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【正文】 nts profit. The Newsboy Model (Continuous Demands) Notation: = Cost per unit of positive inventory remaining at the end of the period (known as the overage cost). = Cost per unit of unsatisfied demand. This can be thought of as a cost per unit of negative ending inventory (known as the underage cost). The demand is a continuous nonnegative random variable with density function and cumulative distribution function . The decision variable is the number of units to be purchased at the beginning of the period. ocucD? ?xf? ?xFQThe Newsboy Model (Continuous Demands) Determining the optimal policy: The cost function The optimal solution equation ? ?uoucccQF??*? ? ? ? ? ? ? ? ? ??? ? ???? QuQo dxxfQxcdxxfxQcQG 0The Newsboy Model (Continuous Demands) Determining the optimal policy: 02468101210 6 2 1000 200 300 400(Thousands)G(Q)OQ*QThe Newsboy Model (Continuous Demands) Example 2 (continued): Normally distributed with mean = and standard deviation = . Since Mac purchases the magazines for 25 cents and can salvage unsold copies for 10 cents, his overage cost is = 25 ? 10 = 15 cents. His underage cost is the profit on each sale, so that = 75 ? 25 = 50 cents. ??ocucThe Newsboy Model (Continuous Demands) Example 2 (continued): The critical ratio is = Purchase enough copies to satisfy all of the weekly demand with probability . The optimal is the 77th percentile of the demand distribution. *Q? ?uou ccc ?The Newsboy Model (Continuous Demands) Example 2 (continued): Ar e a = Q* xf ( x )The Newsboy Model (Continuous Demands) Example 2 (continued): Using the data of the normal distribution we obtain a standardized value of = . The optimal is Hence, he should purchase 15 copies every week. z Q* ??????? ?? zQThe Newsboy Model (Discrete Demands) Optimal policy for discrete demand: The procedure for finding the optimal solution to the newsboy problem when the demand is assumed to be discrete is a natural generalization of the continuous case. The optimal solution procedure is to locate the critical ratio between two values of and choose the corresponding to the higher value. That is ? ?QF Q? ? ? ?uouQx cccQFxp?????*0*The Newsboy Model (Discrete Demands) Example 2: Q ? ?Qf ? ?QF Q ? ?Qf ? ?QF 0 1/52 1/52 ( 0 . 0 19 2) 12 4/52 30 /52 ( 0 .5 76 9) 1 0 1/52 ( 0 .0 19 2) 13 1 /52 3 1 /52 ( 0 .5 96 2) 2 0 1 /52( 0 .0 1 92 ) 1 4 5/52 36 /52 ( 0 .6 92 3) 3 0 1 /52( 0 .0 19 2) 1 5 5/52 4 1 /52 ( 0 .7 88 5) 4 3/52 4/52 ( 0 .0 76 9) 1 6 1 /52 42 /52 ( 0 .8 07 7) 5 1/52 5/52 ( 0 .0 96 2) 1 7 3/52 45 /52 ( 0 .8 65 4) 6 2/52 7/52 ( 0 .1 34 6) 1 8 3/52 48 /52 ( 0 .9 23 1 ) 7 2/52 9/52 ( 0 .1 73 1 ) 1 9 3/52 5 1 /52 ( 0 .9 80 8) 8 4/52 1 3/52 ( 0 .2 50 0) 20 0 5 1 /52 ( 0 .9 80 8) 9 6/52 1 9/52 ( 0 .3 65 4) 2 1 0 5 1 /52 ( 0 .9 80 8) 1 0
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