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Unformatted text preview: IEOR 162, Spring 2010 Suggested Solution to Homework 05 Problem 3.Review.11 First, we label the two refinery at Los Angeles and Chicago as refinery 1 and 2 and the two distribution points at Houston and New York City as distribution points 1 and 2. Then let the decision variables be x i = million barrels of capacity created for refinery i , i = 1 , 2, and y ij = million barrels of oil shipped from refinery i to distribution point j , i = 1 , 2, j = 1 , 2. Also we define the following parameters: P ij is the profit (in thousands) per million barrels of oil shipped from refinery i to distribution point j , C i is the unit cost (in thousands) of expanding capacity for one million barrel in refinery i , K i is the current capacity (in million barrel) in refinery i , and D j is the demand size (in million barrels) at distribution point j for all i = 1 , 2 and j = 1 , 2. Specifically, we have P = • 20 15 18 17 ‚ , C = • 120 150 ‚ , K = • 2 3 ‚ , D = £ 5 5 / . With the definitions of variables and parameters, we formulate the problem as max 10 2 X i =1 2 X j =1 P ij y ij- 2 X i =1 C i x i s.t. 2 X i =1 y ij ≤ D j ∀ j = 1 , 2 2 X j =1 y ij ≤ K i + x i ∀ i = 1 , 2 x i ,y ij ≥ ∀ i = 1 , 2 ,j = 1 , 2 . The objective function consists of two parts, the 10-year total profit and the one-time expansion cost. The first constraint ensures that the total sales at each distribution point is at most the demand size. The second constraint ensures that the total production quantity at each refinery does not excess the (post-expansion) capacity. The last constraint is the nonnegativity constraint. Problem 3.Review.21 First we label products A and B as 1 and 2. Let the decision variables be x i = price of product i , i = 1 , 2. With the definition of variables, we formulate the problem as max 1000 x 1 + 1500 x 2 s.t. 10- x 1 ≥ 8- x 2 , 10- x 1 ≥ 15- x 2 ≥ 12- x 1 , 15- x 2 ≥ x i ≥ ∀ i = 1 , 2 . The objective function maximizes the total sales revenue because group 1 members purchase A and group 2 members purchase B. The first and second constraints ensure that group 1 members will purchase A rather than purchasing B or nothing. The Third and fourth constraints ensure that group 2 members will purchase B rather than purchasing A or nothing. The last constraint is the nonnegativity constraint. 1 Problem 3.Review.29 First, we label the premium and regular juice by product 1 and 2 and the grade 6 and 3 orange by material 1 and 2. In other words, the one with higher quality is labeled as 1 while the other one is labeled as 2. This kind of consistency in defining indices is helpful.kind of consistency in defining indices is helpful....
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- Spring '07
- Optimization, Constraint, Distribution Point, resource limitation constraint, nonnegativity constraint