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DMOR-Midterm-04 - UF-ESI-6314 Problem Statement page 1 of...

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UF-ESI-6314 204be18ecfd6b68601feb157f4e7d43d4bd5b7d2.xls Problem Statement page 1 of 11 package page _ of __ Luther Setzer 1 NASA Pkwy E Stop NEM3, Kennedy Space Center, FL 32899 (321) 544-7435 Question 4 (25 points): Row Basis VRS z RHS Ratio 0 z 1.00 0.00 0.00 0.00 a b 3.00 e 1 0.00 1.00 0.00 0.00 c 2.00 -3.00 d 2 0.00 0.00 1.00 0.00 -4.00 1.00 4.00 2.00 3 0.00 0.00 0.00 1.00 -1.00 3.00 1.00 3.00 Marginal Net Gain Pivot Column Suppose that I have optimized some maximization problem by the simplex method, and have obtained the following tableau. The values a , b , c , d , and e in the tableau are parameters that I have not specified. Warning: for this question, you must show me the work or rationale you used to arrive to your answer. No credit is given for simply writing the answer. Hint: Be careful when you say something like " a is positive" or " a > 0." If you mean strictly positive, then that's fine. But there’s a difference between " a ≥ 0" and " a > 0." For instance, if I asked, "When is this a feasible solution?" the answer would be " d ≥ 0" not " d > 0." x 1 x 2 x 3 x 4 x 5 x 6 Pivot Row x 1 x 2 x 3
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UF-ESI-6314 204be18ecfd6b68601feb157f4e7d43d4bd5b7d2.xls Part A page 2 of 11 package page _ of __ Luther Setzer 1 NASA Pkwy E Stop NEM3, Kennedy Space Center, FL 32899 (321) 544-7435 Tableau 1 Row Basis VRS z RHS Ratio 0 z 1.00 0.00 0.00 0.00 a b 3.00 e 1 0.00 1.00 0.00 0.00 c 2.00 -3.00 d 2 0.00 0.00 1.00 0.00 -4.00 1.00 4.00 2.00 3 0.00 0.00 0.00 1.00 -1.00 3.00 1.00 3.00 Marginal Net Gain Pivot Column Otherwise, the tableau would generate a pivot column and drive another iteration. To assure this tableau remains optimal, these variables must remain nonnegative. a 0 b 0 Part A (4 points): For what values of a and b is this an optimal tableau? (Assume d 0.) x 1 x 2 x 3 x 4 x 5 x 6 Pivot Row x 1 x 2 x 3 By the rules of optimization, the marginal net gain must remain either zero or negative in that row of the tableau.
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UF-ESI-6314 204be18ecfd6b68601feb157f4e7d43d4bd5b7d2.xls Part B page 3 of 11 package page _ of __ Luther Setzer 1 NASA Pkwy E Stop NEM3, Kennedy Space Center, FL 32899 (321) 544-7435 Row Basis VRS z RHS Ratio 0 z 1.00 0.00 0.00 0.00 a b 3.00 e 1 0.00 1.00 0.00 0.00 c 2.00 -3.00 d 2 0.00 0.00 1.00 0.00 -4.00 1.00 4.00 2.00 -0.50 3 0.00 0.00 0.00 1.00 -1.00 3.00 1.00 3.00 -3.00 Marginal Net Gain -a -b -3.00 Pivot Column Therefore, Tableau 2 becomes unbounded under the following conditions. a < 0 b > a c < 0 Part B (4 points): For what values of a , b , and c can we say that this problem is unbounded by looking at the current tableau? (You again need to have d ≥ 0.) If a parameter can take any value, please say so.
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