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# GE330_lect7 - Lecture 6 LP Sensitivity Analysis Sensitivity...

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Lecture 6 LP Sensitivity Analysis February 12, 2009

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Sensitivity Analysis Study the impact of the changes in input data to the optimal value (solution). In general, there are three different types of changes in input data. - Changes in the coefficients in the objective function - Changes in the right-hand side constants of the constraints - Changes in the left-hand side coefficients of the con- straints Sensitivity analysis regarding the last type is very hard, and therefore will not be covered in this course. 2
Graphical Sensitivity Analysis An Example: JOBCO produces two products on two machines. A unit of product 1 requires 2 hours on machine 1 and 1 hour on machine 2. For product 2, a unit requires 1 hour on machine 1 and 3 hours on machine 2. The revenues per unit of products 1 and 2 are \$30 and \$20, respectively. The total daily processing time available for each machine is 8 hours. LP Model: Let x 1 and x 2 be the daily number of units of products 1 and 2, respectively. max z = 30 x 1 + 20 x 2 s.t. 2 x 1 + x 2 8 (Machine 1) x 1 + 3 x 2 8 (Machine 2) x 1 , x 2 0 Optimal Solution: x 1 = 3 . 2, x 2 = 1 . 6, z = 128. 3

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Change in the Right-Hand Side Question: If we increase the daily capacity of machine 1 from 8 hours to 9 hours, how does the optimal value change? Look at the graph: The optimal solution moved from point C to point G and the optimal value changes from 128 to 142. The change in the optimal ob- jective value per unit change in the availability of machine 1 capacity (the shadow price or dual price of machine 1 capac- ity) is: 142 - 128 9 - 8 = \$14 / hr 4
Change in the Right-Hand Side Question: Is the calculated shadow price always valid? What is the range in which it is valid? ( the feasibility range ) As long as the changes in ma- chine 1 capacity move its con- straint parallel to itself to any point on the line segment BF , the shadow price is valid. Therefore, the feasibility range is: Minimum (at B ): 8 3 hr Maximum (at F ): 16hr 8 3 hr Machine 1 Cap 16 hr 5

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Change in the Right-Hand Side Similarly, we can get the shadow price and the feasibility range of Machine 2 capacity: Shadow price of Machine 2 capacity: 2\$/hr Feasibility range: 4 hr Machine 2 Capacity 24 hr Now we can answer the following questions: If we can increase the capacity of both machines, which one should receive higher priority? (Machine 1, because of its higher shadow price.) Should we suggest to increase the capacity of Machine 1 at the additional cost of \$10/hr? (Yes, 14-10 = 4 > 0.) If the capacity of Machine 1 increases from 8 hours to 13 hours, how will this affect the optimal revenue? (14 × (13 - 8) = 70. The optimal revenue will increase \$70.) NOTE : If the change in the right-hand side exceed the corre- sponding feasibility range, then the shadow price is no longer valid. 6
Change in the Objective Coefficients Question: Is it possible that we can change the objective coef- ficients without affecting the optimal value? What is the range of this kind of change?

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GE330_lect7 - Lecture 6 LP Sensitivity Analysis Sensitivity...

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