# lect17 - (e Suppose that the number of hours available in...

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ISE 536–Fall03: Linear Programming and Extensions November 3, 2003 Lecture 17: Review Duality and Sensitivity Lecturer: Fernando Ord´o˜nez Problem 5.8 max 51 E + 102 C + 66 P 1 66 P 2 89 B s . t . 10 E + 15 C + 10 P 1 + 10 P 2 + 20 B 130 E + 2 C + 2 P 1 + P 2 + B 13 3 E + C + 6 P 1 + 6 P 2 + 3 B 45 2 E + 4 C + 2 P 1 + 5 P 2 + 3 B 23 P 1 - P 2 = 0 Solution: Optimal Reduced Objective Allowable Allowable value cost coeﬃcient increase decrease E 0 -3.571 51 C 2 102 16.667 12.5 P 1 0 66 37.571 P 2 0 -37.571 66 37.571 B 5 0 89 47 12.5 Slack Dual Constr. Allowable Allowable value variable RHS increase decrease Clay 130 1.4 130 23.33 43.75 Enamel 9 13 4 Dry 17 0 45 28 Klin 20.1 23 5.6 3.5 Prim. 0 11.4 0 3.5 0 (0) Fill the solution tables (a) Optimal quantity of each product, what is the total proﬁt? 1

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(b) Economical interpretation of the optimal dual variables appearing in the sensitivity report, for each constraint. (c) Should we buy 20 lb. of Clay at \$1.1 per pound? (d) We would like to introduce a new product that requires 10 Clay, 2 Enamel, 2 Dry room, and 4 Kiln, with a price such that the proﬁt per unit is \$80. Should we produce this product? What is the minimum proﬁt that makes this product attractive?
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Unformatted text preview: (e) Suppose that the number of hours available in the dry room decreases by 30. Give a bound for the decrease in the total proﬁt. (f) If we replace the last constraint with P 1-P 2 ≥ 0, would the amount of P 1 produced in the optimal solution be positive? Problem 4.26 Let A be a given matrix. Show that exactly one of the following alternatives must hold. (a) There exists some x 6 = 0 such that Ax = 0, x ≥ 0. (b) There exists some p such that p t A > t . proof: Assume that both (a) and (b) hold. 2 Now ﬁnish the proof by showing that if (a) doesn’t hold, then (b) holds. Why is this enough? Problem 4.14 Give an example in which the primal problem has a degenerate optimal basic feasible solution, but the dual has a unique optimal solution. 3...
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## This note was uploaded on 02/13/2012 for the course ISE 536 taught by Professor Yy during the Spring '05 term at South Carolina.

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lect17 - (e Suppose that the number of hours available in...

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