## Linear Programming and ExtensionsIn 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. 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

1 | |

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 |

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617 | |