hw9_sol - Homework 9 Solution Exercise 3 page 352 Suppose...

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Homework 9 Solution April 14, 2009 Exercise 3, page 352 Suppose you have 7 full water bottles, 7 half-full, and 7 empty. You would like to divide the 21 bottles among three individuals so that each receives 7 bottles with the same quantity of water. Express the problem as integer programming and find an optimal solution. Solution : Let x ij denote the number of bottles of type i assigned to individual j = 1 , 2 , 3, where i = 1 denotes a full bottle, i = 2 denotes a half-full and i = 3 denotes an empty bottle. We have x ij 0 and integer for all i and j . The fact that there are 7 bottles of each type is modeled as follows: x i 1 + x i 2 + x i 3 = 7 for all i = 1 , 2 , 3 . The total available amount of water is 10 and 1 / 2, which gives a share of 3 and 1 / 2 bottle with water to each. The fact that we want to divide the amount of water equally among the individuals is modeled as x 1 j + x 2 j / 2 = 3 . 5 for all j = 1 , 2 , 3 . The fact that each individual receives 7 bottles is modeled as follows: x 1 j + x 2 j + x 3 j = 7 for all j = 1 , 2 , 3 . To complete the formulation, we may introduce the dummy objective of minimizing 3 i,j =1 0 x ij , so the problem is actually integer programming feasibility problem. The solution is no unique. One possible solution is that individual 1 receives 3 full bottles, 1 half-full and 3 empty bottles. Individual 2 receives the same, i.e., 3 full bottles, 1 half-full and 3 empty bottles. Individual 3 receives 1 full, 5 half-full and 1 empty bottle. Exercise 7, page 352 You have the following three-letter words: AFT, FAR, TVA, ADV, JOE, FIN, OSF, and KEN. Each letter in the alphabet is assigned a value, staring with A=1 to Z=26. Each word is scored by summing the numeric values of its letters. For example, AFT has score of 1+6+20=27. You are to select five of the given eight words that yield the maximum total score. The selected five words should satisfy the following constraints: sum of letter 1 scores < sum of letter 2 scores < sum of letter 3 scores . 1
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Formulate the problem as integer programming (IP). Solution : Given the score assignment to each letter, we can organize the words in the table as follows; Index j Word Letter 1 score Letter 2 score Letter 3 score Word score 1 AFT 1 6 20 27 2 FAR 6 1 18 25 3 TV A 20 22 1 43 4 ADV 1 4 22 27 5 JOE 10 15 5 30 6 FIN 6 9 14 29 7 OSF 15 19 6 40 8 KEN 11 5 14 30 We introduce x j variable denoting the selection of word j , j = 1 , 2 , . . . , 8, where x j = 0 when word j is not selected and x j = 1 when word j is selected. The objective is to maximize the word score: max 27 x 1 + 25 x 2 + 43 x 3 + 27 x 4 + 30 x 5 + 29 x 6 + 40 x 7 + 30 x 8 .
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