ROBUST OPTIMIZATION OF LINEAR OPTIMIZATION PROBLEMS AND AN APPROXIMATION APPROACH TO SOLVE ROBUST KNAPSACK PROBLEM By Blake Smith Lakmali Weerasena Aniekan Ebiefung Assistant Professor of Mathematics Professor of Mathematics (Major Advisor) (Committee Member) Ossama Saleh Sumith Gunasekera Professor of Mathematics Associate Professor of Statistics (Committee Member) (Committee Member) ROBUST OPTIMIZATION OF LINEAR OPTIMIZATION PROBLEMS AND AN APPROXIMATION APPROACH TO SOLVE ROBUST KNAPSACK PROBLEM By Blake Smith A Thesis Submitted to the University of Tennessee at Chattanooga in Partial Fulfillment of the Requirements of the Degree of Master of Science: Mathematics The University of Tennessee at Chattanooga Chattanooga, Tennessee May 2019 ii . Copyright c 2019 By Blake James Smith All Rights Reserved iii ABSTRACT The goal of classical KP, is to find a subset of items whose total weight does not exceed the knapsack capacity, and whose profit is a maximum. In the robust KP the goal is to find a subset of items whose total weight does not exceed the knapsack capacity, and remains near maximum for the worst scenario. Solving the robust KP exactly is difficult due to this data uncertainty and combinatorial structure of the problem. In this research, a polynomial-time algorithm is proposed to approximately obtain a near optimal solution for the robust KP with a provable quality. The quality is described by an error term and it is derived for the proposed algorithm. It is shown that the error depends on the characteristics of the problem. We verify the accuracy of the algorithm theoretically and computationally. rithm. It is shown that the error depends on the characteristics of the problem. We verify the accuracy of the algorithm theoretically and computationally. iv DEDICATION This Thesis is dedicated to my parents, Jim and Julie Smith, who have been an incredible source of love and encouragment. v ACKNOWLEDGEMENTS I would first like to thank the Department of Mathematics at the University of Tennessee at Chattanooga for giving me the opportunity to pursue a graduate degree in mathematics. The sup- port of the students and faculty in the past two years has been incredibly helpful in my journey. I especially want to thank my thesis advisor, Dr. Lakmali Weerasena for her guidance through grad- uate school, in particular, throughout the research process. Without your support and insight in the past two years this thesis would not have been written. I would also like to thank my committee members, Prof. Aniekan Ebiefung, Prof. Ossama Saleh, and Dr. Sumith Gunasekera for your critiques and advice. A special thanks also goes to the SIMCenter-Center of Excellence in Applied Computational Science and Engineering at University of Tennessee Chattanooga for providing funds to give me more time to work on this research. I would also like to thank my professors Dr. Debra Gladden and Dr. Caroline Boulis, from Lee University, without whom I would never have had the confidence to pursue a graduate degree in the first place. To my friends, too many to thank individually, thank you for being there for me to constantly encourage me to continue to work hard in my studies. Finally and dearly, I would like to thank my family for your unending support through the years. vi TABLE OF CONTENTS ABSTRACT............................................................................iv DEDICATION . v ACKNOWLEDGMENTS . vi LIST OF TABLES . ix LIST OF ABBREVIATIONS. .xi LIST OF SYMBOLS . xii CHAPTER 1. INTRODUCTION . 1 2. A COMPARISON BETWEEN CLASSICAL AND ROBUST OPTIMIZATION . 5 2.1 Preliminaries and Basic Definitions . 5 Theorem1.....................................................................7 2.2 Robust Optimization of Uncertain Linear Optimization Problems . 7 2.2.1 Illustrative Example: Geometric Procedure . 11 2.2.2 Practical Application . 19 3. KNAPSACK PROBLEM . 25 3.1 Problem Formulation . 25 3.2 Approximation for classical KP. .29 3.3 An 2-approximation algorithm to classical Knapsack Problem . 29 Theorem 2 . 31 4. APPROXIMATING THE KP WITH FINITE NUMBER OF SCENARIOS . 34 4.1 Formulation of KP with finite number of scenarios . 34 vii 4.2 The algorithm for Robust KP with finite scenarios . 36 4.3 Theoretical Performance of the Algorithm 1 . 42 Theorem 3 . 42 Corollary 1 . 44 Theorem 4 . 44 Corollary 2 . 46 5. COMPUTATIONAL WORK . .48 5.1 Demonstration of the algorithm using educational example ......................... 48 5.1.1 Improvement Procedure . .52 5.2 Simulation Study . 52 6. OVERVIEW, SUMMARY, AND FUTURE RESEARCH. .54 6.1 Overview . 54 6.2 Summary . 55 6.3 Future research . 56 REFERENCES . 57 VITA .................................................................................59 viii LIST OF TABLES 1 Corner Points for Example 1. .12 2 Corner Points for Scenario I . .14 3 Corner Points for Scenario II . .14 4 Corner Points for Robust Scenario . 16 5 Corner Points for Discrete Scenarios . 18 6 Drug Production . .19 7 Contents of Raw Materials . 19 8 Resources . .19 9 Perturbations of x1 ..............................................................22 10 Perturbations of x2 ..............................................................22 11 Joint perturbations of x1 and x2 .................................................23 12 Feasibility with respect to the nominal data given in Problem 8 . .24 13 Possible Comination to load 4 items . 27 14 Number of Combinations for given n .............................................28 15 - approximation algorithm for KP . 33 16 Input data: profits for 4 scenarios and weights . 48 17 Ratios for the test problem . 50 18 Ratios Sorted from Greatest to Least . 50 ix 19 Order that items are selected . 51 20 Optimal profit vs. approximated profit .................................................................... 51 21 Improvement Procedure . 52 22 Simulation Results for Robust KP . 53 x LIST OF ABBREVIATIONS CO, Cominatorial Optimization KP, Knapsack Problem LPP, Linear Programing Problem RLLP, Robust Linear Programing Problem pmf, probability mass function MLE, maximum likelihood estimate EF, Efficiency Matrix xi LIST OF SYMBOLS z(x), Objective function value c, Coefficient vector for objective function A, Coefficient matrix for constraints b, Coefficient vector for right hand side of constraints δ, Amount of perturbation for coefficients, c, A, and b a¯ij, Uncertainty for value aij in matrix A ¯bi, Uncertainty for value bi in vector b c¯j, Uncertainty for value cj in vector C UA, Uncertainty set for matrix A Ub, Uncertainty set for vector b Uc, Uncertainty set for vector c , Set of uncertainty set’s U ,U ,U U A b c x¯, Optimal solution of uncertainty model x∗, Optimal solution for a problem xs∗, optimal solution for a scenario s wi, Weight of item i xii pi, Profit added to the objective function for item i W , Total weight capacity for the knapsack problem xiii CHAPTER 1 INTRODUCTION Deterministic optimization is a one of the main branches in optimization, the process in which a best value of a given function satisfying some conditions is obtained. Linear, nonlinear, convex, and discrete optimizations are common areas of the classical deterministic optimization. Although, the deterministic optimization problems provide the best solution for a given scenario the most important assumption, when solving deterministic optimization problem, is that all input data of the problem.