# Linear Programming and Extensions

Princeton University Press, Aug 10, 2016 - Mathematics - 656 pages

In real-world problems related to finance, business, and management, mathematicians and economists frequently encounter optimization problems. In this classic book, George Dantzig looks at a wealth of examples and develops linear programming methods for their solutions. He begins by introducing the basic theory of linear inequalities and describes the powerful simplex method used to solve them. Treatments of the price concept, the transportation problem, and matrix methods are also given, and key mathematical concepts such as the properties of convex sets and linear vector spaces are covered.

George Dantzig is properly acclaimed as the "father of linear programming." Linear programming is a mathematical technique used to optimize a situation. It can be used to minimize traffic congestion or to maximize the scheduling of airline flights. He formulated its basic theoretical model and discovered its underlying computational algorithm, the "simplex method," in a pathbreaking memorandum published by the United States Air Force in early 1948. Linear Programming and Extensions provides an extraordinary account of the subsequent development of his subject, including research in mathematical theory, computation, economic analysis, and applications to industrial problems.

Dantzig first achieved success as a statistics graduate student at the University of California, Berkeley. One day he arrived for a class after it had begun, and assumed the two problems on the board were assigned for homework. When he handed in the solutions, he apologized to his professor, Jerzy Neyman, for their being late but explained that he had found the problems harder than usual. About six weeks later, Neyman excitedly told Dantzig, "I've just written an introduction to one of your papers. Read it so I can send it out right away for publication." Dantzig had no idea what he was talking about. He later learned that the "homework" problems had in fact been two famous unsolved problems in statistics.

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### Contents

 CHAPTER 1 THE LINEAR PROGRAMMING CONCEPT 1 CHAPTER 2 ORIGINS AND INFLUENCES 12 CHAPTER 3 FORMULATING A LINEAR PROGRAMMING MODEL 32 CHAPTER 4 LINEAR EQUATION AND INEQUALITY SYSTEMS 69 CHAPTER 5 THE SIMPLEX METHOD 94 CHAPTER 6 PROOF OF THE SIMPLEX ALGORITHM AND THE DUALITY THEOREM 120 CHAPTER 7 THE GEOMETRY OF LINEAR PROGRAMS 147 CHAPTER 8 PIVOTING VECTOR SPACES MATRICES AND INVERSES 173
 CHAPTER 17 NETWORKS AND THE TRANSSHIPMENT PROBLEM 352 CHAPTER 18 VARIABLES WITH UPPER BOUNDS 368 CHAPTER 19 MAXIMAL FLOWS IN NETWORKS 385 CHAPTER 20 THE PRIMALDUAL METHOD FOR TRANSPORTATION PROBLEMS 404 CHAPTER 21 THE WEIGHTED DISTRIBUTION PROBLEM 413 CHAPTER 22 PROGRAMS WITH VARIABLE COEFFICIENTS 433 CHAPTER 23 A DECOMPOSITION PRINCIPLE FOR LINEAR PROGRAMS 448 CHAPTER 24 CONVEX PROGRAMMING 471

 CHAPTER 9 THE SIMPLEX METHOD USING MULTIPLIERS 210 CHAPTER 10 FINITENESS OF THE SIMPLEX METHOD UNDER PERTURBATION 228 CHAPTER 11 VARIANTS OF THE SIMPLEX ALGORITHM 240 CHAPTER 12 THE PRICE CONCEPT IN LINEAR PROGRAMMING 254 CHAPTER 13 GAMES AND LINEAR PROGRAMS 277 CHAPTER 14 THE CLASSICAL TRANSPORTATION PROBLEM 299 CHAPTER 15 OPTIMAL ASSIGNMENT AND OTHER DISTRIBUTION PROBLEMS 316 CHAPTER 16 THE TRANSSHIPMENT PROBLEM 335
 CHAPTER 25 UNCERTAINTY 499 CHAPTER 26 DISCRETE VARIABLE EXTREMUM PROBLEMS 514 AN EXAMPLE OF FORMULATION AND SOLUTION 551 CHAPTER 28 THE ALLOCATION OF AIRCRAFT TO ROUTES UNDER UNCERTAIN DEMAND 568 BIBLIOGRAPHY 592 INDEX 617 Copyright

### About the author (2016)

George B. Dantzig is Professor Emeritus in the Department of Engineering-Economic Systems and Operations Research at Stanford University.