APPLIED INTEGER PROGRAMMING

Transcription

1 APPLIED INTEGER PROGRAMMING Modeling and Solution DER-SAN CHEN The University of Alabama ROBERT G. BATSON The University of Alabama YU DANG Quickparts.com, Inc. WILEY A JOHN WILEY & SONS, INC., PUBLICATION

2 PREFACE xvii PART I MODELING 1 1 Introduction Integer Programming, Standard Versus Nonstandard Forms, Combinatorial Optimization Problems, Successful Integer Programming Applications, Text Organization and Chapter Preview, Notes, Exercises, 18 2 Modeling and Models Assumptions an Mixed Integer Programs, Modeling Process, Project Selection Problems, Knapsack Problem, Capital Budgeting Problem, Production Planning Problems, Uncapacitated Lot Sizing, Capacitated Lot Sizing, Just-in-Time Production Planning, 34 vii

3 viii CONTENTS 2.5 Workforce/Staff Scheduling Problems, Scheduling Full-Time Workers, Scheduling Full-Time and Part-Time Workers, Fixed-Charge Transportation and Distribution Problems, Fixed-Charge Transportation, Uncapacitated Facility Location, Capacitated Facility Location, Multicommodity Network Flow Problem, Network Optimization Problems with Side Constraints, Supply Chain Planning Problems, Notes, Exercises, 48 3 Transformation Using 0-1 Variables Transform Logical (Boolean) Expressions, Truth Table of Boolean Operations, Basic Logical (Boolean) Operations on Variables, Multiple Boolean Operations on Variables, Transform Nonbinary to 0-1 Variable, Transform Integer Variable, Transform Discrete Variable, Transform Piecewise Linear Functions, Arbitrary Piecewise Linear Functions, Concave Piecewise Linear Cost Functions: Economy of Scale, Transform 0-1 Polynomial Functions, Transform Functions with Products of Binary and Continuous Variables: Bundle Pricing Problem, Transform Nonsimultaneous Constraints, Either/Or Constraints, p Out of m Constraints Must Hold, Disjunctive Constraint Sets, Negation of a Constraint, If/Then Constraints, Notes, Exercises, 73 4 Better Formulation by Preprocessing Better Formulation, Automatic Problem Preprocessing, Tightening Bounds on Variables, Bounds on Continuous Variables, Bounds on General Integer Variables, Bounds on 0-1 Variables, 90

4 ix Variable Fixing, Redundant Constraints, and Infeasibility, Preprocessing Pure 0-1 Integer Programs, Fixing 0-1 Variables, Detecting Redundant Constraints And Infeasibility, Tightening Constraints (or Coefficients Reduction), Generating Cutting Planes from Minimum Cover, Rounding by Division with GCD, Decomposing a Problem into Independent Subproblems, Scaling the Coefficient Matrix, Notes, Exercises, Modeling Combinatorial Optimization Problems I Introduction, Set Covering and Set Partitioning, Set Covering Problem, Set Partitioning and Set Packing, Set Covering in Networks, Applications of Set Covering Problem, Matching Problem, Matching Problems in Network, Integer Programming Formulation, Cutting Stock Problem, One-Dimensional Case, Two-Dimensional Case, Comparisons for Above Problems, Computational Complexity of COP, Problem Versus Problem Instance, Computational Complexity of an Algorithm, Polynomial Versus Nonpolynomial Function, Notes, Exercises, Modeling Combinatorial Optimization Problems II Importance of Traveling Salesman Problem, Transformations to Traveling Salesman Problem, Shortest Hamiltonian Paths, TSP with Repeated City Visits, Multiple Traveling Salesmen Problem, Clustered TSP, Generalized TSP, Maximum TSP, 139

5 6.3 Applications of TSP, Machine Sequencing Problems in Various Manufacturing Systems, Sequencing Problems in Electronic Industry, Vehicle Routing for Delivery/Dispatching, Genome Sequencing for Genetic Study, Formulating Asymmetrie TSP, Subtour Elimination by Dantzig Fulkerson- Johnson Constraints, Subtour Elimination by Miller Tucker Zemlin (MTZ) Constraints, Formulating Symmetrie TSP, Notes, Exercises, 149 PART II REVIEW OF LINEAR PROGRAMMING AND NETWORK FLOWS Linear Programming Fundamentals Review of Basic Linear Algebra, Euclidean Space, Linear and Convex Combinations, Linear Independence, Rank of a Matrix, Basis, Matrix Inversion, Determinant of a Matrix, Upper and Lower Triangular Matrices, Uses of Elementary Row Operations, Finding the Rank of a Matrix, Calculating the Inverse of a Matrix, Converting to a Triangular Matrix, Calculating the Determinant of a Matrix, Solving a System of Linear Equations, The Dual Linear Program, The Linear Program in Standard Form, Formulating the Dual Problem, Economic Interpretation of the Dual, Importance of the Dual, Relationships Between Primal and Dual Solutions, Relationships Between All Primal and All Dual Feasible Solutions, Relationship Between Primal and Dual Optimum Solutions, 172

6 xi Relationships Between Each Complementary Pair of Variables at Optimum, Notes, Exercises, Linear Programming: Geometric Concepts Geometric Solution, Objective Function, Solution Space, Requirement Space, Convex Sets, Convex Sets and Polyhedra, Directions of Unbounded Convex Sets, Convex and Polyhedral Cones, Convex and Concave Functions, Describing a Bounded Polyhedron, Representation by Extreme Points, Example Application of Representation Theorem, Describing Unbounded Polyhedron, Finding Extreme Direction Algebraically, Representing by Extreme Points and Extreme Directions, Example of Representation Theorem, Faces, Facets, and Dimension of a Polyhedron, Describing a Polyhedron by Facets, Correspondence Between Algebraic and Geometric Terms, Notes, Exercises, Linear Programming: Solution Methods Linear Programs in Canonical Form, Basic Feasible Solutions and Reduced Costs, Basic Feasible Solution, Adjacent Basic Feasible Solution, Reduced Costs, The Simplex Method, Better and Feasible Solution, Updating Simplex Tableau by Pivoting, Optimality Test, Initial Basic Feasible Solution, Interpreting the Simplex Tableau, Entire Simplex Tableau, Rows of Simplex Tableau, Columns of Simplex Tableau, Pivot Column and Pivot Row, 219

7 xii CONTENTS Predicting the New Objective Value Before Updating, Geometric Interpretation of the Simplex Method, Basic Feasible Solution Versus Extreme Point, Explanation of "Simplex Method" Nomenclature, Identifying an Extreme Ray in a Simplex Tableau, The Simplex Method for Upper Bounded Variables, The Dual Simplex Method, The Revised Simplex Method, Notes, Exercises, Network Optimization Problems and Solutions Network Fundamentals, A Ciass of Easy Network Problems, The Minimum Cost Network Flow Problem, Formulating the Transportation Assignment Problem as an MCNF Problem, Formulating the Transshipment Problem as an MCNF Problem, Formulating the Maximum Flow Problem as an MCNF Problem, Formulating the Shortest Path Problem as an MCNF Problem, Totally Unimodular Matrices, Definition, Sufficient Condition for a Totally Unimodular Matrix, Some Properties of Totally Unimodular Matrices, Matrix Structure of the MCNF Problem, Lower Triangular Matrix and Forward Substitution, Naturally Integer Solution for the MCNF Problem, The Network Simplex Method, Feasible Spanning Trees Versus Basic Feasible Solutions, The Network Algorithm, Numerical Example, Solution via LINGO, Notes, Exercises, 265 PART III SOLUTIONS Classical Solution Approaches Branch-and-Bound Approach, Basic Concepts, Branch-and-Bound Algorithm, 278

8 xiii 11.2 Cutting Plane Approach, Dual Cutting Plane Approach, Fractional Cutting Plane Method, Mixed Integer Cutting Plane Method, Group Theoretic Approach, Group Theory Terminology, Deriving the Group (Minimization) Problem, Formulating a Group Problem, Solving Group Problem as a Shortest Route Problem, Solving the Original Integer Program, Geometric Concepts, Various Polyhedrons in Original Space, Corner Polyhedron in Solution Space of Nonbasic Variables, Notes, Exercises, Branch-and-Cut Approach Introduction, Basic Concept, Branch-and-Cut Algorithm, Generating Valid Cuts and Preprocessing, Valid Inequalities, Valid Inequalities for Linear Programs, Valid Inequalities for Integer Programs, Types of Valid Inequalities, Cut Generating Techniques, Rounding Technique, Disjunction Technique, Lifting Technique, Cuts Generated from Sets Involving Pure Integer Variables, Gomory Fractional Cut, Chvätal Gomory Cut, Pure Integer Rounding Cut, Objective Integrality Cut, Cuts Generated from Sets Involving Mixed Integer Variables, Gomory Mixed Integer Cut, Mixed Integer Rounding Cut, Cuts Generated from 0-1 Knapsack Sets, Knapsack Cover, Lifted Knapsack Cover, GUB Cover, Cuts Generated from Sets Containing 0-1 Coefficients and 0-1 Variables, 324

9 xiv CONTENTS 12.8 Cuts Generated from Sets with Special Structures, Flow Cover from Fixed-Charge Flow Network, Plant/Facility Location (Fixed-Charge Transportation), Notes, Exercises, Branch-and-Price Approach Concepts of Branch-and-Price, Dantzig Wolfe Decomposition, Generalized Assignment Problem, Conventional Formulation, Column Generation Formulation, Initial Solution, GAP Example, GAP Branching Scheme, Tailing-Off Effect of Column Generation, Treatment of Identical Machines, Branch-and-Price Algorithm, Other Application Areas, Notes, Exercises, Solution via Heuristics, Relaxations, and Partitioning Introduction, Overall Solution Strategy, Better Formulation by Preprocessing, LP-Based Branch-and-Bound Framework, Heuristics for Tightening Lower Bounds, Relaxations for Tightening Upper Bounds, Strong Cuts for Tightening Solution Polyhedron, Primal Solution via Heuristics, Local Search Approaches, Artificial Intelligence Approaches, Dual Solution via Relaxation, Linear Programming Relaxation, Combinatorial Relaxation, Lagrangian Relaxation, Lagrangian Dual, Lagrangian Dual in LP, Lagrangian Dual in IP, Properties of the Lagrangian Dual, Primal Dual Solution via Benders' Partitioning, Notes, Exercises, 383

10 xv 15 Solutions with Commercial Software Introduction, Typical IP Software Components, Solvers, Presolvers, Modeling Languages, User' s Options/Intervention, Data and Application Interfaces, The AMPL Modeling Language, Components of the AMPL Modeling Language, An AMPL Example: the Diet Problem, Enhanced AMPL Modeling Techniques, AMPL Compatible MIP Solvers, LINGO Modeling Language, Prescription of Tolerances, Presolver Automatic Problem Reduction, Solvers for Linear/Integer Programming, Interfacing with the User, LINGO Modeling Conventions, LINGO Model for the Diet Problem, MPL Modeling Language, MPL Modeling Conventions, MPL Model for the Diet Problem, MPL Compatible MIP Solvers, 409 REFERENCES 411 APPENDIX: ANSWERS TO SELECTED EXERCISES 423 INDEX 459