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LP - LINEAR PROGRAMMING A Concise Introduction Thomas S...

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LINEAR PROGRAMMING A Concise Introduction Thomas S. Ferguson Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Standard Maximum and Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Diet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Transportation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Activity Analysis Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Optimal Assignment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Dual Linear Programming Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 The Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Equilibrium Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Interpretation of the Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3. The Pivot Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4. The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The Simplex Tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The Pivot Madly Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Pivot Rules for the Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 The Dual Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5. Generalized Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 The General Maximum and Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Solving General Problems by the Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . 29 Solving Matrix Games by the Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1
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6. Cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A Modification of the Simplex Method That Avoids Cycling . . . . . . . . . . . . . . . 33 7. Four Problems with Nonlinear Objective Function . . . . . . . . . . . . . . . . . . . 36 Constrained Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 The General Production Planning Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Minimizing the Sum of Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Minimizing the Maximum of Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Chebyshev Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Linear Fractional Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Activity Analysis to Maximize the Rate of Return . . . . . . . . . . . . . . . . . . . . . . . . . 40 8. The Transportation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Finding a Basic Feasible Shipping Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Checking for Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 The Improvement Routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Related Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2
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LINEAR PROGRAMMING 1. Introduction. A linear programming problem may be defined as the problem of maximizing or min- imizing a linear function subject to linear constraints . The constraints may be equalities or inequalities. Here is a simple example. Find numbers x 1 and x 2 that maximize the sum x 1 + x 2 subject to the constraints x 1 0, x 2 0, and x 1 + 2 x 2 4 4 x 1 + 2 x 2 12 x 1 + x 2 1 In this problem there are two unknowns, and five constraints. All the constraints are inequalities and they are all linear in the sense that each involves an inequality in some linear function of the variables. The first two constraints, x 1 0 and x 2 0, are special. These are called nonnegativity constraints and are often found in linear programming problems. The other constraints are then called the main constraints . The function to be maximized (or minimized) is called the objective function . Here, the objective function is x 1 + x 2 . Since there are only two variables, we can solve this problem by graphing the set of points in the plane that satisfies all the constraints (called the constraint set) and then finding which point of this set maximizes the value of the objective function. Each inequality constraint is satisfied by a half-plane of points, and the constraint set is the intersection of all the half-planes. In the present example, the constraint set is the five- sided figure shaded in Figure 1. We seek the point ( x 1 , x 2 ), that achieves the maximum of x 1 + x 2 as ( x 1 , x 2 ) ranges over this constraint set. The function x 1 + x 2 is constant on lines with slope 1, for example the line x 1 + x 2 = 1, and as we move this line further from the origin up and to the right, the value of x 1 + x 2 increases. Therefore, we seek the line of slope 1 that is farthest from the origin and still touches the constraint set. This occurs at the intersection
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