HW4Sol - Fall 2010 IEOR 160 Industrial Engineering...

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Unformatted text preview: Fall 2010 IEOR 160 Industrial Engineering & Operations Research September 25, 2010 Page 1 of 4 HOMEWORK 4 SOLUTIONS Chapter 12.6 2. We wish to maximize f ( q 1 ,q 2 ) = 2( q 1 / 3 1 + q 2 / 3 2 )- q 1- 1 . 5 q 2 . To find the stationary point(s) for f ( q 1 ,q 2 ), f q 1 = 2 q- 2 / 3 1 / 3- 1 = 0 f q 2 = 4 q- 1 / 3 2 / 3- 1 . 5 = 0 So q 1 = (2 / 3) 3 / 2 , and q 2 = (4 / 4 . 5) 3 . The objective function is the sum of concave functions, so we know that the above values for q 1 and q 2 will maximize profit. 3. Let f ( q 1 ,q 2 ) be the profit when company 1 sells q 1 units and company 2 sells q 2 units. Then f ( q 1 ,q 2 ) = ( q 1 + q 2 )(200- q 1- q 2 )- q 1- . 5 q 2 2 = 199 q 1 + 200 q 2- q 2 1- 2 q 1 q 2- 1 . 5 q 2 2 To find the stationary point(s), f q 1 = 199- 2 q 1- 2 q 2 = 0 f q 2 = 200- 3 q 2- 2 q 1 = 0 So q 1 = 98 . 5 , and q 2 = 1. Since H =- 2- 2- 2- 3 The first leading pricipal minor H 1 =- 2 < 0 and the second leading principal minor H 2 = 2 > 0. Thus q 1 = 98 . 5 , and q 2 = 1 are local max. Since f ( q 1 ,q 2 ) is a concave function, we know that (98 . 5 , 1) maximizes the profit. 5. Let f ( p 1 ,p 2 ) be the profit if the company charges p 1 and p 2 for product 1 and 2. Then f ( p 1 ,p 2 ) = ( p 1- 25)(60- 3 p 1 + p 2 ) + ( p 2- 72)(80 + p...
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HW4Sol - Fall 2010 IEOR 160 Industrial Engineering...

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