CH5_Sensitivity - Chapter 5 Sensitivity Analysis presented...

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MIT and James Orlin © 2003 1 Chapter 5. Sensitivity Analysis presented as FAQs Points illustrated on a running example of glass manufacturing.
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MIT and James Orlin © 2003 2 Glass Example x 1 = # of cases of 6-oz juice glasses (in 100s) x 2 = # of cases of 10-oz cocktail glasses (in 100s) x 3 = # of cases of champagne glasses (in 100s) max 5 x 1 + 4.5 x 2 + 6 x 3 ($100s) s.t 6 x 1 + 5 x 2 + 8 x 3 60 (prod. cap. in hrs) 10 x 1 + 20 x 2 + 10 x 3 150 (wareh. cap. in ft 2 ) x 1 8 (6-0z. glass dem.) x 1 0, x 2 0, x 3 0
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MIT and James Orlin © 2003 3 FAQ. Could you please remind me what a shadow price is? Let us assume that we are maximizing. A shadow price is the increase in the optimum objective value per unit increase in a RHS coefficient, all other data remaining equal. The shadow price is valid in an interval.
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MIT and James Orlin © 2003 4 FAQ. Of course, I knew that. But can you please provide an example. Certainly. Let us consider the glass example. Let’s look at the objective function if we change the production time from 60 and keep all other values the same. Production hours Optimal obj. value difference 60 51 3/7 61 52 3/14 11/14 62 53 11/14 63 53 11/14 11/14 The shadow Price is 11/14.
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MIT and James Orlin © 2003 5 More changes in the RHS Production hours Optimal obj. value difference 64 54 4/7 11/14 65 55 5/14 11/14 66 56 1/11 * 67 56 17/22 15/22 The shadow Price is 11/14 until production = 65.5
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MIT and James Orlin © 2003 6 FAQ. What is the intuition for the shadow price staying constant, and then changing? Recall from the simplex method that the simplex method produces a “basic feasible solution.” The basis can often be described easily in terms of a brief verbal description. Glass Example
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MIT and James Orlin © 2003 7 The verbal description for the optimum basis for the glass problem: 1. Produce Juice Glasses and cocktail glasses only 2. Fully utilize production and warehouse capacity z = 5 x 1 + 4.5 x 2 6 x 1 + 5 x 2 = 60 10 x 1 + 20 x 2 = 150 x 1 = 6 3/7 x 2 = 4 2/7 z = 51 3/7
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MIT and James Orlin © 2003 8 The verbal description for the optimum basis for the glass problem: 1. Produce Juice Glasses and cocktail glasses only 2. Fully utilize production and warehouse capacity z = 5 x 1 + 4.5x 2 6 x 1 + 5 x 2 = 60 + 10 x 1 + 20 x 2 = 150 x 1 = 6 3/7 + 2 /7 x 2 = 4 2/7 – /7 z = 51 3/7 + 11/14 For = 5.5, x 1 = 8, and the constraint x 1 8 is binding.
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MIT and James Orlin © 2003 9 FAQ. How can shadow prices be used for managerial interpretations? Let me illustrate with the previous example. How much should you be willing to pay for an extra hour of production? Glass Example
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MIT and James Orlin © 2003 10 FAQ. Does the shadow price always have an economic interpretation? The answer is no, unless one wants to really stretch what is meant by an economic interpretation. Consider ratio constraints
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MIT and James Orlin © 2003 11 Apartment Development x 1 = number of 1-bedroom apartments built x 2 = number of 2-bedroom apartments built x 3 = number of 3-bedroom apartments build x 1 /(x 1 + x 2 + x 3 ) .5 x 1 .5x 1 + .5x 2 + .5x 3 .5x 1 – .5x 2 - .5x 3 0
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