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Unformatted text preview: Lecture 13: Sensitivity analysis of Linear programs September 2224, 2010 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 righthand side constants of the constraints Changes in the lefthand 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 Optimal Solution: x 1 = 3 . 2, x 2 = 1 . 6, z = 128. 3 Change in the RightHand 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 = $14 / hr 4 Change in the RightHand Side Question: Is the calculated shadow price always valid? What is the range in which it is valid? ( the feasibility range ) As long as the changes in ma chine 1 capacity move its con straint parallel to itself to any point on the line segment BF , the shadow price is valid. Therefore, the feasibility range is: Minimum (at B ): 8 3 hr Maximum (at F ): 16hr 8 3 hr Machine 1 Cap 16 hr 5 Change in the RightHand Side Similarly, we can get the shadow price and the feasibility range of Machine 2 capacity: Shadow price of Machine 2 capacity: 2$/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, 1410 = 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....
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 Spring '08
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