Linear-Fractional Programming Theory, Methods, Applications and Software

By: E.B. Bajalinov

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Hardcover | 30 November 2003

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456 Pages

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23.5 x 15.88 x 2.54

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This is a book on Linear-Fractional Programming (here and in what follows we will refer to it as "LFP"). The field of LFP, largely developed by Hungarian mathematician B. Martos and his associates in the 1960's, is concerned with problems of op timization. LFP problems deal with determining the best possible allo cation of available resources to meet certain specifications. In particular, they may deal with situations where a number of resources, such as people, materials, machines, and land, are available and are to be combined to yield several products. In linear-fractional programming, the goal is to determine a per missible allocation of resources that will maximize or minimize some specific showing, such as profit gained per unit of cost, or cost of unit of product produced, etc. Strictly speaking, linear-fractional programming is a special case of the broader field of Mathematical Programming. LFP deals with that class of mathematical programming problems in which the relations among the variables are linear: the con straint relations (i.e. the restrictions) must be in linear form and the function to be optimized (i.e. the objective function) must be a ratio of two linear functions.

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"Although several books on general (nonlinear) fractional programming are available, this is the first monograph entirely devoted to linear-fractional problems. As the title indicates, it deals with both the theoretical and the algorithmic aspects of the subject, including the description of a program package for Windows." (Juan-Enrique Martinez-Legaz, Zentralblatt MATH, Vol. 1067 (18), 2005)

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Non-Fiction Computing & I.T.Computer Programming & Software Development Mathematics Optimisation Linear Programming Applied Mathematics

List of Figures	p. xv
List of Tables	p. xix
Preface	p. xxiii
Acknowledgments	p. xxvii
Introduction	p. 1
Subject of the book	p. 1
Description of the content	p. 3
What is new in this book?	p. 5
Required knowledge and skills	p. 5
How to use the book for courses	p. 6
Basic Linear Algebra	p. 7
Matrices and their Properties	p. 7
Vectors and their Properties	p. 14
Linear Independence and Dependence	p. 17
Determinants	p. 17
The Inverse of Matrix	p. 19
Matrices and Systems of Linear Equations	p. 22
The Gaussian Elimination	p. 24
Elementary Row Operations	p. 24
Main Steps	p. 25
Forward Substitution	p. 29
Pivoting	p. 31
The Gauss-Jordan Elimination	p. 32
Multiple RHS's and Inverses	p. 37
Discussion Questions and Exercises	p. 38
Introduction to LFP	p. 41
What is a Linear-Fractional Problem?	p. 41
Main Definitions	p. 43
Relationship with Linear Programming	p. 43
Main Forms of the LFP Problem	p. 45
The Graphical Method	p. 48
The Single Optimal Vertex	p. 48
Multiple Optimal Solutions	p. 50
Mixed cases	p. 51
Asymptotic cases	p. 51
Charnes & Cooper's Transformation	p. 54
Dinkelbach's Algorithm	p. 59
LFP models	p. 62
Main Economic Interpretation	p. 62
A Maritime Transportation Problem	p. 63
Product Planning	p. 64
A Financial Problem	p. 65
A Transportation Problem	p. 66
A Blending Problem	p. 68
A Location Problem	p. 70
Discussion Questions and Exercises	p. 72
The Simplex Method	p. 75
Main Definitions and Theorems	p. 76
Criteria of Optimality	p. 79
General Scheme of the Simplex Method	p. 83
Simplex Tableau	p. 86
Connection Between Iterations	p. 87
Theoretical Background	p. 87
Pivot Transformation	p. 89
Initialization of the Simplex Method	p. 90
The Big M Method	p. 93
The Two-Phase Simplex Method	p. 100
Compact Form of the Simplex Tableau	p. 104
Rules of Entering and Dropping Variables	p. 108
Entering Rules	p. 109
Dropping Rules	p. 111
Degeneracy and Cycling	p. 112
Unrestricted-In-Sign Variables	p. 116
Bounded Variables	p. 117
Discussion Questions and Exercises	p. 126
Duality Theory	p. 129
Short overview	p. 129
Gol'stein-type Lagrangian	p. 133
Main Theorems	p. 142
Computational Relations Between Primal and Dual Problems	p. 154
Connection with Linear Programming	p. 158
Dual Variables in Stability Analysis	p. 160
Comparative Analysis of Dual Variables in LP and LFP	p. 168
Discussion Questions and Exercises	p. 174
Sensitivity Analysis	p. 177
Graphical Introduction to Sensitivity Analysis	p. 178
Change in RHS Vector b	p. 180
Change in Numerator Vector p	p. 187
Change in Numerator Constant p[subscript 0]	p. 192
Change in Denominator Vector d	p. 194
Change in Denominator Constant d[subscript 0]	p. 199
Discussion Questions and Exercises	p. 201
Interconnection Between LFP and LP	p. 205
Preliminaries	p. 205
Primal Problems	p. 206
Stability	p. 209
Dual Problems	p. 211
Economic Interpretation	p. 213
Numeric Example	p. 215
Discussion Questions and Exercises	p. 218
Integer Linear-Fractional Programming	p. 219
LFP Models with Integer Variables	p. 221
The Knapsack Problem	p. 221
Capital Budgeting Problems	p. 222
Set Covering Problems	p. 223
The Traveling Salesperson Problem	p. 225
The Branch-and-Bound Method	p. 226
The Cutting Plane Method	p. 233
Formulating discrete LFP Problems	p. 240
Converting Problems	p. 240
Practical Situations	p. 241
Discussion Questions and Exercises	p. 243
Special LFP Problems	p. 245
The Transportation Problem	p. 245
Formulation and Preliminaries	p. 245
The Transportation Simplex Method	p. 248
Determining Initial BFS	p. 257
Numerical Example	p. 267
Duality Theory for the Transportation Problem	p. 274
The Transshipment Problem	p. 278
The Assignment Problem	p. 282
Discussion Questions and Exercises	p. 284
Advanced Methods and Algorithms in LFP	p. 287
The Dual Simplex Method in LFP	p. 287
The Criss-Cross Method	p. 293
The Interior-Point Methods	p. 298
Discussion Questions and Exercises	p. 301
Advanced Topics in LFP	p. 303
Generalized LFP	p. 303
Multi-objective LFP	p. 307
Computational Aspects	p. 311
Scaling LFP Problems	p. 313
RHS Vector b to [rho]b	p. 314
Column A[subscript j] to [rho]A[subscript j]	p. 317
Row a[subscript i] to [rho]a[subscript i]	p. 320
Numerator Vector p to [rho]p	p. 321
Denominator Vector d to [rho]d	p. 322
Scaling Factors	p. 323
Numeric examples	p. 326
Factorization of Basis Matrix	p. 330
LU-factorization	p. 331
LU-factorization and Gaussian Elimination	p. 338
Updating LU-factorization	p. 343
Other Types of Factorization	p. 358
Re-using Basis	p. 365
Iterative Refinement of a Solution	p. 369
Sparse matrices	p. 370
Sparse Vectors	p. 371
Coordinate Scheme	p. 372
Collection of Sparse Vectors	p. 374
The Linked List	p. 377
Discussion Questions and Exercises	p. 379
The Wingulf Package	p. 381
Program Overview and Background	p. 382
The Editor	p. 385
Problems with Continuous Variables	p. 387
Input and Main Options	p. 387
Output	p. 389
Interpreting an Optimal Solution	p. 390
An LP Example	p. 394
An LFP Example	p. 397
Problems with Integer Variables	p. 401
Input and Main Options	p. 401
Output	p. 402
An Integer Example	p. 404
Future Developments	p. 405
References	p. 409
Index	p. 421
Table of Contents provided by Ingram. All Rights Reserved.

Linear-Fractional Programming Theory, Methods, Applications and Software

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