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Unformatted text preview: IEOR 162, Spring 2012 Suggested Solution to Homework 04 Problem 1 (a) The manufacturers problem is max 0 . 4( p 1 10 q 1 ) + 0 . 6( p 2 10 q 2 ) s.t. 20 q 1 p 1 (IR1) 15 q 2 p 2 (IR2) 20 q 1 p 1 20 q 2 p 2 (IC1) 15 q 2 p 2 15 q 1 p 1 . (IC2) (b) Adding (IC1) and (IC2), we get 20 q 1 + 15 q 2 20 q 2 15 q 1 5 q 1 5 q 2 q 1 q 2 . (c) To show that (IR1) is redundant, we apply (IC1) and (IR2) as follows: 20 q 1 p 1 20 q 2 p 2 (IC1) 15 q 2 p 2 (Because 20 > 15) . (IR2) (d) To show that (IC1) is binding at any optimal solution, consider the manufacturers problem without (IR1). Now, to maximize her expected profit, the manufacturer would like to make p 1 as large as possible. If there is no (IR1), then (IC1) is the only constraint that has p 1 at the LHS. The manu facturer will keep increasing p 1 until (IC1) is binding (i.e., when the LHS hits the RHS). Therefore, (IC1) must be binding at any optimal solution. Note that when the manufacturer increases p 1 , (IC2) will be relaxed as the RHS becomes smaller. We thus do not need to worry about (IC2)....
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 Spring '07
 Zhang

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