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practice_6_solution

practice_6_solution - Practice#6 Solutions Linear...

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Practice #6 Solutions Linear Programming Sensitivity Analysis BUAD311 – Operations Management Fall 2010 1. Solved Problem 2 on p49 of the textbook. 2. Consider the following LP: Max 3x 1 + 4x 2 + 6x 3 + 10x 4 Subject to: x 1 + x 2 + 3x 4 120 x 1 + x 2 + 3x 3 + x 4 150 x 1 + 2x 2 + x 3 + 2x 4 100 x 1 0, x 2 0, x 3 0, x 4 0 After solving the problem using Excel solver, the following sensitivity repost was generated: Microsoft Excel 10.0 Sensitivity Report Adjustable Cells d d Final Reduced Objective Allowable Allowable Cell Nam e Value Cost Coefficient Increase Decrease $A$1 d X1 0 -2.2 3 2.2 1E+30 $B$1 d X2 0 -6 4 6 1E+30 $C$1 d X3 40 0 6 24 1 $D$1 d X4 30 0 10 2 5.499999999 Constraints d d Final Shadow Constraint Allowable Allowable Cell Nam e Value Price R.H. Side Increase Decrease $E$3 d 90 0 120 1E+30 30 $E$4 d 150 0.4 150 150 50 $E$5 d 100 4.8 100 16.66666667 50
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(a) What is the value of optimal objective function? 6 * 40 + 10 * 30 = 540 (b) Suppose the first constraint instead was: x 1 + x 2 + 3x 4 100. What is the new optimal objective function value? Explain. 540. The change (-20) is less than the allowable decrease and the shadow price is 0. (c) Suppose the second constraint instead was: x 1 + x 2 + 3x 3 + x 4 190. What is the new optimal objective function value? Explain. The increase is 40 and is less than the allowable increase (150). Therefore, we can use the shadow price 0.4 to calculate the new objective value. It will be 540 + 40 * 0.4 = 556 (d) Suppose the coefficient multiplying x 3 in the objective function was 10 instead of 6. What is the new optimal objective function value? Why? The increase is 4 and is less than the allowable increase (24). The optimal decision variables stay the same. The new objective’s value is 10*40+10*30=700 (e) Suppose the coefficient multiplying x 4 in the objective function was 13 instead of 10. What is the new optimal objective function value? Why? The increase is 3 and is more than the allowable increase (2). The optimal decision variables will change. We need to rerun solver to find the new objective’s value.
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3. Consider the following LP minimize 2x 1 + 5x 2 + 6x 3 + 12x 4 subject to: x 1 + x 2 + 3x 3 ≥ 50 x 1 + x 2 + x 3 + 6x 4 ≥ 155 x 1 + x 2 + 8x 4 ≥ 100 x 1 ≥0, x 2 ≥0, x 3 ≥0, x 4 ≥ 0 Using Excel solver, we have the following sensitivity report Adjustable Cells . . Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase
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