSNs [36]. Since the discretization of the space will cause missing some deployment information inevitably, we should carefully select the size of the grid. While in the continuous space-based deployment, nodes can be deployed at any place in the predefined region, which ensures that the optimal solution must be in the search space. However, the searching cost of this method is higher than the discrete space-based deployment. In Refs. [43, 44], the virtual force algorithm is adopted to deal with the issues in the continuous space-based deployment. The total forces from other nodes, protected objects, and boundaries are used to adjust the locations of the nodes. It prevents the nodes from congesting in a local area and keeps the nodes in a certain density with avoidance of blind area. 1.3.1.2

Deployment Formation

Deployment formation refers to the form of the nodes in the deployment space. Generally, the formations of the deployed nodes can be divided into four types: ring deployment, sector deployment, line deployment, and hybrid deployment [52]. In ring deployment, the nodes are deployed around some key points: one ring, two rings, and multiple rings. So the targets from any direction can be covered by nodes. Its advantage is to strengthen the coverage for the main direction, but also has the omnidirectional capability. Therefore the reliability and the efficiency are high. However, the number of nodes required under the same conditions is greater. In Refs. [39, 47], grids are used for ring deployment. Although it is continuous space-based deployment in Refs. [43, 44], considering the targets are coming from all directions, the final deployment schemes are still in ring deployment formation. In sector deployment, only the main directions are considered. The deployed nodes will form like a sector. It is mainly used when the number of nodes is insufficient and terrain constraints cannot form a ring deployment. The advantage is to save the number of nodes, but the reliability is not high. If the target changes the direction, the deployment is easy to lose effectiveness. In line deployment, the nodes are deployed along with the key points

Introduction  15

as a line, with a large width of interception front, but a small depth. It is suitable for the situation where the target comes in many directions, and the front is wide. This deployment is always seen in the barrier coverage problem. These three types of deployment formations are relatively typical, but the actual deployment scenarios are complicated. Only one type of formation cannot deal with the actual complex environment. The ring deployment, sector deployment, and line deployment should be selected according to the actual terrain conditions. Two or three formations are combined to form a hybrid deployment formation, which is also called group deployment [53]. 1.3.2 Constraints

The resource deployment problem is a constrained optimization problem. The constraints limit the range of solutions and ensure that the planning results are reasonable and effective. In the real-world deployment problem, the feasible solution is limited by the deployment space, and accurate geographic information is analyzed to determine the deployable and non-deployable areas. For example, lakes, depressions, forests, or locations where the slope is too large or the terrain is too low are not suitable for deploying nodes. It is often difficult to mathematically describe non-deployable areas (especially irregular non-deployable areas). Such constraints will be dealt with in a specific manner during the optimization process. Except for the constraints in the deployment space, other features and demands of the nodes should be considered. The number of the nodes is the first consideration when deploying nodes. The number of available nodes in different areas is set as constraints. The nodes’ cost can also be a constraint since some of the nodes may be very expensive. In Ref. [51], the coverage demands for some important directions are set as constraints. Besides, the minimum distance between nodes is set as a constraint to prevent electromagnetic interference. Meanwhile, the distance between nodes and the protected objects should satisfy some requirements. These are typical constraints in the case of static node deployment. In dynamic node deployment, the region for deployment changes according to the nodes and targets. As a result, the corresponding deployment space and constraints should be adjusted. In Ref. [44], some deployed nodes are malfunctioning, and the corresponding redeployment problem needs to be solved based on the updated constraints.

16  Metaheuristics for Resource Deployment

1.3.3 Objective Functions

The mathematical model of node deployment is presented as follows. min [f1 (x), f2 (x), ..., fi (x)]

(1.1)

s.t. gk (x) ≤ 0, k = 1, . . . , l

(1.2)

hk (x) = 0, k = l + 1, . . . , p

(1.3)

x∈X

(1.4)

where X ⊆ Rn denotes the set of all feasible solutions. fi (x) denotes the ith performance metrics for evaluating the deployment. hk (x) denotes the kth constraint for deployment space requirement. gk (x) denotes the kth other constraint. x denotes the decision variable, which is a vector x = [x1 , x2 , ..., xn ]. In the discrete space-based deployment, the candidate location for deployment is denoted as p = [p1 , p2 , ..., pm ], where xi denotes the ith node assigned with the jth location p. In the continuous space-based deployment, xi denotes the position of the ith node. The critical issue is how to describe the performance of the nodes in different locations. Therefore, the objective function can be used to evaluate the performance of the deployment. There are many different types of node deployment problems, which involve different optimization goals. We mainly consider the objective functions in WSNs and air defense. 1.3.3.1

Node Deployment in Wireless Sensor Networks

In this subsection, an overview of the optimization objectives in WSNs is provided. There are many performance metrics to be optimized in WSNs. For example, coverage, connectivity, lifetime, and energy consumption are important concerns for maintaining the quality-of-service (QoS) in WSNs [37]. Meanwhile, these metrics usually conflict with each other. For example, more energy consumed reduces the lifetime of the network. • Coverage. In WSNs, coverage is the most widely used metric to evaluate the QoS. WSN coverage can be classified into three types: area coverage, target (point) coverage, and barrier coverage. In the area coverage problem, the entire region needs to be covered. In the target coverage problem, a finite set of discrete points are to be covered. The barrier coverage problem deals with the detection of movements across a barrier of sensors [36]. The area coverage problem is the most studied in WSNs. Two main problems

Introduction  17

related to area coverage are achieving satisfactory coverage and maximum coverage, both NP-hard problems [36, 37]. Satisfactory coverage means finding the minimum number of sensors that any point in the covered region of interest (RoI) can be detected with a probability that is equal or exceed a predefined threshold. Another method to describe satisfactory coverage is k-coverage, which means any point in the RoI is within the sensing range of at least k different sensor nodes [54]. Maximum coverage means to find the optimal locations of a set of nodes to achieve maximum coverage. • Network connectivity. Connectivity requires that the location of any deployed node should within the communication range of one or more active nodes, so that all active nodes can form a connected communication network. Two sensor nodes are directly connected if the distance of them is smaller than the communication range Rc . The communication range Rc should be greater than the sensing range Rs to achieve network connectivity. For a set of nodes that cover a convex region, the network remains connected if Rc ≥ 2Rc [37]. • Network lifetime. Lifetime is an important metric in WSNs. The energy source of the sensor nodes is limited by the battery. It is also crucial to maximize the lifetime of the sensor nodes with limited resources. The networks’ lifetime is calculated by the time duration from the activation of the network to the time when any sensor node fails by depleted of energy. • Energy consumption. Since the sensor nodes are equipped with limited battery power, the energy consumption problem in WSN should be taken into consideration. The total energy consumption is calculated by the sum of the energy expended at the node along the path for data acquisition, processing, and transmission phases. 1.3.3.2

Node Deployment in Air Defense

This subsection provides an overview of the optimization objectives in air defense. The strength and depth of the coverage provided by the deployed weapons show the ability in air defense. The strength refers to multiple coverages, and the depth refers to the covered area. Multiple factors should be considered for deployment evaluation. The main factors are the deployed and protected resources and the targets for air defense.

18  Metaheuristics for Resource Deployment

The ability of the deployed resources and the condition of the protected resources should be analyzed. Meanwhile, the moving direction, position, and number of targets should also be considered. Similar to the area coverage and target coverage in WSNs, two types of the node deployment problem in air defense are usually studied, which are area air defense and point air defense. • Area air defense. Area air defense represents that the entire area needs to be protected by the deployed nodes (e.g., weapon nodes) [55]. Since the resources are limited compared with the protected area, it is necessary to select several important sampling points for protected resources to evaluate the deployment performance. In Ref. [55], some important protected resources are chosen from the entire area and are used for evaluation. The selection for the protected resources is similar to the discretization method in candidate locations. Both uniform and non-uniform sampling points are used for calculation. To maximize the protection of these resources is belongs to the coverage problem. The strength of the coverage is more suitable for performance evaluation in area air defense. Some studies consider the set covering problem (SCP) and maximal covering location problem (MCLP) to formulate this problem and proposed a 0-1 programming problem [20, 56]. SCP minimizes the deployment cost with satisfying coverage, while MCLP maximizes the coverage of the targets with limited resources. Ref. [57] proposes firing range covering and firing angle covering based on SCP. In Ref. [39], ship sector locations are optimized to provide a robust air defense formation as the sector allocation problem (SAP). The problem is formulated as a 0-1 integer linear programming (ILP) problem, and the coverage provided to the ships is maximized. This is a classic MCLP. Since there are always multiple covering situations for covering the protected resources in the area air defense, an aggregate signal function can be used to describe the cooperative coverage1 [58]. The followings are two methods to deal with it. – Add up the coverage capabilities of different resources directly [39]. 1 The deployed nodes belong to the same type with different capabilities. It is different with the idea in Section 1.5, where heterogeneous nodes are considered.

Introduction  19

– A certain probability is introduced to describe the resource’s coverage capability. Multiple coverages consider calculating the joint probability, which is the maximum expected coverage location problem (MECLP) [20]. Most of the studies consider joint probability to describe the cooperative coverage rate of the deployment [43, 44, 50, 55, 59]. The performances of the node deployment in WSNs and air defense are evaluated by calculating the joint probability of the sampling points. If a sampling point is within the sensing (attacking) range of the sensor (weapon) nodes, the joint probability by sensor (weapon) nodes can be calculated as follows. P =1−

Yn k=1

(1 − pk )

(1.5)

where pk denotes the sensing (killing) probability of the kth node. n denotes the total number of the nodes. P denotes the joint probability, which is the sensing (killing) probability from all n nodes. In the area air defense, most studies of only consider the protected resources and do not consider the threat target. Therefore, the entire region for air defense is sampled by grids as protected resources. However, this coverage evaluation method can only reflect the coverage capability of deployment but cannot directly reflect the effect of deployment in real combat. In Ref. [39], both the size and the direction of the threat are considered and discretized as sampling points as a MECLP. Refs. [60, 61] describe the threat target by queuing theory. In deploying weapon nodes in a multilayer defense, the incoming target in the defense line is regarded as a random service system. The target flow entering the defense area is regarded as Poisson flow. The shooting efficiency of the defense is obtained by calculating the killing probability of each layer of defense and the probability of enemy penetration. Maximizing the defense capability of the weapon nodes is the evaluation metric. This queuing theory method does not consider the sampling of the entire defense area and avoids the calculation difficulty caused by sampling. • Point air defense. In point air defense, there exist some critical resources for protection. The formation of weapon nodes should have enough strength

20  Metaheuristics for Resource Deployment

and depth for protection. The coverage strength and coverage depth are two conflicting objectives. A deployment with too much coverage strength will have less coverage depth. Point air defense formations are usually circular or sector-shaped to ensure a certain depth of coverage. Ref. [62] proposes an objective function related to flight distance in the defense area based on entropy. When the entropy is maximized, the distance difference among all directions is the smallest. It ensures the uniform protection ability of air defense in all directions. In Ref. [51], the coverage of some critical directions should be satisfied, and other directions should be balanced. The sampling point can be determined by calculating the number and position of a single target when it passes through the attacking range. We can maximize the killing probability (minimize the penetrating probability) of the enemy target from different directions [43, 44]. Besides, the queuing theory method can also describe the process of the enemy target entering the defense area [47]. Besides, some other models are proposed, e.g., minimum regret model [63], chance-constraint programming model, dependent-chance programming model [64], and mean-entropy model [65]. 1.3.3.3

Other Types of Optimization Objective

Besides, some studies consider the multi-objective problem (MOP) and dynamic optimization. In MOP, the coverage and the deployment cost are two conflicting objective functions usually considered together. When the environment changes, the previous deployment scheme may not be optimal. Redeployment cost is another objective function that needs to be considered. Since it is a real-time redeployment process, the computing time for solving the redeployment problem should meet the requirement.

1.4

RESEARCH PROGRESS OF NODE DEPLOYMENT METHODS

As node deployment becomes complicated, the manual method to determine the locations may not achieve a satisfying result, thus using computer algorithms in solving the node deployment is necessary. Different methods are proposed to deal with the node deployment problem. The

Introduction  21

solution of deployment location involves encoding, constraint handling, multi-objective optimization handling, and algorithms. 1.4.1 Encoding

Encoding is applied to describe the locations for the deployed nodes. It is the decision variable and related to the deployment space. There are differences between discrete space-based deployment and continuous space-based deployment. In discrete space-based deployment, the decision variable is usually set as an integer representing whether a resource has been deployed. Branch-and-bound [39,66] and genetic algorithm [55] are applied to solve the problem. In continuous space-based deployment, any position in the deployment space can be the location for deployment. This encoding mechanism is usually adopted in the problem solving by particle swarm optimization [32] and virtual force algorithm [43]. 1.4.2 Constraints Handling

The complexity of the constraint increases the difficulty of the problem. Therefore, constraints should be handled to improve the efficiency of the algorithm. There are various techniques to deal with the constraints [67]. The constraints can be used for generating feasible solutions, e.g., repair algorithms. It can replace an infeasible solution with a feasible one. Another popular technique is the penalty functions. This method aims to transform a constrained optimization problem to an unconstrained one by adding a penalty item based on the amount of constraint violation in a particular solution [50]. However, the selection of the penalty function may affect the efficiency of the algorithm since there are multiple constraints [44]. Therefore, an artificial potential field algorithm has been applied to deal with the geographical and coverage constraints and obtain deployment schemes [43, 44]. 1.4.3 Multi-Objective Handling

The problem is mainly transformed into a single-objective optimization problem or tackled by multi-objective evolutionary algorithms to deal with the node deployment with multiple objective functions. Several methods are presented as follows. • Weighting method. The objective functions are combined as a single objective function by weights.

22  Metaheuristics for Resource Deployment

• ε-constraint method. We optimize one of the objective functions, and other objective functions are transformed into constraints. It is better than the weighing method since different weights may obtain the same solution. On the other hand, we may obtain a different solution with the ε-constraint. • Stratified sequencing method. Multiple objective functions are ranked by different levels and solved according to the rank. In [68], network coverage rate, energy consumption in redeployment, and bi-connected network topology are considered objective functions in a sensor redeployment problem. A distributed algorithm based on the virtual forces is applied to solve the constraint of biconnected features. The network coverage is maximized first, and then the energy consumption is minimized. • Others. Multi-objective evolutionary algorithms (MOEAs) have been applied to solve the MOP. In MOP, the objective functions are conflicting, e.g., the coverage and the cost. The solution in the obtained Pareto solutions set may be superior to others in one objective function and inferior in another objective function. Besides, analytic hierarchy process (AHP) [69], technique for order preference by similarity to an ideal solution (TOPSIS) [70], expert systems [69], and risk analysis [71] are applied to deal with the multiple objective functions. 1.4.4 Algorithms

Exact algorithms are applied in node deployment, e.g., enumeration method [72], branch-and-bound [39], etc. Besides, since node deployment is belongs to NP-hard problem, metaheuristic algorithms, e.g., genetic algorithm [55,59], particle swarm optimization algorithm [50], simulated annealing algorithm [73], etc. are also applied. These algorithms are nature-inspired and have a stochastic searching mechanism. 1.4.4.1

Exact Algorithm

The exact algorithm provides a precise and repeatable solution. In Ref. [39], a branch-and-bound approach is proposed to deal with the sector allocation problem based on a linear MCLP. Tight lower and upper bounds are proposed to speed up computation time better than CPLEX. Ref. [66] proposes a new reformulation for the generalized partial cov-

Introduction  23

ering problem with lower bounding strategies. The problem is solved by the branch-and-bound approach. In Ref. [46], the nonlinear objective functions in MECLP are decomposed into linear summation by setting the maximum covering times and then solved by using LINGO. The number of nodes and candidate locations are small in these problems. As the size of the problem increases, the problem will become hard to solve with a larger searching space. 1.4.4.2

Metaheuristic Algorithm

In metaheuristic algorithms or intelligent optimization algorithms, stochastic variables are introduced during the optimization process. The obtained solutions are not optimal and not the same in each run. It is a population-based algorithm. The best parameters of the algorithms can be selected appropriately. The metaheuristic algorithms used in the node deployment problem are as follows. • Simulated annealing (SA). SA is based on the annealing in metallurgy. It can tackle tough computational optimization problems with nonlinear, discontinuous, and non-differentiable functions, where exact algorithms fail. It can achieve an approximate solution to the global optimum by controlling the cooling schedule. Refs. [45, 73] adopt simulated annealing to deal with the node deployment problem in air defense. • Genetic algorithm (GA). GA simulates the natural selection of biological evolution and the biological evolution process in genetics for the optimal solution [74]. In Ref. [59], GA is proposed with a new generation method of the initial population. Ref. [75] proposes the parallel gene combination genetic algorithm to solve the air defense deployment problem. The initial population is generated by problem-specific knowledge and random selection. This method overcomes the phenomenon of non-unique encoding and has a good performance on the convergence time and solution quality. Refs. [64, 65] use GA to solve the multilayer node deployment for air defense in fuzzy environment. • Particle swarm optimization (PSO). PSO is widely used in recent decades [76]. The global best and personal best particles are used for updating the position. It is easy to be applied in node deployment problem [50]. However, the PSO algorithm will easily

24  Metaheuristics for Resource Deployment

converge to local minima and cause premature convergence. This algorithm needs to be improved according to the corresponding problem. Ding et al. [32] proposed a disturbance PSO (d-PSO) with a Gaussian perturbation to update the velocity, which shows fast convergence. Parallel particle swarm optimization (PPSO), which divides the sensing area and the sensors equally into several parts, and it is used when there are large numbers of sensors to be deployed [77]. Thus, the searching space is partitioned, and the computation time is saved. Soleimanzadeh et al. [78] proposed a PSO-LA (learning automata) algorithm, and the velocity is changed by using learning automata. Tang et al. [79] established a three-dimensional sensor network model and provided an improved PSO algorithm. Wang et al. [80] proposed a resampled PSO (RPSO) to maximize the coverage and energy efficiency of the WSN. In Ref. [81], the Voronoi diagram is combined with PSO to obtain the best coverage. In this algorithm, PSO is used to find an optimal position, while the Voronoi diagram is used for evaluating the fitness of the solution. • Artificial potential field (APF) algorithm. APF algorithm is also called virtual force algorithm (VFA), which considers the attractive and repulsive force between nodes for optimal deployment [82]. The positions of the nodes will converge under the force from VFA. Chen et al. [43] used virtual force between nodes, environment, and obstacles to generate solutions and use GA to optimize the parameters in APF under different deployment conditions. Zhou et al. [83] proposed two adaptive distribution algorithms based on virtual force to tackle the different obstacles and moving obstacles in WSN. VFA has also been combined with other metaheuristics. Wang et al. [30] presented an improved co-evolutionary PSO algorithm that combines virtual force and PSO with a co-evolutionary mechanism to solve the dynamic sensor deployment problem. Liang et al. [84] used a virtual force-based coverage algorithm to achieve area coverage. Four different forces caused by neighbor sensors and uncovered regions are exerted on directional sensors. Binh et al. [85] combined the heuristic initialization and modified virtual force algorithm with both GA and PSO to deal with the maximization of obstacles constrained area coverage problem. • Memetic algorithm (MA). MA is an algorithm framework combining global search and local search. This hybrid algorithm can

Introduction  25

be a unique one with different search strategies [86]. Therefore, it is easy to develop an improved version of it by this mechanism. Chen et al. [51] used GA for global search and neighborhood search for local search in sector-shaped point air defense problem. Liu et al. [47] used GA for global search and hill-climbing algorithm for local search in ring-shaped point air defense problem. Li et al. [87] used PSO for global search and SA for local search in naval fleet formation problem. • Multi-objective evolutionary algorithm (MOEA). MOEA is a population-based evaluation process to obtain Pareto solutions. Wu et al. [88] used a non-dominated sorting genetic algorithm with an elitist strategy (NSGA-II) to minimize the deployment cost and penetrating probability. Liu et al. [89] proposed a group divided dimensional reduction algorithm with strength Pareto evolutionary algorithm (SPEA2) in sector-shaped point air defense problem. Zhang et al. [90] proposed a problem specific decomposition-based MOEA for barrier coverage with wireless sensors considering the power, reliability, and fairness of the deployment. A 2-tuple encoding scheme is adopted, and a cover-shrink algorithm is proposed to produce feasible and relatively optimal solutions. Besides, local search is performed with problem-specific knowledge. The above analysis shows that metaheuristics are commonly used in node deployment problems. Many researchers propose improved and hybrid intelligent optimization algorithm to tackle the problem. Besides, most of the current algorithms are based on static and deterministic problems. There are lacking algorithms for dynamic and uncertain node deployment problems.

1.5 MAIN ISSUES AND CHALLENGES Based on the above analysis, the following will be the main issues and challenges in the node deployment problem. • Node deployment with multi-objective functions. In order to deal with the multi-objective node deployment problem, different methods are considered based on the decision maker’s (DM’s) preferences, which are listed as follows [91].

26  Metaheuristics for Resource Deployment

– A priori methods: The global preference information of the problem is provided by the DM. Thus, weighted methods can be applied. – A posteriori methods: An approximation of the Pareto front is obtained, and the most preferred one is selected. ε-constraint method and MOEA can be adopted. – Interactive methods: The DM specifies preferences progressively during the solution process to guide the search towards his/her preferred regions. This can be a promising research direction that needs further investigation. • Node deployment under uncertainty. In node deployment modeling, there are uncertainties with the number and state of the nodes and targets. However, most current works are deterministic. According to the features of the parameters, it can be divided into accurate information, information subject to a certain probability distribution, information whose distribution is unknown but belongs to a certain interval, fuzzy information, and real-time dynamic information. The latter four kinds of information types can be formulated as follows. – Stochastic node deployment: The parameters in this problem are uncertain, but their statistic distribution can be obtained by historical data. The expectation operator is usually used to transfer the original problem into a deterministic one. – Robust node deployment: The distributions of the parameters in this problem are unknown but discrete or continuous. The discrete events can be described by scenarios, while the continuous ones can be described by intervals. The optimal deployment scheme under different conditions has significant limitations, so it is necessary to design a robust deployment method. The deployment scheme provided by this method has a good deployment effect for various complex situations. Min-max, min-max regret operator, etc. are commonly used to transfer the original problem into a deterministic one. – Fuzzy node deployment: The parameters in this problem are uncertain under certain fuzzy distributions, e.g., trapezoidal fuzzy distribution, triangular fuzzy distribution, etc. Fuzzy simulation [92] is usually used to transfer the original problem into a deterministic one.

Introduction  27

– Dynamic node deployment: The parameters may change during the whole optimization process. Some dynamic optimization algorithms are applied, e.g., diversity and multi swarms are considered in improved PSO to improve the adaptability to the dynamic environment. • Real-time performance of the algorithms. Since the static node deployment problem belongs to the planning problem, real-time performance is not required. However, it is a critical issue in the dynamic node deployment problem since the conditions of the deployment will change. Therefore, redeployment is needed. It is meaningless to obtain a deployment scheme with much time, so a satisfactory solution must be obtained in a limited time. Therefore, we need to balance the trade-off problem between the precision and computation time of the deployment method. • Node deployment with cooperative coverage. As a node deployment problem, different types of nodes should be considered. For example, heterogeneous resources, which are sensors and actuators providing different tasks. For example, in unmanned systems, both UAVs and UGVs are deployed for sensing and attacking tasks. Therefore, the coverage problem of these two nodes becomes cooperative coverage, which brings new challenges [93]. • Node deployment with target assignment. Node deployment is the basis of the target assignment. The location of the nodes may affect the number of nodes to be assigned to the targets and limit the efficiency of the assignment algorithm. Therefore, the whole process of deployment operation needs to consider the target assignment. The relationship between these two problems needs further investigation. In this book, we mainly deal with the challenges related to uncertainty in the node deployment problem. Meanwhile, both single-objective and multi-objective problems are considered.

1.6 BOOK OUTLINE The research structure of this book is shown in Figure 1.9. The following chapters are arranged as follows: Chapters 2 and 3 investigate the stochastic node deployment for area

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The research structure of this book.

coverage and target coverage, respectively. Chapters 4 and 5 focus on the cooperative coverage problem, where uncertainty sets and fuzzy variables are considered, respectively. A simulation-based evaluation method for node deployment with the decision maker’s risk preference and the uncertainty of the targets is established in Chapter 6. Chapter 7 concludes the overview and future directions of the book.

CHAPTER

2

Stochastic Node Deployment for Area Coverage Problem

Area coverage of node deployment is a fundamental issue in wireless sensor networks (WSNs). This chapter discusses an area coverage optimization problem under uncertainty. There exist uncertainties in the detection of sensors, and the uncertainties are modeled as stochastic parameters. An improved particle swarm optimization (PSO) is introduced. The framework of virtual force directed co-evolutionary particle swarm optimization (VFCPSO) is improved by adding perturbation to the particle evolution process. Therefore, the exploration ability of the algorithm could be improved significantly. The experiment results have demonstrated the effectiveness of the proposed algorithm compared with other algorithms.

2.1 INTRODUCTION Wireless sensor networks (WSNs) are formed by small, inexpensive, lowpowered sensors. WSNs have recently become a popular research area for many applications such as environmental monitoring, battlefield surveillance, target tracking, intrusion detection, seismic detection, and transportation [94–96]. The objective is to monitor the environment and communicate information with each sensor. This chapter focuses on the area coverage problem. The main objective is to determine the location of the sensor network in order to monitor DOI: 10.1201/9781003202653-2

29

30  Metaheuristics for Resource Deployment

an area. There are typically two types of sensor node deployment [97]: random deployment and deterministic deployment. Random deployment is realized by scattering sensor nodes from aircraft for deployment, while in deterministic deployment, details of the region are known in advance. The coverage problem in WSNs is nondeterministic polynomial complete (NP-complete) [98]. Many researchers use metaheuristics to deal with this problem. Among the algorithms, particle swarm optimization (PSO) is most frequently used in solving the problem. Besides, most of the studies consider problems with parameters in an exact value. However, so many parameters are uncertain in the real environment, such as the sensor’s detection probability. The sensor type considered here is probabilistic rather than binary. If the distribution of the detection probability is known, the problem belongs to stochastic programming. The uncertainty of the sensors can be tackled by using a scenario-based approach. We address the following problem: given the target area and a set of WSNs, we have to determine the deployment formation of WSNs, which have better sensor quality in covering the target area. Since the detection probability of the sensor is an uncertain value, expected values are calculated to approximate the real value. Therefore, we formulate this problem as the maximum area coverage problem. We summarize our contributions as follows: 1. We present a scenario-based area coverage model for sensor deployment under uncertainty and prove its NP-hardness. 2. Different scenarios are generated by the uncertainty of the sensors’ parameters with probabilistic sensors. 3. Perturbation is added to the VFCPSO framework to improve the exploration of the algorithm. The rest of this chapter is organized as follows. In Section 2.2, we describe the preliminaries and formulation of the stochastic node deployment for area coverage. Section 2.3 presents the algorithm based on perturbation under the VFCPSO framework and other compared algorithms. Experiments and discussion are given in Section 2.4. Finally, conclusions are given in Section 2.5.

Stochastic Node Deployment for Area Coverage Problem  31

2.2 PROBLEM FORMULATION In this study, we analyze the area coverage problem based on probabilistic sensors under uncertainty using scenarios. First, we describe the detection model and area detection deployment scenarios based on the probabilistic sensors. Then, we define the maximum area coverage problem based on the probabilistic detection model. Our goal is to maximize the expected covered area coverage rate with the given probabilistic sensors. For convenience, the notations used in this chapter are summarized in Table 2.1. TABLE 2.1

Summary of notations.

Symbol Description S N r re cxy (si ) cxy (S) Ξ K

Sensor set Number of sensors Sensing range Uncertainty of the detection Detection probability by sensor i Collaborative detection probability by sensor Scenarios that corresponds to the uncertainty of the detection, ξ ∈ Ξ Number of scenarios

2.2.1 Detection Models

Sensor detection models describe the sensing ability and quality of the geometric relation between a target point and a sensor node [95]. In WSNs, there are two sensor detection models: the binary detection model and the probabilistic detection model [82]. 2.2.1.1

Binary Detection Model

The disk sensing model is usually used as its simplicity in coverage calculation [99]. In this model, the target is only covered within the sensing range. 0 if r < d(si , P ) 1 if d(si , P ) ≤ r

(

cxy (si ) =

(2.1)

where d(si , P ) denotes the Euclidean distance between sensor node si and point P , the constant r is the sensing range.

32  Metaheuristics for Resource Deployment

2.2.1.2

Probabilistic Detection Model

The detection probability of the sensor may be uncertain due to some noise and obstacles. Therefore, this model is more practical. The detection probability decays by distance from the sensor [100].

cxy (si ) =

   0

e

if r + re ≤ d(si , P ) β

β

(−α1 λ1 1 /λ2 2 +α2 )

  1

if r − re < d(si , P ) < r + re if d(si , P ) ≤ r − re

(2.2)

where d(si , P ) denotes the Euclidean distance between sensor node si and point P , the constant r is the sensing range, re (re < r) measure the uncertainty of the detection, λ1 = re − r + d(si , P ), and λ2 = re + r − d(si , P ). α1 , α2 , β1 , and β2 are detection probabilities parameters. These values vary depending on the sensor’s types and characteristics. In our work, we use the probabilistic sensor detection model. 2.2.2 Network Model

The WSNs are introduced with the following basic assumptions: (1) detection radius of the sensors are the same; (2) all of the sensors can communicate with each other; (3) mobile sensor nodes can move to the scheduled positions. There are N sensors deployed in a L × L 2D plane. The target area L L × grid grids where each grid size is equal to grid2 . is divided to grid Each grid can be monitored by multiple sensors nearby in a collaborative manner. Let S denote the set of sensor nodes. Let sensor node si be the sensor located at point (xi , yi ), point P represent any gird point (x, y). For any grid point only covered by one sensor, it will have low coverage. Therefore, the grid covered by multiple sensors may have a better coverage rate. This chapter considers discrete space-based deployment, that is, the sensor nodes can only be placed in the grid network. The probability for P (x, y) to be covered by a set of sensors S is defined as: cxy (S) = 1 −

N Y

(1 − cxy (si ))

(2.3)

i=1

According to Figure 2.1, the coverage rate is affected by the grid size. Therefore, the grid size should be chosen carefully in order to balance the calculation time and accuracy.

Stochastic Node Deployment for Area Coverage Problem  33

(a)

(b)

Three sensor nodes deployed in gridded ROI: (a) grid size: 0.5×0.5; (b) grid size: 0.25×0.25. Figure 2.1

2.2.3 Problem Statement

The area coverage problem under uncertainty in this chapter use expectation to deal with the uncertainty in different scenarios. Therefore, we define the maximum area coverage problem: Given a set of sensors S and L L target area with np = grid × grid grids, the expected value of the covered rate in the area is required to be maximized. We aim to decide the position of different sensors in the area. The uncertainty of the detection re is considered as an uncertain value with a known discrete distribution. The scenario-based approach is adopted to deal with the uncertainty. Next, we define the area coverage rate and the expected value of area coverage rate, which are used as metrics for performance evaluation of the node deployment of WSNs. Definition 2.1 (Area coverage rate). Area coverage rate (ACR) of a sensor node set S is defined as: Area(S) ACR(S) = = L×L

Pnp

j=1 cxj yj (S)

np

(2.4)

where Area(S) is the covering area of the sensor node set S, L × L is the total area of the target area region. A good sensor deployment scheme should have a higher ACR. Thus, the large Area(S) is the better the quality of the sensor deployment. Since the detection re is considered as an uncertain value with known discrete distribution, different scenarios are generated by random Monte

34  Metaheuristics for Resource Deployment

Carlo sampling. The expectation is an effective way to approximate the real value. With the area coverage rate, we have the following definition: Definition 2.2 (Expectation of area coverage rate). The expectation of area coverage rate is an effective metric to evaluate the performance of the node deployment when uncertainty is considered. A finite set of scenarios ξ ∈ Ξ with the total number K is used to formulate different realization of the sensor network parameters. The expectation of area coverage rate of a sensor node set S is defined as: E(ACR(S, ξ)) =

K X

pk ACR(S, ξk )

(2.5)

k=1

where ξk denotes the uncertain parameters of the WSNs in the kth scenario, pk is the probability of scenario ξk . Similarly, A good sensor deployment scheme should have a higher Expectation of ACR. Therefore, we proposed the maximum area coverage problem: max E(ACR(S, ξ))

(2.6)

We consider the value pk can all be set to 1/K, so that each scenario has the same probability. This problem can be easily transformed as follows: K 1 X max K k=1

Pnp

j=1 cxj yj (S)

np

(2.7)

2.2.4 NP-Hardness Proof

In this subsection, the maximum area coverage problem is proven to be NP-hard. Theorem 2.1 The maximum area coverage problem is NP-hard. Proof 2.1 Actually, in Ref. [101], the NP-hardness proof of generic sensor deployment problem is given. Therefore, finding an optimal sensor placement to cover a grid area and achieve maximum coverage is an NPcomplete problem. Furthermore, expectation-based maximum area coverage problem should be harder to solve. Therefore, the maximum area coverage problem must also be NP-hard.

Stochastic Node Deployment for Area Coverage Problem  35

2.3 SOLUTION ALGORITHMS Since the maximum area coverage problem is NP-hard, there is no polynomial-time algorithm to obtain the exact solution. Nature-inspired metaheuristic algorithms are applied to solve this problem with an approximated solution. Particle swarm optimization (PSO) has been widely used to deal with the sensor deployment problem. PSO is a swarm-based intelligent method inspired by the social behavior of a flock of birds that was developed by Kennedy and Eberhart [76]. The motion of the particles is interpreted as birds flying. The particles move in the searching space according to their former speed, their experience, and the experience of their surrounding neighbors. However, there are some drawbacks in PSO: 1. Since the particle of the swarm is updated based on its personal best particle and the global best particle. This algorithm will quickly converge to local minima and cause premature convergence. As a result, the diversity of the particles is hard to maintain. The optimal global solution is hard to obtain, especially for those large-scale, high-dimensional problems. 2. PSO lacks the ability to jump out of local minima when all the particles are converged to the same suboptimal solution. In order to overcome the drawbacks, our approach first utilizes a Gaussian-perturbation technique with improved particle swarm optimization. This algorithm combines the disturbance particle swarm optimization (d-PSO) with the virtual force directed co-evolutionary particle swarm optimization (VFCPSO) to obtain d-VFCPSO [30, 32]. 2.3.1 D-VFCPSO

d-PSO and VFCPSO are two effective algorithms to deal with the sensor deployment problem. In this section, we combine these two algorithms and obtain d-VFCPSO. The only difference is that we adopt the velocity updating in d-PSO to update VFCPSO. Therefore, d-VFCPSO uses Equations 2.8 and 2.9 to update velocity and position of a particle. t+1 t vid =c0 ∗ ni +c1 ∗ r1i ∗ (ptid − xtid )+c2 ∗ r2i ∗ (ptgd − xtid ) + c3 ∗ r3i ∗ gid (2.8)

36  Metaheuristics for Resource Deployment t+1 t xt+1 id =xid + vid

(2.9)

t+1 th where vid and xt+1 particle in id are the velocity and position of the i t t dimension d at time t + 1. pid and pgd are the best position of the ith particle and the best position of the whole particles in dimension d at time t. c0 is the disturbance factor that controls the effect of the disturbance. c1 and c2 are acceleration constants and respectively the cognitive factor and social factor that control the motion of the particle to its personal best position and global best position. c3 is acceleration constant that controls the particle by the virtual force. r1 , r2 , and r3 are random values with uniform distribution [0, 1]. ni is a Gaussian distribution with mean 0 and variance 1. The dimension of the particle represents the number of variables that need to be optimized. The number of the particle is set in advance, and the initial position and velocity of the particles are rant domly set according to certain constraints. gid is the motion suggested th by virtual force of the i particle in dimension d. It is defined as follows:

t gid =

 −1 (i,(j+1)/2)  (i,(j+1)/2)  Fx Fxy  × M axStep × e  (i,(j+1)/2)  Fxy    j = 1, 3, 5 . . . , 2N − 1 −1

(i,j/2)  (i,j/2) Fy  Fxy   (i,j/2) × M axStep × e  F  xy   j = 2, 4, 6 . . . , 2N

(2.10)

where the superscript of Fx , Fy , and Fxy are the index of particles and index of sensor nodes with virtual force exerts on. M axStep is the predefined single maximum moving distance. Fx and Fy are x- and ycoordinate forces, respectively. As for the virtual force exerts on each particle, this force can be an attracting or a repelling force. The total force on sensor i can be expressed as N X → − − → Fi = Fij

(2.11)

j=1,j6=i

where N is the total number of sensor nodes. The virtual force between sensor i and sensor j can be express as an

Stochastic Node Deployment for Area Coverage Problem  37

attractive or repulsive force. It is given as follows:   0      (wA (dij − dth ), αij ) − → Fij =  0         wR 1 − 1 , αij + π dij dth

if dij ≥ C if C > dij > dth if dij = dth

(2.12)

if dij < dth

where C is the communication range, αij is the orientation of the line from sensor i to sensor j, and wA (wR ) describes the effect of the attractive (repulsive) force. dij is the distance between sensor i and sensor j, and dth is a predefined threshold. The position of the particle is renewed by the velocity. The algorithm will stop when the maximum number of iterations is met. Since it is a discrete space-based deployment, the position of the particle is continuous space-based and will be rounded to the nearest grid point for fitness function evaluation. The best position is defined by a fitness function, which evaluates the position quality of a particle, and pid and pgd are replaced according to it. For the coverage problem, the fitness function is the coverage rate. It should be noted that all the particles have the ability to memorize their personal best positions and the best positions of their neighbors.

(a) Without disturbance Figure 2.2

(b) With disturbance

Searching space comparison.

Figure 2.2 describes the difference of the position tendency in the searching space with disturbance or not. This disturbance is a Gaussian perturbation used to overcome the drawback of local minima. Here, c1 and c2 are set to 1, c3 is set to 0. c0 is set according to the number of sensors, the sensing range, and the space size. Actually, PSO, VFPSO,

38  Metaheuristics for Resource Deployment

and VFCPSO adopt the former one while d-PSO and d-VFCPSO adopt the latter one. These five algorithms will be used for comparison. From Figure 2.2, it is obvious that the searching space in the PSO algorithm without disturbance is smaller than the PSO algorithm with disturbance. The shadow area in Figure 2.2(a) only considers more about the direction of its velocity. However, in Figure 2.2(b), the shadow area contains parts of the shadow in Figure 2.2(a), but has spaces away from its original direction as well. This unique feature ensures that the result will not lead to a local optimal, as in the PSO algorithm, because the personal best and the global best position may be a suboptimal position. The disturbance in Figure 2.2(b) makes it possible for the particle to jump away from the suboptimal position. It should be noted that the searching probability of the outer shadow introduced by the disturbance is not uniformly distributed since the disturbance is normally distributed. The outer shadow is 3c0 in width, since for N (0, 1), P (−3 < x < 3) = 2Φ(3) − 1 = 99.7%. The factor c0 plays a vital role in influencing particles to converge to the globally optimized solution. If this factor is extremely large or small, this algorithm will perform poorly. Besides adopting the velocity and position updating Equations 2.8 and 2.9, d-VFCPSO uses a co-evolutionary manner, that is, the solution vector of a particle is divided into small values. This is the key idea from co-evolutionary particle swarm optimization (CPSO) [102]. Therefore, a 2N -dimensional vector is split into a 1-D problem for 2N swarms. The pseudo-code of d-VFCPSO is presented in Algorithm 2.1. Readers can refer to Refs. [30, 32, 102] for more details. 2.3.2 Other PSO-Based Algorithm for Area Coverage Problem

In order to evaluate the performance of d-VFCPSO, the following four algorithms are considered for comparison. • PSO [76] is the standard algorithm framework compared in this chapter. • VFPSO [30] combines virtual force with the particle swarm optimization. • d-PSO [32] uses the Gaussian perturbation in the velocity updating in the PSO evolution process.

Stochastic Node Deployment for Area Coverage Problem  39

Algorithm 2.1 d-VFCPSO Input: N P : Particle size D: Swarm size (dimension size) f : Objective function b(k, z) ≡ (pg1 , pg2 , pg3 , . . . , pg(k−1) , pgz , pg(k+1) , . . . , pgD ) Output: p∗g : Final solution (global best particle) 1: Initial the particles. 2: while Stopping criteria not satisfied do 3: for j = 1 to D do 4: for i = 1 to N P do 5: Update particles by Equations 2.8 and 2.9. 6: Update pij by min{f (b(j, xij ), f (b(j, pij ))}. 7: Update pgj by min{f (b(j, pij ), f (pg ))}. 8: end for 9: end for 10: end while 11: p∗g = pg . 12: return • VFCPSO [30] combines virtual force and co-evolutionary particle swarm optimization. 2.3.3 Complexity Analysis

The time complexity of PSO, d-PSO, and VFPSO are O(N P · D). The time complexity of VFCPSO and d-VFCPSO are both O(N P · D2 ). N P is the particle size, and D is the dimension size. The time complexity order of these algorithms is VFPSO=VFCPSO=d-VFCPSO≥PSO=dPSO.

2.4 EXPERIMENTS AND DISCUSSION The deployment scenario considered in this chapter is to deploy sensor nodes for area coverage. The sensors need to provide coverage for the entire region. Figure 2.3 shows an illustration of a wireless sensor network performing area coverage.

40  Metaheuristics for Resource Deployment

6HQVRU

Figure 2.3

An illustration of a wireless sensor network performing area

coverage. 2.4.1 Test Instances

The experiments are conducted to analyze the advantages of the proposed d-VFCPSO with other PSO-based algorithms. Test instances are generated with a different number of sensors N . The parameters of different sensors are shown in Table 2.2. The uncertainty value detection re was selected from the set ∈ {0.1r, 0.3r, 0, 5r}, the number of scenarios K was set to 5. TABLE 2.2

Parameters of different sensors.

Number of sensors N Sensing range (m)

10 20 30 8

5

4

2.4.2 Parameter Setting

In WSNs, since there are N sensors, the position for one sensor is described using a coordinate system as (xi , yi ). Therefore, a particle for N sensors can be represented as (x1 , y1 , x2 , y2 , x3 , y3 , . . . , xN , yN ). The particles are 2N -dimensional for N sensors. Other parameters are as follows: The area for coverage was set to 40×40m2 . There were 20 particles in the entire algorithm. w was selected linear decreasing from 0.9 to 0.4 with t for PSOs without disturbance. The acceleration constants c1 and c2 were set to 1.4962 [103], and c3 was set to 1 [30]. The maximum iteration was set to 700. As for the parameters of the sensor model, α1 , α2 , β1 , and β2 was set to 1, 0, 1, and 0.5, respectively. The parameters for virtual force were set as

Stochastic Node Deployment for Area Coverage Problem  41

wA = 1, wR = 5, dth = 2r, C = 3r, and M axStep = 0.5r [104]. The size of grid was set to 1 × 1m2 , so there were 1600 grids to determine coverage rate. Since the disturbance factor c0 in d-VFCPSO and d-PSO has a significant effect on the performance. We experimented with 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. By empirical testing, we found that c0 = 0.4 gives the best overall performance. Thus, we used c0 = 0.4 in our experimentation. 2.4.3 Analysis of Results

In order to test the proposed algorithm, the simulation is implemented on an Intel(R) Core(TM) i7-7700HQ (2.8 GHz) computer with 16 GB RAM using MATLAB R2018b. The independent runs for each algorithm on each instance is set to 20. Table 2.3 shows that d-VFCPSO has the best average rank in area coverage rate in different cases, while d-PSO and VFCPSO are with the same rank. In small-scaled instance (Case 1), d-PSO is better than other PSO-based algorithms, while in larger-scaled instances (Case 2 and Case 3), d-VFCPSO is better than other PSO-based algorithms. Figure 2.4 shows that d-VFCPSO has better convergence speed with good quality than other algorithms. Although d-PSO shows good convergence than other algorithms at the beginning of the process, VFCPSO and d-VFCPSO perform better in global optimal searching ability, especially in Figures 2.4(b) and 2.4(c). Finally, based on the above results and analysis. It can be concluded that d-VFCPSO is the most suitable algorithm for the area coverage problem of WSNs in this chapter, while d-PSO performs well in instance with small size.

2.5

CONCLUSION

This chapter aims at stochastic node deployment for area coverage problem. Discrete space-based deployment is considered in the WSNs model. An improved PSO algorithm named d-VFCPSO is introduced by adding disturbance in a co-evolutionary manner. The velocity updating mechanism of the proposed algorithm is changed by a Gaussian distribution perturbation. In order to show its superiority, d-VFCPSO is compared with standard PSO, d-PSO, VFPSO, and VFCPSO. Numerical experiments demonstrate that d-VFCPSO performs the best among the compared algorithms.

Instance Case 1 Case 2 Case 3 Average Rank

PSO

The comparison of expectation of area coverage rate.

VFPSO

d-PSO

VFCPSO

d-VFCPSO

0.7404±0.0214[5] 0.7546±0.0166[4] 0.8197±0.0036[1] 0.8187±0.0048[3] 0.8195±0.0051[2] 0.6341±0.0180[5] 0.6862±0.0061[4] 0.7334±0.0049[3] 0.7375±0.0027[2] 0.7381±0.0035[1] 0.6348±0.0135[5] 0.6851±0.0113[4] 0.7315±0.0080[3] 0.7502±0.0019[2] 0.7532±0.0029[1] 5.0000

4.0000

2.3333

2.3333

1.3333

42  Metaheuristics for Resource Deployment

TABLE 2.3

Stochastic Node Deployment for Area Coverage Problem  43

0.82 0.8 0.78

coverage rate (%)

0.76 0.74 0.72 0.7 PSO VFPSO d-PSO VFCPSO d-VFCPSO

0.68 0.66 0.64 0.62 0

5

10

15

20

25

30

time (s)

(a) 10/8m 0.74 0.72 0.7

coverage rate (%)

0.68 0.66 0.64 0.62 PSO VFPSO d-PSO VFCPSO d-VFCPSO

0.6 0.58 0.56 0.54 0

10

20

30

40

50

60

time (s)

(b) 20/5m 0.76 0.74 0.72

coverage rate (%)

0.7 0.68 0.66 0.64 0.62 PSO VFPSO d-PSO VFCPSO d-VFCPSO

0.6 0.58 0.56 0

10

20

30

40

50

60

70

time (s)

(c) 30/4m Figure 2.4

Comparation with a different number of sensors/sensing range.

44  Metaheuristics for Resource Deployment

In the future, we will consider distributions other than Gaussian and adopt an adaptive selection method. Besides, some problem-specific knowledge will be applied in the population’s initialization, and we will consider some mechanism, e.g., local search and restart strategy.

CHAPTER

3

Stochastic Dynamic Node Deployment for Target Coverage Problem

The previous chapter discusses the area coverage problem in WSNs, which is a stochastic programming problem with one objective. This chapter discusses a bi-objective stochastic dynamic node deployment problem for target coverage in air defense. There are uncertainties that exist in the targets and nodes. A bi-objective optimization problem is proposed to minimize the threat of the targets and minimize the average distance of redeployment. Pareto optimal solutions are obtained by using an improved multi-objective particle swarm optimization (MOPSO) algorithm. Different global best selection methods (crowding distance and adaptive grids) and perturbation methods (rapidly decreasing and elitist learning strategy (ELS)) for MOPSO are researched and compared with a non-dominated sorting genetic algorithm with an elitist strategy (NSGA-II). Since the algorithms’ parameters have significant effects on their performance, the Taguchi method with a novel response value is used to tune the parameters of the proposed algorithms. Numerical results show that the proposed MOPSO-GRID-ELS with ELS operator outperforms other algorithms in solving the problem.

DOI: 10.1201/9781003202653-3

45

46  Metaheuristics for Resource Deployment

3.1 INTRODUCTION Dynamic node deployment is a location problem related to uncertainty. The dynamic of the deployment is considered, which means the deployment decision is made throughout the whole process. In air defense, the location of the pre-deployed nodes should be adjusted based on nodes’ state, battle situation, etc., in order to improve the performance. In the dynamic environment, the location of the nodes before redeployment is known, while the position and threat of the targets, the kill probability of the nodes to targets, and the failure probability of nodes are unknown. The nodes should be redeployed based on these disruptions. Therefore, the node deployment problem in this chapter is a stochastic programming problem for dynamic node deployment. The stochastic dynamic node deployment for target coverage studied in this chapter considers the threat degree of the target and the moving distance of the node. Therefore, it belongs to the multi-objective optimization problem. Few studies consider the moving distance of the node and the energy consumed during node movement in air defense deployment. There will be less energy consumed if the moving distance of the node is small, and the time for finishing redeployment will be less. Ning et al. [105] described a discrete and multi-swarm PSO to solve the dynamic deployment of WSN, maximizing the coverage while minimizing the moving distance. The two objectives are combined to obtain a weighted objective. It belongs to the NP-hard problem. The objectives considered in this chapter are minimizing the threat of the targets and minimizing the moving distance of the nodes, which are similar to Ref. [105]. These two objectives are optimized separately, and the Pareto optimal front is obtained. We summarize our contributions as follows: 1. Unlike many researches in which a single objective node deployment model is considered, a novel bi-objective dynamic node deployment model under uncertainty is presented in this chapter. The first objective function aims to minimize the threat of the targets, and the second objective function, minimize the average distance of redeployment. 2. Different scenarios are generated by the uncertainty of the nodes (kill probability to targets, normal working probability) and the targets (position and threat). 3. Several multi-objective evolutionary algorithms (MOEAs) based

Stochastic Dynamic Node Deployment for Target Coverage Problem  47

on NSGA-II and MOPSO are adopted to solve the bi-objective stochastic dynamic node deployment model. We consider some improved strategies in MOPSO. They are global best selection methods (crowding distance and adaptive grids) and perturbation methods (rapidly decreasing and elitist learning strategy). 4. Since the metaheuristic algorithms are sensitive to the parameters, a Taguchi method is conducted to calibrate the parameters of the algorithms with a novel metric that uses a combination of convergence and diversity as the response value. This chapter is organized as follows. Section 3.2 formulates the stochastic dynamic node deployment for target coverage. Several MOEAs based on NSGA-II and MOPSO are proposed in Section 3.3. Experiments and discussion are provided in Section 3.4. Section 3.5 summarizes the chapter.

3.2 PROBLEM FORMULATION In this chapter, we introduce a bi-objective stochastic dynamic node deployment problem for target coverage. There are N nodes deployed in a Ax × Ay 2D plane. The attacking range of each node is R. The deployment decision is described as [(x1 , y1 ), (x2 , y2 ), ..., (xN , yN )], where (xi , yi ) denotes the position of the ith node. There are T targets randomly generated with uncertainty. During the engagement in the air defense, nodes will be damaged by the targets and malfunctioned. As a result, the previous deployment decision will not satisfy the demand of the defense. Therefore, redeployment will be conducted with new locations for dynamic deployment. The location of the deployed nodes determines the protection of the area in air defense. Therefore, two objectives are considered: minimizing the remaining threat of the targets and minimizing the average moving distance of the nodes’ redeployment. Obviously, the first objective ensures the fully utilized of the nodes, and the second objective eliminates the redeployment cost of the nodes. Different from the previous chapter, continuous space-based deployment is considered in this chapter. We have the following assumptions: • Each target can be attacked by multiple nodes simultaneously. • Each node can attack multiple targets simultaneously.

48  Metaheuristics for Resource Deployment

• The kill probability is different between nodes to targets. • The position and threat of targets are stochastic variables with known distribution. • The kill probability is a stochastic variable with known distribution. • The node may be malfunctioning. The normal working probability of the node is a stochastic variable with known distribution. Table 3.1 summarizes all the notations that are used throughout this chapter. TABLE 3.1

Summary of notations.

Symbol Indices i∈N j∈T Parameters N T Ax Ay Ri pij pvoid Vj (xtj , yjt ) (x0i , yi0 ) Decision variables (xi , yi ) Auxiliary variable si fij dij

Description Indices for nodes Indices for targets Number of nodes Number of targets Range of deployment space in x-axis Range of deployment space in y-axis Attacking range of node i Kill probability of node i to target j Failure probability of node i Threat of target j Position of target j Position of node i before redeployment Position of node i after redeployment 1, if node i is working normally; 0, otherwise 1, if node i is able to attack target j; 0, otherwise distance between node i and target j

Stochastic Dynamic Node Deployment for Target Coverage Problem  49

3.2.1 Mathematical Model

The bi-objective stochastic node dynamic deployment problem can be stated as follows:  T X min Z1 = E  Vj j=1



N P

si   i=1 min Z2 = E  

N Y

!

(1 − pij )fij 

(3.1)

i=1

q

(xi −

x0i )2 N P

+ (yi −



yi0 )2 

si

(3.2)

  

i=1

s.t. 0 ≤ xi ≤ Ax

∀i ∈ N

(3.3)

0 ≤ y i ≤ Ay

∀i ∈ N

(3.4)

∀i ∈ N, j ∈ T

(3.5)

∀i ∈ N

(3.6)

∀i ∈ N, j ∈ T

(3.7)

dij fij ≤ si Ri xi , yi ∈ R fij ∈ {0, 1}

where Equation 3.1 is to minimize the expected value of the remaining threat. Equation 3.2 is to minimize the expected value of the average distance of redeployment. In these two objectives, E(·) denotes the expectation operator. Equations 3.3 and 3.4 guarantee the location range of the nodes. Equation 3.5 guarantees the feasibility of the nodes to attack targets. Equations 3.6 and 3.7 restrict the decision variables to be real numbers and auxiliary variable to be 0 or 1. 3.2.2 Scenario-Based Model Reformulation

Since the objectives proposed are minimizing the remaining threat and minimizing the average distance of redeployment. Both of these objectives are expectation related models. It is hard to calculate the expected values due to its complexity. Therefore, a scenario-based reformulation is proposed to address the issue. The uncertainty parameters are stochastic variables with known distribution. Therefore, we can use random sampling to generate different scenarios based on their probability distribution. Each combination of uncertain parameters lead to one scenario, representing one possibility of the uncertainty. Therefore, the collection of the various scenarios are generated by random sampling will be used for approximating the expected value. This method is called Monte Carlo simulation, and it has

50  Metaheuristics for Resource Deployment

been utilized as a basic method to deal with many stochastic programming problems [106–108]. The reformulation is presented as follows: S X T N Y 1X (k) (k) (k) min Z1 = Vj (1 − pij )fij S k=1 j=1 i=1 N P

S 1X i=1 min Z2 = S k=1

(k)

si

(3.8)

q

(xi − x0i )2 + (yi − yi0 )2 N P i=1

(k)

(3.9)

si

s.t. (3.3), (3.4), (3.6) (k) (k)

!

(k)

dij fij ≤ si Ri (k)

fij ∈ {0, 1}

(3.10) ∀i ∈ N, j ∈ T, k ∈ S (3.11) ∀i ∈ N, j ∈ T, k ∈ S (3.12)

where ·(k) denotes the specific values of the uncertain parameters in the kth scenario, and S is the total number of scenarios generated by random sampling. The expected objective values are equal to the true objective values if the number of replications tends to be infinite. Therefore, one needs to balance the tradeoff between optimality and computational cost.

3.3 SOLUTION ALGORITHMS Two kinds of MOEAs are reviewed and analyzed in this section. They are NSGA-II and MOPSO. They are employed to find the Pareto solutions. For a bi-objective problem to be optimized, f1 and f2 are the two objective functions. We say x1 dominate x2 if the following two conditions satisfied: 1. f1 (x1 ) ≤ f1 (x2 ) and f2 (x1 ) ≤ f2 (x2 ) 2. f1 (x1 ) < f1 (x2 ) or f2 (x1 ) < f2 (x2 ) The solutions that cannot dominate each other are called Pareto front (PF). 3.3.1 NSGA-II

NSGA-II is one of the most popular multi-objective evolutionary algorithms and has been applied in many engineering applications. The main

Stochastic Dynamic Node Deployment for Target Coverage Problem  51

feature of this algorithm is the elitist non-dominated sorting method [109]. Crowding distance is used as a ranking criterion when sorting the non-dominated solutions. The pseudo-code of NSGA-II is shown in Figure 3.1. Initialize population. Do non-dominated sorting with crowding distance to rank the individuals. Iter = 0. while Iter < M axIter do Get offspring population from tournament selection, crossover, and mutation. Merge parent population and offspring population. Do non-dominated sorting with crowding distance to rank the individuals. Select individuals by the order within the population size. Iter = Iter + 1. end while Report results in the population.

Figure 3.1

The pseudo-code of NSGA-II.

The chromosomes in the population are represented as possible solutions for the problem. The crowding distance is used to evaluate solutions with the same non-dominated rank by measuring the relative density of the individuals. As a result, we have the rank of all the individuals in the population. Elitism is guaranteed by combining the parent population with the offspring population and selected by the crowding distancebased non-dominated sorting. The pseudo-code of calculating crowding distance is shown in Figure 3.2. By adopting crowding distance, the individual i in the population can be easily sorted with the crowded operator [109] with two attributes: non-dominated rank (irank ) and crowding distance (idistance ). The order of two individuals i and j can be as follows: • i ≺ j when (irank < jrank ) or (irank = jrank and idistance > jdistance ).

52  Metaheuristics for Resource Deployment

n: size of the non-dominated solutions in S. m: number of the objectives. CD[i]: the crowding distance of the ith individual. f [i, j]: the jth objective value of the ith individual Initialize population. for i = 1 to n do CD[i] = 0. end for for j = 1 to m do Sort the solution S with objective j in ascending order. CD[1] = CD[n] = inf. for i = 2 to n − 1 do CD[i] = CD[i] + (f [i + 1, j] − f [i − 1, j])/(max(f [∗, j]) − min(f [∗, j])). end for end for

Figure 3.2

The pseudo-code of calculating crowding distance.

3.3.2 MOPSO

Here we introduce how PSO is applied to solve multi-objective optimization problems. Since the basic PSO for single-objective optimization is introduced in Section 2.3, we only introduce how to deal with multiobjective problems using PSO. In order to apply PSO in solving multi-objective optimization problems, there are some issues to be discussed [110]: (1) How to select particles as leaders in the updating process; (2) How to maintain the non-dominated solutions to obtain the optimal Pareto front; (3) How to prevent the particles from converging to local optimal and maintain the diversity of the swarm. 3.3.2.1

Personal Best Selection

When selecting the personal best solution for MOPSO, the pbest is replaced by the new particle if it dominates the previous pbest or the two are non-dominated with each other. After all the particles in the swarm have updated their pbest , they can be used for the global best selection.

Stochastic Dynamic Node Deployment for Target Coverage Problem  53

3.3.2.2

Non-Dominated Solutions Maintaining and Global Best Selection

An external archive is introduced to store the non-dominated solutions found by MOPSO. At each generation of the swarm, the new particles are used to updates the external archive. Any dominated solutions are deleted from the archive. The size of the archive will increase during the updating process, which adds to the computation cost. Therefore, the archive size should be fixed. We have different methods to maintain the external archive based on the global best selection. For a particle in the swarm, the global best selection means selecting a non-dominated solution from an external archive as its gbest . In this chapter, we discuss two methods for global best selection: (1) crowding distance; (2) adaptive grids. MOPSO-CD [111] uses the crowding distance (CD) to measure the relative density of the non-dominated solutions in the external archive. It is similar to NSGA-II, which use crowding distance as criteria for nondominated sorting of the individuals. For any particle in the swarm, the gbest is selected randomly from a predefined top part of the archive based on non-dominate rank and crowding distance. Adaptive grids method (GRID) proposed in Ref. [112] is another way for updating the external archive in MOPSO. The objective space of the non-dominated solutions is divided into hypercubes. Each hypercube may have a different amount of solutions. Those hypercubes with fewer solutions are more likely to remain in the archive. A fitness value is assigned to each hypercube in inverse proportion to the number of nondominated solutions in it. Roulette-wheel selection is applied to select hypercube with certain particles. Then, we randomly select a particle, which is the gbest . 3.3.2.3

Diversity Maintaining

In order to avoid early convergence of the particle to an optimal local solution and maintain the diversity of the particles in the swarm, some mutation techniques are adopted. It is an effective way to avoid convergence to a single solution in MOPSO [113]. In this chapter, we consider two methods: rapidly decreasing [112] and elitist learning strategy (ELS) [114]. Rapidly decreasing mutation method is first developed by Coello et al. [112]. This method is widely used in most of the algorithms based on MOPSO. The number of the particles being mutated is decreasing rapidly with respect to the number of the iteration. Only one dimension

54  Metaheuristics for Resource Deployment

(randomly chosen) of the particle is mutated. The mutation operator may affect all the particles in the swarm at first, with the mutation range covers the entire searching space. As the number of the iteration increases, the mutation range of the particle will decrease. The rapidly decreasing can be performed as: Pd = Pd +

(Xdub

−

Xdlb )

g × 1− G 

1/mu

(3.13)

where Pd is the dth dimension of the particle, d is randomly selected among the all the dimension of the decision space, Xdub and Xdlb are the upper and lower bound of the particle in dth dimension. g is the index of the current iteration, and G is the number of the iterations. mu is the mutation rate. The ELS is similar to the rapidly decreasing in mutation operator with a different function. In rapidly decreasing, the mutation range is controlled by a nonlinear function, while the ELS adopts a Gaussian perturbation. Gaussian mutation adds more diversity. Because at the iteration g, the mutation range in ELS is not stable, while the mutation range in rapid decreasing is stable. It is very effective for global optimization using PSO [32]. Besides, the ELS only deals with the gbest in the swarm. The global best position is an important factor in leading the particles to the optimal position. Therefore, for MOPSO, the particles in the external archives are mutated. The elitist learning is performed as: gBestd = gBestd + (Xdub − Xdlb ) × Gaussian(µ, σ) g σ = σmax − (σmax − σmin ) × G

(3.14a) (3.14b)

where gBestd is the dth dimension of a gBest, the Gaussian(µ, σ) is a random number of a Gaussian distribution with mean µ and standard deviation σ. σ is also called “elitist learning rate” with is linearly decreasing with respect to the number of the iteration. σmax and σmin are the upper bound and lower bound of σ. The values of the bounds are set to σmax = 1.0 and σmin = 0.1, respectively, by empirical study. With the new particles mutated from ELS, the external archive itself updates. The pseudo-code of the ELS procedure is shown in Figure 3.3. The pseudo-code of the provided MOPSO algorithms is shown in Figure 3.4. We divide the MOPSO into four algorithms with different strategies using CD or GRID (MOPSO-CD or MOPSO-GRID) with rapidly

Stochastic Dynamic Node Deployment for Target Coverage Problem  55

D: size of searching space. n: size of the external archive. g[i]: the ith solution in the external archive. g mu[i]: the mutated ith solution in the external archive. for i = 1 to n do d = random(1, D). σ = σmax − (σmax − σmin ) × g/G. g mu[i] = g[i] + (Xdub − Xdlb ) × Gaussian(µ, σ). Keep g mu[i] within the range [Xdlb , Xdub ]. Evaluate the objectives with the new solution. end for Update the external archive with the mutated solution g mu.

Figure 3.3

The pseudo-code of elitist learning strategy.

decreasing method, and elitist learning strategy (MOPSO-CD-ELS or MOPSO-GRID-ELS). We will run these algorithms together with NSGA-II in the next section and compare them by using some popular performance metrics. 3.3.3 Complexity Analysis

Table 3.2 summarizes the time complexity of the primary operations in MOPSO. The total time complexity of MOPSO is min  O(m · N 2 ), O(m · S 2 ) , where m is the number of objectives, N is the population size, and S is the size of the external archive. The time complexity of NSGA-II is O(m · N 2 ). The main computational cost of MOPSO is selecting non-dominated solutions and updating the external archive. TABLE 3.2

Time complexity analysis of MOPSO.

Procedure

Worst-case time complexity

Generate a new solution Select non-dominated solution Update external archive Rapidly decreasing method Elitist learning strategy

O(m) O(m · N 2 ) O(m · S 2 ) O(m · N ) O(m · S)

56  Metaheuristics for Resource Deployment

Initialize the particles in the swarm (velocity and position). Evaluate each of the particles in the swarm. Initialize external archive by non-dominated solution maintaining. Iter = 0. while Iter < M axIter do for each particle do Update the velocity and position of the particle. if rand() < (1 − g/G)1/mu and using rapidly decreasing mutation then A rapidly decreasing mutation strategy is applied to the particle. end if Evaluate the new particle. Update pbest . end for Update the external archive by non-dominated solution maintaining with the new particles. if rand() < σ and using elitist learning strategy then An elitist learning strategy is applied to the external archive. end if Control the archive size and select gbest for the swarm from the external archive by crowding distance or adaptive grids. Iter = Iter + 1. end while Report results in the external archive.

Figure 3.4

The pseudo-code of the MOPSO algorithms.

3.4 EXPERIMENTS AND DISCUSSION The deployment scenario considered in this chapter is to deploy weapon nodes for target coverage. The weapons need to provide coverage for the targets. There also exists malfunctioning weapons. Figure 3.5 shows an illustration of weapons performing target coverage. 3.4.1 Test Instances

In order to evaluate the five MOEAs (NSGA-II, MOPSO-CD, MOPSOCD-ELS, MOPSO-GRID, and MOPSO-GRID-ELS), several test in-

Stochastic Dynamic Node Deployment for Target Coverage Problem  57

:HDSRQ 7DUJHW

0DOIXQFWLRQLQJ ZHDSRQ

Figure 3.5

An illustration of weapons performing target coverage.

stances are generated for computational experiments in this section. The parameters of the proposed bi-objective model are shown in Table 3.3, where the threat of targets, position of targets, the kill probability of nodes, and the position of nodes before redeployment are uniformly distributed, and the working state of nodes is binomially distributed. TABLE 3.3

Parameters of the model.

Parameter

Value

Ri /km pij pvoid Vj xtj yjt x0j yj0 si

5 ∼ U (0.7, 0.9) 0.05 ∼ U (0.8, 1.2) ∼ U (0.1Ax , 0.9Ax ) ∼ U (0.1Ay , 0.9Ay ) ∼ U (Ax , Ax ) ∼ U (Ay , Ay ) ∼ B(1, 1 − pvoid )

Eight test instances are considered, including four small-scale instances and four large-scale instances. The specific information of the test instances is shown in Table 3.4. 3.4.2 Performance Metrics

To analyze the performance of NSGA-II and the proposed MOPSOs, we use the following popular performance metrics. Set coverage (C − metric): This measure was proposed by E. Zitzler and L. Thiele [115]. Consider two PFs P and Q, C(P, Q) is defined as

58  Metaheuristics for Resource Deployment TABLE 3.4

Parameters of the test instances on a different scale.

Instance Number of scenarios (N, T, Ax /km, Ay /km) Small-scale

Large-scale

P1 P2 P3 P4 P5 P6 P7 P8

20 20 20 20 20 20 20 20

(5, 10, 20, 20) (10, 10, 30, 50) (10, 15, 30, 50) (15, 15, 30, 50) (20, 20, 30, 50) (20, 50, 50, 50) (50, 50, 50, 100) (100, 100, 100, 100)

the percentage of the solutions in Q dominated by at least one solution in P , as in the follows: C(P, Q) =

|{q ∈ Q|∃p ∈ P : p dominates q}| |Q|

(3.15)

where |X| is the size of the PF X. Consider the true PF P ∗ and an approximation PF P , the lower the value of C(P ∗ , P ), the better the solutions in P . Spacing (SP ): This value measures the distribution of the nondominated solutions along the approximation front. The definition of the metric is as follows [116]: v u n u1 X ¯2 SP = t (di − d)

n

(3.16)

i=1

where n is the number of the non-dominated solutions in the approximation front, di = minj

m P

k=1

|fki − fkj |, i, j = 1, 2, 3, . . . , n, m is the number

n P of the objectives, d¯ = di /n. The solutions distributed uniformly when i=1

the value is close to zero. Number of non-dominated solutions (N S): This metric represents the number of non-dominated solutions in the set. The solution quality is better when we have a large value of N S. Inverted generational distance (IGD): This metric reflects both the convergence and diversity of the solutions. Consider a uniformly distributed along with the true PF P ∗ and an approximation PF P . We

Stochastic Dynamic Node Deployment for Target Coverage Problem  59

have the IGD as follows [117]: P

IGD(P, P ) = ∗

v∈P

d(v, P ∗ ) |P ∗ |

(3.17)

where d(v, P ∗ ) is the minimum Euclidean distance between v and all the points in P ∗ . The smaller value of IGD is, the better solution quality of P is. To get a lower IGD, the set P should be close to the true PF in any part. Hypervolume (HV ) [115]: This metric is obtained by calculating the hypervolume of the approximation PF with a reference point. It measures both closeness and diversity [118]. The calculation of HV is as follows: HV (P ) = {

[

a(xi )|∀xi ∈ P }

(3.18)

i

where xi is an individual in a PF P , and a(xi ) is a rectangle area bounded by a reference point and f (xi ). The reference point adopted here is [max(f1 ), max(f2 )]. The solution set P with a higher value of HV have better performance. Since the true PF is unknown, in this chapter, the P ∗ is obtained by merging all the approximation PFs found by all the proposed algorithms. 3.4.3 Parameter Turning

As mentioned previously, NSGA-II and MOPSO are proposed to find the non-dominated solutions of the bi-objective stochastic dynamic node deployment model. For NSGA-II, the parameters are population size (P S), the number of generation (N OG), the crossover probability (CP ), and the mutation probability (M P ). The MOPSO factors are: population size (P S), number of generation (N OG), acceleration coefficient c1 , and acceleration coefficient c2 . Since the metaheuristic algorithms are very sensitive to the parameters, a Taguchi method [119] is utilized to tune the parameters of the algorithms. This method uses a set of arrays called orthogonal arrays, which contains the full information of the factors that affect the performance of the algorithms. The factors are categorized into two groups: (1) controllable or signal factors and (2) noise factors. Based on the robustness concept, the method seeks to minimize the effect of noise and determine the optimal level of signal factors. The signal to noise ratio (S/N ), which calculates the amount of variation involved in the response,

60  Metaheuristics for Resource Deployment TABLE 3.5

NSGA-II parameter ranges and levels.

Parameters

Range

PS CP MP N OG

50–100 0.6–0.8 0.01–0.2 100–300

Low(1) Medium(2) High(3) 50 0.6 0.01 100

75 0.7 0.1 200

100 0.8 0.2 300

is used. In this research, the goal is to minimize S/N . The S/N is given as ! S(Y 2 ) S (3.19) = −10 × log N n where, Y and n are the response value and the number of orthogonal arrays, respectively. S(Y 2 ) is the summation of the response Y 2 . In MOEAs, two main goals (convergence and diversity) are considered simultaneously. Among the metrics mentioned in Section 3.4.2, both IGD and N S are suitable metrics. However, the Taguchi method only deals with one response function. Therefore, a combination of the performance measures should be defined. In this study, a new metric is introduced as IGD C.R. = (3.20) NS This new metric is called the combinatorial ratio (C.R.) and acts as the response variable of the Taguchi method. In order to utilize the Taguchi method, the levels of the factors are provided in Tables 3.5 and 3.6 for the proposed algorithms. In each of the algorithms, three levels are considered for each factor. The L9 (33 ) design is the best to tune the parameters of the algorithms. This has been a common approach in some studies [26, 120–123]. The orthogonal arrays of these designs, along with experimental results, are shown in Tables 3.7 and 3.8. For each algorithm, the effect plots of S/N ratio are given in Figure 3.6. Using these plots, the optimal values of the parameters for each algorithm are obtained in Table 3.9. 3.4.4 Analysis of Results

In the simulation experiment, the proposed algorithm is programmed and simulated by MATLAB R2016b on a computer with an Intel Xeon E5 CPU with 2.60 GHz speed and 64 GB of RAM. For each instance, 20

Stochastic Dynamic Node Deployment for Target Coverage Problem  61 TABLE 3.6

MOPSO parameter ranges and levels.

Parameters

Range

Low(1) Medium(2) High(3)

PS c1 c2 N OG

50–100 1.0–2.0 1.0–2.0 100–300

TABLE 3.7

Calibration process of NSGA-II.

50 1.0 1.0 100

75 1.5 1.5 200

Run order Algorithm parameters P S CP 1 2 3 4 5 6 7 8 9

1 1 1 2 2 2 3 3 3

1 2 3 1 2 3 1 2 3

MP 1 2 3 2 3 1 3 1 2

100 2.0 2.0 300

Response

N OG NSGA-II 1 2 3 3 1 2 2 3 1

0.1412 0.0967 0.1247 0.0541 0.1017 0.0356 0.0653 0.0123 0.0540

independent runs are performed using each algorithm. The simulation results are shown in Tables 3.10 and 3.11. In Table 3.10, the symbols X, Y, Z, U, and V represent the proposed NSGA-II, MOPSO-CD, MOPSOCD-ELS, MOPSO-GRID, and MOPOS-GRID-ELS. C-metric measures the percentage of two solution sets that are dominated by each other. Therefore, the algorithm with a larger C-metric is better. The symbols (‘+’, ‘−’, and ‘∼’) denote the statistical results of two algorithms by Wilcoxon rank-sum test, representing that the first algorithm is significantly better, significantly worse, and significantly equivalent to the compared algorithm. From Table 3.10, we can find that NSGA-II and MOPSO-GRID-ELS perform better than other algorithms in terms of C-metric on all the test instances. The average C-metric of NSGA-II with respect to MOPSOCD-ELS and MOPSO-GRID-ELS are 0.8475 and 0.4265. Moreover, the W-test results show that NSGA-II is statistically better than MOPSOCD-ELS but statistically equivalent to MOPSO-GRID-ELS. Besides, the MOPSOs with ELS show better results in C-metric than those without ELS. For example, the average C-metric of MOPSO-CD-ELS with respect to MOPSO-CD is 0.9457, and the average C-metric of MOPSO-

Calibration process of MOPSOs.

Run order

Algorithm parameters P S CP

1 2 3 4 5 6 7 8 9

1 1 1 2 2 2 3 3 3

1 2 3 1 2 3 1 2 3

MP 1 2 3 2 3 1 3 1 2

Response

N OG MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS 1 2 3 3 1 2 2 3 1

0.5677 0.4350 0.3775 0.3057 0.4582 0.4932 0.2820 0.3818 0.4826

0.3253 0.1634 0.1094 0.0964 0.2543 0.1806 0.1247 0.1052 0.2754

0.4602 0.2049 0.8260 0.1010 0.7643 0.1546 0.2648 0.1027 0.2664

0.3823 0.0968 0.2077 0.0412 0.4067 0.1322 0.1259 0.0495 0.2041

62  Metaheuristics for Resource Deployment

TABLE 3.8

Stochastic Dynamic Node Deployment for Target Coverage Problem  63 TABLE 3.9

Optimal values of the parameters.

Algorithms NSGA-II

Parameters Optimal value

PS CP MP N OG MOPSO-CD PS c1 c2 N OG MOPSO-CD-ELS PS c1 c2 N OG MOPSO-GRID PS c1 c2 N OG MOPSO-GRID-ELS P S c1 c2 N OG

100 0.7 0.01 300 100 1.0 2.0 300 100 1.0 2.0 300 100 1.0 1.5 200 100 1.5 1.5 300

GRID-ELS with respect to MOPSO-GRID is 0.8464. Meanwhile, the W-test results shown that MOPSOs with ELS are statistically better than MOPSOs without ELS. The experimental results of five algorithms in terms of SP , N S, IGD, HV, and Time are shown in Table 3.11. The best average results for each test instance are shown in boldface. Table 3.11 also show the one-way Analysis of Variance (ANOVA) of the performance indices SP , N S, IGD, HV, and Time at 95% confidence level along with the values of the corresponding p-values. In terms of SP , NSGA-II is better than other algorithms on test instances P2, P4, P6, while MOPSO-GRID-ELS is better on P1, P3, P5, and MOPSO-GRID is better on P7, P8. The one-way ANOVA of SP shows there are significant differences among algorithms except on P3. Figure 3.7 shows the boxplots of SP on eight instances by the five algorithms. In general, NSGA-II and MOPSO-GRID-ELS outperform their compared algorithms in terms of SP , while MOPSO-GRID has an advantage in large-scale instances.

Pi

C(X,Z)

C(Z,X)

W-test

C(X,V)

C(V,X)

W-test

C(Y,Z)

C(Z,Y)

W-test

C(Z,V)

C(V,Z)

W-test

C(U,V)

C(V,U)

W-test

P1 P2 P3 P4 P5 P6 P7 P8

0.6533 0.7601 0.7657 0.8853 0.7372 0.9781 1.0000 1.0000

0.2049 0.0762 0.1533 0.0277 0.0428 0.0034 0.0000 0.0000

+ + + + + + + +

0.2053 0.3849 0.4988 0.5848 0.1995 0.2670 0.7028 0.5685

0.6506 0.4735 0.3357 0.2112 0.5381 0.4885 0.1812 0.2801

− ∼ ∼ + − − + +

0.0436 0.0020 0.1442 0.0065 0.1199 0.0000 0.0000 0.0000

0.9340 0.9929 0.7992 0.9708 0.8687 1.0000 1.0000 1.0000

− − − − − − − −

0.1019 0.0624 0.1505 0.0844 0.0516 0.0000 0.0000 0.0000

0.8045 0.8765 0.7949 0.8736 0.9282 0.9851 0.9972 1.0000

− − − − − − − −

0.1833 0.1326 0.1525 0.1619 0.0233 0.0133 0.0000 0.0155

0.7196 0.7577 0.7783 0.7341 0.9426 0.9353 0.9691 0.9346

− − − − − − − −

Average/(+, −, ∼)

0.8475

0.0635

(8,0,0)

0.4265

0.3949

(3,3,2)

0.0395

0.9457

(0,8,0)

0.0564

0.9075

(0,8,0)

0.0853

0.8464

(0,8,0)

64  Metaheuristics for Resource Deployment

Comparison of five algorithms via C-metric (X: NSGA-II, Y: MOPSO-CD, Z: MOPSO-CD-ELS, U: MOPSO-GRID, V: MOPSO-GRID-ELS). TABLE 3.10

TABLE 3.11

Comparison of five algorithms via SP , N S, IGD, HV, and time. Small-scaled

Large-scaled

P2

P3

P4

P5

P6

P7

P8

SP

NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS ANOVA p-value

0.0402 0.1065 0.0452 0.0573 0.0358 0.0000

0.0671 0.2549 0.1103 0.0959 0.0673 0.0000

0.0768 0.0956 0.0803 0.0731 0.0676 0.8920

0.0805 0.2446 0.1181 0.0925 0.0810 0.0000

0.0764 0.2875 0.2027 0.0787 0.0726 0.0002

0.1096 0.3521 0.2319 0.1254 0.1896 0.0000

0.1885 0.5621 0.4468 0.0599 0.2146 0.0115

0.3707 0.5515 0.2875 0.0347 0.2442 0.0000

NS

NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS ANOVA p-value

80.15 60.95 100.15 98.85 128.45 0.0000

75.60 26.90 65.80 54.85 76.75 0.0000

77.95 64.05 77.70 61.55 81.05 0.0000

76.50 32.20 56.60 43.60 78.00 0.0000

49.10 34.75 33.85 36.25 41.50 0.0002

73.10 29.25 39.35 48.90 75.35 0.0000

38.40 34.80 34.70 34.70 37.50 0.7279

37.15 40.20 55.60 27.75 29.30 0.0000

IGD

NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS ANOVA p-value

0.1351 0.4870 0.1634 0.1833 0.1090 0.0000

0.3735 3.6155 0.5105 0.6205 0.2560 0.0000

0.3380 0.8242 0.5057 0.7007 0.3188 0.0000

0.5020 3.0437 0.9286 1.2458 0.5393 0.0000

0.9234 3.5832 1.3423 2.7909 0.4784 0.0000

1.3642 10.6964 2.9632 3.5223 0.9194 0.0000

1.7613 17.0924 7.7682 10.1959 2.4680 0.0000

2.9851 23.9480 15.5776 15.6447 4.1780 0.0000

HV

NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS ANOVA p-value

0.4453 0.3679 0.4398 0.4167 0.4636 0.0000

0.2379 0.1230 0.2638 0.1790 0.2725 0.0000

0.2285 0.1792 0.2209 0.1504 0.2460 0.0000

0.2596 0.1716 0.2786 0.1270 0.3133 0.0000

0.1913 0.1619 0.2626 0.0715 0.2310 0.0000

0.1414 0.0993 0.1802 0.0482 0.2179 0.0000

0.0822 0.0990 0.0887 0.0056 0.1067 0.0000

0.0650 0.0799 0.0733 0.0002 0.0247 0.0000

Time/s

NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS ANOVA p-value

88.98 48.19 236.62 34.14 342.13 0.0000

141.94 73.77 199.06 50.65 246.17 0.0000

190.86 99.17 261.11 67.02 295.47 0.0000

266.93 136.32 267.95 92.17 300.10 0.0000

439.21 222.51 328.06 149.56 327.53 0.0000

1034.61 527.35 660.44 347.74 730.99 0.0000

2512.43 1259.06 1562.36 840.91 1461.99 0.0000

9686.80 4850.39 6158.15 3238.43 5303.46 0.0000

Stochastic Dynamic Node Deployment for Target Coverage Problem  65

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(e) MOPSO-GRID-ELS Figure 3.6

Taguchi ratios for all proposed algorithms.

As for N S, NSGA-II is better than other algorithms on test instances P2, P5, P7, while MOPSO-GRID-ELS is better on P1, P3, P4, P6, and MOPSO-CD-ELS is better on P8. The one-way ANOVA of N S shows there are significant differences among algorithms. Figure 3.8 shows the boxplots of N S on eight instances by the five algorithms. In general, MOPSO-GRID-ELS outperforms their compared algorithms in terms of N S. For IGD, NSGA-II is better than other algorithms on test instances P4, P7, and P8, while MOPSO-GRID-ELS is better on P1, P2, P3, P5, and P6. The one-way ANOVA of IGD shows there are significant differences among algorithms. Figure 3.9 shows the boxplots of IGD on

Stochastic Dynamic Node Deployment for Target Coverage Problem  67

Boxplot of SP

Boxplot of SP 0.9

0.4

0.8

0.35

0.7 0.3 0.6

SP

SP

0.25 0.2

0.5 0.4

0.15

0.3

0.1

0.2

0.05

0.1 0 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(a)

(b)

Boxplot of SP

Boxplot of SP

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

0.7

0.7

0.6

0.6

0.5

0.5

SP

SP

0.8

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(c)

-CD-ELS

(d)

Boxplot of SP

Boxplot of SP

1.2 1.2 1 1

0.8

SP

SP

0.8

0.6

0.4

0.6

0.4

0.2

0.2

0

0 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(e)

-CD-ELS

(f)

Boxplot of SP

Boxplot of SP

1.2

4.5 1

4 3.5

0.8

SP

SP

3 2.5

0.6

2 0.4

1.5 1

0.2 0.5 0

0 NSGA-II

-CD

-CD-ELS

(g) Figure 3.7

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(h)

Boxplot of SP on eight instances by the five algorithms.

68  Metaheuristics for Resource Deployment

Boxplot of NS

Boxplot of NS 90

140 80 120

70 60

NS

NS

100

50

80 40 60

30 20

40 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(a)

(b)

Boxplot of NS

Boxplot of NS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

100 100 90 90 80 80

NS

NS

70 70

60

60

50

50

40 30

40

NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(c)

-CD-ELS

(d)

Boxplot of NS

Boxplot of NS

70

110 100

60

90 80

50

NS

NS

70 40

60 50

30

40 30

20

20 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(e)

-CD-ELS

(f)

Boxplot of NS

Boxplot of NS

90

70 80 60

70 60

50

NS

NS

50 40

40 30

30 20

20 10 10

0 NSGA-II

-CD

-CD-ELS

(g) Figure 3.8

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(h)

Boxplot of N S on eight instances by the five algorithms.

Stochastic Dynamic Node Deployment for Target Coverage Problem  69

Boxplot of IGD

Boxplot of IGD

0.9 6

0.7

5

0.6

4

IGD

IGD

0.8

0.5

3

0.4 2

0.3 0.2

1

0.1 0 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(a)

-CD-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

(b)

Boxplot of IGD

Boxplot of IGD 6

1.2 1.1

5 1 0.9

4

IGD

IGD

0.8 0.7

3

0.6 0.5

2

0.4 1

0.3 0.2

0 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(c)

-CD-ELS

(d)

Boxplot of IGD

Boxplot of IGD

9 12 8 10

7

8

5

IGD

IGD

6

4 3

6

4

2 2

1 0

0 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(e)

-CD-ELS

(f)

Boxplot of IGD

Boxplot of IGD

20 25 18 16 20 14

IGD

IGD

12 10

15

8 10 6 4 5

2 0 NSGA-II

-CD

-CD-ELS

(g) Figure 3.9

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(h)

Boxplot of IGD on eight instances by the five algorithms.

70  Metaheuristics for Resource Deployment

eight instances by the five algorithms. In general, MOPSO-GRID-ELS outperforms their compared algorithms in terms of IGD. In terms of HV, MOPSO-GRID-ELS is better than other algorithms on test instances P1, P2, P3, P4, P6, P7, while MOPSO-CD-ELS is better on P5 and MOPSO-CD is better on P8. The one-way ANOVA of HV shows there are significant differences among algorithms. Figure 3.10 shows the boxplots of HV on eight instances by the five algorithms. In general, MOPSO-GRID-ELS outperforms their compared algorithms in terms of HV. As for Time, MOPSO-GRID is the best among the algorithms. The one-way ANOVA of Time shows there are significant differences among algorithms. Figure 3.11 shows the boxplots of Time on eight instances by the five algorithms. In general, MOPSO-GRID outperforms their compared algorithms in terms of Time. In order to demonstrate the experimental results more clearly, Figure 3.12 illustrates the distributions of the non-dominated solution sets on all the test instances. We can find out that the solution sets obtained by MOPSO-GRID-ELS have better approximation and distribution than other MOPSOs, while compared with NSGA-II, MOPSO-GRID-ELS is better in some instances and similar in other instances. Finally, by observing these performance results, we can see that MOPSO-GRID-ELS is suitable for the bi-objective stochastic dynamic node deployment problem proposed in this chapter, while NSGA-II can also be an alternative algorithm. Meanwhile, MOPSOs with ELS operator outperform the other MOPSOs in terms of different metrics. Due to the information-sharing mechanism in MOPSO. It is concluded that the ELS mutation operator has significant improvements in MOPSOs. There are two reasons: 1. ELS deals with the global best particle, while rapid decreasing deals with the whole particles in the swarm. Therefore, ELS is more likely to converge to the optimal position. 2. The perturbation in ELS is a stochastic function (Gaussian perturbation) rather than a stable function in rapid decreasing. Stochastic perturbation is better than stable perturbation since it provides more diversity.

Stochastic Dynamic Node Deployment for Target Coverage Problem  71

Boxplot of HV

Boxplot of HV

0.5

0.3

0.45

0.25

0.4

HV

HV

0.55

0.35

0.2

0.15

0.3

0.1

0.25

0.05 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(a)

(b)

Boxplot of HV

Boxplot of HV

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

0.4

0.3

0.35 0.3 0.25

0.2

HV

HV

0.25

0.2 0.15

0.15 0.1

0.1

0.05 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(c)

(d)

Boxplot of HV

0.4

-CD-ELS

Boxplot of HV 0.3

0.35 0.25

0.3

0.2

HV

HV

0.25 0.2

0.15

0.15 0.1 0.1 0.05

0.05 0

0 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(e)

-CD-ELS

(f)

Boxplot of HV

Boxplot of HV

0.16 0.12 0.14 0.1 0.12 0.08

HV

HV

0.1 0.08

0.06

0.06 0.04 0.04 0.02

0.02

0

0 NSGA-II

-CD

-CD-ELS

(g) Figure 3.10

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(h)

Boxplot of HV on eight instances by the five algorithms.

72  Metaheuristics for Resource Deployment

Boxplot of time

Boxplot of time

400 250

350 300

200

time

time

250 200

150

150 100 100 50 50 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(a)

-CD-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

-GRID

-GRID-ELS

(b)

Boxplot of time

Boxplot of time

350 300

300

250

time

time

250

200

200

150 150 100 100 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(c)

-CD-ELS

(d)

Boxplot of time

Boxplot of time 1100

450

1000 400 900 350

time

time

800 300

700 600

250

500

200

400 150 NSGA-II

-CD

-CD-ELS

-GRID

-GRID-ELS

NSGA-II

-CD

(e)

(f)

Boxplot of time

2600

-CD-ELS

Boxplot of time 10000

2400 9000 2200 8000 2000 7000

time

time

1800 1600

6000

1400 5000 1200 4000

1000 800

3000 NSGA-II

-CD

-CD-ELS

(g) Figure 3.11

-GRID

-GRID-ELS

NSGA-II

-CD

-CD-ELS

(h)

Boxplot of time on eight instances by the five algorithms.

Stochastic Dynamic Node Deployment for Target Coverage Problem  73

7

12 True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

6

5

True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

10

8

Z2

Z2

4 6

3 4 2 2

1

0 1

1.5

2

2.5

3

3.5

4

4.5

5

0 2.5

5.5

3

3.5

4

4.5

5

Z1

Z1

(a)

(b)

10

5.5

6

6.5

7

15 True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

9 8 7

True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

10

Z2

Z2

6 5 4 5 3 2 1 0

0 4

5

6

7

8

9

10

2

3

4

5

Z1

Z1

(c)

(d)

14

6

7

8

20 True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

12

10

True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

18 16 14 12

Z2

Z2

8

6

10 8 6

4

4 2 2 0 2

2.5

3

3.5

4

4.5

5

5.5

0 14

6

16

18

20

22

Z1

Z1

(e)

(f)

30

26

28

30

45

True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

25

24

True PF NSGA-II MOPSO-CD MOPSO-CD-ELS MOPSO-GRID MOPSO-GRID-ELS

40

35

Z2

Z2

20 30

15 25

10 20

5 10

12

14

16

18

20

22

15 25

30

35

40

45

Z1

Z1

(g)

(h)

50

55

60

65

Final nondominated solutions on eight instances by the five algorithms. Figure 3.12

74  Metaheuristics for Resource Deployment

3.5

CONCLUSION

This chapter aims at bi-objective stochastic dynamic node deployment for the target coverage problem. Continuous space-based deployment is considered for air defense. A bi-objective optimization problem minimizing the threat of the targets and minimizing the moving distance of the nodes is proposed. Two kinds of MOEAs are provided to deal with the problem, i.e., NSGA-II and MOPSO, and MOPSO are improved by using ELS as a diversity maintaining strategy. Taguchi method is implemented to tune the algorithms’ parameters with a novel response value considering the convergence and diversity of the solutions. Experimental results have demonstrated that both NSGA-II and MOPSOGRID-ELS are an efficient way of solving the problem, and MOPSOGRID-ELS shows significantly better performance among the proposed algorithms (NSGA-II, MOPSO-CD, MOPSO-CD-ELS, MOPSO-GRID, and MOPSO-GRID-ELS). The results can help engineers and decision makers make a reasonable decision when multi-objective and uncertainty are concerned with the node deployment problem for target coverage. In the future, some self-adaptive mutation mechanisms on ELS and embedding problem-specific knowledge in MOPSO can be applied to improve their performance.

CHAPTER

4

Robust Node Deployment for Cooperative Coverage Problem

The former two chapters discuss the stochastic node deployment problem for area coverage and target coverage, respectively. In this chapter, two types of nodes are concerned and will be deployed cooperatively. Firstly, this chapter proposes a deterministic two-level cooperative set covering problem (TLCSCP), which considers two types of facilities cooperatively serving each demand node with minimal cost. Different from the stochastic node deployment problem, the uncertainty concerned in this chapter is the coverage probability, which is a parameter without known distribution. Then the robust uncertain two-level cooperative set covering problem (RUTLCSCP) is formulated. The constraints are strong, robust, and nonlinear. A linear approximation robust counterpart version of RUTLCSCP (RUTLCSCP-LA-RC) is developed by linear approximation of the constraints. The solution for RUTLCSCP-LA-RC, ε-underapproximate solution, can also be the solution for RUTLCSCP in some conditions. Exact method is applied to obtain the ε-under-approximate solution. There are 333 instances (10125 instances in total) with 12 types that violate the constraints of RUTLCSCP. Both constructive heuristic methods and improved self-adaptive differential evolution (SaDE) are used to obtain the optimal solutions. The solution of the heurisDOI: 10.1201/9781003202653-4

75

76  Metaheuristics for Resource Deployment

tic method is also used for population initialization. Results on these instances demonstrate that the proposed SaDE is effective and outperforms other competitors in most of the cases, which indicates that our approach is promising in dealing with RUTLCSCP.

4.1 INTRODUCTION Robust node deployment is a selection of locations robust to a various environment with different parameters. The parameters considered are uncertain within a known range. The problem analyzed in this chapter is a variant of the set covering problem. The set covering problem (SCP) is one of the most studied combinatorial optimization problems. In the SCP, a set I = {1, . . . , m} of m demand nodes, a set J = {1, . . . , n} of n potential facility location sites and their building costs cj are given. The 0-1 matrix A = [aij ]m×n indicates whether a location j ∈ J is able to cover a demand node i ∈ I. The goal of SCP is to find a minimum cost cover of the demand nodes by x where xj is a binary value whether site j is selected. It is proven to be NP-complete [124]. min

X

(4.1)

cj xj

j∈J

s.t.

X

aij xj ≥ 1

∀i ∈ I

(4.2)

∀j ∈ J

(4.3)

j∈J

xj ∈ {0, 1}

The SCP has widely been used in many real-world applications, especially in facility location [20], where both exact and heuristic algorithms are proposed to deal with it. Daskin [125] considers the facility may not be working with probability, and it can be applied in many applications, e.g., node deployment in wireless sensor networks [32], weapon platforms [38], etc. Beraldi et al. [126] proposed the probabilistic setcovering aiming at covering constraint satisfied with a predefined probability. Aardal et al. [127] considered more than one facility type, and proposed a two-level uncapacitated facility location problem. Berman et al. [58] first proposed the cooperative cover model with one facility type. Pereira et al. [128] proposed the robust SCP with uncertain cost coefficients within predefined interval. In real-world applications, node deployment usually consists of different types of nodes. We take node deployment in Command and Control

Robust Node Deployment for Cooperative Coverage Problem  77

Systems as an example. The revolution of network-centric warfare has systematically organized the initially separated combat platforms, thus achieving a high level of information sharing and increasing the chance of more efficient operations. However, the collaborative node deployment of multi-platforms remains an urgent problem that needs to be solved. Sensor platforms and weapon platforms are two essential parts in completing the overall combat mission, e.g., air defense. As a result, when deploying these two types of nodes, cooperations are considered. Xin et al. [129] discussed the sensor-weapon-target assignment problem as a collaborative task assignment of sensor and weapon platforms. The probability of successful engagement is modeled as the product of the interceptor’s probability of kill and the sensor’s probability of detection. Moreover, an efficient marginal-return-based constructive heuristic is proposed. The current related works lack collaborative node deployment of sensor and weapon platforms. The information obtained from the sensor platforms is shared and interacted with weapon platforms based on the change in the battlefield environment and situation, which are important for the deployment of the weapon platforms. This information provides favorable conditions for weapon platform deployment. Thus, the advantage of cooperation is realized. The complexity of the battlefield and the increasing number of nodes bring great challenges in decision making. Meanwhile, uncertainty is inevitable and should also be focused. This chapter considers the uncertainty parameters with unknown distribution but a known range, which is different from the previous two chapters. To the best of our knowledge, the robust set covering problem with probabilistic and cooperative covering by two types of facilities has not previously been analyzed. Although Xin et al. [129] discussed a collaborative task assignment of sensor and weapon platforms and the probability of capturing the target is similar to the cooperative covering in this paper. The former is regarded as the objective function. We summarize our contributions as follows: 1. A compact mixed-integer linear programming (MILP) formulation is proposed by utilizing robust optimization and constraint relaxation. 2. The proposed formulation is analyzed on a large set of test cases with 10125 different instances. 3. A majority of the under-approximate solutions are proven to be

78  Metaheuristics for Resource Deployment

optimal, while few of them slightly violate the constraints and provide an efficient lower bound. 4. A subproblem dealing technique is proposed to deal with the robust uncertain constraints without linear approximation. 5. Heuristic and SaDE are utilized to obtain the optimal solutions for the instances with ε-under-approximate solution. This chapter is organized as follows. Section 4.2 formulates the robust node deployment for cooperative coverage. The solution algorithms are proposed in Section 4.3. Experiments and discussion are provided in Section 4.4. Section 4.5 summarizes the chapter.

4.2 PROBLEM FORMULATION In this section, we first propose a novel two-level cooperative set covering problem, which considers two types of facilities cooperatively serving each demand node. 4.2.1 The Deterministic and Uncertain Two-Level Cooperative Set Covering Problem 4.2.1.1

Two-Level Cooperative Set Covering Problem

In the Two-Level Cooperative Set Covering Problem (TLCSCP), a set I = {1, . . . , m} of m demand nodes, a set J = {1, . . . , n1 } of n1 potential y-facility location sites and a set K = {1, . . . , n2 } of n2 potential z-facility location sites are given. The 0-1 matrix A = [aij]m×n1 or B = [bik ]m×n2 indicates whether a location j ∈ J or k ∈ K is able to cover a demand node i ∈ I. c1j represents the costs of building y-facility located at site j, and c2k represents the costs of building z-facility located at site k. Both yj and zk are binary value, which means whether building a y-facility at site j and z-facility at site k. The objective is to find two subsets C 1 ⊆ J P P and C 2 ⊆ K with minimal cost c(C 1 , C 2 ) := j∈C 1 c1j + k∈C ∈ c2k covering all the demand nodes, i.e., for each demand node i ∈ I, there exists at least one y-facility j ∈ C 1 and z-facility k ∈ C 2 that ensures aij = 1 and bik = 1 simultaneously. A standard binary nonlinear programming formation of Two-Level Cooperative Set Covering Problem is defined as min

X j∈J

c1j yj +

X k∈K

c2k zk

(4.4)

Robust Node Deployment for Cooperative Coverage Problem  79





s.t. 

!

∀i ∈ I

(4.5)

yj ∈ {0, 1}

∀j ∈ J

(4.6)

zk ∈ {0, 1}

∀k ∈ K.

(4.7)

X

X

aij yj  ·

j∈J

bik zk

≥1

k∈K

where Equation 4.4 minimizes the building cost of two kinds of facilities. Equation 4.5 ensures that for each demand node, it is covered at least one y-facility and z-facility simultaneously. Equations 4.6 and 4.7 ensure decision variables are binary value. Since aij, bik , yj and zk are binary value, TLCSCP is equivalent to the following integer linear programming formulation: min

X

c1j yj +

j∈J

s.t.

X

X

c2k zk

k∈K

aij yj ≥ 1

∀i ∈ I

(4.8)

bik zk ≥ 1

∀i ∈ I

(4.9)

j∈J

X k∈K

yj ∈ {0, 1}

∀j ∈ J

zk ∈ {0, 1}

∀k ∈ K,

where Equations 4.8 and 4.9 linearize Equation 4.5. And similar to Set Covering Problem [130], Two-Level Cooperative Set Covering Problem is also a NP-hard combinatorial optimization problem. 4.2.1.2

Generalized Uncertain Two-Level Cooperative Set Covering Problem

The Generalized Uncertain Two-Level Cooperative Set Covering Problem (GUTLCSCP) is formulated based on TLCSCP, which introduces uncertainty into covering model. Like Generalized Uncertain Set Covering Problem (GUSCP) [130], aij and bik are independent random binary variable: with a probability of 1 − pij when aij = 1 and pij when aij = 0; with a probability of 1 − qik when bik = 1 and qik when bik = 0. Since the probabilities are assumed to be independent, the probability of two sets C 1 and C 2 cooperatively covering demand node i is as follows: 

P

 X j∈C 1

aij ≥ 1 = 1 −

Y j∈C 1

pij .

80  Metaheuristics for Resource Deployment





bik ≥ 1 = 1 −

X

P

Y

k∈C 2



P 

 

aij ≥ 1 · 

X

j∈C 1



bik ≥ 1

X

k∈C 2



=P 

 X

aij ≥ 1,

j∈C 1

X

 X

bik ≥ 1

k∈C 2



=P 

qik .

k∈C 2





aij ≥ 1 · P 

X

bik ≥ 1

k∈C 1

j∈C 1



 



= 1 −

pij  · 1 −

Y

Y

qik  .

k∈C 2

j∈C 1

Then, the GUTLCSCP can be formulated as a binary model given by min

c1j yj +

X j∈J

X

c2k zk

k∈K





s.t. P 

X

aij yj ≥ 1,

j∈J

X

bik zk ≥ 1 ≥ α

∀i ∈ I

(4.10)

k∈K

yj ∈ {0, 1}

∀j ∈ J

zk ∈ {0, 1}

∀k ∈ K.

When a solution y ∗ ∈ {0, 1}n1 and z ∗ ∈ {0, 1}n2 is feasible for the GUTLCSCP, Equation 4.10 is equivalent to 

P

 X

aij yj∗ ≥ 1,

j∈J

bik zk∗ ≥ 1

k∈K



= 1 −

X

  Y j∈C 1 (y ∗ )

pij  · 1 −

 Y

qik  ≥ α

(4.11)

k∈C 2 (z ∗ )

for all i ∈ I with C 1 (y ∗ ) = {j ∈ J |y ∗ = 1} and C 2 (z ∗ ) = {k ∈ K|z ∗ = 1}. The sets C 1 (y ∗ ) and C 2 (z ∗ ) satisfying Equation 4.11 are referred to as two-level-cooperative α-cover. Note that these covering probabilities are assumed as independent variables (see Ref. [129]). A more realistic model

Robust Node Deployment for Cooperative Coverage Problem  81

with dependent probabilities may be further analyzed (which is out of this paper’s scope). Therefore, GUTLCSCP can be reformulated as: min

X

c1j yj +

j∈J

X

c2k zk

k∈K





s.t. 1 −

Y

y pijj 

!

· 1−

j∈J

Y

zk qik

≥α

∀i ∈ I

(4.12)

k∈K

yj ∈ {0, 1}

∀j ∈ J

zk ∈ {0, 1}

∀k ∈ K.

where, Equation 4.12 is a nonlinear constraint. A linear approximation method is given as follows: Q Q y zk In Equation 4.12, set mi = j∈J pijj, ni = k∈K qik , (1 − mi )(1 − ni ) ≥ α for all i ∈ I. The original Equation 4.12 can be reformulated as:  Q yj   mi =Q j∈J pij zk ni = k∈K qik   (1 − m )(1 − n ) ≥ α i i  P  ln(mi ) = ln(pij )yj    j∈J  P ⇐⇒ ln(ni ) = ln(qik )zk   k∈K   (1 − m )(1 − n ) ≥ α i

(4.13)

(4.14)

,

i

where for all i ∈ I with mi , ni , α ∈ [0, 1]. Therefore, GUTLCSCP can be reformulated as: min

X

c1j yj +

j∈J

s.t. ln(mi ) =

X

c2k zk

k∈K

ln(pij )yj

∀i ∈ I

(4.15)

ln(qik )zk

∀i ∈ I

(4.16)

(1 − mi )(1 − ni ) ≥ α

∀i ∈ I

(4.17)

X j∈J

ln(ni ) =

X k∈K

yj ∈ {0, 1}

∀j ∈ J

zk ∈ {0, 1}

∀k ∈ K

0 ≤ mi ≤ 1

∀i ∈ I

(4.18)

82  Metaheuristics for Resource Deployment

0 ≤ ni ≤ 1

∀i ∈ I.

(4.19)

The key issue is to deal with the Equation 4.17. Since mi , ni , α ∈ [0, 1], Equation 4.17 can be reformulated as follows: (

1 − mi ≥ α 1 − ni ≥ α

(4.20)

P  ln(pij )yj ≤ ln(1 − α)  ⇐⇒ j∈J P  ln(qik )zk ≤ ln(1 − α) 

(4.21)

k∈K

for all i ∈ I. According to the Equations 4.15 and 4.16, ln(mi ) and ln(ni ) are linear functions with respect to yj and zk . Besides, we have obtained constraint 4.21. As a result, we can construct constraints like β ln(mi ) + γ ln(ni ) ≤ ln(Fi (α, β, γ)),

(4.22)

where β + γ = 1, Fi (α, β, γ) is a function with respect to α, β and γ for all i ∈ I. In order to determine the parameters of the Equation 4.22, we can find the function tangent to Equation 4.17. The Equation 4.22 can be reformulated as follows: β ln(mi ) + γ ln(ni ) ≤ ln(Fi (α, β, γ)) ⇐⇒mβi nγi ≤ Fi (α, β, γ).

(4.23)

Set (1−mi )(1−ni ) = α, then mi = 1−α/(1−ni ). We can substitute it into Equation 4.23 and obtain α f (ni ) = 1 − 1 − ni 

β

nγi .

(4.24)

Then by determining the first derivative of Equation 4.24, which is f 0 (ni ) = 0, the tangency function is obtained. The solution to f 0 (ni ) = 0 is as follows:   ni1 = 0      ni2 = 1 − α  ni3 =      n = i4

√

2γ+αβ−αγ−

√ 2γ+αβ−αγ+

α(4βγ+αβ 2 +αγ 2 −2αβγ) 2γ α(4βγ+αβ 2 +αγ 2 −2αβγ) 2γ

.

Robust Node Deployment for Cooperative Coverage Problem  83

The corresponding tangency function is obtained when ni3 is selected. Figure 4.1(a) shows comparison between Equation 4.22 after linear approximation and nonlinear Equation 4.17 when α = 0.9, β = 0.5, and γ = 0.5. The ranges for x-axis and y-axis are determined by Equation 4.20 with [0, 0.1]. Figure 4.1(a) shows that there exist regions between these two constraints with one pair of β/γ. Therefore, multiple combinations of β/γ are needed. Figure 4.1(b) shows the comparison when β = [0.1 0.3 0.5 0.7 0.9]. Figure 4.1(c) combines the multiple β/γ together to get an intersection of those constraints. Obviously, the linear approximate constraints show great similarity to the original nonlinear constraint. To be specific, the decision space under the linear approximate constraints is slightly bigger than under nonlinear constraint. Therefore, a relaxation method is obtained. Relaxing the problem in Equation 4.12 leads to the following linear approximation formulation of the GUTLCSCP (GUTLCSCP-LA): min

X

c1j yj +

j∈J

s.t.

X

X

c2k zk

k∈K

ln(pij )yj ≤ ln(1 − α)

∀i ∈ I

ln(qik )zk ≤ ln(1 − α)

∀i ∈ I

j∈J

X k∈K

β

X

ln(pij )yj + γ

j∈J

≤ ln

X

ln(qik )zk

k∈K

"

α 1− 1−δ

β

#

nγi

∀i ∈ I

yj ∈ {0, 1}

∀j ∈ J

zk ∈ {0, 1}

∀k ∈ K,

√ 2γ+αβ−αγ−

α(4βγ+αβ 2 +αγ 2 −2αβγ)

where δ = 2γ or vectors with β + γ = 1.

. β, γ ∈ [0, 1] are constants

Remark 4.1 The linear approximation (LA) method used here actually transforms the original problem to a constraint relaxed problem as an integer linear programming program. Exact methods are enabled to deal with this problem. All the problems in this chapter with LA are the constraint relaxed version of the original problems.

84  Metaheuristics for Resource Deployment

0.06

0.06

ni

0.08

ni

0.08

0.04

0.04 (1-m i )(1-n i )>0.9

0.02

(1-m i )(1-n i )>0.9

0.02

mβi n γi 0 then i∗ := arg mini∈Q F (Qi ). Delete solutions in P , which are greater in the objective value than the solutions in Qi . end if if |P | > PN then Delete the extra solutions in P . end if end while i∗ := arg mini∈Q F (Qi ). Calculate y, z, F (y, z) based on Qi∗ . return

4.3.2.2

Complexity Analysis of MRBCH-k

The time complexity of the proposed algorithm can be approximated by that of the lookup operations embedded in the heuristic. A desirable

96  Metaheuristics for Resource Deployment

lookup algorithm has the complexity of O(l), where l is the size of the array for lookup. The size of the array is varying with the increasing of iterations. The size of the array at the first iteration is L = |J | + |K|. The size decreases by one after each iteration. As for MRBCH-1, only one solution is used to update archive P . Thus, the lookup operation runs at most L · (L + 1)/2. Therefore, the worst-case time complexity of MRBCH-1 can be expressed by O(L2 ). When it comes to MRBCH-k (k > 1), the worst-case time complexity for the enumeration method is O(2L ), and there are k solutions used to update archive P . Therefore, the operation in each iteration decreases by L − k − i, where i is the current number of iterations. Thus, the total operation runs at most 2L − L · (L − k + 1)/2, which increases exponentially. The size of the archive P is considered as PN . Therefore, the worst-case time complexity of MRBCH-k (k > 1) can be expressed by O(PN · L2 ). 4.3.3 Proposed SaDE for RUTLCSCP

Differential evolution (DE), is first proposed by Storn and Price [133]. Three operations are included, they are mutation, crossover, and selection. The final solution is obtained by continuously repeating these three operations during the evolution process. Different trial vector generation strategies can be selected, as well as three control parameters: crossover rate (CR), scaling factor (F), and population size (NP). Self-adaptive differential evolution (SaDE) is an algorithm both trial vector generation strategies (K strategies in total), and their associated control parameter values can be gradually self-adapted according to their previous experience of generating solutions [134]. More details of SaDE are described in Ref. [134]. There are two issues for SaDE to solve RUTLCSCP: 1) How to encode the solution for SaDE. 2) How to deal with the maximum subproblems in the nonlinear constraints (4.28), so as to ensure the solution belongs to a feasible region. In Section 4.3.1, we give the process to deal with the maximum operator. The following subsections will tackle these two issues.

Robust Node Deployment for Cooperative Coverage Problem  97

4.3.3.1

Encoding

Binary variables are used to describe the decision variables in RUTLCSCP. However, SaDE is a typical algorithm for continuous space optimization. Therefore, an encoding scheme should be developed. When the solution in SaDE satisfies xi < 0.5, the corresponding decision variable in RUTLCSCP is 0, that is, there is no y-facility or z-facility selected at the site i. If the solution in SaDE satisfies xi ≥ 0.5, the corresponding decision variable in RUTLCSCP is 1, that is, there is a y-facility or z-facility selected at the site i. A mapping between discrete space and continue space is realized by this encoding method. This encoding method ensures continuity of the solutions in SaDE, which is important for effective mutation. It is more reliable than SaDE with a solution encoded by binary variables 0-1. 4.3.3.2

Constraints Handling

In this section, four different constraints handling methods are proposed, including a commonly used constraints handling method and three methods based on the knowledge of RUTLCSCP using a repair operator. 1. Feasibility rules-based SaDE Deb [135] proposed a straightforward approach based on three feasibility rules (FR) to handle constraints in constrained optimization. 1) Any feasible solution is preferred to an infeasible one. 2) Among two feasible solutions, the one having better objective value is preferred. 3) Among two infeasible solutions, the one having a smaller constraint violation is preferred. The total constraint violations φ(X) of a solution X [yj , zk ]1×(|J |+|K|) is measured as follows: φ(X) =

X



h

i h

i

H α − 1 − βi1 (y 2 , Γi ) · 1 − βi2 (z 2 , Γi )

,

=

(4.36)

i∈I

where H(·) is equal to the value of its argument if the argument is positive, and zero otherwise. It is obvious that the more total constraint violations of the solution X, the larger value φ(X) is.

98  Metaheuristics for Resource Deployment

Algorithm 4.3 Random repair operator Input: Unfeasible solution X, parameters |I|, |J |, |K|, βi1 (y, Γi ), 2 βi (z, Γi ), α Output: Feasible solution X ∗ 1: while X satisfies the nonlinear constraints do 2: Find the sets for potential facilities to be selected: J ∗ , K∗ . 3: f := f ind(X < 0.5). 4: if |J ∗ | + |K∗ | = 0 then 5: break. 6: end if 7: i := brand(0, 1) × (|J ∗ | + |K∗ |)c. 8: Xf (i) := 1 − Xf (i) . 9: end while 10: X ∗ = X. 11: return Actually, FR is equivalent to a static penalty function method. The equivalent penalty function for RUTLCSCP is shown as F R(X) = F (y, z) + (F (1, 1) + φ(X) − F (y, z)) · G(φ(X)),

(4.37)

where G(·) is equal to 1 if the argument is positive, and zero otherwise. As a result, this FR based SaDE is called SaDE-FR. 2. Repair operator based SaDE Coello [67] indicated that using a repair operator can efficiently solve the combinational optimization problem. The infeasible individuals obtained are repaired to make it feasible. There is no standard heuristic for repair operators; greedy operator, random operator, or other heuristic are the most commonly used repairing operators [67]. In this subsection, three different repair operators are developed. 1) Random repair operator The random repair (RR) operator randomly selects a new candidate position of y-facility or z-facility continuously until nonlinear constraints are satisfied. Algorithm 4.3 presents the procedure of a random repair operator. 2) Ordered repair operator The difference between this ordered repair (OR) operator and random repair operator is that the new candidate position is no randomly selected. It depends on the order of the solutions obtained by SaDE. The

Robust Node Deployment for Cooperative Coverage Problem  99

Algorithm 4.4 Ordered repair operator Input: Unfeasible solution X, parameters |I|, |J |, |K|, βi1 (y, Γi ), 2 βi (z, Γi ), α Output: Feasible solution X ∗ 1: while X satisfies the nonlinear constraints do 2: Find the sets for potential facilities to be selected: J ∗ , K∗ . 3: f := f ind(X < 0.5). 4: i := arg maxi∈{J ∗ ∪K∗ } X. 5: if |J ∗ | + |K∗ | = 0 then 6: break. 7: end if 8: Xf (i) := 1 − Xf (i) . 9: end while 10: X ∗ = X. 11: return candidate position, which is closest to 0.5 will be preferred. Therefore, the ordered repair operator can be described in Algorithm 4.4. 3) Marginal-return-based repair operator The marginal-return-based repair (MRR) operator is derived from MRBCH-1. The difference is MRBCH-1 starts from solution 0 with no candidate position selected, while the marginal-return-based repair operates on a constraint violated solution with some positions selected until constraints satisfied. Algorithm 4.5 presents the procedure of the marginal-return-based repair operator. Actually, these three constraints handling methods add to the computational complexity of SaDE to a different extent. There is an extra lookup operation in the ordered repair operator than in the random repair operator. The extra operation in marginal-return-based repair operator is calculating the marginal return of each possible candidate position. The marginal-return-based repair operator is effective but timeconsuming. Therefore, one needs to balance the total running time and performance of the algorithm. These SaDE algorithms with constraints handling methods are called SaDE-FR, SaDE-RR, SaDE-OR, and SaDE-MRR, respectively.

100  Metaheuristics for Resource Deployment

Algorithm 4.5 Marginal-return-based repair operator Input: Unfeasible solution X, parameters |I|, |J |, |K|, βi1 (y, Γi ), 2 βi (z, Γi ), α Output: Feasible solution X ∗ 1: while X satisfies the nonlinear constraints do 2: Find the sets for potential facilities to be selected: J ∗ , K∗ . 3: f := f ind(X < 0.5). 4: if |J ∗ | + |K∗ | = 0 then 5: break. 6: end if 7: r := f ind(y < 0.5); w := f ind(z < 0.5).    8: v1 (i) := 1 − βi1 (y 1 , Γi ) · 1 − βi2 (z 1 , Γi ) . 9: Find the set of demand nodes with nonlinear constraint violations: I ∗ . 10: for m=1 to |J ∗ | + |K∗ | do F 11: Update δm using Equation 4.32.    12: v2 (i, m) := 1 − βi1 (y 2 , Γi ) · 1 − βi2 (z 2 , Γi ) . P C LHS := [v2 (i, m) − v1 (i)]. 13: δm i∈I ∗

14: 15: 16: 17: 18: 19: 20: 21: 22: 23:

end for P|J ∗ |+|K∗ | C LHS if m=1 δm = 0 then C LHS δm := 1. end if M R = ∆C LHS./∆F . i := arg maxi∈{J ∗ ∪K∗ } M R. Xf (i) := 1 − Xf (i) . end while X ∗ = X. return

4.3.3.3

Complexity Analysis of SaDE

The time complexity of SaDE in this section can be calculated by the time complexity of the original SaDE with constraint handling. The time complexity of the original SaDE is O(K · N P · D) [134], where K is the total number of strategies. As for constraint handling, there are at most L · (L + 1)/2 operations in RR, OR and MRR, where L = |J | + |K|. Therefore, the worst-case time complexity for RR, OR, and MRR methods are O(L2 ). Note that there is extra computation cost in MMR, since there are nonlinear constraints to deal with. As a result,

Robust Node Deployment for Cooperative Coverage Problem  101

the worst-case time complexity of SaDE with constraint handling can be expressed by O((K + L2 ) · N P · D).

4.4

EXPERIMENTS AND DISCUSSION

This section is devoted to the performance investigation of the proposed algorithm. At first, we present a RUTLCSCP test-case generator, which can produce instances of different scales. Then, we solve the problem under different methods, which includes exact solutions for RUTLCSCPLA-RC and approximate solutions for RUTLCSCP. We present the comparative results of these methods and make an analysis. All experiments were carried out on a PC with an Intel Xeon E5 CPU 2.60GHz and 64 GB internal memory. RUTLCSCP-LA-RC problems were implemented in MATLAB R2016a using YALMIP as the modeling language and CPLEX 12.5 with default parameter settings [136]. RUTLCSCP problems were implemented in MATLAB R2016b. We can apply the model to a deployment scenario, which is to deploy sensor nodes and weapon nodes for cooperative coverage. While in RUTLCSCP, the two types of facilities can be sensor nodes and weapon nodes. An effectively damage to the target occurs when the sensor nodes and the weapon nodes can cover the target simultaneously. The weapons need to provide coverage for the targets. Figure 4.2 shows an illustration of sensors and weapons performing cooperative coverage.

:HDSRQ 7DUJHW

6HQVRU

Figure 4.2

coverage.

An illustration of sensors and weapons performing cooperative

102  Metaheuristics for Resource Deployment TABLE 4.1

The test-case for RUTLCSCP.

Instance

(|I|, |J |, |K|)

P1.1–P1.5 (20, 20, 20) P2.1–P2.5 (25, 25, 25) P3.1–P3.5 (30, 30, 30) P4.1–P4.5 (40, 40, 40) P5.1–P5.5 (50, 50, 50) P6.1–P6.5 (60, 60, 60) P7.1–P7.5 (80, 80, 80) P8.1–P8.5 (100, 100, 100) P9.1–P9.5 (120, 120, 120) P10.1–P10.5 (140, 140, 140)

(yr/km, zr/km) (Ax /km, Ay /km) (10, 5) (10, 5) (10, 5) (14, 7) (14, 7) (14, 7) (20, 10) (20, 10) (20, 10) (20, 10)

(25, 25) (25, 25) (25, 25) (50, 50) (50, 50) (50, 50) (100, 100) (100, 100) (100, 100) (100, 100)

4.4.1 Test Instances

Due to the lack of benchmark instances for the RUTLCSCP in literature, we consider the following parameter setting. The fixed costs coefficients building y-facility c1j and z-facility c2k were randomly generated by sampling from a uniform distribution in [0, 100]. The nominal value of probabilities p¯ij and q¯ik were both obtained by sampling from a uniform distribution in [0.9, 1.0]. Deviations for the default probability pˆij and qˆik were both taken from a uniform distribution in [0, 0.1]. Besides, we consider two covering ranges, yr and zr, for these two kinds of facilities. If the Euclidean distance of the demand node and facility location is greater than the covering range, the corresponding probability pij or qik is 0. Each demand node serves as a candidate location site for y-facility and z-facility, i.e., I = J = K. The positions of the demand nodes are randomly generated within the region Ax × Ay. All the RUTLCSCP formulations were solved for the parameters α ∈ {0.8, 0.85, 0.9} and Γ ∈ {0, . . . , |I|}. 10 cases were considered. For each case, we randomly generated five different instances. In total, 10125 derived RUTLCSCP instances were generated. The detailed information of these instances is in Table 4.1. 4.4.2 Analysis of Results 4.4.2.1

Solving RUTLCSCP-LA-RC through CPLEX

First, we use exact solver CPLEX to deal with RUTLCSCP-LARC, which obtains the ε-under-approximate solutions for RUTLCSCP. Three different α = {0.8, 0.85, 0.9} are considered. For each α,

Robust Node Deployment for Cooperative Coverage Problem  103

the results for deterministic GUTLCSCP-LA and robust RUTLCSCPLA-RC are given. We found that the approximation accuracy of the constraints is related to the amount of the β/γ pairs. If we use more pairs, the approximation will be better, which increases the total running time of the algorithm. Therefore, one needs to balance these two conflicts. Here we considered the combination β = [0.001 0.01 0.05 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.85 0.9 0.95 0.99 0.999] based on empirical testing, where γ = 1 − β. The results for RUTLCSCP are presented in Table 4.2 with the following statistics: • Proof of opt: The proportion of instances in which the solution was proven to be optimal. • Time: Arithmetic mean of run times in seconds. • Constraint violation (CV): The proportion of violated constraints in RUTLCSCP with feasible ε-under-approximate solution. • Degree of feasibility: The ratio of feasible solutions without any violated constraint in RUTLCSCP-LA-RC and the total number of instances. From Table 4.2, there exist infeasible solutions for RUTLCSCP-LARC since the degree of feasibility is less than 100%. These instances are especially those with α = 0.9 and Γ ≥ 1. For α = 0.8/0.85, the degree of feasibility is 100%. As a result, the corresponding instances of RUTLCSCP have no solution. Besides, for some instances, the solutions violate the original nonlinear constraints but feasible to RUTLCSCPLA-RC, which are the ε-under-approximate solutions. These instances are shown in Table 4.3 with 12 instances types. φ represents the total constraint violations, and # stands for the proportion of violations with total nonlinear constraints. The solutions for the remaining instances are also the solutions for the original problem RUTLCSCP. Most of the φ of the approximate solutions are at a level of E-4 to E-6, and with only one violated constraint, which means excellent approximation. The corresponding objective value is closely lower than the optimal value, which is an efficient under-approximation and a lower bound. For α = 0.8 and α = 0.85, the objective values are the same as Γ ≥ 1. However, for α = 0.9, the objective values are different under different Γ. Most instances have the same objective value when Γ ≥ 2. Note that in P8.3 (α = 0.9, Γ ≥ 2) marked with † , the objective values are still varying

Instance P1.1–P1.5 P2.1–P2.5 P3.1–P3.5 P4.1–P4.5 P5.1–P5.5 P6.1–P6.5 P7.1–P7.5 P8.1–P8.5 P9.1–P9.5 P10.1–P10.5

Computational results for RUTLCSCP.

α = 0.8

α = 0.85

α = 0.9

Opt. (%) Time CV (%) Opt. (%) Time CV (%) Opt. (%) Time CV (%) Degree of feasibility (%) 100.00 100.00 100.00 100.00 100.00 100.00 80.25 100.00 100.00 100.00

0.14 0.21 0.24 0.32 0.56 1.03 1.48 3.00 4.59 7.29

0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.00 0.00 0.00

100.00 100.00 100.00 100.00 100.00 100.00 100.00 80.00 100.00 80.14

0.17 0.25 0.33 0.48 0.76 1.01 1.60 3.76 5.98 8.33

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.20 0.00 0.14

100.00 98.75 99.20 100.00 100.00 100.00 100.00 96.10 99.18 100.00

0.10 0.20 0.37 0.30 0.44 0.44 0.74 2.81 4.58 5.68

0.00 0.05 0.03 0.00 0.00 0.00 0.00 0.04 0.01 0.00

61.90 61.54 80.65 21.95 41.18 21.31 1.23 40.59 40.50 20.57

104  Metaheuristics for Resource Deployment

TABLE 4.2

Robust Node Deployment for Cooperative Coverage Problem  105 TABLE 4.3

Instance types with constraint violation.

No. Instance 1 2 3 4 5 6 7 8 9 10 11 12

P2.2 P3.1 P8.1 P8.2 P8.2 P8.3 P9.2 P9.3 P7.1 P8.3 P10.3 P8.3

α 0.9 0.9 0.9 0.85 0.9 0.9 0.9 0.9 0.8 0.85 0.85 0.9

Γ

Obj.

φ

0 463.94 7.40E-06 0 309.00 1.29E-05 0 1258.93 1.13E-05 0 1514.56 4.21E-04 0 1603.00 8.01E-06 0 1409.12 1.93E-04 0 1501.84 3.91E-04 0 1312.91 3.97E-06 1+ 1464.12 3.79E-04 1+ 1444.47 2.51E-04‡ 1+ 1408.94 3.44E-04 2+ 1942.41† 2.12E-04‡

# 1/20 1/25 1/100 1/100 1/100 1/100 1/120 1/120 1/80 1/100 1/140 1/100

when Γ ≥ 2. When Γ = 2, 3, 6, 18, 19, the corresponding objective values are 1942.41. The values for the rest are 1947.82. The total constraint violations marked with ‡ means that they vary with different Γ. For example, in P8.3 (α = 0.85, Γ ≥ 1), φ = 2.51E − 04 when Γ = 1; while φ = 6.30E − 04 when Γ ≥ 2. In P8.3 (α = 0.9, Γ ≥ 2), φ = 2.12E − 04 when Γ = 2, 3, 6, 18, 19; while the rest are constraint satisfied. Note that CPLEX can efficiently solve RUTLCSCP-LA-RC, with the computation time less than 10 seconds. Details of the objective value for each instance are presented in Tables 4.4 and 4.5. Therefore, we consider those instances with constraint violations to be solved with other algorithms. The settings for these instances are shown in Table 4.3. For instance, P8.3 (α = 0.90), we only consider two conditions with Γ = 0, 2. We would like to obtain the optimal solution for these instances only with ε-under-approximate solutions. We consider using heuristic methods and evolutionary algorithms to deal with these instances. In summary, a set of 10125 instances are generated and solved with good quality and acceptable time. Up to 74.10% (7502 instances) are solved to optimality, 3.29% (333 instances) are under-approximation, and 22.62% (2290 instances) are with no solution. 4.4.2.2

Comparisons of MRBCH-k with Different k

In this section, we compare the heuristic method mentioned in Section 4.3.2 with different k. The methods to be compared are MRBCH-1 and

Performance of GUTLCSCP-LA and RUTLCSCP-LA-RC using CPLEX (α = 0.8/0.85).

TABLE 4.4 No.

α = 0.8

Instance

α = 0.85

φ

RUTLCSCP-LA-RC (Γ ≥ 1)

φ

GUTLCSCP-LA

φ

RUTLCSCP-LA-RC (Γ ≥ 1)

φ

P1

1 2 3 4 5

474.3629 294.1487 318.8355 489.7330 294.2856

– – – – –

474.3629 339.6178 320.8358 489.7330 324.3616

– – – – –

474.3629 339.6178 320.8358 489.7330 324.3616

– – – – –

519.0568 375.2627 330.1110 522.8859 378.8608

– – – – –

P2

1 2 3 4 5

365.9290 438.4438 606.4277 542.6907 292.5251

– – – – –

376.8503 438.4438 609.2639 563.0972 342.5618

– – – – –

379.8592 452.5023 617.8557 553.3575 299.4029

– – – – –

425.9394 488.6202 630.7410 618.2490 429.2588

– – – – –

P3

1 2 3 4 5

267.7788 451.8846 307.5857 379.0953 329.8849

– – – – –

267.7788 451.8846 307.5857 394.6540 329.8849

– – – – –

271.1718 451.8846 307.5857 380.7506 329.8849

– – – – –

297.0408 498.1496 353.9348 488.6825 371.5057

– – – – –

P4

1 2 3 4 5

705.9167 701.6367 616.4309 612.8232 538.9510

– – – – –

720.5978 701.9407 616.4309 648.1614 579.6432

– – – – –

705.9167 701.9407 622.8818 648.6132 545.4873

– – – – –

801.8550 794.8652 685.3590 700.2201 715.1338

– – – – –

P5

1 2 3 4 5

706.0680 774.4805 869.7677 884.0075 652.1882

– – – – –

736.8343 818.9574 880.0969 884.0075 695.9986

– – – – –

729.8987 788.1456 875.2519 903.0965 662.9431

– – – – –

785.6699 894.5507 962.4296 952.3835 747.2542

– – – – –

P6

1 2 3 4 5

842.0472 579.1228 769.0421 662.4698 914.2975

– – – – –

870.9580 605.0930 845.7655 713.5558 936.2814

– – – – –

849.0636 604.8495 792.9650 737.2067 936.2814

– – – – –

944.3655 661.0475 931.6256 747.6536 1013.9736

– – – – –

P7

1 2 3 4 5

1438.5101 1473.7892 1201.9448 1226.9650 1295.1443

– – – – –

1464.1213 1525.4206 1240.1806 1284.2830 1381.9111

3.7884E-04[1] – – – –

1439.4150 1520.8276 1239.8658 1281.2451 1361.8785

– – – – –

1589.7715 1708.7334 1306.6820 1411.1067 1526.7702

– – – – –

P8

1 2 3 4 5

1147.3186 1411.9737 1259.0093 1512.2938 1707.3206

– – – – –

1183.6658 1476.4988 1306.6879 1520.5349 1738.3552

– – – – –

1186.8944 1514.5623 1309.0383 1550.5642 1738.3552

– 4.2076E-04[1] – – –

1251.9008 1579.9770 1444.4682 1661.2268 1924.4945

– – 2.5109E-04[1]‡ – –

P9

1 2 3 4 5

1387.0334 1332.7672 1158.4098 1407.3265 1073.9292

– – – – –

1448.6764 1341.6886 1193.1532 1417.2579 1088.4732

– – – – –

1387.0334 1367.7394 1190.3716 1453.4219 1093.1843

– – – – –

1590.2737 1535.5245 1299.4727 1645.1386 1175.5917

– – – – –

P10

1 2 3 4 5

1454.3153 1606.0438 1273.0686 1010.7023 1408.6633

– – – – –

1474.6562 1647.4659 1288.6015 1023.2883 1453.3857

– – – – –

1480.6911 1670.0778 1307.5378 1027.1229 1439.7978

– – – – –

1578.4766 1756.9067 1408.9394 1121.8003 1597.3150

– – 3.4461E-04[1] – –

106  Metaheuristics for Resource Deployment

GUTLCSCP-LA

TABLE 4.5 No.

Performance of GUTLCSCP-LA and RUTLCSCP-LA-RC using CPLEX (α = 0.9). α = 0.9

Instance GUTLCSCP-LA

φ

RUTLCSCP-LA-RC (Γ = 1)

φ

Γ≥2

φ

1 2 3 4 5

568.4536 448.0245 332.9167 572.3486 378.8608

– – – – –

655.0008 558.8849 – 703.9558 –

– – – – –

655.4195 558.8849 – 703.9558 –

– – – – –

P2

1 2 3 4 5

425.5046 463.9355 642.7522 618.2490 404.3393

– 7.3998E-06[1] – – –

533.4476 – 789.4080 774.2621 –

– – – – –

533.4476 – 789.4080 774.2621 –

– – – – –

P3

1 2 3 4 5

308.9978 507.3845 361.3366 482.9324 362.3792

1.2934E-05[1] – – – –

501.4917 639.8534 552.2350 – 530.7765

– – – – –

505.0157 639.8534 552.2350 – 531.6609

– – – – –

P4

1 2 3 4 5

807.9073 835.6031 754.6463 714.1892 683.7051

– – – – –

– – 904.4409 – –

– – – – –

– – 907.9853 – –

– – – – –

P5

1 2 3 4 5

863.5646 853.0171 958.7317 980.0904 755.0429

– – – – –

1066.0526 – – – 1144.4302

– – – – –

1073.0273 – – – 1144.4302

– – – – –

P6

1 2 3 4 5

924.9980 692.3795 926.8233 744.7785 986.6679

– – – – –

– – 1106.0948 – –

– – – – –

– – 1106.0948 – –

– – – – –

P7

1 2 3 4 5

1648.7879 1734.7229 1337.1787 1493.1017 1554.6877

– – – – –

– – – – –

– – – – –

– – – – –

– – – – –

P8

1 2 3 4 5

1258.9281 1602.9778 1409.1186 1704.1390 1937.5750

1.1348E-05[1] 8.0089E-06[1] 1.9253E-04[1] – –

– – 1942.4060 – 2405.6326

– – – – –

– – 1942.4065† – 2466.0752

– – 2.1165E-04[1]‡ – –

P9

1 2 3 4 5

1591.0830 1501.8406 1312.9137 1631.7036 1172.8485

– 3.9051E-04[1] 3.9680E-06[1] – –

2350.6057 – – 2166.7789 –

– – – – –

2362.7264 – – 2166.7789 –

– – – – –

P10

1 2 3 4 5

1607.3151 1753.6170 1416.1396 1174.3795 1638.1271

– – – – –

– – – – 2242.2650

– – – – –

– – – – 2242.2650

– – – – –

Robust Node Deployment for Cooperative Coverage Problem  107

P1

108  Metaheuristics for Resource Deployment TABLE 4.6

Performance of heuristics with different rules.

No.

MRBCH-1

MRBCH-2

1

Performance CPU time/s

767.3710[2] 0.2797

528.1744[1] 7.8603

2

Performance CPU time/s

371.5865[2] 0.1034

322.5433[1] 9.7410

3

Performance CPU time/s

1495.3306[2] 2.1275

1475.5867[1] 339.8587

4

Performance 1790.1620[1] CPU time/s 2.2045

5

Performance CPU time/s

1799.5735[2] 2.1021

1736.0346[1] 345.0099

6

Performance CPU time/s

2929.5617[2] 2.8782

2905.8785[1] 491.6630

7

Performance 2602.0448[1] CPU time/s 3.8646

8

Performance CPU time/s

1706.4869[2] 3.3891

1516.6453[1] 525.4174

9

Performance CPU time/s

2978.2781[2] 9.3145

2890.9753[1] 1615.6249

10

Performance CPU time/s

2919.9311[2] 18.2805

2898.4995[1] 3318.9452

11

Performance 1612.0967[1] CPU time/s 35.4430

12

Performance CPU time/s

Mean rank

1985.0408[2] 399.2773

2669.0125[2] 699.0834

1699.9648[2] 7068.7360

3211.9261[2] 19.5085

3119.1563[1] 3359.2429

1.7500

1.2500

MRBCH-2. We consider the instances with ε-under-approximate solutions provided in Section 4.4.2.1 to test these methods. Parameter PN is set to 200. Since heuristic methods are exact methods, that is, solutions are stable compared to evolutionary algorithms. We only run these methods one time, and the results are shown in Table 4.6. For these 12 instances, the mean rank can be seen in Table 4.6. In terms of method performance, MRBCH-2 shows significant advantages over MRBCH-1. As for the computation time, MRBCH-1 is significantly

Robust Node Deployment for Cooperative Coverage Problem  109

faster than MRBCH-2. For the large-scale cases, e.g., No. 11, the one time running CUP time for MRBCH-1 is only 35.4430s. It is better than the breadth k greater than 1. It is because only one solution is obtained by heuristic through depth-first search. Since PN = 200, the computation time of MRBCH-2 is about 200 times to MRBCH-1. In No. 10, 11, and 12, the computation time of MRBCH-2 is around 3000s, 7000s, and 3000s. These instances are relatively hard to solve. Therefore, MRBCH-1 is selected when efficiency is considered; otherwise, MRBCH2 is a better choice among these two heuristics. 4.4.2.3

Comparisons of SaDE and Its Variants

Various SaDE variants (SaDE-FR, SaDE-RR, SaDE-OR, and SaDEMRR) are compared in this section. We consider the instances with ε-under-approximate solutions provided in Section 4.4.2.1 to test these methods. The size of the population (N P ) is set to 100. The maximum function evaluations (F ES) is set to 1000N P (100000). Since there is a lot of computation cost in SaDE-MMR, 1/10F ES is set. The independent runs for each algorithm on each instance are set to 30. The results are shown in Tables 4.7 and 4.8 with “mean value ± standard derivation”. Table 4.7 shows that SaDE-OR performs better than SaDE-FR, SaDE-RR, and SaDE-MRR in terms of objective value. SaDE-OR performs the best in No. 4, 6, and 9–12, which shows its advantages in large scale instances with uncertainty. SaDE-MRR performs the best in No. 1, 3, 5, 7, and 8. SaDE-RR performs the best in No. 1 and 2, which shows its advantages in small scale instances. As reflected in Table 4.8, SaDE-FR shows advantages in the computation time among the compared algorithms. Although the F ES in SaDE-MRR is 1/10 of other algorithms, the computation time is the largest among the compared algorithms. The computation time of SaDE-FR is the smallest. 4.4.2.4

Comparisons on RUTLCSCP

This section analyses the algorithms for RUTLCSCP proposed in Section 4.3, including MRBCH-1/2, SaDE-OR, and the SaDE-OR, with an initial solution provided by MRBCH-1 called SaDE-OR-MRBCH-1. The independent runs for each algorithm on each instance are set to 30. We also consider the instances with ε-under-approximate solutions provided in Section 4.4.2.1 to test these methods. Experimental results are presented in Table 4.9.

Performance of different SaDE variants on the objective value.

No.

SaDE-FR

SaDE-RR

SaDE-OR

SaDE-MRR

1 2 3 4 5 6 7 8 9 10 11 12

470.0150±0.0000[1] 311.1725±1.4382[2] 2009.9152±82.2111[4] 2087.6061±113.5483[4] 2337.7006±81.6049[4] 1909.7948±89.7016[4] 2708.2080±152.2753[4] 2345.6924±136.4253[4] 1742.9644±88.5231[4] 1943.0055±100.0086[3] 2513.0352±137.4808[3] 2480.9007±82.5474[3]

470.0150±0.0000[1] 310.8315±1.5972[1] 1849.4947±89.9197[3] 1984.8705±61.3581[3] 2197.1436±69.5882[3] 1813.9908±74.7431[2] 2545.8655±134.1215[3] 2140.6050±103.5733[3] 1689.2255±52.9304[2] 1840.4838±69.8993[2] 2363.1439±157.7812[2] 2294.9488±69.9599[2]

470.7445±2.2259[4] 312.0493±3.2767[3] 1754.6315±133.2989[2] 1862.0502±113.6931[1] 2093.6224±124.3532[2] 1732.3302±102.1497[1] 2407.4260±213.1517[2] 2004.1130±133.1352[2] 1611.4767±45.5028[1] 1755.6506±83.2208[1] 2050.6533±228.5131[1] 2112.1477±72.4984[1]

470.0150±0.0000[1] 318.9879±3.9791[4] 1644.3597±70.5243[1] 1905.5688±64.6383[2] 2027.2915±91.9501[1] 1885.1313±69.3455[3] 2245.1181±156.5490[1] 1998.4898±107.2639[1] 1717.1791±66.6805[3] 2245.4586±140.7304[4] 3142.2738±242.6339[4] 3046.9504±130.7568[4]

Mean rank

3.3333

2.2500

1.7500

2.4167

110  Metaheuristics for Resource Deployment

TABLE 4.7

Performance of different SaDE variants on the computation time (unit: s)

No. 1 2 3 4 5 6 7 8 9 10 11 12 Mean rank

SaDE-FR

SaDE-RR

SaDE-OR

SaDE-MRR

13.8847±0.5139[1] 29.9770±3.1120[4] 24.3051±1.6439[2] 29.4965±2.4290[3] 14.8590±0.6010[1] 35.5464±4.1293[3] 28.3865±3.1374[2] 38.0035±2.8508[4] 25.5652±0.4356[1] 430.2697±20.3793[3] 381.7874±36.9957[2] 637.1826±41.1209[4] 26.1644±0.7093[1] 465.3567±30.4429[3] 399.6213±39.8753[2] 648.5964±36.6387[4] 26.5587±0.6141[1] 471.5666±16.4303[3] 420.4063±26.3148[2] 714.2774±31.6474[4] 25.0985±0.3922[1] 433.3224±21.2567[3] 360.2605±39.8552[2] 631.5482±24.9610[4] 28.1480±0.3419[1] 640.2019±19.1917[3] 559.1861±47.3903[2] 926.1545±52.9884[4] 28.0964±0.2910[1] 630.4208±21.9324[3] 557.2418±38.8817[2] 916.8041±43.9620[4] 95.6913±1.8267[1] 2028.8059±76.0179[3] 1570.3560±257.0513[2] 2807.1188±124.9703[4] 130.0330±2.8577[1] 3185.4410±113.3820[3] 2655.8181±302.9315[2] 3618.8813±12.6941[4] 230.8249±2.1501[1] 8141.5441±291.1042[4] 6471.5499±825.0634[2] 8043.9807±32.5771[3] 135.3849±0.9814[1] 3426.1670±165.6427[3] 2432.9870±322.4640[2] 3622.8481±14.2747[4] 1.0000

3.1667

2.0000

3.8333

Robust Node Deployment for Cooperative Coverage Problem  111

TABLE 4.8

TABLE 4.9

Overall performances of the proposed algorithms. Algorithm

Performance

CPU time/s

Best

Worst

Mean±Std.

Gap (%)

1

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

767.3710 528.1744 470.0150 470.0150

767.3710 528.1744 477.3099 470.0150

767.3710±0.0000 528.1744±0.0000 470.7445±2.2259 470.0150±0.0000

39.5422[4] 12.1624[3] 1.4464[2] 1.2935[1]

0.2116±0.0964[1] 6.3715±2.1054[2] 24.3051±1.6439[3] 25.5329±0.9943[4]

2

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

371.5865 322.5433 310.2801 310.2801

371.5865 322.5433 320.8281 325.6428

371.5865±0.0000 322.5433±0.0000 312.0493±3.2767 318.3404±4.6934

16.8436[4] 4.1996[3] 0.9779[1] 2.9348[3]

0.0860±0.0246[1] 8.0281±2.4225[2] 28.3865±3.1374[4] 28.1589±1.4869[3]

3

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

1495.3306 1475.5867 1457.9738 1277.9238

1495.3306 1475.5867 1995.1444 1290.5028

1495.3306±0.0000 1475.5867±0.0000 1754.6315±133.2989 1278.9156±3.2031

15.8094[3] 14.6829[2] 28.2511[4] 1.5628[1]

2.1404±0.0181[1] 358.8240±26.8210[3] 381.7874±36.9957[4] 180.1532±6.5480[2]

4

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

1790.1620 1985.0408 1677.6381 1585.8900

1790.1620 1985.0408 2053.3145 1592.0808

1790.1620±0.0000 1985.0408±0.0000 1862.0502±113.6931 1588.4487±1.3606

15.3952[2] 23.7012[4] 18.6616[3] 4.6515[1]

2.2378±0.0471[1] 409.9210±15.0525[4] 399.6213±39.8753[3] 195.3734±11.9864[2]

5

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

1799.5735 1736.0346 1861.9379 1648.1208

1799.5735 1736.0346 2311.9028 1686.2862

1799.5735±0.0000 1736.0346±0.0000 2093.6224±124.3532 1659.6658±16.6839

10.9246[3] 7.6644[2] 23.4352[4] 3.4156[1]

2.1367±0.0489[1] 357.3242±17.4151[3] 420.4063±26.3148[4] 203.2409±7.5865[2]

6

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

2929.5617 2905.8785 1525.5343 1441.1626

2929.5617 2905.8785 1969.7462 1472.0863

2929.5617±0.0000 2905.8785±0.0000 1732.3302±102.1497 1459.1333±7.6654

51.9000[4] 51.5080[3] 18.6576[2] 3.4277[1]

2.8886±0.0146[1] 508.5328±23.8575[4] 360.2605±39.8552[3] 200.9335±26.2492[2]

7

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

2602.0448 2669.0125 1956.2105 1546.6149

2602.0448 2669.0125 2828.6473 1589.7109

2602.0448±0.0000 2669.0125±0.0000 2407.4260±213.1517 1560.6992±11.7283

42.2823[3] 43.7305[4] 37.6163[2] 3.7713[1]

4.1144±0.3533[1] 752.4439±75.4632[4] 559.1861±47.3903[3] 272.0032±30.9750[2]

8

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

1706.4869 1516.6453 1729.9629 1347.8271

1706.4869 1516.6453 2223.2540 1380.5171

1706.4869±0.0000 1516.6453±0.0000 2004.1130±133.1352 1355.5439±9.8453

23.0634[3] 13.4330[2] 34.4890[4] 3.1449[1]

3.7488±0.5087[1] 565.0749±56.0841[4] 557.2418±38.8817[3] 265.6452±26.6899[2]

9

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

2978.2781 2890.9753 1537.4357 1488.4377

2978.2781 2890.9753 1716.7260 1516.8433

2978.2781±0.0000 2890.9753±0.0000 1611.4767±45.5028 1498.5761±8.0898

50.8400[4] 49.3555[3] 9.1441[2] 2.2992[1]

8.7783±0.7584[1] 1510.5732±148.5656[3] 1570.3560±257.0513[4] 1043.4591±135.1675[2]

10

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

2919.9311 2898.4995 1615.1811 1488.5673

2919.9311 2898.4995 1965.1948 1528.0656

2919.9311±0.0000 2898.4995±0.0000 1755.6506±83.2208 1504.9202±12.1526

50.5307[4] 50.1650[3] 17.7246[2] 4.0170[1]

17.2004±1.5274[1] 3072.0004±349.2327[4] 2655.8181±302.9315[3] 1629.9096±181.8571[2]

11

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

1612.0967 1699.9648 1636.7624 1431.5429

1612.0967 1699.9648 2391.4088 1438.8330

1612.0967±0.0000 1699.9648±0.0000 2050.6533±228.5131 1431.8970±1.4443

12.6021[2] 17.1195[3] 31.2931[4] 1.6033[1]

34.2527±1.6833[1] 6718.7628±494.9368[4] 6471.5499±825.0634[3] 2986.9002±108.8711[2]

12

MRBCH-1 MRBCH-2 SaDE-OR SaDE-OR-MRBCH-1

3211.9261 3119.1563 2002.5262 1985.9519

3211.9261 3119.1563 2307.9934 2016.7551

3211.9261±0.0000 3119.1563±0.0000 2112.1477±72.4984 1996.0763±9.3111

39.5252[4] 37.7265[3] 8.0364[2] 2.6888[1]

19.6786±0.2406[1] 3192.9615±235.1573[4] 2432.9870±322.4640[3] 1969.4904±221.5084[2]

112  Metaheuristics for Resource Deployment

No.

Robust Node Deployment for Cooperative Coverage Problem  113

The overall rank of the four algorithms is shown in Table 4.10. SaDEOR-MRBCH-1 has the best performance. As for the computation time, MRBCH-1 is better than other algorithms due to its effectiveness in generating a solution based on a heuristic. Using this method with random initialization in SaDE-OR-MRBCH-1 will obtain a better start of the evolutionary process without compromising the diversity of the population and consuming much time. There is a significant improvement in performance and computation time. The gaps for SaDE-OR-MRBCH1 are all below 5%, where the gap is a ratio computed as (f − f ∗ )/f (f is the objective value obtained by the proposed algorithms and f ∗ is the under-approximation of the optimal objective value). Among the instances, SaDE-OR performs better than SaDE-OR-MRBCH-1 in No. 2, while SaDE-OR-MRBCH-1 outperforms the other algorithms in the rest instances. Meanwhile, the gaps obtained by the other algorithms are much larger. As a result, SaDE-OR-MRBCH-1 is an effective algorithm to solve RUTLCSCP. Discussion: Why the computation time of MRBCH-k and the proposed SaDE is much longer than CPLEX? Why SaDE-OR-MRBCH-1 has a shorter computation time than other SaDE variants? The reasons are as follows: (1) There are nonlinear constraints with two subproblems in RUTLCSCP, while the corresponding RUTLCSCP-LA-RC is an integer linear programming problem with constraints relaxed by linear approximation. The constraints are no longer hard to solve. (2) The initial population with MRBCH-1 will provide great help to SaDE-OR in searching for the feasible space. The unfeasible region with constraints violation is prohibited, which decreases a large amount of computation in repairing the infeasible individuals. TABLE 4.10

Overall rank.

Algorithm/metric

MRBCH-1

MRBCH-2

SaDE-OR

SaDE-OR-MRBCH-1

Performance CPU time

3.3333 1.0000

2.9167 3.4167

2.6667 3.3333

1.0833 2.2500

4.5 CONCLUSION This chapter aims at the robust node deployment for cooperative coverage problem. An extension of the set covering problem (SCP) called the

114  Metaheuristics for Resource Deployment

robust uncertain two-level cooperative set covering problem (RUTLCSCP) by integrating uncertainty in covering demand nodes is proposed. The concepts of probabilistic, robust optimization, and cooperative covering are combined. A compact mixed-integer linear programming formulation for the RUTLCSCP is proposed by utilizing the strong duality theorem and constraint relaxation. This model can be applied in node deployment with two types of nodes with discrete space-based deployment. The uncertainty of the covering probability is considered and controlled by the budget of uncertainty. CPLEX is used to solve the relaxed formulation. For those instances which cannot be solved optimally, subproblem dealing, heuristic, and an improved SaDE are proposed. Computational experiments demonstrate that the RUTLCSCP can be efficiently solved by CPLEX with optimal solutions and a few under-approximate solutions. SaDE-OR-MRBCH-1 is an effective method to obtain nearoptimal solutions that are close to under-approximate solutions. In the future, over-approximate solutions with more constraints, and less feasible regions are likely to be investigated. Besides, new exact, heuristic algorithms, and new reformulation can be considered. Meanwhile, the proposed model can be applied in many real-world applications, e.g., collaborative task assignment [137], joint allocation of heterogeneous stochastic resources [138], etc. These assignment problems considering deployment can be formulated as a bi-level programming problem. The deployment scheme decides the feasibility of the assignment. Thus, this is also a challenging research direction. The uncertainty considered in this chapter is a known interval, while distributionally robust optimization (RDO) constructs an uncertainty set of probability distributions from uncertainty data through statistical inference and big data analytics [139]. It is a new data-driven optimization paradigm, and the node deployment version is needed.

CHAPTER

5

Fuzzy Node Deployment for Cooperative Coverage Problem

The cooperative coverage problem in the previous chapter is formulated through robust optimization. However, the fuzzy variable is another way to describe uncertainty, and the problem in this chapter is formulated as a fuzzy programming problem. In this chapter, a two-level cooperative node deployment problem with two objectives is formulated with a fuzzy target threat. This model is formulated as the fuzzy two-level cooperative covering node deployment problem aiming at minimizing the fuzzy conditional value-at-risk of the target threat and minimizing the deployment cost. Two state-of-the-art decomposition-based multiobjective evolutionary algorithms (i.e., MOEA/D and DMOEA-εC) are utilized to solve the proposed bi-objective problem. Since the number of deployed nodes is unknown a priori, the length of each individual in the population should be varied during the evolution process. New mutation and crossover operators are designed for this variable-length optimization problem. Computational results between two improved algorithms with variable individual size (VIS), i.e., MOEA/D-VIS and DMOEAεC-VIS, demonstrate that they have successfully solved the problem and MOEA/D-VIS shows better performance.

DOI: 10.1201/9781003202653-5

115

116  Metaheuristics for Resource Deployment

5.1 INTRODUCTION The target threat is typically an uncertain value or fuzzy variable. Therefore, we cannot have an exact value about the threat of the targets. Credibility theory [140] provides ways to solve this kind of problem. Different from the uncertainty variables in Section 3 and 4, fuzzy variable is used to represent uncertainty in this chapter. Refs. [64, 65] analyze the node deployment problem with fuzzy variables, including the height, speed of the target, and the distance between the target and the node. Fuzzy chance-constraint programming model, fuzzy dependent-chance programming model, and fuzzy mean-entropy model are proposed. Besides, most of the decision makers for node deployment are riskaverse. Therefore, it is necessary to formulate a location model with risk control. The existing risk measurements under fuzzy environment includes mean-variance [141], mean-semivariance [142], mean-entropy [143], mean-variance-skewness [144], and fuzzy value-at-risk (VaR) [145]. In this chapter, fuzzy conditional value-at-risk (CVaR) [146] is applied to build the node deployment model under fuzzy environment with fuzzy remaining target threat. In this chapter, a two-level cooperative node deployment problem with fuzzy target threats is proposed. It is a bi-objective problem with target threat and deployment cost to be minimized. Decompositionbased multi-objective evolutionary algorithms (MOEAs) are applied with fuzzy simulation, and variable individual size is utilized in solution representation. Simulation experiments have shown the effectiveness of the proposed algorithms, which are novel methods for the cooperative node deployment problem. We summarize our contributions as follows: 1. A novel bi-objective two-level cooperative node deployment problem under fuzzy environment is formulated. The first objective function aims to minimize the fuzzy CVaR of the target threat and the second objective function minimizes the deployment cost. The node deployment problem is formulated as a variable-length optimization problem. 2. Two decomposition-based multi-objective evolutionary algorithms (i.e., MOEA/D and DMOEA-εC) are adopted to solve the biobjective fuzzy node deployment model. The algorithms adopt a variable individual size to represent the possible deployment solution and provide a new method for solution updating. This chapter is organized as follows. In Section 5.2, we describe the

Fuzzy Node Deployment for Cooperative Coverage Problem  117

preliminaries and formulation of the fuzzy node deployment for cooperative coverage. Several decomposition-based multi-objective evolutionary algorithms based on MOEA/D and DMOEA-εC are proposed in Section 5.3. Experiments and discussion are provided in Section 5.4. Section 5.5 summarizes the chapter.

5.2

PROBLEM FORMULATION

Fuzzy random variables are the basis to describe the fuzzy uncertainty, and fuzzy set theory has been applied in various engineering problems with fuzzy uncertainty recently. In this section, the possibility, necessity, and credibility measurements of fuzzy variables are introduced. Meanwhile, before the introduction of the fuzzy Conditional Value-at-Risk, we briefly review another risk measurement, which is, Value-at-Risk. For more detailed introduction of the fuzzy variables, one may refer to Refs. [92, 145, 147, 148]. Definition 5.1 [149] Suppose ξ is a fuzzy variable whose membership function is µξ , and r is a real number. Then the credibility function of event ξ ≤ r can be expressed as: Cr{ξ ≤ r} =

1 [Pos{ξ ≤ r} + Nec{ξ ≤ r}] 2

(5.1)

where Pos{.} and Nec{.} represent the possibility and necessity measurements [149], defined as: Pos{ξ ≤ r} = sup µξ (t)

(5.2)

Nec{ξ ≤ r} = 1 − sup µξ (t)

(5.3)

t≤r

t>r

The credibility measure is a self-dual set function, i.e., Cr{ξ ≤ r} = 1 − Cr{ξ > r}. Definition 5.2 [147] Suppose that ξ is a fuzzy variable which represents the loss of a particular investment, then the Value-at-Risk (VaR) of ξ under the confidence level (1 − β) can be written as: VaR1−β [ξ] = sup{λ|Cr(ξ ≥ λ) ≥ β}

(5.4)

where, β ∈ (0, 1). Equation 5.4 tells that the greatest loss of the investment under confidence level (1 − β) is λ.

118  Metaheuristics for Resource Deployment

5.2.1 Fuzzy Conditional Value-at-Risk

The difference between Conditional Value-at-Risk (CVaR) and VaR is that CVaR considers the average value exceed VaR at the confidence level (1 − β). The definition of CVaR is: Definition 5.3 Suppose that ξ is a fuzzy variable which represents the loss of a particular investment, then the Conditional Value-at-Risk of ξ under the confidence level (1 − β) can be expressed as CVaR1−β [ξ] = E[ξ|ξ ≥ VaR1−β [ξ]]

(5.5)

where, β ∈ (0, 1). Meanwhile, CVaR is equivalent to Expected Shortfall (ES) [150]. Therefore, CVaR can also be written as: CVaR1−α [ξ] =

1 α

Z 0

α

VaR1−β [ξ]dβ

(5.6)

where α ∈ (0, 1), β ∈ (0, 1). 5.2.2 Mathematical Model

This section provides the mathematical model of a bi-objective fuzzy node deployment for cooperative coverage problem. There are at most R sensor nodes and W weapon nodes deployed in a Ax × Ay 2D plane. The deployment decision is described r w w w r r as [(xr1 , y1r ), (xr2 , y2r ), . . . , (xrR , yR ), (xw 1 , y1 ), . . . , (xW , yW )], where (xj , yj ) w denotes the position of the jth sensor node and (xw k , yk ) denotes the position of the kth weapon node. There are |I| targets randomly generated. The target will be damaged effectively if the sensor node and the weapon node can cover the target simultaneously. Therefore, two objectives are considered, which are minimizing the fuzzy CVaR of the target threat and minimizing the deployment cost. Obviously, the first objective ensures the fully utilized of two types of the nodes, and the second objective eliminates the deployment cost of the nodes. Continuous space-based deployment is considered in this chapter. We have the following assumptions: • The target can be detected by multiple sensor nodes. • The target can be attacked by multiple weapon nodes.

Fuzzy Node Deployment for Cooperative Coverage Problem  119

• Each sensor node can detect multiple targets simultaneously. • Each weapon node can attack multiple targets simultaneously. • The detection probability and the deployment cost are the same among the sensor nodes. • The kill probability and the deployment cost are the same among the weapon nodes. • An effective attack to the target from the weapon node is under the effective guidance of the sensor node, which is a cooperative covering. The target should be within the sensing range of the sensor node and the attacking range of the weapon node. • The threat of target is a fuzzy variable with known distribution. Table 5.1 summarizes all the notations that are used throughout this chapter. CVaR-based fuzzy two-level cooperative node deployment problem (CVaR-FTLCNDP) is given as follows: "

min F1 (x, y) = CVaR1−β

#

zi ξi

(5.7)

f wk

(5.8)

X i∈I

min F2 (x, y) =

X

f rj +

j∈J



X k∈K

"

s.t. zi = 1 − 1 −

Y j∈J

(1 − pij ) 1 −

# Y

(1 − qik )

∀i ∈ I

(5.9)

k∈K

|J | ≤ R

(5.10)

|K| ≤ W

(5.11)

0 ≤ xri ≤ Ax 0 ≤ yir ≤ Ay 0 ≤ xw i ≤ Ax w 0 ≤ y i ≤ Ay w xri , yir , xw i , yi

∈R

∀i ∈ I

(5.12)

∀i ∈ I

(5.13)

∀i ∈ I

(5.14)

∀i ∈ I

(5.15)

∀i ∈ I

(5.16)

where Equation 5.7 is to minimize the fuzzy CVaR of the remaining threat. Equation 5.8 is to minimize the deployment cost of the node deployment, including the sensor and weapon nodes. The zi in Equation 5.9 denotes the survival probability of the target i. Equations 5.10

120  Metaheuristics for Resource Deployment TABLE 5.1

Summary of notations.

Symbol Indices i∈I j∈J k∈K Parameters I J K Ax Ay R W rr wr f rj f wk pij qik ξi (xti , yit ) drij dwik

Description Indices for targets Indices for sensor nodes Indices for weapon nodes Set of targets Set of sensor nodes Set of weapon nodes Range of deployment space in x-axis Range of deployment space in y-axis Total number of sensor nodes Total number of weapon nodes Sensing range of sensor node Attacking range of weapon node Deployment cost of sensor node j Deployment cost of weapon node k Detection probability of sensor node j to target i Kill probability of weapon node k to target i under effective guidance Threat of target i Position of target i Distance between target i and sensor node j Distance between target i and weapon node k

Decision variables (xrj , yjr ) Position of sensor node j w (xw , y ) Position of weapon node k j j

and 5.11 restrict the total number of sensor and weapon nodes, respectively. Equations 5.12–5.15 guarantee the location range of the sensor and weapon nodes. Equation 5.16 restricts the decision variables to be real numbers.

Fuzzy Node Deployment for Cooperative Coverage Problem  121

5.2.3 Some Properties on CVaR-FTLCNDP

In some special conditions, when all target threats can be taken as fuzzy variables that follow the same type of distribution (e.g., trapezoidal or triangular distribution), the theorem below enables us to solve CVaRFTLCNDP easily. Theorem 5.1 Suppose that the threats of all the targets are independent trapezoidal fuzzy variables as ξi = (ai , bi , ci , di ). Then, for any zi , when β ≤ 0.5, VaR1−β

" X

#

zi ξi =

zi [(1 − 2β)di + 2βci ]

(5.17)

zi [(ci − di )β + di ]

(5.18)

zi [(2 − 2β)bi + (2β − 1)ai ]

(5.19)

i∈I

i∈I

CVaR1−β

X

" X

#

zi ξi =

i∈I

X i∈I

Otherwise, when β > 0.5, "

VaR1−β

# X

zi ξi =

i∈I

CVaR1−β

" X i∈I

X i∈I

#

zi ξi =

X

zi [(ai − bi )β + 2bi − ai ]

(5.20)

i∈I

Proof 5.1 The VaR part of Theorem 5.1 can be proved according to Ref. [145]. Note that in Ref. [145], the loss function is −(x1 ξ1 + x2 ξ2 + P · · · + xn ξn ), while the loss function in this model is i∈I zi ξi . Therefore, ai and bi should be −di and −ci when β ≤ 0.5, and ci and di should be −bi and −ai when β > 0.5. As for CVaR, it can be calculated by Equation 5.6. Remark 5.1 In Theorem 5.1, when bi = ci , the trapezoidal fuzzy variables become the triangular ones. Therefore, another similar theorem can be obtained when the threats of all the targets are independent triangular fuzzy variables.

122  Metaheuristics for Resource Deployment

5.2.4 Linear Approximation of CVaR-FTLCNDP

In this section, the nonlinear objective function F1 is linearized. Actually, we only need to linear approximate zi in Equation 5.9. Equation 5.9 is reformulated as follows: 

"

zi = 1 − 1 −

Y

#

(1 − pij ) 1 −

j∈J

=

Y

(1 − pij ) +

j∈J

(1 − qik )

Y

(5.21)

k∈K

Y

(1 − qik ) −

Y

(1 − pij ) ·

j∈J

k∈K

Y

(1 − qik )

(5.22)

k∈K

Set mi = j∈J (1 − pij ), ni = k∈K (1 − qik ), then zi = mi + ni − mi ni for all i ∈ I. Taking logarithms on both sides, then we can obtain P P ln(mi ) = j∈J ln(1 − pij ), and ln(ni ) = k∈K ln(1 − qik ). Set m0i = 0 0 − ln(mi ), n0i = − ln(ni ), then e−mi = mi , e−ni = ni . Given above, we can obtain: Q

Q

0

0

0

0

zi = e−mi + e−ni − e−mi −ni

(5.23)

Then, the objective function in Equation 5.7 can be reformulated as: F1 (x, y) = CVaR

" X



−m0i

ξi e

−n0i

+e

−m0i −n0i

−e



#

(5.24)

i∈I

Therefore, the key issue to approximate the objective function F1 is to approximate the exponential function e−x (x ≥ 0). We use piecewiselinear convex function to obtain the lower bound function e−x Lower and upper bound function e−x . U pper 1. The lower bound function e−x Lower can be calculated as follows: We draw the tangents of e−x at a series of values (0 ≤ x1 < x2 < · · · < xn < +∞) and obtain the tangents f1L (x, y, x1 ), f2L (x, y, x2 ), . . . , fnL (x, y, xn ). The ith tangent line function is y − e−xi = −e−xi (x − xi ) fiL (x, y, xi )

:y=e

−xi

−xi

−e

(5.25)

(x − xi )

(5.26)

Therefore, the lower bound function e−x Lower can be expressed as: e−x Lower =

max

i∈{1,...,n}



e−xi − e−xi (−x − xi )



(5.27)

2. The upper bound function e−x U pper can be calculated as follows:

Fuzzy Node Deployment for Cooperative Coverage Problem  123

We draw the secants of e−x at every two consecutive points (0 ≤ x1 < x2 < · · · < xn < +∞) and obtain the secants f2U (x, y, x1 , x2 ), f3U (x, y, x2 , x3 ), . . . , fnU (x, y, xn−1 , xn ). The ith secant line function is e−xi − e−xi−1 (x − xi−1 ) xi − xi−1 e−xi − e−xi−1 fiL (x, y, xi−1 , xi ) : y = e−xi−1 + (x − xi−1 ) xi − xi−1 y − e−xi−1 =

(5.28) (5.29)

Therefore, the upper bound function e−x U pper can be expressed as: )

(

e−x U pper

=

max

i∈{2,...,n}

e

−xi−1

e−xi − e−xi−1 (−x − xi−1 ) + xi − xi−1

(5.30)

As a result, the lower and upper bound of F1 are expressed as: "

F1L (x, y) = CVaR

X

ξi



−m0i eLower

+

−n0i eLower

+

−n0i eU pper

−

−m0i −n0i eLower



−

−m0i −n0i eU pper



#

(5.31)

i∈I

"

F1U (x, y)

= CVaR

X

ξi



−m0i eU pper

#

(5.32)

i∈I

Figure 5.1 shows the under-approximation and upper-approximation of e−x by piecewise-linear convex functions. e−x is evenly divided into n segments (e.g., n = 5, 10, and 20) and x ∈ [0, 8]. The more segments, the better approximation of the piecewise-linear convex function, which brings higher computation cost. Remark 5.2 Using logarithms and linear segments to approximate a function like Equation 5.21 has been widely used [151]. The difference in this section is that the function to be approximate is more complex and both lower and upper bound of the corresponding function are obtained.

5.3 SOLUTION ALGORITHMS Actually, when the target threats are fuzzy variables with different distributions, and they may not always be independent of each other. It is not possible to calculate the fuzzy CVaR using the theorem in Section 5.2.3. Therefore, fuzzy simulation is used to approximate CVaR, and improved decomposition-based MOEAs are proposed to deal with the proposed model.

124  Metaheuristics for Resource Deployment n=5

1

n = 10

1 e-x Lower

0.8

e

e-x Lower

0.8

-x

e

e-x Upper

-x

e-x Upper

y

0.6

y

0.6 0.4

0.4

0.2

0.2

0

0 0

2

4

6

8

0

2

x

4

6

8

x

(a) n = 5

(b) n = 10 n = 20

1

e-x Lower

0.8

e

-x

e-x Upper

y

0.6 0.4 0.2 0 0

2

4

6

8

x

(c) n = 20

The under-approximation and upper-approximation of e−x with different number of segments when x ∈ [0, 8]. Figure 5.1

5.3.1 Fuzzy Simulation

In Section 5.2, the calculation of fuzzy CVaR for any single fuzzy variables is discussed, and some properties for the variables with the same distribution are provided. However, the fuzzy variables we will deal with are under different distributions, e.g., ξi ∈ {ξ1 , ξ2 , . . . , ξn }, where i = 1, 2, . . . , n. It is hard to solve through the theorem. The fuzzy simulation proposed by Liu [92] made it possible to do some complex calculations through fuzzy variables under different distributions. The essence of fuzzy simulation is to approximate the membership distribution of the fuzzy variable ξi by a series of discrete fuzzy vector ζ, so that the corresponding CVaR in the objective function is obtained [148]. Suppose that ξi (i = 1, 2, . . . , n) is a series of fuzzy variables with Q supports ni=1 [Li , Ui ], where Li and Ui are the lower and upper bounds of ξi . The fuzzy CVaR is calculated as follows [148].

Fuzzy Node Deployment for Cooperative Coverage Problem  125

1) Divide each fuzzy variable ξi into l parts r ζir = Li + (Ui − Li ) l

(5.33)

where r and l are integers, and 0 ≤ r ≤ l. 2) Calculate the membership degree corresponds to each ζir in order to approximate the membership function of ξi n

o

µξi → µ(ζi1 ), . . . , µ(ζir ), . . . , µ(ζil ) . 3) Simulate the membership distribution of

P

i∈I

(5.34)

zi ξi

a) Calculate the possible combinations of i∈I zi ξi , i.e., randomly select ζir to describe ξi and compute P

oj = z1 ζ1r + · · · + zi ζir + · · · + zn ζnr

(5.35)

where 0 ≤ r ≤ l and 1 ≤ j ≤ N . The membership degree of oj is µ(oj ) = µ(ζ1r ) ∧ · · · ∧ µ(ζir ) ∧ · · · ∧ µ(ζnr )

(5.36)

b) Record all oj and the related membership degrees. If some oj are all equal t, then the one with the largest membership degree is selected to construct the membership distribution P of i∈I zi ξi , i.e., µ(t) = sup µ(oj )

(5.37)

where oj = t. 4) Calculate the possibility and credibility measurements of based on Equations 5.1–5.3. 5) Calculate the fuzzy CVaR of

P

i∈I

P

i∈I

zi ξi

zi ξi using Equation 5.5.

Remark 5.3 It should be noted that selecting larger l and N contributes P to a better approximation of i∈I zi ξi , while the computational cost will be increased exponentially. Therefore, the selection of l and N should be set specially to the specific problem. Besides, there is a slight difference with the Ref. [148], where N is selected as ln . It means that all the P possible combinations of i∈I zi ξi are calculated, which will lead to a tremendous computational burden in the proposed model. Therefore, we randomly select N possible combinations.

126  Metaheuristics for Resource Deployment

5.3.2 Improved Decomposition-Based Multi-Objective Evolutionary Algorithms

Decomposition-based MOEAs are widely used in recent years. Two stateof-the-art MOEAs (i.e., MOEA/D and DMOEA-εC) are proposed to deal with CVaR-FTLCNDP. Optimization through decomposition is an effective way to deal with multi-objective optimization problems (MOPs). The most widely used decomposition-based MOEA is the multi-objective evolutionary algorithm based on decomposition (MOEA/D). MOEA/D decomposes a MOP into a set of scalar subproblems by using aggregated functions. Commonly used aggregated functions used in MOEA/D include the weighted sum method, the Tchebycheff method, boundary intersection method, and the ε-constraint method. The first three methods are utilized in MOEA/D, and ε-constraint method is adopted in the decomposition-based multi-objective evolutionary algorithm with the εconstraint framework (DMOEA-εC). In decomposition-based MOEAs, all subproblems are optimized simultaneously by only using information from neighboring subproblems. Readers can refer to Refs. [117, 152] for more information. One of the objective functions in CVaR-FTLCNDP is minimizing the deployment cost of the nodes. Actually, this objective value is closely related to the total number of deployed sensor and weapon nodes. Therefore, how to determine an encoding method that can reflect the number of the deployed nodes via solution representation? We adopt the idea from Ref. [153] and proposed the corresponding encoding, crossover, and mutation operators for the improved MOEA/D and DMOEA-εC with variable individual size (VIS). Thus two new algorithms are proposed, i.e., MOEA/D-VIS and DMOEA-εC-VIS. For MOEA/D-VIS, Tchebycheff method is selected. 5.3.2.1

Encoding

Since the objective (i.e., deployment cost) of the MOP is related to the total number of deployed sensor and weapon nodes. Each individual represents an entire deployment, and the total number of deployed sensor and weapon nodes in the individuals are different. Therefore, the proposed MOP is a variable-length optimization problem, and the length of each individual can vary during the evolution. A variable individual size encoding mechanism is proposed with different length representing solutions for different subproblems in the decomposition-based MOEA.

Fuzzy Node Deployment for Cooperative Coverage Problem  127

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