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Unformatted text preview: IEOR 150, Fall 2010 Homework 5 1. (25 points) Consider a pair of supply chains with one manufacturer and one retailer in each chain. Retailers are exclusive, i.e., each of them does not buy products from the manufacturer in the other supply chain. The effect of demand substitution exists between the two retailers. In particular, the demand substitution rates are 1 for both retailers. We label the two parties in the first supply chain as manufacturer 1 and retailer 1 and the other two parties as manufacturer 2 and retailer 2. &amp;% $ M 1 &amp;% $ R 1 ? 6 &amp;% $ M 2 &amp;% $ R 2 Let c 1 = c 2 be the unit production costs for manufacturers 1 and 2, r 1 = r 2 = 1 be the unit retail prices for retailers 1 and 2, w 1 = 0 . 5 and w 2 = 0 . 6 be the unit wholesale price offered by manufacturers 1 and 2, and D 1 and D 2 be the demand distributions faced by retailers 1 and 2. We assume D 1 U (0 , 1), D 2 U (0 , 1), and D 1 and D 2 are independent. Let R 1 = D 1 + ( D 2 q 2 ) + and R 2 = D 2 + ( D 1 q 1 ) + be retailers 1 and 2s effective demands, where ( x ) + = max { x, } . The two retailers optimal order quantities q * 1 and q * 2 in equilibrium satisfy q * 1 = argmax q r 1 E h min { q,R 1 } i w 1 q and q * 2 = argmax q r 2 E h min { q,R 2 } i w 2 q. It has been shown that ( q * 1 ,q * 2 ) satisfies P ( R 1 q * 1 ) = 1 w 1 r 1 and P ( R 2 q * 2 ) = 1 w 2 r 2 . In other words, if the order quantity of retailer 2, q 2 , is fixed, then retailer 1 should choose her order quantity q 1 to satisfy P ( R 1 q 1 ) = 1 w 1 r 1 . (1) (a) (2 points) Suppose there is NO demand substitution. Find the optimal order quantities for(a) (2 points) Suppose there is NO demand substitution....
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 Fall '09

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