s02lec06b

s02lec06b - 15.053 Glass Example x1 = of cases of 6-oz...

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1 15.053 February 26, 2002 z Sensitivity Analysis z presented as FAQs Points illustrated on a running example of glass manufacturing. If time permits, we will also consider the financial example from Lecture 2. 2 Glass Example z x 1 = # of cases of 6-oz juice glasses (in 100s) z x 2 = # of cases of 10-oz cocktail glasses (in 100s) z 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 3 FAQ. Could you please remind me what a shadow price is? z 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. z The shadow price is valid in an interval. 4 FAQ. Of course, I knew that. But can you please provide an example. z Certainly. Let us recall the glass example given in the book. Let’s look at the objective function if we change the production time from 60 and keep all other values the same. 11/14 53 11/14 63 11/14 53 62 11/14 52 3/14 61 51 3/7 60 difference Optimal obj. value Production hours The shadow Price is 11/14. 5 More changes in the RHS 15/22 56 17/22 67 * 56 1/11 66 11/14 55 5/14 65 11/14 54 4/7 64 difference Optimal obj. value Production hours The shadow Price is 11/14 until production = 65.5 6 FAQ. What is the intuition for the shadow price staying constant, and then changing? z 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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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 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. 9 FAQ. How can shadow prices be used for managerial interpretations? z Let me illustrate with the previous example. z How much should you be willing to pay for an extra hour of production? Glass Example 10 FAQ. Does the shadow price always have an economic interpretation? z The answer is no, unless one wants to really stretch what is meant by an economic interpretation. z Consider ratio constraints 11 Apartment Development z x 1 = number of 1-bedroom apartments built z x 2 = number of 2-bedroom apartments built z x 3 = number of 3-bedroom apartments build z x 1 /(x 1 + x 2 + x 3 ) .5 Î x 1 .5x 1 + .5x 2 + .5x 3 z Î .5x 1 –5 .x 2 -.5x 3 0 z The shadow price is the impact of increasing the 0 to a 1.
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s02lec06b - 15.053 Glass Example x1 = of cases of 6-oz...

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