Linear Programming

, by Ignizio, James P.; Cavalier, Tom M.

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ISBN: 9780131837577 | 0131837575
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Copyright: 11/21/2019

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Introductory level text on linear programming, thorough, up-to-date, and comprehensive. Also addresses linear integer programming. Provides a basis for the understanding and appreciation of the truly remarkable power of the linear programming method. DLC: Linear programming.

Preface

xix

Introduction and Overview

(9)

The Linear Decision Model

(1)

What is Linear Programming? A Preview

(1)

Purpose

(1)

Applications of Linear Programming: A Preview

(1)

Traditional Applications

(1)

Applications in Artificial Intelligence and Information Technology

(1)

Dealing with Problems of Size and Complexity Beyond the Capabilities of Linear Programming

(1)

Intended Audience and Prerequisites

(1)

What is Different About This Text?

(1)

Overview of Material to Follow

(1)

Course-Structure Recommendations

(2)

PART I Conventional Linear Programming

(303)

The (Conventional) Linear Programming Model

(31)

Chapter Overview

(1)

Models and Model Types

(3)

General Guidelines in Model Building

(1)

Definitions

(2)

Basic Steps in Linear Programming Model Formulation

(2)

Determination of the Decision Variables

(1)

Formulation of the Objective

(1)

Formulation of the Constraints

(1)

The General Form of the Linear Programming Model

(1)

Assumptions of the Linear Programming Model

(1)

Examples of Linear Programming Model Formulation

(11)

Model Validity

(3)

An Evaluation of Model Structure

(1)

An Evaluation of Model Logic

(1)

An Evaluation of the Design and/or Input Data

(1)

An Evaluation of Model Response

(1)

Summary

(1)

Exercises

(7)

Foundations of the Simplex Method

(39)

Chapter Overview

(1)

Converting a Linear Program into Standard Form

(5)

Constraint Conversion

(1)

The Objective Function

(2)

Notation and Definitions

(2)

Graphical Solution of Two-dimensional Linear Programs

(7)

Convex Sets and Polyhedral Sets

(4)

*Extreme Points, Extreme Directions, and Optimality

(10)

Basic Feasible Solutions and Extreme Points

(7)

Summary

(1)

Exercises

(4)

The Simplex Method: Tableaux and Computation

(54)

Chapter Overview

(1)

Algebra of the Simplex Method

(12)

Checking for Optimality

(1)

Determining the Entering Variable

(2)

Determining the Departing Variable

(1)

*Optimality Conditions and Directions

(1)

Checking for an Unbounded Objective

(6)

The Simplex Method in Tableau Form

(9)

Identifying B-1 from the Simplex Tableau

(9)

Finding an Initial Basic Feasible Solution

102

(10)

The Two-Phase Method

103

(7)

The Big-M Method

110

(2)

Unrestricted Variables and Variables with Negative Lower Bounds

112

(2)

Degeneracy and Cycling

114

(8)

*The Lexicographic Minimum Ratio Test and Finite Convergence

118

(4)

Summary

122

(1)

Exercises

122

(12)

*Special Simplex Implementations

134

(33)

Chapter Overview

134

(1)

The Revised Simplex Method

135

(7)

Suboptimization

141

(1)

The Product Form of the Inverse

142

(6)

Operations with Elementary Matrices

144

(4)

The Bounded-Variables Simplex Method

148

(12)

Checking for Optimality

150

(1)

Determining the Entering Variable

151

(1)

Increasing a Nonbasic Variable Xk from its Lower Bound lk

151

(1)

Decreasing a Nonbasic Variable Xk from its Upper Bound Uk

152

(8)

Decomposition

160

(1)

Summary

161

(1)

Exercises

162

(5)

Duality and Sensitivity Analysis

167

(80)

Chapter Overview

167

(1)

Formulation of the Linear Programming Dual

168

(9)

The Canonical Form of the Dual

168

(4)

General Duality

172

(4)

The Standard Form

176

(1)

Relationships in Duality

177

(11)

Primal-Dual Tableau Relationships

182

(2)

Complementary Slackness

184

(4)

*Geometric Interpretation of Optimality Conditions

188

(4)

Economic Interpretation of the Dual

192

(5)

The Dual Variables as Rates of Change

(104)

The Dual Simplex Algorithm

197

(9)

An Extended Dual Simplex Algorithm

202

(4)

Sensitivity Analysis in Linear Programming

206

(11)

Changes in the Objective Coefficients

209

(2)

Changes in the Right-Hand Side

211

(1)

Changes in the Technological Coefficients, aij

212

(2)

Addition of a New Variable

214

(2)

Addition of a New Constraint

216

(1)

Parametric Programming

217

(19)

Systemic Variation of the Cost Vector C

218

(6)

Systemic Variation of the Right-Hand-Side Vector b

224

(4)

Resource Values and Ranges

228

(6)

Objective Coefficients and Ranges

234

(2)

Summary

236

(1)

Exercises

236

(11)

*Alternatives to the Simplex Algorithm

247

(27)

Chapter Overview

247

(1)

Computational Complexity

248

(1)

Khachian's Ellipsoid Algorithm

249

(3)

Affine Scaling Algorithms

252

(16)

A Primal Affine Scaling Algorithm

254

(14)

Finding an Initial Strictly Positive Solution

268

(1)

Karmarkar's Projective Algorithm

268

(3)

Summary

271

(1)

Exercises

271

(3)

Applications of LP in Information Technology

274

(39)

Chapter Overview

274

(1)

Problem Types

275

(1)

Methods in Information Technology

276

(1)

Prediction via Linear Programming

276

(9)

Pattern Classification via LP

285

(7)

An Enhanced Approach to Pattern Classification

292

(10)

Baek and Ignizio Algorithm for Pattern Classification

294

(7)

Other Modifications

301

(1)

Cluster Analysis via Mathematical Programming

302

(4)

Input--Output Analysis and LP

306

(4)

Simulation of Continuous Processing Systems

310

(1)

Summary and Conclusions

310

(1)

Exercises

311

(2)

PART II Network and Integer Models

313

(193)

The Network Simplex Method

313

(48)

Chapter Overview

313

(1)

Network Terminology

314

(2)

Minimum-Cost Network Flow Problems

316

(9)

Bases and Rooted Spanning Trees

320

(4)

Unimodularity

324

(1)

The Network Simplex Method

325

(16)

Brief Review of the Standard Primal Simplex Method

325

(1)

Computing the Flows

326

(3)

Checking for Optimality and Choosing the Entering Arc

329

(2)

Determining the Departing Arc

331

(3)

Updating the Dual Solution

334

(4)

Finding an Initial Basic Feasible Solution

338

(3)

Degeneracy and Cycling

341

(3)

Strongly Feasible Trees and Finite Convergence

342

(2)

Network Flow Problems with Bounds on the Flows

344

(6)

Determining the Entering Variable

345

(1)

Increasing the Flow on a Nonbasic Arc from its Lower Bound

345

(1)

Decreasing the Flow on a Nonbasic Arc from its Upper Bound

346

(4)

Applications

350

(3)

Summary

353

(1)

Exercises

354

(7)

The Transportation and Assignment Problems

361

(41)

Chapter Overview

361

(1)

The Transportation Problem

362

(22)

Properties of the Coefficient Matrix

367

(2)

Feasibility of the Model

369

(1)

Finiteness of the Objective Value

370

(1)

Finding an Initial Basic Feasible Solution

370

(5)

Optimality Conditions

375

(2)

Determining the Dual Solution

377

(2)

Checking for Optimality

379

(1)

Determining the Departing Variable

380

(4)

The Assignment Problem

384

(12)

The Dual Problem and Complementary Slackness

386

(1)

A Basic Primal-Dual Strategy

387

(1)

Choosing an Initial Dual Feasible Solution

388

(2)

Identifying an Assignment Corresponding to a Reduced Matrix

390

(1)

Modifying the Dual Solution and the Reduced Matrix

391

(5)

Summary

396

(1)

Exercises

396

(6)

Integer Programming

402

(55)

Chapter Overview

402

(1)

Introduction

403

(2)

Graphical Solution of Two-Dimensional Integer Programs

405

(2)

Computational Complexity

407

(1)

Formulating Integer Programming Problems

408

(12)

The Knapsack Problem

409

(1)

The Set-Covering Problem

410

(1)

The Fixed-Charge Problem

411

(1)

The Traveling Salesman Problem

412

(2)

Either--Or Constraints

414

(1)

p out of m Constraints

415

(1)

Representing General Integer Variables Using Zero--One Variables

416

(1)

Transforming a Piecewise Linear Function

416

(4)

Branch-and-Bound Enumeration

420

(10)

Branching

420

(1)

Computing Bounds

421

(1)

Fathoming

422

(1)

Search Strategies

423

(7)

Implicit Enumeration

430

(11)

The Zero Completion at a Node

432

(1)

The Infeasibility Test

432

(8)

Refinements

440

(1)

Cutting-Plane Methods

441

(9)

Dual Fractional Cuts for Pure Integer Programs

443

(7)

Summary

450

(1)

Exercises

450

(7)

Heuristic Programming---And AI

457

(49)

Chapter Overview

457

(1)

Terminology and Definitions

458

(2)

Attitudes, Opinions, and AI

460

(2)

Fundamental Heuristics Types

462

(2)

The Add/Drop Heuristic Programming Method for Cluster Analysis/Site Location

464

(7)

Observations and Extensions

471

(1)

The Exchange Heuristic Programming Method for Cluster Analysis/Site Location

471

(3)

Observations and Extensions

473

(1)

A Heuristic Program for Scheduling and Deployment Problems

474

(6)

Extensions to Problems of Deployment

478

(2)

A Heuristic Program for Minimal Conflict Scheduling

480

(6)

Genetic ``Algorithms''

486

(6)

Some Observations

491

(1)

Simulated Annealing

492

(5)

Some Observations

496

(1)

Tabu Search, Expert Systems, and Neural Networks

497

(3)

Tabu Search

497

(1)

Expert Systems

498

(2)

Neural Networks

500

(1)

Assessment of Heuristic Programming Methods

500

(4)

Summary and Conclusions

504

(1)

Exercises

504

(2)

PART III Multiobjective Optimization

506

(116)

Multiobjective Optimization

506

(35)

Chapter Overview

506

(1)

The Familiar vs. the Nonconventional

507

(2)

An Illustrative Example

509

(24)

Conversion of a Linear Program via Objective Function Transformation (or Deletion)

511

(1)

Conversion to a Linear Program via Utility Theory: A Method of Aggregation

512

(3)

Conversion to a Goal Program (GP)

515

(8)

Conversion to a Chebyshev Goal Program

523

(4)

The Generating Method: The (Not So) Perfect Approach

527

(4)

The Lexicographic Vectormax (Vectormin) Approach

531

(2)

Interactive Methods

533

(1)

Myths and Misconceptions

533

(2)

The Multiplex Approach: A Preview

535

(2)

Overview of Material to Follow

537

(1)

Summary

537

(1)

Exercises

537

(4)

Multiobjective Models

541

(33)

Chapter Overview

541

(1)

Terminology

541

(4)

Modeling Basics

545

(11)

The Weighting of Objectives

546

(1)

The Establishment of Aspiration Levels

547

(1)

The Processing of Goals

548

(2)

The Weighting of Unwanted Goal Deviations

550

(3)

The Ranking of Goals and/or Objectives

553

(1)

Scaling and Normalization of Goals

554

(2)

The Development of the Achievement Vector

556

(1)

Multiplex Formulations (Ignizio, 1985b)

556

(12)

Multiplex Model: Linear Programs

557

(2)

Multiplex Model: Archimedean Linear Goal Programs

559

(3)

Multiplex Model: Non-Archimedean Linear Goal Programs

562

(2)

Multiplex Model: Chebyshev and Fuzzy Linear Goal Programs

564

(2)

Multiplex Model: Linear Lexicographic Vectormin Problems

566

(2)

Good and Poor Modeling Practices

568

(3)

Summary

571

(1)

Exercises

571

(3)

Multiplex Algorithm

574

(27)

Chapter Overview

574

(1)

Reduced Form of the Multiplex Model

575

(4)

Basic Solutions and Basic Feasible Solutions

577

(1)

Associated Conditions

578

(1)

Revised Multiplex Algorithm

579

(12)

Algorithm Steps

580

(1)

Explicit Form of the Inverse: Tableau Representation

581

(10)

Sequential Multiplex

591

(6)

The Sequential Algorithm: For Linear Multiplex

591

(5)

Observations

596

(1)

Computational Considerations

597

(1)

Summary

598

(1)

Exercises

599

(2)

Duality and Sensitivity Analysis in Linear Multiplex

601

(18)

Chapter Overview

601

(1)

The Formulation of the Multidimensional Dual

602

(3)

Economic Interpretation

605

(1)

MDD Algorithms

605

(6)

The General MDD Algorithm

606

(2)

The Restricted MDD Algorithm

608

(3)

Sensitivity Analysis in Linear Multiplex

611

(5)

Discrete Sensitivity Analysis

611

(3)

Range Analysis

614

(2)

Sensitivity Analysis in Sequential Linear Multiplex

616

(1)

Summary

616

(1)

Exercises

617

(2)

Extensions of Multiplex

619

(3)

Chapter Overview

619

(1)

Integer Models

619

(1)

Nonlinear Models

620

(1)

Intelligent Interfaces

621

(1)

Summary and Conclusions

621

(1)

APPENDIX REVIEW OF LINEAR ALGEBRA

622

(23)

Vectors

622

(5)

Vector Addition

623

(1)

Multiplication of a Vector by a Scalar

624

(1)

Vector Multiplication

624

(1)

Norm of a Vector

625

(1)

Special Vector Types

626

(1)

Linear Dependence and Independence

626

(1)

Spanning Sets and Bases

627

(1)

Matrices

627

(11)

Matrix Addition

628

(1)

Multiplication by a Scalar

629

(1)

Matrix Multiplication

629

(2)

Special Matrices

631

(1)

Determinants

632

(3)

The Inverse of a Matrix

635

(2)

The Rank of a Matrix

637

(1)

The Solution of Simultaneous Linear Equations

638

(7)

A Unique Solution of Ax = b

640

(3)

An Infinite Number of Solutions of Ax = b

643

(2)

References

645

(10)

Index

655

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Linear Programming

Summary

Table of Contents

Supplemental Materials