1、统计应用案例-报童模型,徐小林 香港中文大学 博士 南京大学 商学院 工商管理系 副教授Tel:25-83621292 ,在你面前有三个门，其中两个门里面是山羊，另外一个是汽车。你当然想得到那辆汽车，而不是臭气轰轰的山羊。主持人要求你选中一个，但是不能打开。此时主持人打开另外一个，是山羊。现在主持人问你，你要不要改成第三个门？,3,谁是更好的采购经理?,如果产品是 贺卡: 利润 = $3.00 成本 = $0.20 ？,设想有俩个经理 管理同样的一产品（超10个品种）的采购。假定条件相同。在季节结束， 张经理 基本 没有 剩余 ； 王经理多个品种有剩余。,报亭老板如何决策？,销售份数 天数1
2、102 203 404 205 10,南大北门的报亭在过去100天里，某报纸的销售记录如左表。报纸销售价1元，进价0.3元。问：报亭老板每天订几份报纸合适？ A. 2 B. 3 C. 4,5,什么是分析能力？,一位物理学家，工程师和数学家在东马徒步旅行， 看到一头黑山羊在山坡上吃草。物理学家 首先发表高见:“所有马国的山羊都是黑色的。”“不对，西蒙。有些马国的山羊是黑色,” 工程师纠正到。数学家最后发言：“从所看到的，我们只能说，在东马，至少有一只山羊至少身体的 一面是黑色的。”,决策思路,边际成本 = 边际收入,7,单周期最优订货量-1,定义：期末库存余量为正时的单位成本（滞销成本）需求未满
3、足导致的成本（机会成本）周期一开始购买的商品数量需求量D的概率密度函数需求量D的概率分布函数,案例：报童模型,Pandora皮衣：单位利润14.50$，因滞销降价而带来损失5.00$/件.,9,单周期最优订货量-2,从A-1看出， 对于面积=0.74， z=0.65 。因此,面积=0.74,需求量， X,f(x),ONeills Hammer 3/2 wetsuit,Hammer 3/2 timeline and economics,Economics: Each suit sells for p = $180 TEC charges c = $110 per suit Discounted
4、suits sell for v = $90,The “too much/too little problem”: Order too much and inventory is left over at the end of the season Order too little and sales are lost. Marketings forecast for sales is 3200 units,Newsvendor model implementation steps,Gather economic inputs: Selling price, production/procur
5、ement cost, salvage value of inventoryGenerate a demand model: Use empirical demand distribution or choose a standard distribution function to represent demand, e.g. the normal distribution, the Poisson distribution. Choose an objective: e.g. maximize expected profit or satisfy a fill rate constrain
6、t.Choose a quantity to order.,The Newsvendor Model: Develop a Forecast,Historical forecast performance at ONeill,Forecasts and actual demand for surf wet-suits from the previous season,Empirical distribution of forecast accuracy,Normal distribution tutorial,All normal distributions are characterized
7、 by two parameters, mean = m and standard deviation = s All normal distributions are related to the standard normal that has mean = 0 and standard deviation = 1. For example: Let Q be the order quantity, and (m, s) the parameters of the normal demand forecast. Probdemand is Q or lower = Probthe outc
8、ome of a standard normal is z or lower, where (The above are two ways to write the same equation, the first allows you to calculate z from Q and the second lets you calculate Q from z.) Look up Probthe outcome of a standard normal is z or lower in the Standard Normal Distribution Function Table.,Con
9、verting between Normal distributions,Start with = 100, = 25, Q = 125,Center the distribution over 0 by subtracting the mean,Rescale the x and y axes by dividing by the standard deviation,Start with an initial forecast generated from hunches, guesses, etc. ONeills initial forecast for the Hammer 3/2
10、= 3200 units. Evaluate the A/F ratios of the historical data:Set the mean of the normal distribution toSet the standard deviation of the normal distribution to,Using historical A/F ratios to choose a Normal distribution for the demand forecast,ONeills Hammer 3/2 normal distribution forecast,ONeill s
11、hould choose a normal distribution with mean 3192 and standard deviation 1181 to represent demand for the Hammer 3/2 during the Spring season.,Empirical vs normal demand distribution,Empirical distribution function (diamonds) and normal distribution function with mean 3192 and standard deviation 118
12、1 (solid line),“Too much” and “too little” costs,Cu = underage cost Co = overage cost The cost of ordering one more unit than what you would have ordered had you known demand. In other words, suppose you had left over inventory (i.e., you over ordered). Co is the increase in profit you would have en
13、joyed had you ordered one fewer unit. For the Hammer 3/2 Co = Cost Salvage value = c v = 110 90 = 20The cost of ordering one fewer unit than what you would have ordered had you known demand. In other words, suppose you had lost sales (i.e., you under ordered). Cu is the increase in profit you would
14、have enjoyed had you ordered one more unit. For the Hammer 3/2 Cu = Price Cost = p c = 180 110 = 70,Balancing the risk and benefit of ordering a unit,Ordering one more unit increases the chance of overage Expected loss on the Qth unit = Co x F(Q) F(Q) = Distribution function of demand = ProbDemand =
15、 Q) but the benefit/gain of ordering one more unit is the reduction in the chance of underage: Expected gain on the Qth unit = Cu x (1-F(Q),As more units are ordered, the expected benefit from ordering one unit decreases while the expected loss of ordering one more unit increases.,Newsvendor expecte
16、d profit maximizing order quantity,To maximize expected profit order Q units so that the expected loss on the Qth unit equals the expected gain on the Qth unit:Rearrange terms in the above equation - The ratio Cu / (Co + Cu) is called the critical ratio. Hence, to maximize profit, choose Q such that
17、 we dont have lost sales (i.e., demand is Q or lower) with a probability that equals the critical ratio,Finding the Hammer 3/2s expected profit maximizing order quantity with the empirical distribution function,Inputs: Empirical distribution function table; p = 180; c = 110; v = 90; Cu = 180-110 = 7
18、0; Co = 110-90 =20 Evaluate the critical ratio:Lookup 0.7778 in the empirical distribution function table If the critical ratio falls between two values in the table, choose the one that leads to the greater order quantity (choose 0.788 which corresponds to A/F ratio 1.3)Convert A/F ratio into the o
19、rder quantity,25,A/F Ratio =实际需求/ 预测需求,33 种产品,26,用历史数据构建的需求分布函数,如果某款新设计的产品预测需求为3,200,Q = 需求量,Hammer 3/2s expected profit maximizing order quantity using the normal distribution,Inputs: p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20; critical ratio = 0.7778; mean = m = 3192; standard deviation = s = 1181 Look up critical ratio in the Standard Normal Distribution Function Table: If the critical ratio falls between two values in the table, choose the greater z-statistic Choose z = 0.77 Convert the z-statistic into an order quantity:,
