Linear Programming and its Applications

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Springer Science & Business Media, Aug 15, 2007 - Business & Economics - 380 pages
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Based on earlier work by a variety of authors in the 1930s and 1940s, the simplex method for solving linear programming problems was developed in 1947 by the American mathematician George B. Dantzig. Helped by the computer revolution, it has been described by some as the overwhelmingly most significant mathematical development of the last century. Owing to the simplex method, linear programming (or linear optimization, as some would have it) is pervasive in modern society for the planning and control of activities that are constrained by the availability of resources such as manpower, raw materials, budgets, and time. The purpose of this book is to describe the field of linear programming. While we aim to be reasonably complete in our treatment, we have given emphasis to the modeling aspects of the field. Accordingly, a number of applications are provided, where we guide the reader through the interactive process of mathematically modeling a particular practical situation, analyzing the consequences of the model formulated, and then revising the model in light of the results from the analysis.
 

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Contents

A1 Matrix Algebra
1
A2 Systems of Simultaneous Linear Equations
5
A3 Convexity
23
B1 Algorithms and Time Complexity Functions
31
B2 Examples of Time Complexity Functions
37
B3 Classes of Problems and Their Relations
41
11 A Short History of Linear Programming
45
12 Assumptions and the Main Components of Linear Programming Problems
48
53 Column Generation
219
6 POSTOPTIMALITY ANALYSES
225
61 Graphical Sensitivity Analysis
227
62 Changes of the RightHand Side Values
232
63 Changes of the Objective Function Coefficients
240
64 Sensitivity Analyses in the Presence of Degeneracy
245
65 Addition of a Constraint
248
66 Economic Analysis of an Optimal Solution
252

13 The Modeling Process
53
14 The Three Phases in Optimization
57
15 Solving the Model and Interpreting the Printout
60
21 The Diet Problem
67
22 Allocation Problems
71
23 Cutting Stock Problems
75
24 Employee Scheduling
80
25 Data Envelopment Analysis
82
26 Inventory Planning
85
27 Blending Problems
89
28 Transportation Problems
91
29 Assignment Problems
102
A Case Study
107
31 Graphical Concepts 311 The Graphical Solution Technique
129
312 Four Special Cases
138
321 The Algebraic Solution Technique
143
322 Four Special Cases Revisited
158
41 The Fundamental Theory of Duality
166
42 PrimalDual Relations
183
43 Interpretations of the Dual Problem
198
5 EXTENSIONS OF THE SIMPLEX METHOD
203
52 The Upper Bounding Technique
212
7 NONSIMPLEX BASED SOLUTION METHODS
261
71 Alternatives to the Simplex Method
262
72 Interior Point Methods
273
81 Reformulations of Variables 811 Lower Bounding Constraints
295
812 Variables Unrestricted in Sign
296
82 Reformulations of Constraints
298
831 Minimize the Weighted Sum of Absolute Values
301
832 Bottleneck Problems
306
833 Minimax and Maximin Problems
313
834 Fractional Hyperbolic Programming
320
9 MULTIOBJECTIVE PROGRAMMING
325
91 Vector Optimization
327
921 The Weighting Method
337
922 The Constraint Method
339
93 Models with Exogenous Achievement Levels
341
931 Reference Point Programming
342
932 Fuzzy Programming
346
933 Goal Programming
351
94 Bilevel Programming
359
REFERENCES
363
SUBJECT INDEX
377
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Page 371 - Research 117 (3) 565577. Jensen, A., 1980. Traffic, Operational Research, Futurology, North-Holland, Amsterdam. Powell, MJD, 1991. A view of nonlinear optimization. In: Lenstra, JK, Rinnooy Kan, AHG, Schrijver, A. (Eds.), History of Mathematical Programming, Elsevier Publishers, Amsterdam, pp.
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About the author (2007)

Marianne and H.A. Eiselt have spent many years walking trails in North America and Europe, and they have co-written several guides on New Brunswick, including the first and second editions of their popular hiking guide to New Brunswick. In between hiking seasons they teach at the University of New Brunswick in Fredericton.

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