cho x0 0 sản phẩm, lợi nhuận là cho hàm mục tiêu này được xác định bằngsau đó trong nhà tối ưu lợi nhuận cho một x0 nhất định là sau đó giả định rằng nhà sản xuất có lợi nhuận trong điều kiện ban đầu không có | 148 3 MODELING IN A MULTI-PERIOD FRAMEWORK the replenishment period the greater the replenishment period T the ÕsRT T n larger the base-stock level nT_ 0. ÕT Proof First note that neither Jd nor its derivative which is the left-hand side of equation denoted by B explicitly depends on snT . Furthermore according to Proposition no matter what base-stock level snT we choose the quantity that retailer n orders has the same distribution which depends only on demand. Thus given replenishment period T fQ . does not depend on the base-stock policy snT employed. This is to say that Jd and B are independent on snT . However if B does not depend on snT then the ÕT distributors best response 1 1R sn1 does not depend on snT 0 . ÕsnT The retailer s best response is determined with the standardized basestock level snT sRT T Tu 4tunsn see Proposition and thus sRt T 1 ÕT Mn as stated in the proposition. u s 0 T n n There are two important conclusions related to Proposition . The first conclusion is concerned with the supply chain s performance and thereby the corresponding centralized supply chain. If the supply chain is vertically integrated with one decision-maker responsible for setting both a replenishment period and base-stock level for each retailer then the centralized objective function is a summation of all costs involved J T Jđ-Â Jnr . The distributor s cost Jd is independent of the base-stock level as shown in Proposition . Therefore applying the first-order optimality condition to J T with respect to either q or snT we obtain equation . This implies that the condition for the Nash base-stock level is identical to the system-wide optimality condition. Next to find the system-wide optimality condition for the replenishment period we differentiate J T with respect to T which when taking into account and results in ÕJ T _ õC T ệ _C T ệ z dz ÕT J TÕT T2 f z z _ u Jhn s _z H z dz_ Jh s _z H z dz 0. T3 J 2y 1 _ s ò n REPLENISHMENT
