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IEOR162_Midterm1_sol

IEOR162_Midterm1_sol - IEOR 162 Fall 2011 Suggested...

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IEOR 162, Fall 2011 Suggested Solution to Midterm 1 1. Let x i = production quantity of product i , i = 1 , 2. To produce x 1 units of product 1, we need x 1 units of component 1 and 2 x 1 units of component 2. To produce x 2 units of product 2, we need 2 x 2 units of component 1 and 3 x 2 units of component 2. Therefore, in total we need x 1 + 2 x 2 units of component 1 and 2 x 1 + 3 x 2 units of component 2. This requires 50( x 1 + 2 x 2 ) + 10(2 x 1 + 3 x 2 ) = 70 x 1 + 130 x 2 dollars. To maximize the sales revenue (not profit), we formulate the problem as min 120 x 1 + 200 x 2 s.t. 70 x 1 + 130 x 2 1000 (budget constraint) x 1 + 2 x 2 10 (supply limitation of component 1) x i 0 i = 1 , 2 . 2. (a) To remove x 3 , note that the third constraint must be binding at the optimal solution: If it is not, we may decrease x 3 and improve our solution. Therefore, the third constraint can be rewritten as - x 1 + x 2 - x 3 = 0, or x 3 = - x 1 + x 2 . Then we may remove the third constraint and replace x 3 in the objective function by - x 1 + x 2 . This results in the following equivalent LP
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