## Linear ProgrammingFor upper-division/graduate courses in operations research/management science, mathematics, and computer science, this text covers basic theory, selected applications, network flow problems, and advanced techniques. |

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### Contents

Introduction | 3 |

How the Simplex Method Works | 13 |

Pitfalls and How to Avoid Them | 27 |

How Fast Is the Simplex Method? | 45 |

The Duality Theorem | 54 |

Gaussian Elimination and Matrices | 71 |

The Revised Simplex Method | 97 |

Solutions by the Simplex Method | 118 |

Approximating Data by Linear Functions | 213 |

Matrix Games | 228 |

Systems of Linear Inequalities | 240 |

Finding All Vertices of a Polyhedron | 271 |

The Network Simplex Method | 291 |

Applications of the Network Simplex Method | 320 |

UpperBounded Transshipment Problems | 353 |

Maximum Flows Through Networks | 367 |

Theorems on Duality and Infeasibility | 137 |

Sensitivity Analysis | 148 |

Selected Applications | 169 |

Efficient Allocation of Scarce Resources | 171 |

Scheduling Production and Inventory | 188 |

The CuttingStock Problem | 195 |

The PrimalDual Method | 390 |

Updating a Triangular Factorization of the Basis | 405 |

The DantzigWolfe Decomposition Principle | 425 |

The Ellipsoid Method | 443 |

465 | |

### Common terms and phrases

algorithm arcs ij augmenting path auxiliary problem basic feasible solution basic solution basic variables bipartite graph Chapter coefficients column vector components computing constraints convex convex hull convex sets corresponding defined entering arc entering column entering the basis entering variable eta column example feasible tree solution finals of width finite Gaussian elimination Hence identity matrix integer leaving the basis leaving variable linear inequalities linear programming linear programming problem LP problems maximize cx subject maximum-flow problem mixed strategy network simplex method node nonbasic variables nonzero number of iterations objective function obtain optimal solution original problem path payoff matrix permutation permutation matrices pivot polyhedron problem maximize cx procedure proof Prove pure strategies replace resulting revised simplex method right-hand side satisfies slack variables solvable Solving the system Step subject to Ax system Ax systems of linear Theorem transshipment problem triangular factorization update upper bound zero