hw10solutions - IEOR 162 Linear Programming Spring 2007...

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IEOR 162 Linear Programming Spring 2007 Homework 10 Solutions 6.3.1 (3 points) For the Dakota problem, the original optimal tableau is given by: z x 1 x 2 x 3 s 1 s 2 s 3 RHS 1 0 5 0 0 10 10 280 0 0 - 2 0 1 2 - 8 24 0 0 - 2 1 0 2 - 4 8 0 1 1 . 25 0 0 - . 5 1 . 5 2 Now we want to consider changes in c 3 , so that we now have c 3 = 20 + Δ. Let Δ c B = (0 , Δ , 0). We need to recompute the reduced costs for the non-basic variables and determine the range of Δ for which they remain non-positive. ¯ c N = c N - ( c B c B ) A - 1 B A N = ( - 5+2Δ , - 10 - , - 10+4Δ) (0 , 0 , 0) ⇒ - 5 Δ 2 . 5 15 c 3 22 . 5 If c 3 = 21, then Δ = 1, which implies that the current basis remains optimal based on the previous computations. Thus, x * = (2 , 0 , 8) and z * = 280 + Δ c B A - 1 B b = 288. If c 3 = 25, then Δ = 5, which implies that the current basis is not optimal. We update our tableau and perform the simplex algorithm to find the new optimal solution. Note that the first tableau below is the same as the original optimal tableau with the top row adjusted for the change in c 3 . z x 1 x 2 x 3 s 1 s 2 s 3 RHS 1 0 - 5 0 0 20 - 10 320 0 0 - 2 0 1 2 - 8 24 0 0 - 2 1 0 2 - 4 8 0 1 1 . 25 0 0 - . 5 1.5 2 z x 1 x 2 x 3 s 1 s 2 s 3 RHS 1 20 3 10 3 0 0 50 3 0 1000 3 0 16 3 14 3 0 1 - 2 3 0 104 3 0 8 3 4 3 1 0 2 3 0 40 3 0 2 3 5 6 0 0 - 1 3 1 4 3 This tableau is optimal, so our optimal solution is x * = (0 , 0 , 40 3 ) and z * = 1000 3 . 6.3.6 (6 points) a. c 1 = 3 + Δ. Note that x 1 is a nonbasic variable, so as Δ increases, so does the reduced cost for x 1 . Clearly Δ 3 in order for the current basis to remain optimal since ¯ c 1 = - 3+Δ. If c 1 = 7, then Δ = 4 and the current basis is not optimal. Use the simplex method starting from the current basic feasible solution: z x 1 x 2 x 3 s 1 s 2 RHS 1 - 1 0 0 4 1 300 0 .5 0 1 1 . 5 - . 5 25 0 . 5 1 0 - . 5 . 5 25 1
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z x 1 x 2 x 3 s 1 s 2 RHS 1 0 0 2 7 0 350 0 1 0 2 3 - 1 50 0 0 1 - 1 - 2 1 0 This implies that x * = (50 , 0 , 0) and z * = 350. b.
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This homework help was uploaded on 04/02/2008 for the course IEOR 162 taught by Professor Zhang during the Spring '07 term at University of California, Berkeley.

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hw10solutions - IEOR 162 Linear Programming Spring 2007...

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