IEOR162_hw10_sol

IEOR162_hw10_sol - IEOR 162 Fall 2011 Suggested Solution to...

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Unformatted text preview: IEOR 162, Fall 2011 Suggested Solution to Homework 10 Problem 1 (Problem 9.2.10) Let z be a binary variable such that z = ‰ 0 if x + y ≤ 3 is satisfied and 1 if 2 x + 5 y ≤ 12 is satisfied . Let M be a very large number, then the following two constraints x + y- 3 ≤ Mz 2 x + 5 y- 12 ≤ M (1- z ) ensures that at least one of x + y ≤ 3 and 2 x + 5 y ≤ 12 is satisfied. The condition that both x and y are integers is not important. Note that because there is no information regarding the possible values of x and y , there is no way for us to find a specific value for M . Problem 2 (Problem 9.2.11) First, note that “if x ≤ 2 then y ≤ 3” is equivalent to “ x > 2 or y ≤ 3”. Before we apply the technique of modeling “either-or” requirements, note that it is not allowed to have strict inequalities in an LP or IP formulation. Therefore, we must apply the condition that x is an integer to convert x > 2 into a weak inequality. To do this, note that x > 2 is equivalent to x ≥ 3 if x is an integer. Therefore, all we need to do is to write constraints so that “ x ≥ 3 or y ≤ 3”. Let z be a binary variable such that z = ‰ 0 if x ≥ 3 is satisfied and 1 if y ≤ 3 is satisfied . Let M be a very large number, then the following two constraints 3- x ≤ Mz y- 3 ≤ M (1- z ) ensures that at least one of x ≥ 3 and y ≤ 3 is satisfied, i.e., if x ≤ 2 then y ≤ 3. Note that because there is no information regarding the possible values of x and y , there is no way for us to find a specific value for M . Problem 3 (Problem 9.2.12) Let New York, Los Angeles, Chicago, and Atlanta be city 1, 2, 3, and 4. We then define x ij = units shipped from city i to region j , i = 1 ,..., 4, j = 1 ,..., 3 and y i = ‰ 1 if city i is chosen 0 otherwise , i = 1 ,..., 4 as our decision variables. We also define F = (400 , 500 , 300 , 150) as the fixed cost vector for cities, D = (80 , 70 , 40) as the demand vector for regions, and C = 20 40 50 48 15 26 26 35 18 24 50 35...
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This note was uploaded on 02/25/2012 for the course IEOR 162 taught by Professor Zhang during the Fall '07 term at Berkeley.

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IEOR162_hw10_sol - IEOR 162 Fall 2011 Suggested Solution to...

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