Integer Programming and Network Modeis

Transcription

1 H.A. Eiselt C.-L. Sandblom Integer Programming and Network Modeis With Contributions by K. Spielberg, E. Richards, B.T. Smith, G. Laporte, B.T. Boffey With 165 Figures and 43 Tables &m Springer

2 CONTENTS Introduction: Basic Definitions and Results 1 a Linear Programming 3 a.l Fundamental Concepts and the Simplex Method 3 a.2 Duality and Postoptimality Analysis 8 a.3 Problems with Special Structures 11 b Analysis of Algorithms 13 b.l Algorithms and Time Complexity Functions 13 b.2 Examples of Time Complexity Functions 19 b.3 Classes of Problems and Their Relations 26 c Graph Theory 35 c.l Basic Definitions and Examples 35 c.2 Representation and Storage of Graphs 43 c.3 Reachability and Connectivity 51 c.4 Graphs with Special Structures 57 d Dynamic Programming 65 d.l Basic Ideas 65 d.2 A General Algorithm 68 d.3 Various Examples 73

3 VIII Part I: Integer Programming 87 1 The Integer Programming Problem and its Properties Definitions and Basic Concepts Relaxations of Integer Programming Problems Polyhedral Combinatorics Formulations in Logical Variables The Modeling of Discrete Variables The Modeling of Fixed Charges Disjunctive Variables Constraint Selection Imposing a Sequence on Variables Imposing a Sequence on Constraints Absolute Values of Functions and Nonconcave Objectives A Problem with Collective Absolute Values A Problem with Individual Absolute Values A Problem with a Nonconcave Objective Piecewise Linear Functions Semicontinuous Variables Applications and Special Structures Applications A Distribution-Location Problem A Cutting Stock Problem Examination Timetabling Forestry Harvesting Technology Choice : Political Districting Apportionment Problems Open Pit Mining Bin Packing and Assembly Line Planning Problems with Special Structures Knapsack Problems Set Covering, Set Packing, and Set Partitioning Problems Reformulation of Problems Streng and Weak Formulations 161

4 Contents IX 4.2 Model Strengthening and Logical Processing Single Constraint Procedures Multiple Constraint Procedures Aggregation Disaggregation Cutting Plane Methods Dantzig's Cutting Plane Method Gomory's Cutting Plane Methods Cutting Plane Methods for Mixed Integer Programming Branch and Bound Methods Basic Principles Search Strategies Node Selection Branch Selection A General Branch and Bound Procedure Difficult Problems Integer Programming Duality and Relaxation Lagrangean Decomposition., Heuristic Algorithms Neighborhood Search Simulated Annealing Tabu Search Genetic Algorithms Other Approaches 256 Part II: Network Path Models Tree Networks Minimal Spanning Trees Definitions and Examples Solution Techniques 264

5 X Contents 1.2 Extensions of Minimal Spanning Tree Problems Node-Constrained Minimal Spanning Trees Edge-Constrained Minimal Spanning Trees Alternative Objective Functions Connectivity and Reliability The Steiner Tree Problem Shortest Path Problems The Problem and its Formulation Applications of Shortest Paths Most Reliable Paths Equipment Replacement Functional Approximation Matrix Chain Multiplications Solution Methods Dijkstra's Method The Bellman-Ford-Moore Algorithm The Floyd-Warshall Algorithm Extensions ofthe Basic Problem The ^-Shortest Paths Problem The Minimum Cost-to-Time Ratio Problem The Resource-Constrained Shortest Path Problems Traveling Salesman Problems and Extensions The Problem and its Applications Applications Integer Linear Programming Formulations Exact Algorithms Heuristic Algorithms Vehicle Routing Problems Are Routing Euler Graphs and Cycles Constructing Eulerian Graphs, Rural Postman Problems The Capacitated Are Routing Problem 356

6 Contents XI Part III: Network Flow and Network Design Models Basic Principles of Network Models The Problem and its Formulation Transformations of Flow Problems Duality and Optimality Conditions Some Fundamental Results Applications of Network Flow Models Building Evacuation Flow Sharing Problems A Worker Allocation Problem Airline Crew Assignment Allocation of Representatives to Committees Computer Program Testing Distributed Computing Matrix Balancing Problems Matrix Rounding Problems Network Flow Algorithms Maximal Flow Algorithms The Method of Ford and Fulkerson Karzanov's Preflow Algorithm Feasible Flow Problems Cost-Minimal Flow Problems An Augmenting Path Construction Algorithm The Primal Improvement Algorithms of Klein The Primal-Dual Out-of-Kilter Algorithm The Network Simplex Method Multicommodity Network Flows The Model, ist Formulation and Properties Solution Methods Price-Directive Decomposition Resource Directive Decomposition Network Design Problems 452

7 XII Contents 5 Networks with Congestion System-Optimal and User-Optimal Network Flows Solving Flow Assignment Problem Discrete Route Assignment Network Design Problems Continuous Network Design..' Discrete Network Design Combined Routing and Discrete Link-Size Determination 476 References 479 Subject Index 501