Clemson University TigerPrints All Dissertations Dissertations 8-2018 Mathematical Models and Algorithms for Network Flow Problems Arising in Wireless Sensor Network Applications Robert M. Curry Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Recommended Citation Curry, Robert M., "Mathematical Models and Algorithms for Network Flow Problems Arising in Wireless Sensor Network Applications" (2018). All Dissertations. 2226. https://tigerprints.clemson.edu/all_dissertations/2226 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. Mathematical Models and Algorithms for Network Flow Problems Arising in Wireless Sensor Network Applications A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Industrial Engineering by Robert M. Curry August 2018 Accepted by: Dr. J. Cole Smith, Committee Chair Dr. Warren Adams Dr. Sandra D. Eksioglu Dr. Amin Khademi Abstract We examine multiple variations on two classical network flow problems, the maximum flow and minimum-cost flow problems. These two problems are well-studied within the optimization community, and many models and algorithms have been presented for their solution. Due to the unique characteristics of the problems we consider, existing approaches cannot be directly applied. The problem variations we examine commonly arise in wireless sensor network (WSN) applications. A WSN consists of a set of sensors and collection sinks that gather and analyze environmental conditions. In addition to providing a taxonomy of relevant literature, we present mathematical programming models and algorithms for solving such problems. First, we consider a variation of the maximum flow problem having node-capacity restrictions. As an alternative to solving a single linear programming (LP) model, we present two alternative solution techniques. The first iteratively solves two smaller auxiliary LP models, and the second is a heuristic approach that avoids solving any LP. We also examine a variation of the maximum flow problem having semicontinuous restrictions that requires the flow, if positive, on any path to be greater than or equal to a minimum threshold. To avoid solving a mixed-integer programming (MIP) model, we present a branch-and-price algorithm that significantly improves the computational time required to solve the problem. Finally, we study two dynamic network flow problems that arise in wireless sensor networks under non-simultaneous flow assumptions. We first consider a dynamic maximum flow problem that requires an arc to transmit a minimum amount of flow each time it begins transmission. We present an MIP for solving this problem along with a heuristic algorithm for its solution. Additionally, we study a dynamic minimum-cost flow problem, in which an additional cost is incurred each time an arc begins transmission. In addition to an MIP, we present an exact algorithm that iteratively solves a relaxed version of the MIP until an optimal solution is found. ii Acknowledgments All thanks and praise to God for his grace, mercy, and provision. I hope my work will always reflect his goodness and creativity. Thank you to my best friend and wonderful wife, Lauren. Without her support, I would never have completed my Ph.D. She has sacrificially helped and cared for me during these last four years. I cannot express enough thanks to my friend, mentor, and advisor, Dr. Cole Smith, for his guidance and wisdom. Countless hours discussing research and writing have made me a better researcher and human being. I am also thankful for the countless miles run together discussing everything under the sun. Thank you to the rest of my committee for helping me to hone my research skills and ideas through my coursework, comprehensive exam, and dissertation defense. To my parents, thank you for your continued love and encouragement. This was made possible because of your hard work and sacrifice. Finally, thank you to all the students and faculty that have supported and accompanied me during my academic journey. iii Table of Contents Title Page ............................................ i Abstract ............................................. ii Acknowledgments ....................................... iii List of Tables .......................................... vi List of Figures..........................................vii 1 Introduction......................................... 1 1.1 Background and Contribution............................... 1 1.2 Literature Review and Application Areas ........................ 3 1.3 Dissertation Organization................................. 6 2 Survey of Maximum Lifetime Maximization Problems in WSN Applications.. 8 2.1 Introduction......................................... 8 2.2 Mathematical Optimization Models for the Fundamental Lifetime Maximization Problem 13 2.3 Extensions to the Fundamental Lifetime Maximization Problem............ 18 2.4 Extensions Based on Sink Characteristics ........................ 32 2.5 Alternative Optimization Metrics............................. 44 2.6 Future Challenges ..................................... 51 3 Augmenting-flow Algorithms for Solving a Class of Maximum Flow Problems Having Node-capacity Restrictions ...........................54 3.1 Introduction and Problem Statement........................... 54 3.2 Augmenting Path and Cycle Algorithm ......................... 57 3.3 Heuristic APC Algorithm ................................. 71 3.4 Computational Results................................... 78 4 Models and Algorithms for Solving Maximum Flow Problems Having Semicon- tinuous Path-flow Restrictions in Simultaneous Flow Settings...........83 4.1 Introduction and Problem Description.......................... 83 4.2 Problem Definition and Formulations........................... 84 4.3 Computational Results................................... 94 5 Dynamic Network Flow Problems in Non-simultaneous Flow Settings . 98 5.1 Dynamic Flow Stability .................................. 99 5.2 Minimum-cost Flow Problem Having Arc-activation Costs . 108 6 Conclusions .........................................121 iv Appendices ...........................................124 Bibliography...........................................132 v List of Tables 2.1 Section 2.2 summary.................................... 14 2.2 Section 2.3.1 summary................................... 19 2.3 Section 2.3.2 summary................................... 20 2.4 Section 2.3.3 and 2.3.4 summary ............................. 27 2.5 Section 2.4.1 summary................................... 33 2.6 Section 2.4.2 summary, where sink travel times are negligible unless otherwise specified 34 2.7 Section 2.4.3 summary, where sink travel times are negligible and routing cannot be delayed ........................................... 43 2.8 Section 2.5.1 and 2.5.2 summary ............................. 45 2.9 Section 2.5.3 summary................................... 50 3.1 Average NCMFP results for b = 50............................ 79 3.2 Average NCMFP results over networks having 800 nodes ............... 80 3.3 Average APC results over networks having 1000 nodes................. 80 3.4 Average h-APC results over networks having 100 nodes ................ 81 4.1 Average MFP-S results over networks having 10 relay nodes.............. 95 4.2 Average MFP-S results over networks having 15 relay nodes.............. 96 4.3 Average MFP-S results over networks having 20 relay nodes.............. 96 4.4 Average MFP-S results using the B&P approach where ` = 3............. 97 5.1 Results for solving the MFP-S and the MFP-D..................... 108 5.2 Average results for the MFP-D heuristic for networks having 20 relay nodes . 108 5.3 Average MCF-A results over five networks having 10 nodes . 118 5.4 Average MCF-A results over five networks having 12 nodes . 118 5.5 Average MCF-A results over five networks having 15 nodes . 119 5.6 Average bounding procedure results over five networks having 20 nodes . 119 vi List of Figures 2.1 Wireless sensor network routing.............................. 9 2.2 WSN multi-hop communication example......................... 10 2.3 WSN taxonomy....................................... 11 2.4 Static WSN taxonomy................................... 12 2.5 Mobile WSN taxonomy .................................. 12 2.6 Single-sink WSN distances................................. 15 2.7 Optimal data flows per hour................................ 16 2.8 WSN clustering method.................................. 22 2.9 Square grid topology.................................... 26 2.10 Hexagon grid topology................................... 28 2.11 WSN backbone....................................... 28 2.12 Load-balanced virtual backbone.............................. 30 2.13 Schedule transition graph ................................. 30 2.14 Virtual scheduling graph, in which EA = 3, EB = 2, EC = 1, and ED = 1 . 31 2.15 Two-sink WSN example.................................. 33 2.16 Optimal data flows per hour................................ 35 2.17 Distances (in km) for the single mobile-sink WSN example............... 41 2.18 Maximum WSN lifetime as a function of maximum tolerable delay.......... 42 2.19 Locating critical sensors.................................