linear and integer programming

LINEAR AND INTEGER PROGRAMMING Theory and Practice Second Edition

Gerard Sierksma University of Groningen Groningen, The Netherlands

MARCEL

MARCEL DEKKER, INC.

N E W YORK • BASEL

CONTENTS

PREFACE CHAPTER 1. Linear Programming; Basic Concepts 1.1 History and Introduction 1.2 The Company 'Dovetail' 1.2.1 Formulating Model 'Dovetail' 1.2.2 The Graphical Solution Method 1.3 The Definition of LP-Model 1.3.1 The Standard Form of an LP-model 1.3.2 Types of Optimal Solutions and Feasible Regions 1.4 Basic Feasible Solutions 1.4.1 Slack Variables and Binding Constraints 1.4.2 Basic Feasible Solutions and Degeneracy 1.4.3 The Row-column Complementarity Theorem 1.5 Adjacency and Optimality 1.5.1 Hyperplanes and Halfspaces 1.5.2 Basic Feasible Solutions and Vertices of the Feasible Region 1.5.3 Degenerate Vertices and Basic Feasible Solutions 1.5.4 Adjacent Vertices; Optimal Vertex Theorem 1.6 Alternatives of the Standard LP-model 1.6.1 Reductions to Standard Form 1.6.2 Basic Feasible Solutions for Models with Equality Constraints 1.7 Exercises

1 1 2 2 3 7 7 10 12 12 13 18 21 21 24 28 31 36 36 37 38

CHAPTER 2. Dantzig's Simplex Method

43

ix

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Contents 2.1 2.2 2.3 2.4 2.5 2.5.1 2.5.2 2.6 2.7 2.7.1 2.7.2 2.7.3 2.8

From Vertex to Vertex to an Optimum The Simplex Algorithm Simplex Tableaus; Simplex Adjacency Graphs Cycling; the Perturbation Procedure Initialization The Big-M Procedure The Two-phase Procedure Multiple and Unbounded Optimal Solutions The Revised Simplex Method Formulating the Algorithm The Product Form of the Inverse Applying the Revised Simplex Algorithm Exercises

43 46 52 55 61 62 65 68 73 74 76 77 79

CHAPTER 3. Duality and Optimality 3.1 Introduction 3.2 The Companies 'Dovetail' and 'Salmonnose' 3.2.1 Formulating the Dual Model 3.2.2 Economic Interpretation 3.3 Duality and Optimality 3.3.1 Dualizing the Standard LP-model 3.3.2 Dualizing Nonstandard LP-models 3.3.3 Optimality and Optimal Dual Basic Feasible Solutions 3.4 Complementary Slackness Relations; Farkas' Lemma 3.4.1 Complementary Dual Variables 3.4.2 Strong Complementary Slackness 3.4.3 Determining the Optimality of a Given Solution 3.5 Infeasibility and Unboundedness 3.6 The Simplex Method and the Dual Model 3.7 Exercises

87 87 87 88 89 90 90 92 95 99 99 103 108 109 112 113

CHAPTER 4. Sensitivity Analysis 4.1 Sensitivity of the Model Parameters 4.1.1 Perturbing the Objective Coefficients 4.1.2 Perturbing Right-hand Sides of Constraints; Shadow Prices The Nondegenerate Case 4.1.3 Perturbation on the Nonnegativities; Shadow Costs The Nondegenerate Case 4.1.4 Perturbation of the Technology Matrix 4.2 Sensitivity Analysis for the Degenerate Case 4.2.1 Duality between Multiple and Degenerate Optimal Solutions 4.2.2 Left- and Right-shadow Prices/Costs 4.3 Shadow Prices/Costs and Redundancy of Equality Constraints

117 117 118 126 133 139 142 142 147 159

Contents 4.4

Exercises

XI

163

CHAPTER 5. Karmarkar's Interior Path Method 5.1 Introduction 5.1.1 Some History 5.1.2 The Lagrange Multiplier Method 5.2 The Interior Path 5.2.1 The Karush-Kuhn-Tucker Conditions 5.2.2 The Logarithmic Barrier Function and the Interior Path 5.2.3 Monotonicity and Duality 5.3 Formulation of the Interior Path Method 5.3.1 The Interior Path as an Invisible Hand 5.3.2 Projections on Null Space and Row Space 5.3.3 Dikin's Affine Scaling Procedure 5.3.4 Determining the Search Direction 5.4 Convergence to the Interior Path; Maintaining FeasibiUty 5.4.1 The Convergence Measure 5.4.2 Approximations of Interior Paths 5.4.3 Maintaining Feasibility; the Interior Path Algorithm 5.5 Termination and Initialization 5.5.1 Termination and the Duality Gap 5.5.2 Initialization 5.6 Exercises

175 175 175 176 178 178 180 185 186 186 187 189 191 195 195 197 198 199 199 201 204

CHAPTER 6. Integer Linear Programming 6.1 Introduction 6.1.1 The Rounding-off Procedure 6.1.2 The Company 'Cheemi' 6.2 The Branch-and-bound Method 6.2.1 Applying the Branch-and-bound Method 6.2.2 The General Form of the Branch-and-bound Method 6.2.3 The Knapsack Problem 6.2.4 A Machine Scheduling Problem 6.3 Linearizing Logical Forms with Binary Variables 6.3.1 The Binary Variables Theorem 6.3.2 Introduction to the Theory of Logical Variables 6.3.3 Logical Forms Represented by Binary Variables 6.3.4 A Decentralization Problem 6.4 Special Methods for Integer and Mixed Integer Models 6.4.1 Gomory's Cutting-plane Method for ILP-modeis 6.4.2 Gomory's Mixed-integer Cutting-plane Method 6.4.3 Benders' Decomposition Method for MILP-modeis 6.5 Exercises

209 209 210 211 213 213 217 220 224 229 230 231 233 239 241 242 246 248 254

XU

Contents

CHAPTER 7. Linear Network Models 7.1 LP-Models with Integer Solutions; Total Unimodularity 7.1.1 Total Unimodularity and Integer Vertices 7.1.2 Total Unimodularity and Incidence Matrices 7.1.3 ILP-models having Totally Unimodular Matrices 7.2 The Network Simplex Method 7.2.1 The Transshipment Problem 7.2.2 Basis Network Matrices and Feasible Tree Solutions 7.2.3 Formulation of the Network Simplex Method 7.3 Exercises

269 269 269 273 277 288 288 290 293 307

CHAPTER 8. Computational Complexity Issues 8.1 Introduction to Computational Complexity 8.2 Computational Aspects of Dantzig's Simplex Method 8.3 Polynomiality of the Interior Path Method 8.4 Computational Aspects of the Branch-and-bound Method 8.5 Exercises

317 317 320 324 326 329

CHAPTER 9. Model Building, Case Studies, and Advanced Techniques 9.1 Linear Programming Applications 9.1.1 The Art of Building and Implementing Mathematical Models 9.1.2 Nine Steps of the Decision Process 9.2 Production Planning; a Single Product Case 9.2.1 The Parameters of the Model 9.2.2 Regulär Employees and Regulär Working-hours 9.2.3 Overtime 9.2.4 Sensitivity Analysis 9.3 The Production of Several Designs of Coffee Machines 9.3.1 The Parameters and the Input Data of the Model 9.3.2 Minimizing Shortages 9.3.3 Old and Recent Shortages 9.3.4 Füll Week Productions 9.3.5 Sensitivity Analysis 9.4 Minimizing Trim-loss when Cutting Cardboard 9.4.1 Formulating the Problem 9.4.2 Gilmore-Gomory's Solution Method 9.4.3 Transformation into a 0-1 Knapsack Problem 9.4.4 Calculating an Optimal Solution 9.5 Designing a Reservoir for Irrigation 9.5.1 The Parameters and the Input Data 9.5.2 Maximizing the Irrigation Area 9.5.3 Changing the Input Parameters of the Model 9.6 Routing Helicopters for Crew Exchanges on Off-shore Locations

331 331 333 333 336 336 339 342 346 348 349 350 353 357 358 361 361 364 367 370 372 373 377 379 382

Contents

xm

9.6.1 9.6.2 9.6.3 9.6.4 9.6.5

383 385 387 390

9.6.6 9.6.7 9.6.8 9.7 9.7.1 9.7.2 9.7.3 9.7.4 9.8 9.8.1 9.8.2 9.8.3 9.8.4 9.8.5 9.8.6 9.8.7 9.8.8 9.8.9 9.9 9.9.1 9.9.2 9.9.3 9.9.4 9.9.5 9.10

Problem Description The Off-shore Transportation Problem as Vehicle Routing Problem Decreasing the Number of Platform Visits Integer Linear Programming Formulation The Column Generation Procedure; a Knapsack and Traveling Salesman Problem Dual Values as Price Indicators for Crew Exchanges A Rounding-off Procedure for Determining an Integer Solution Computational Experiments The Catering Service Problem The Formulation of the Problem The Transshipment Problem Formulation Applying the Network Simplex Method Sensitivity Analysis Conflicting Objectives: Producing or Importing Problem Description and Input Data Modeling the Two Conflicting Objectives; Pareto-optimal Points Goal Programming for Conflicting Objectives The LP-model of the Goal Programming Problem The Computer Solution for Conflicting Objectives Soft and Hard Constraints; Computer Solutions Sensitivity Analysis Alternative Solution Techniques A Comparison of the Solutions Coalition Formation and Profit Distribution The Farmers Cooperation Problem Game Theory; Linear Production Games How to Distribute the Total Profit among the Farmers Profit Distribution in the Case of p-Replica Games Sensitivity Analysis Exercises

Solutions to Selected Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9

391 396 399 401 402 402 405 408 412 415 416 417 420 421 423 424 426 429 434 435 435 436 440 443 446 449

461 465 481 487 505 515 529 537 539

Contents

XIV

Appendix Appendix Appendix Appendix

Linear Algebra Convexity Graph Theory Computer Package INTPM

Bibliography List of Symbols Index

553 571 581 593 603 611 615