IEOR150F10_hw05_sol

IEOR150F10_hw05_sol - IEOR 150 Fall 2010 Homework 5 1(a Let...

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Unformatted text preview: IEOR 150, Fall 2010 Homework 5 1. (a) Let ¯ q 1 and ¯ q 2 be the optimal order quantities of retailers 1 and 2, we have P ( D 1 ≤ ¯ q 1 ) = ¯ q 1- 1- = 1- w 1 r 1 = 0 . 5 ⇒ ¯ q 1 = 0 . 5 . Similarly, ¯ q 2 = 1- w 2 r 2 = 0 . 4. (b) If retailer 2 has no stock-out ( D 2 ≤ q 2 ), then R 1 = D 1 . On the other hand, if retailer 2 does has stock out ( D 2 < q 2 ), then R 1 = D 1 + D 2- q 2 . Therefore, P ( R 1 ≤ q 1 ) = P ( R 1 ≤ q 1 ,D 2 ≤ q 2 ) + P ( R 1 ≤ q 1 ,D 2 > q 2 ) = P ( D 1 ≤ q 1 ,D 2 ≤ q 2 ) + P ( D 1 + D 2- q 2 ≤ q 1 ,D 2 > q 2 ) Note that P ( D 1 ≤ q 1 ,D 2 ≤ q 2 ) = P ( D 1 ≤ q 1 ) P ( D 2 ≤ q 2 ) = q 1 q 2 because D 1 and D 2 are independent. Also note that P ( D 1 + D 2- q 2 ≤ q 1 ,D 2 > q 2 ) = Z 1 q 2 P ( D 1 + y- q 2 ≤ q 1 ) 1 1- dx (integrating over all possible realization of D 2 ) = Z 1 q 2 P ( D 1 ≤ q 1 + q 2- y ) dx = Z 1 q 2 ( q 1 + q 2- y ) dx = ( q 1 + q 2 )(1- q 2 )- 1 2 (1- q 2 2 ) ....
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IEOR150F10_hw05_sol - IEOR 150 Fall 2010 Homework 5 1(a Let...

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