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Unformatted text preview: LP Example Stanley B. Gershwin Massachusetts Institute of Technology Consider the factory in Figure 1 that consists of three parallel machines. It makes a single product which can be produced using any one of the machines. The possible material ﬂows are indicated. Assume that the cost ($/part) of using machine M i is c i , and that the maximum rate that M i can operate is µ i . Assume that c 3 > c 2 > c 1 > and let µ 3 = . The total demand is D . Problem: How should the demand be allocated among the machines to minimize cost? Intuitive answer: We want to use the least expensive machine as much as possible, and the most expensive machine as little as possible. Therefore If D µ 1 , x 1 = D, x 2 = x 3 = cost = c 1 D If µ 1 < D µ 1 + µ 2 , x 1 = µ 1 , x 2 = D − µ 1 , x 3 = cost = c 1 µ 1 + c 2 ( D − µ 1 ) If µ 1 + µ 2 < D, x 1 = µ 1 , x 2 = µ 2 , x 3 = D − µ 1 − µ 2 cost = c 1 µ 1 + c 2 µ 2 + c 3 ( D − µ 1 − µ 2 ) LP formulation: min c 1 x 1 + c 2 x 2 + c 3...
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