GE330_lect8 - Lecture 8 Algebraic Sensitivity Analysis...

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Lecture 8 Algebraic Sensitivity Analysis February 17, 2009
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Algebraical Sensitivity Analysis TOYCO Model: TOYCO assembles three types of toys–trains, trucks and cars–using three operations. The daily limits on the available times for the three operations are 430,460, and 420 minutes, respectively, and the revenues per unit of toy train, truck, and car are $3, $2, and $5, respectively. The assembly times per train at the three operations are 1, 3, 1 minutes, re- spectively. The corresponding times per train and per car are (2,0,4) and (1,2,0) minutes (a zero time indicates that the op- eration is not used.) LP Model: Let x 1 , x 2 , and x 3 be the daily number of units assembled of trains, trucks, and cars, respectively. max z = 3 x 1 + 2 x 2 + 5 x 3 s.t. 2 x 1 + x 2 + x 3 430 (Operation 1) x 1 + 2 x 3 460 (Operation 2) x 1 + 4 x 2 420 (Operation 3) x 1 ,x 2 ,x 3 0 9
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TOYCO Model: Optimal Tableau Introduce slack variables x 4 , x 5 , and x 6 , the initial tableau is: Basic x 1 x 2 x 3 x 4 x 5 x 6 Solution z -3 -2 -5 0 0 0 0 x 4 1 2 1 1 0 0 430 x 5 3 0 2 0 1 0 460 x 6 1 4 0 0 0 1 420 The optimal tableau is: Basic x 1 x 2 x 3 x 4 x 5 x 6 Solution z 4 0 0 1 2 0 1350 x 2 - 1 4 1 0 1 2 - 1 4 0 100 x 3 3 2 0 1 0 1 2 0 230 x 6 2 0 0 -2 1 1 20 10
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After adding slack variables, rewrite the constraints as follows: x 1 + 2 x 2 + x 3 = 430 - x 4 (Operation 1) 3 x 1 + 2 x 3 = 460 - x 5 (Operation 2) x 1 + 4 x 2 = 420 - x 6 (Operation 3) We can say that a one-unit decrease in the slack variables is equivalent to a one-unit increase in the resource (operations time). On the other hand, from the optimal tableau we know that: z + 4 x 1 + x 4 + 2 x 5 + 0 x 6 = 1350 , which is equivalent to z = 1350 - 4 x 1 + 1 × (increase in operation 1 time) + 2 × (increase in operation 2 time) + 0 × (increase in operation 3 time) Note: 1. In this case, the shadow price for a resource is actually the corresponding z -row coefficient in the optimal tableau! 2. The shadow price of operation 3 is zero, this is reasonable because this resource is already abundant (the slack is positive). 11
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This note was uploaded on 08/31/2010 for the course IESE GE 330 taught by Professor Nedich during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE330_lect8 - Lecture 8 Algebraic Sensitivity Analysis...

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