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IEOR162_hw02_sol - IEOR 162 Fall 2011 Suggested Solution to...

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IEOR 162, Fall 2011 Suggested Solution to Homework 02 Problem 1 (Problem 3.5.5) To formulate this problem, we label the four shifts as in the following table. Shift Number 1 2 3 4 Time 0 - 6 6 - 12 12 - 18 18 - 24 The decision variables are x ij = number of officers working at shifts i and j , i = 1 , ..., 4, j = i + 1 , ..., 4. The problem can then be formulated as min 144( x 12 + x 23 + x 34 + x 14 ) + 216( x 13 + x 24 ) s.t. x 12 + x 13 + x 14 15 (Demand in shift 1) x 12 + x 23 + x 24 5 (Demand in shift 2) x 13 + x 23 + x 34 12 (Demand in shift 3) x 14 + x 24 + x 34 6 (Demand in shift 4) x i 0 i = 1 , ..., 4 . Here 144 = 12 × 12 is the per worker wage for one who works in two consecutive shifts. Similarly, 216 = 18 × 12 is that for one who works in two nonconsecutive shifts. 1 The objective function minimize the total payments while the constraints guarantee that there are enough workers for every shifts. 2 Problem 2 (Problem 3.8.2) Let x 11 = number of pounds of grade 9 oranges sold in bags, x 12 = number of pounds of grade 9 oranges used to produce juice, x 21 = number of pounds of grade 6 oranges sold in bags, and x 22 = number of pounds of grade 6 oranges used to produce juice.
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