# Opti_1to50 by UPAD.pdf - Introduction to Mathematical...

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Introduction to Mathematical Programming by Russell C. Walker Department of Mathematical Sciences Carnegie Mellon University Solutions to the Even Numbered Exercises, the Cases in Chapter 10 and Errata c January, 2006 All rights reserved.

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Contents 1 Introduction to the Problems 1 1.3 Sample problems . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Graphical solution of linear programs . . . . . . . . . . . . . . 2 2 Vectors and Matrices 5 2.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The span of a set of vectors . . . . . . . . . . . . . . . . . . . 5 2.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Linear independence . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Systems of equations . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 The inverse of a matrix . . . . . . . . . . . . . . . . . . . . . 9 3 Linear Programming 10 3.2 Slack variables . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 The simplex algorithm . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Basic feasible solutions and extreme points . . . . . . . . . . 15 3.5 Formulation examples . . . . . . . . . . . . . . . . . . . . . . 16 3.6 General constraints and variables . . . . . . . . . . . . . . . . 18 4 Duality and Post Optimal Analysis 23 4.2 The dual and minimizing problems . . . . . . . . . . . . . . . 23 4.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 The matrix setting for the simplex algorithm . . . . . . . . . 28 4.5 Adding a variable . . . . . . . . . . . . . . . . . . . . . . . . . 28 5 Network Models 29 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 The transportation problem . . . . . . . . . . . . . . . . . . . 31 5.3 The critical path method . . . . . . . . . . . . . . . . . . . . 38 ii
Contents iii 5.4 Shortest path models . . . . . . . . . . . . . . . . . . . . . . . 45 5.5 Minimal spanning trees . . . . . . . . . . . . . . . . . . . . . 46 5.6 The maximum flow problem . . . . . . . . . . . . . . . . . . . 47 6 Unconstrained Extrema 48 6.2 Locating extrema . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.3 The economic lot size model and convexity . . . . . . . . . . 49 6.4 Location of extrema in two variables . . . . . . . . . . . . . . 51 6.5 Least squares approximation . . . . . . . . . . . . . . . . . . 52 6.6 The n -variable case . . . . . . . . . . . . . . . . . . . . . . . . 54 6.7 Numerical search . . . . . . . . . . . . . . . . . . . . . . . . . 54 7 Constrained Extrema 55 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.2 Two variable problems . . . . . . . . . . . . . . . . . . . . . . 56 7.3 More variables; more constraints . . . . . . . . . . . . . . . . 57 7.4 Problems having inequality constraints . . . . . . . . . . . . . 57 7.5 The convex programming problem . . . . . . . . . . . . . . . 58 7.6 Linear programming revisited . . . . . . . . . . . . . . . . . . 63 8 Integer Programming 64 8.2 The knapsack problem . . . . . . . . . . . . . . . . . . . . . . 64 8.3 The dual simplex algorithm . . . . . . . . . . . . . . . . . . . 65 8.4 Adding a constraint . . . . . . . . . . . . . . . . . . . . . . . 66 8.5 Branch and bound for integer programs . . . . . . . . . . . . 67 8.6 Basic integer programming models . . . . . . . . . . . . . . . 69 8.7 The traveling salesman problem . . . . . . . . . . . . . . . . . 73 9 Introduction to Dynamic Programming 75 9.1 Introduction to recursion . . . . . . . . . . . . . . . . . . . . 75 9.2 The longest path . . . . . . . . . . . . . . . . . . . . . . . . . 75 9.3 A fixed cost transportation problem . . . . . . . . . . . . . . 76 9.4 More examples . . . . . . . . . . . . . . . . . . . . . . . . . . 77 10 Solutions to the Cases 78 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 10.2 A furniture sales opportunity . . . . . . . . . . . . . . . . . . 78 10.3 Building storage lockers . . . . . . . . . . . . . . . . . . . . . 82 10.4 The McIntire farm . . . . . . . . . . . . . . . . . . . . . . . . 84

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iv Contents 10.5 Cylinders for beverages . . . . . . . . . . . . . . . . . . . . . 86 10.6 Books by the holidays . . . . . . . . . . . . . . . . . . . . . . 89 10.7 Into a blind trust . . . . . . . . . . . . . . . . . . . . . . . . . 95 10.8 Max’s taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10.9 A supply network . . . . . . . . . . . . . . . . . . . . . . . . . 101

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Chapter 1 Introduction to the Problems 1.3 Sample problems 2. Let x 1 be the number of acres of potatoes, x 2 the number of acres of wheat, and x 3 the amount borrowed. Then the problem for putting the amount borrowed into start-up costs is Maximize : 40 x 1 + 120 x 2 . 10 x 3 Subject to : x 1 + x 2 100 x 1 + 4 x 2 160 10 x 1 + 20 x 2 x 3 1 , 100 x 3 300 x 1 0 , x 2 0 , x 3 0 To consider hiring additional labor, omit x 3 from the third constraint and re- place the second constraint by x 1 + 4 x 2 1 20 x 3 160 . One might also want to consider the case where the amount borrowed is considered the sum of two amounts – one going into input costs, and one into additional labor.
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