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IEOR162_hw03_sol - IEOR 162 Spring 2012 Suggested Solution...

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IEOR 162, Spring 2012 Suggested Solution to Homework 03 Problem 1 (Modified from Problems 3.8.11) Note. This is actually NOT a typical blending problem. If you define 12 decision variables, each for a pair of chemical and ingredient, then it is quite possible that the formulation is wrong. Let the decision variables be x i = pounds of chemical i used, i = 1 , ..., 4. Then the data from Table 19 allows us to formulate the three quantity constraints. With the constraint that at least 100 lb of chemical 2 must be used, the complete formulation of this problem is min 8 x 1 + 10 x 2 + 11 x 3 + 14 x 4 s.t. x 1 + x 2 + x 3 + x 4 = 1000 (Total amount produced) - 0 . 05 x 1 - 0 . 02 x 2 + 0 . 02 x 3 + 0 . 04 x 4 0 (Quality: ingredient A) - 0 . 02 x 1 - 0 . 01 x 3 + 0 . 05 x 4 0 (Quality: ingredient B) - 0 . 01 x 1 - 0 . 01 x 2 + 0 . 02 x 3 + 0 . 02 x 4 0 (Quality: ingredient C) x 2 100 (The least amount of chemical 2) x i 0 i = 1 , ..., 4 . In this formulation, the first constraint makes sure that 1000 lb are produced. The three constraints on A, B, and C are then formulated by making the percentage enough. For example, for A it must have 0 . 03 x 1 + 0 . 06 x 2 + 0 . 1 x 3 + 0 . 12 x 4 x 1 + x 2 + x 3 + x 4 0 . 08 , (1) which is equivalent to - 0 . 05 x 1 - 0 . 02 x 2 + 0 . 02 x 3 + 0 . 04 x 4 0. As we emphasized in the discussion session, please remove the nonlinear formulations like (1) in your submitted paper . The objective function minimizes the total cost.
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