a Maximize Z x 3 b Maximize Z x 1 2 x 3 51 20 Consider the three variable

A maximize z x 3 b maximize z x 1 2 x 3 51 20

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(a) Maximize Z x 3 . (b) Maximize Z x 1 2 x 3 . 5.1-20. Consider the three-variable linear programming problem shown in Fig. 5.2. (a) Explain in geometric terms why the set of solutions satisfying any individual constraint is a convex set, as defined in Ap- pendix 2. (b) Use the conclusion in part ( a ) to explain why the entire feasi- ble region (the set of solutions that simultaneously satisfies every constraint) is a convex set. 5.1-21. Suppose that the three-variable linear programming prob- lem given in Fig. 5.2 has the objective function Maximize Z 3 x 1 4 x 2 3 x 3 . Without using the algebra of the simplex method, apply just its geometric reasoning (including choosing the edge giving the max- imum rate of increase of Z ) to determine and explain the path it would follow in Fig. 5.2 from the origin to the optimal solution. 5.1-22. Consider the three-variable linear programming problem shown in Fig. 5.2. (a) Construct a table like Table 5.4, giving the indicating variable for each constraint boundary equation and original constraint. (b) For the CPF solution (2, 4, 3) and its three adjacent CPF so- lutions (4, 2, 4), (0, 4, 2), and (2, 4, 0), construct a table like Table 5.5, showing the corresponding defining equations, BF solution, and nonbasic variables. (c) Use the sets of defining equations from part ( b ) to demonstrate that (4, 2, 4), (0, 4, 2), and (2, 4, 0) are indeed adjacent to (2, 4, 3), but that none of these three CPF solutions are adja- cent to each other. Then use the sets of nonbasic variables from part ( b ) to demonstrate the same thing. 5.1-23. The formula for the line passing through (2, 4, 3) and (4, 2, 4) in Fig. 5.2 can be written as (2, 4, 3) [(4, 2, 4) (2, 4, 3)] (2, 4, 3) (2, 2, 1), where 0 1 for just the line segment between these points. After augmenting with the slack variables x 4 , x 5 , x 6 , x 7 for the re- spective functional constraints, this formula becomes (2, 4, 3, 2, 0, 0, 0) (2, 2, 1, 2, 2, 0, 0). Use this formula directly to answer each of the following ques- tions, and thereby relate the algebra and geometry of the simplex 224 5 THE THEORY OF THE SIMPLEX METHOD x 1 1 0 1 2 3 4 2 3 4 5 (2, 5) (4, 5) x 2 5.2-1. Consider the following problem. Maximize Z 8 x 1 4 x 2 6 x 3 3 x 4 9 x 5 , subject to x 1 2 x 2 3 x 3 3 x 4 x 5 180 (resource 1) 4 x 1 3 x 2 2 x 3 x 4 x 5 270 (resource 2) x 1 3 x 2 2 x 3 x 4 3 x 5 180 (resource 3)
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and x 1 0, x 2 0, x 3 0. Let x 4 , x 5 , and x 6 denote the slack variables for the respective con- straints. After you apply the simplex method, a portion of the fi- nal simplex tableau is as follows: and x j 0, j 1, . . . , 5. You are given the facts that the basic variables in the optimal so- lution are x 3 , x 1 , and x 5 and that 1 2 1 7 . (a) Use the given information to identify the optimal solution. (b) Use the given information to identify the shadow prices for the three resources. I 5.2-2.* Work through the revised simplex method step by step to solve the following problem.
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