News Vendor: Betting on Uncertainty The Newsvendor Problem Managing Fashion Goods Profit Maximizing Order Quantity Fill Rate and In Stock Probability Case: Managing Style Goods 不 能 依 賴 歷 史 銷 售 數 據 來 預 測 需 求 委 外 代 工 的 流 行 性 商 品 必 須 在 銷 售 季 節 開 始 前 數 個 月 決 定 採 購 量 銷 售 季 節 短, 少 有 再 補 貨 的 機 會 單 一 機 會 的 採 購 經 常 因 進 貨 不 足 而 提 前 結 束 銷 售 單 一 機 會 的 採 購 經 常 因 庫 存 過 剩 而 必 須 mark down Zara 利 用 限 量 款 式 的 行 銷 消 除 預 測 與 存 貨 管 理 的 需 要 1
1. The Newsvendor Problem Only one procurement opportunity. Stochastic demand leads to lost sales or leftover. There are losses of profit and goodwill for each unsatisfied customer. There is no salvage value for any leftover. How to balance cost of ordering too much vs. cost of ordering too little? 3 Single Period Inventory Control 適 用 於 無 庫 存 價 值 之 商 品 採 購 航 空 超 額 訂 位 1. 報 紙 需 求 D 為 常 態 分 布 平 均 值 =90 papers 標 準 差 =10 papers D~N(E(D), s d 2 ) 進 貨 90 papers P(stockout)=P(D>90)=P(D>E(D))=50% x Goal: P(stockout)<20% P(no stockout)>80% P(D<x)>80% P(D<E(D)+z s d )>80% Decision: 進 貨 90+0.84162(10)=99 papers 4 2
Optimal Quantity for Newspaper Problem 進 貨 高 估 需 求 的 單 位 成 本 =C o 報 紙 的 進 價 =0.20 進 貨 低 估 需 求 的 單 位 成 本 =C u 報 紙 的 銷 售 利 潤 =0.30 Question: 目 前 進 貨 90 份 報 紙, 提 高 進 貨 量 是 否 能 增 加 利 潤? 不 缺 貨 機 率 P =P( 增 加 進 貨 後 不 會 賣 光 )=P( 需 求 <90)=0.5 潛 在 利 潤 (1 P) C u =0.5(0.30) > 潛 在 損 失 P C o =0.5(0.20) Key: 增 加 進 貨 量 直 到 Cu P C C o u Optimal service level 如 果 C u << C o, 我 們 應 如 何 調 整 進 貨 量? 5 Case : Order Management at Sport Obermeyer Klaus Obermeyer founded Obermeyer in 1947, when he was among the first ski instructors on Aspen Mountain. Customer service, marketing, design & research, accounting in Colorado Rockies. Contract manufacturers in Hong Kong and China. Long lead time, short sales period Increasing product variety, more marked downs 3
Case: Forecasting at Sport Obermeyer Demands depend on weather, fashion trend, economy. Forecasts based on Panel Consensus. Dominant members have stronger influence on the outcome of a consensus forecast. Independent forecasts can provide an indicator of the forecast accuracy for each style. 7 Case: Collaborating with Retailers Obermeyer invites key customers to place early orders (20% of total sales) to get market information. Forecasts are updated based on those early orders. 8 4
Case: Order Planning at Sport Obermeyer Panel forecasts Early bird orders Revised forecasts 1st shipment 2nd shipment Phase 1 min. orders Phase 2 revised orders Summer extra orders and expensive styles Selling season 9 2. Forecast and Single Period Inventory Control Two selling seasons: Spring and Fall. Surf suits are fashion products. Production in Taiwan requires a three month lead time. Order quantity based on past sales of similar products and human judgment. Cost=110, price=180, salvage value=90 O Neill s Hammer 3/2 wetsuit 10 5
Ordering Timeline and Economics Half of the actual demands deviate from initial forecasts by at least 25%. 兩 個 月 後 的 更 新 預 測 較 為 準 確, 但 是 來 不 及 修 改 訂 單 Generate forecast of demand and submit an order to TEC Spring selling season Nov Dec Jan Feb Mar Apr May Jun Jul Aug Receive order from TEC at the end of the month Left over units are discounted 11 Creating a Demand Forecast 因 為 設 計 不 同, 前 幾 季 舊 款 式 的 銷 售 不 適 合 用 來 預 測 新 款 式 的 需 求 O Neill surveyed individuals in the organization and took the average (=3200) as the initial forecast. 決 定 進 貨 量 時, 不 能 只 考 慮 需 求 的 預 測 值, 還 要 評 估 需 求 的 變 異 程 度 設 進 貨 量 為 Q, 需 要 評 估 需 求 低 於 或 超 過 Q 的 可 能 性 F(Q)=Prob{Demand Q} 12 6
3. How to Estimate Demand Uncertainty? 預 測 誤 差 預 測 能 力 不 足? 需 求 的 不 確 定 性? 7000 上 一 季 各 款 式 的 預 測 與 實 際 銷 售 之 對 比 Actual demand. 6000 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Forecast 13 Forecasts vs. Actual Demand (Previous Season) A/Fratio actualdemand forecast 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 average A/F ratio 1 過 去 預 測 沒 有 普 遍 高 估 或 低 估 14 7
Sorted A/F Ratios 假 設 今 年 預 測 的 準 確 度 與 去 年 相 當, 而 去 年 的 預 測 誤 差 可 用 來 評 估 今 年 需 求 的 變 異 程 度 去 年 共 有 33 種 款 式,A/F ratio 最 小 值 為 0.25 3% chance that the demand is only 25% of the forecast. Product description Forecast Actual demand A/F Ratio* Rank Percentile** ZEN-ZIP 2MM FULL 470 116 0.25 1 3.0% ZEN 3/2 3190 1195 0.37 2 6.1% ZEN 4/3 430 239 0.56 3 9.1% WMS ELITE 3/2 650 364 0.56 4 12.1% WMS EPIC 4/3 1060 1552 1.46 29 87.9% JR EPIC 4/3 380 571 1.50 30 90.9% EVO 3/2 380 587 1.54 31 93.9% JR ZEN FL 3/2 90 140 1.56 32 97.0% EPIC 3/2 2190 3504 1.60 33 100.0% 15 Empirical Distribution Function for the Hammer 3/2 預 測 值 =3200, 利 用 過 去 的 預 測 誤 差 估 計 今 年 需 求 的 可 能 變 化 A/F Ratio Q F(Q ) A/F Ratio Q F(Q ) A/F Ratio Q F(Q ) 0.25 800 0.0303 0.81 2592 0.3636 1.23 3936 0.6970 0.37 1184 0.0606 0.82 2624 0.3939 1.25 4000 0.7273 0.56 1792 0.0909 0.86 2752 0.4242 1.27 4064 0.7576 0.56 1792 0.1212 0.96 3072 0.4545 1.30 4160 0.7879 0.57 1824 0.1515 0.97 3104 0.4848 1.36 4352 0.8182 0.59 1888 0.1818 0.98 3136 0.5152 1.42 4544 0.8485 0.64 2048 0.2121 1.02 3264 0.5455 1.46 4672 0.8788 0.67 2144 0.2424 1.08 3456 0.5758 1.50 4800 0.9091 0.69 2208 0.2727 1.15 3680 0.6061 1.54 4928 0.9394 0.72 2304 0.3030 1.17 3744 0.6364 1.56 4992 0.9697 0.80 2560 0.3333 1.19 3808 0.6667 1.60 5120 1.0000 Q = A/F ratio times the initial sales forecast, 3200 units F (Q ) = the probability demand is less than or equal to the quantity Q 16 8
Using A/F Ratios to Estimate Demand Uncertainty Empirical Distribution Initial forecast is 3200. 3% chance that the demand is no more than 800. 9.1% chance that the demand is at least 4800. Note. If average A/F ratio < 1, reduce the current forecast. Normal Distribution Expected demand = average A/F ratio forecast = 0.9976 3200 = 3192 Std. deviation of demand = S.D. of A/F ratio forecast = 0.369 3200 = 1181 17 Empirical vs. Normal Demand Distribution 1.00 0.90 0.80 Probability. 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 N(3192, 1181 2 ) 0 1000 2000 3000 4000 5000 6000 Quantity 18 9
Using Distribution to Estimate Demand Uncertainty O Neill 可 假 設 Hammer 3/2 的 需 求 ~N(3192, 1181 2 ) If the order quantity is increased to 4000, probability of not stocking out is 4000 3192 z 0.68 ( z) 0.7517 1181 x z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 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 19 4. Forecast vs. Order Quantity 由 於 預 測 不 會 百 分 百 準 確, 訂 購 量 不 必 等 於 預 測 值, 應 根 據 利 潤 殘 值 需 求 變 異 進 行 調 整 If the profit is good, there is an incentive to order more. If the salvage value is low, we order less to control losses. 20 10
The Profit-Maximizing Order Quantity 80 Expected gain or loss. 70 60 Expected gain 50 40 30 Expected loss 20 10 0 0 800 1600 2400 3200 4000 4800 5600 6400 Q th unit ordered 21 Too much and Too little Costs C o = overage cost 產 品 賣 不 完 而 賠 本 出 清 的 損 失 For Hammer 3/2 C o = cost salvage value = c v = 110 90 = 20 C u = underage cost 產 品 不 夠 賣 而 損 失 的 潛 在 利 潤 For Hammer 3/2 C u = price cost = p c = 180 110 = 70 22 11
Finding Optimal Order Quantity F(Q) = P(D<Q) = in stock probability ( 期 末 仍 有 庫 存 ) To maximize expected profit, order Q units so that expected loss on the Q th unit equals expected gain on the Q th unit: C F( Q) C 1 F Rearrange the above equation C u / (C o +C u ) is called the critical ratio. o Q 最 佳 訂 購 量 Q* : 使 不 缺 貨 機 率 P(D< Q*) critical ratio. u Cu F( Q) C C o u 23 Hammer 3/2 s Optimal Order Quantity p = 180; c = 110; v = 90; C u = 180 110; C o = 110 90 Evaluate the critical ratio: Cu C C Lookup 0.7778 in the empirical distribution table o u 70 0.7778 20 70 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% Convert A/F ratio into the order quantity Q Forecast * A / F 3200 *1.3 4160. 24 12
5. Performance Measures D ~ N(3192, 1181 2 ) 假 設 訂 購 量 為 3456 expected demand expected sales = expected lost sales 3192 expected left over inventory = Q expected sales 3456 expected profit (price cost) expected sales (cost salvage value) expected left over expected fill rate = fraction of demand that is satisfied = expected sales / expected demand in stock probability: P(demand < Q) 25 Order Quantity and Expected Lost Sales Expected Lost Sales 3500 3000 2500 2000 1500 1000 500 0 If 0 units are ordered, all sales are lost, so expected lost sales equals mean demand, 3192 0 1000 2000 3000 4000 5000 Order quantity If 5000 units are ordered, expected lost sales will be nearly zero. 26 13
Example: How to Calculate Expected Lost Sales Demand > Order quantity=120 0.12 (Loss = 10) x (Prob D = 130) 0.10 0.08 0.06 0.04 0.02 + (Loss = 20) x (Prob D = 140) +... + (Loss = 80) x (Prob D = 200) 0.00 0 20 40 60 80 100 120 140 160 180 200 220 expected lost sales = 10 x P{D=130} + 20 x P{D=140} + + 80 x P{D=200} 27 Hammer 3/2 s Expected Loss Sales Table Assume the demand follows the empirical distribution function: Q F (Q ) L (Q ) Q F (Q ) L (Q ) Q F (Q ) L (Q ) 800 0.0303 2392 2592 0.3636 841 3936 0.6970 190 1184 0.0606 2020 2624 0.3939 821 4000 0.7273 170 1792 0.0909 1448 2752 0.4242 743 4064 0.7576 153 1792 0.1212 1448 3072 0.4545 559 4160 0.7879 130 1824 0.1515 1420 3104 0.4848 542 4352 0.8182 89 1888 0.1818 1366 3136 0.5152 525 4544 0.8485 54 2048 0.2121 1235 3264 0.5455 463 4672 0.8788 35 2144 0.2424 1160 3456 0.5758 376 4800 0.9091 19 2208 0.2727 1111 3680 0.6061 281 4928 0.9394 8 2304 0.3030 1041 3744 0.6364 256 4992 0.9697 4 2560 0.3333 863 3808 0.6667 232 5120 1.0000 0 Q = order quantity F (Q ) = probability demand is less than or equal to the order quantity L (Q ) = loss function (the expected amount demand exceeds Q ) 28 14
Measures Based on Empirical Distribution expected sales = expected demand expected lost sales = 3192 376 = 2816 expected left Over Inventory = Order Quantity Expected Sales = 3456 2816 = 640 expected profit = (price cost) expected sales (cost salvage value) expected left over = $70 2816 $20 640 = $184320 expected sales expected fill rate= 2816 88.2% expected demand 3192 Order quantity=3456 in stock probability = F(Q) = 29 In-Stock Probability and Fill Rate for Hammer 3/2 In Stock probability 只 考 慮 是 否 發 生 缺 貨, 可 能 明 顯 低 於 fill rate 100% 90% 80% Expected fill 70% 60% 50% In-stock probability 40% 30% 20% 10% 0% 0 1000 2000 3000 4000 5000 6000 7000 Order quantity 30 15
6. Retail Discounting Model S = current selling price D = discount price P = profit margin on cost (% markup as decimal) Y = average number of years to sell entire stock of dogs at current price (total years to clear stock divided by 2) N = inventory turns (in 1 year) Loss per item = Gain from revenue S D = D(PNY) D S ( 1 PNY) 31 流 行 性 商 品 必 須 在 銷 售 季 節 前 決 定 採 購 量 單 一 機 會 的 採 購 經 常 因 庫 存 過 剩 而 必 須 賠 售 出 清 預 測 與 銷 售 之 間 的 誤 差 可 來 自 於 需 求 的 不 確 定 性 利 用 前 幾 季 的 預 測 誤 差 來 估 算 未 來 需 求 的 變 化 兩 個 預 測 需 求 相 當 的 品 項 可 能 有 明 顯 不 同 的 需 求 變 異 性, 導 致 採 購 量 的 差 異 最 佳 採 購 量 取 決 於 缺 貨 成 本 與 降 價 出 清 的 損 失 32 16