hw11s - ENGRI 1101 Engineering Applications of OR Fall 08...

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ENGRI 1101 Engineering Applications of OR Fall ’08 Homework 11 Homework 11 Solutions 1. Let’s number the empty squares in the grid as follows: The first instinct in formulating this problem as an IP is to define 10 variables to represent the values assigned to each of the 10 empty squares on the grid. However, we will then have to impose constraints that say that the assigned values are all different — not something that can be done with linear constraints on variables at hand! So, instead, we will define the following decision variables: x ij = ± 1 if number i is assigned to square j, 0 otherwise , for i = 0 , 1 , . . . , 9 and j = 1 , . . . , 10. We need constraints so that each number is assigned exactly to one square: 10 X j =1 x ij = 1 for each i = 0 , 1 , . . . , 9 , and each square must get exactly one number: 9 X i =0 x ij = 1 for each j = 1 , . . . , 10 . Now, we also need constraints that ensure that the numbers in each row and column add up to 22. To do that, we need to be able to evaluate the actual number assigned to a particular square of the grid. Consider square
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This note was uploaded on 12/25/2009 for the course ENGRI 1101 at Cornell University (Engineering School).

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hw11s - ENGRI 1101 Engineering Applications of OR Fall 08...

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