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9780131837577

Linear Programming

by ;
  • ISBN13:

    9780131837577

  • ISBN10:

    0131837575

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2019-11-21
  • Publisher: Pearson

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Summary

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.

Table of Contents

Preface xix
Introduction and Overview
1(9)
The Linear Decision Model
1(1)
What is Linear Programming? A Preview
2(1)
Purpose
3(1)
Applications of Linear Programming: A Preview
4(1)
Traditional Applications
4(1)
Applications in Artificial Intelligence and Information Technology
5(1)
Dealing with Problems of Size and Complexity Beyond the Capabilities of Linear Programming
5(1)
Intended Audience and Prerequisites
6(1)
What is Different About This Text?
7(1)
Overview of Material to Follow
7(1)
Course-Structure Recommendations
8(2)
PART I Conventional Linear Programming 10(303)
The (Conventional) Linear Programming Model
10(31)
Chapter Overview
10(1)
Models and Model Types
10(3)
General Guidelines in Model Building
13(1)
Definitions
14(2)
Basic Steps in Linear Programming Model Formulation
16(2)
Determination of the Decision Variables
16(1)
Formulation of the Objective
17(1)
Formulation of the Constraints
18(1)
The General Form of the Linear Programming Model
18(1)
Assumptions of the Linear Programming Model
19(1)
Examples of Linear Programming Model Formulation
19(11)
Model Validity
30(3)
An Evaluation of Model Structure
31(1)
An Evaluation of Model Logic
32(1)
An Evaluation of the Design and/or Input Data
32(1)
An Evaluation of Model Response
33(1)
Summary
33(1)
Exercises
34(7)
Foundations of the Simplex Method
41(39)
Chapter Overview
41(1)
Converting a Linear Program into Standard Form
42(5)
Constraint Conversion
42(1)
The Objective Function
43(2)
Notation and Definitions
45(2)
Graphical Solution of Two-dimensional Linear Programs
47(7)
Convex Sets and Polyhedral Sets
54(4)
*Extreme Points, Extreme Directions, and Optimality
58(10)
Basic Feasible Solutions and Extreme Points
68(7)
Summary
75(1)
Exercises
76(4)
The Simplex Method: Tableaux and Computation
80(54)
Chapter Overview
80(1)
Algebra of the Simplex Method
81(12)
Checking for Optimality
82(1)
Determining the Entering Variable
83(2)
Determining the Departing Variable
85(1)
*Optimality Conditions and Directions
86(1)
Checking for an Unbounded Objective
87(6)
The Simplex Method in Tableau Form
93(9)
Identifying B-1 from the Simplex Tableau
93(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
93(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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