报童模型

Slide 1 ， The Newsvendor Model 报童模型 Slide 2 ， O’Neill’s Hammer 3/2 wetsuit Slide 3 ， Hammer 3/2 timeline and economics Nov Dec Jan Feb Mar Apr May Jun Jul Aug Generate forecast of demand and submit an order to TEC Receive order from TEC at the end of the month Spring selling season Left over units are discounted Economics: • Each suit sells for p = $180 • TEC charges c = $110 per suit • Discounted suits sell for v = $90 ƒ 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. ƒ Marketing’s forecast for sales is 3200 units. Slide 4 ， Newsvendor model implementation steps ƒ Gather economic inputs: − Selling price, production/procurement cost, salvage value of inventory ƒ Generate a demand model: − Use empirical demand distribution or choose a standard distribution function to represent demand, e.g. the normal distribution, the Poisson distribution. ƒ Choose an objective: − e.g. maximize expected profit or satisfy a fill rate constraint. ƒ Choose a quantity to order. Slide 5 ， The Newsvendor Model: Develop a Forecast 需求预测 Slide 6 ， Historical forecast performance at O’Neill 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 Forecast A c t u a l d e m a n d . Forecasts and actual demand for surf wet-suits from the previous season Slide 7 ， Empirical distribution of forecast accuracy Empirical distribution function for the historical A/F ratios. 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 A/F ratio P r o b a b i l i t y Product description Forecast Actual demand Error* A/F Ratio** JR ZEN FL 3/2 90 140 -50 1.56 EPIC 5/3 W/HD 120 83 37 0.69 JR ZEN 3/2 140 143 -3 1.02 WMS ZEN-ZIP 4/3 170 163 7 0.96 HEATWAVE 3/2 170 212 -42 1.25 JR EPIC 3/2 180 175 5 0.97 WMS ZEN 3/2 180 195 -15 1.08 ZEN-ZIP 5/4/3 W/HOOD 270 317 -47 1.17 WMS EPIC 5/3 W/HD 320 369 -49 1.15 EVO 3/2 380 587 -207 1.54 JR EPIC 4/3 380 571 -191 1.50 WMS EPIC 2MM FULL 390 311 79 0.80 HEATWAVE 4/3 430 274 156 0.64 ZEN 4/3 430 239 191 0.56 EVO 4/3 440 623 -183 1.42 ZEN FL 3/2 450 365 85 0.81 HEAT 4/3 460 450 10 0.98 ZEN-ZIP 2MM FULL 470 116 354 0.25 HEAT 3/2 500 635 -135 1.27 WMS EPIC 3/2 610 830 -220 1.36 WMS ELITE 3/2 650 364 286 0.56 ZEN-ZIP 3/2 660 788 -128 1.19 ZEN 2MM S/S FULL 680 453 227 0.67 EPIC 2MM S/S FULL 740 607 133 0.82 EPIC 4/3 1020 732 288 0.72 WMS EPIC 4/3 1060 1552 -492 1.46 JR HAMMER 3/2 1220 721 499 0.59 HAMMER 3/2 1300 1696 -396 1.30 HAMMER S/S FULL 1490 1832 -342 1.23 EPIC 3/2 2190 3504 -1314 1.60 ZEN 3/2 3190 1195 1995 0.37 ZEN-ZIP 4/3 3810 3289 521 0.86 WMS HAMMER 3/2 FULL 6490 3673 2817 0.57 * Error = Forecast - Actual demand ** A/F Ratio = Actual demand divided by Forecast Slide 8 ， Normal distribution tutorial ƒ All normal distributions are characterized by two parameters, mean = μ and standard deviation = σ ƒ All normal distributions are related to the standard normal that has mean = 0 and standard deviation = 1. ƒ For example: − Let Q be the order quantity, and (μ, σ) the parameters of the normal demand forecast. − Prob{demand is Q or lower} = Prob{the outcome 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 Prob{the outcome of a standard normal is z or lower} in the Standard Normal Distribution Function Table. orQz Q zμ μ σσ −= = + × Slide 9 ， ƒ Start with an initial forecast generated from hunches, guesses, etc. − O’Neill’s initial forecast for the Hammer 3/2 = 3200 units. ƒ Evaluate the A/F ratios of the historical data: ƒ Set the mean of the normal distribution to ƒ Set the standard deviation of the normal distribution to Using historical A/F ratios to choose a Normal distribution for the demand forecast Forecast demand Actual ratio A/F = Forecast ratio A/F Expected demand actual Expected ×= Forecast ratios A/F of deviation Standard demand actual of deviation Standard × = Slide 10 ， O’Neill’s Hammer 3/2 normal distribution forecast 3192320099750 =×= . demand actual Expected 118132003690 =×= . demand actual of deviation Standard ƒ O’Neill should choose a normal distribution with mean 3192 and standard deviation 1181 to represent demand for the Hammer 3/2 during the Spring season. Product description Forecast Actual demand Error A/F Ratio JR ZEN FL 3/2 90 140 -50 1.5556 EPIC 5/3 W/HD 120 83 37 0.6917 JR ZEN 3/2 140 143 -3 1.0214 WMS ZEN-ZIP 4/3 170 156 14 0.9176 … … … … … ZEN 3/2 3190 1195 1995 0.3746 ZEN-ZIP 4/3 3810 3289 521 0.8633 WMS HAMMER 3/2 FULL 6490 3673 2817 0.5659 Average 0.9975 Standard deviation 0.3690 Slide 11 ， Empirical vs normal demand distribution 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 1000 2000 3000 4000 5000 6000 Quantity P r o b a b i l i t y . Empirical distribution function (diamonds) and normal distribution function with mean 3192 and standard deviation 1181 (solid line) Slide 12 ， The Newsvendor Model: The order quantity that maximizes expected profit Slide 13 ， “Too much” and “too little” costs ƒ 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 enjoyed had you ordered one fewer unit. − For the Hammer 3/2 Co = Cost – Salvage value = c – v = 110 – 90 = 20 ƒ Cu = underage cost − The 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 have enjoyed had you ordered one more unit. − For the Hammer 3/2 Cu = Price – Cost = p – c = 180 – 110 = 70 Slide 14 ， Balancing the risk and benefit of ordering a unit ƒ 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 = Prob{Demand <= Q) ƒ … 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)) 0 10 20 30 40 50 60 70 80 0 800 1600 2400 3200 4000 4800 5600 6400 Q th unit ordered E x p e c t e d g a i n o r l o s s . Expected gain Expected loss ƒ As more units are ordered, the expected benefit from ordering one unit decreases while the expected loss of ordering one more unit increases. Slide 15 ， Newsvendor expected profit maximizing order quantity ƒ To maximize expected profit order Q units so that the expected loss on the Qth unit equals the expected gain on the Qth unit: ƒ Rearrange terms in the above equation -> ƒ The ratio Cu / (Co + Cu) is called the critical ratio. （临界比或关键比例） ƒ Hence, to maximize profit, choose Q such that we don’t have lost sales (i.e., demand is Q or lower) with a probability that equals the critical ratio ( )( )QFCQFC uo −×=× 1)( uo u CC CQF +=)( Slide 16 ， Product description Forecast Actual demand A/F Ratio Rank Percentile … … … … … … HEATWAVE 3/2 170 212 1.25 24 72.7% HEAT 3/2 500 635 1.27 25 75.8% HAMMER 3/2 1300 1696 1.30 26 78.8% … … … … … … Finding the Hammer 3/2’s expected profit maximizing order quantity with the empirical distribution function ƒ Inputs: − Empirical distribution function table; p = 180; c = 110; v = 90; Cu = 180- 110 = 70; Co = 110-90 =20 ƒ Evaluate the critical ratio: ƒ 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) ƒ Convert A/F ratio into the order quantity 7778.0 7020 70 =+=+ uo u CC C * / 3200 *1.3 4160.Q Forecast A F= = = Slide 17 ， Hammer 3/2’s expected profit maximizing order quantity using the normal distribution ƒ Inputs: p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20; critical ratio = 0.7778; mean = μ = 3192; standard deviation = σ = 1181 ƒ 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 ƒ Convert the z-statistic into an order quantity: 4101118177.03192 =×+= ×+= σμ zQ z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 Slide 18 ， The Newsvendor Model: Performance measures 绩效指标 Slide 19 ， Newsvendor model performance measures ƒ For any order quantity we would like to evaluate the following performance measures: − Expected lost sales（期望销售损失） ‹ The average number of units demand exceeds the order quantity − Expected sales（期望销售）(compared to expected demand) ‹ The average number of units sold. − Expected left over inventory（期望售后剩余库存） ‹ The average number of units left over at the end of the season. − Expected profit − Expected fill rate（期望订单完成率） ‹ The fraction of demand that is satisfied immediately − In-stock probability（存货满足概率） ‹ Probability all demand is satisfied − Stockout probability（缺货概率） ‹ Probability some demand is lost Slide 20 ， formula ƒ Expected sales = μ - Expected lost sales ƒ μ : Expected demand ƒ L(z): ( 标准 excel标准偏差excel标准偏差函数exl标准差函数国标检验抽样标准表免费下载红头文件格式标准下载 正态)损失函数, ƒ Expected Left Over Inventory = Q - Expected Sales ( )Expected lost sales L zσ= × ( )( )zNormsdistzzNormdistzL −−= 1*)0,1,0,()( Slide 21 ， formula ( ) ( ) Expected profit Price-Cost Expected sales Cost-Salvage value Expected left over inventory = ×⎡ ⎤⎣ ⎦ − ×⎡ ⎤⎣ ⎦ 1 μ μ = = = − E x p e c te d s a le s E x p e c te d s a le sE x p e c te d f i l l ra te E x p e c te d d e m a n d E x p e c te d lo s t s a le s In-stock probability = F(Q) = Φ(z) Stockout probability = 1 – F(Q) =1 – In-stock probability Slide 22 ， Expected lost sales of Hammer 3/2s with Q = 3500 ƒ Definition: − e.g., if demand is 3800 and Q = 3500, then lost sales is 300 units. − e.g., if demand is 3200 and Q = 3500, then lost sales is 0 units. − Expected lost sales is the average over all possible demand outcomes. ƒ If demand is normally distributed: − Step 1: normalize the order quantity to find its z-statistic. − Step 2: Look up in the Standard Normal Loss Function Table the expected lost sales for a standard normal distribution with that z-statistic: L(0.26)=0.2824 ‹ or, in Excel − Step 3: Evaluate lost sales for the actual normal distribution: 26.0 1181 31923500 =−=−= σ μQz ( ) 1181 0.2824 334Expected lost sales L zσ= × = × = ( )( )zNormsdistzzNormdistzL −−= 1*)0,1,0,()( Slide 23 ， Measures that follow expected lost sales Expected sales = μ - Expected lost sales = 3192 – 334 = 2858 Expected Left Over Inventory = Q - Expected Sales = 3500 – 2858 = 642 ( ) ( ) ( ) ( )$70 2858 $20 642 $187,221 Expected profit Price-Cost Expected sales Cost-Salvage value Expected left over inventory = ×⎡ ⎤⎣ ⎦ − ×⎡ ⎤⎣ ⎦ = × − × = 28581 3192 89.6% Expected sales Expected salesExpected fill rate Expected demand Expected lost sales μ μ = = = − = = Note: the above equations hold for any demand distribution Slide 24 ， Service measures of performance In-stock probability = F(Q) = Φ(z) Evaluate the z-statistic for the order quantity : Look up Φ(z) in the Std. Normal Distribution Function Table, Φ(0.26) = 60.26% Stockout probability = 1 – F(Q) =1 – In-stock probability = 1 –0.6026 = 39.74% Note: the in-stock probability is not the same as the fill rate 26.0 1181 31923500 =−=−= σ μQz 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 1000 2000 3000 4000 5000 6000 7000 Order quantity In-stock probability Expected fill Slide 25 ， The Newsvendor Model: The target in-stock probability and the target fill- rate objectives for choosing Q in-stock probability：缺货情况，对杂货店较重要 fill-rate：多少顾客满足的情况，对目录零售商较重要 Slide 26 ， Choose Q subject to a minimum in-stock probability ƒ Suppose we wish to find the order quantity for the Hammer 3/2 that minimizes left over inventory while generating at least a 99% in-stock probability. ƒ Step 1: − Find the z-statistic that yields the target in-stock probability. − In the Standard Normal Distribution Function Table we find Φ(2.32) = 0.9898 and Φ(2.33) = 0.9901. − Choose z = 2.33 to satisfy our in-stock probability constraint. ƒ Step 2: − Convert the z-statistic into an order quantity for the actual demand distribution. − Q = μ + z x σ = 3192 + 2.33 x 1181 = 5944 Slide 27 ， Choose Q subject to a minimum fill rate constraint ƒ Suppose we wish to find the order quantity for the Hammer 3/2 that minimizes left over inventory while generating at least a 99% fill rate. ƒ Step 1: − Find the lost sales with a standard normal distribution that yields the target fill rate. ƒ Step 2: − Find the z-statistic that yields the lost sales found in step 1. − From the Standard Normal Loss Function Table, L(1.53)=0.0274 and L(1.54) = 0.0267 − Choose the higher z-statistic, z = 1.54 ƒ Step 3: − Convert the z-statistic into an order quantity for the actual demand distribution. − Q = μ + z x σ = 3192 + 1.54 x 1181 = 5011 ( ) ( ) 0270.099.01 1181 3192 =−⎟⎠ ⎞⎜⎝ ⎛=⎟⎠ ⎞⎜⎝ ⎛= rate Fill-1L(z) σ μ Slide 28 ， Newsvendor model summary ƒ The model can be applied to settings in which … − There is a single order/production/replenishment opportunity. − Demand is uncertain. − There is a “too much-too little” challenge: ‹ If demand exceeds the order quantity, sales are lost. ‹ If demand is less than the order quantity, there is left over inventory. ƒ Firm must have a demand model that includes an expected demand and uncertainty in that demand. − With the normal distribution, uncertainty in demand is captured with the standard deviation parameter. ƒ At the order quantity that maximizes expected profit the probability that demand is less than the order quantity equals the critical ratio: − The expected profit maximizing order quantity balances the “too much- too little” costs. Slide 29 ， 讨论 ƒ 不足成本大时（毛利高），订购量大于期望需求（关键比例大于0.5）；反之 相反。 ƒ 剩余成本可能是可见成本，而销售损失是机会成本，在报表中看不到。 ƒ 剩余成本可能是积压的库存，如未及时清理，造成虚假利润 ƒ 除了利润最大化目标外，很多情况下会选服务水平目标，因为要考虑长期效 应。 The Newsvendor Model O’Neill’s Hammer 3/2 wetsuit Hammer 3/2 timeline and economics Newsvendor model implementation steps The Newsvendor Model: ��Develop a Forecast Historical forecast performance at O’Neill Empirical distribution of forecast accuracy Normal distribution tutorial Using historical A/F ratios to choose a Normal distribution for the demand forecast O’Neill’s Hammer 3/2 normal distribution forecast Empirical vs normal demand distribution The Newsvendor Model: ��The order quantity that maximizes expected profit “Too much” and “too little” costs Balancing the risk and benefit of ordering a unit Newsvendor expected profit maximizing order quantity Finding the Hammer 3/2’s expected profit maximizing order quantity with the empirical distribution function Hammer 3/2’s expected profit maximizing order quantity using the normal distribution The Newsvendor Model: ��Performance measures Newsvendor model performance measures formula formula Expected lost sales of Hammer 3/2s with Q = 3500 Measures that follow expected lost sales Service measures of performance The Newsvendor Model: ��The target in-stock probability and the target fill-rate objectives for choosing Q Choose Q subject to a minimum in-stock probability Choose Q subject to a minimum fill rate constraint Newsvendor model summary 讨论