### Newsvendor store manager issues rain checks to

Newsvendor Model Chapter 9 1 utdallas.

edu/~metin Learning Goals ? Determine the optimal level of product availability – Demand forecasting – Profit maximization ? Other measures such as a fill rate utdallas. edu/~metin 2 O? Neill? s Hammer 3/2 wetsuit utdallas. edu/~metin 3 Hammer 3/2 timeline and economics Generate forecast of demand and submit an order to TEC Economics: • • • Each suit sells for p = $180 TEC charges c = $110/suit Discounted suits sell for v = $90 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 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.

? Marketing? s forecast for sales is 3200 units. 4 utdallas. edu/~metin Newsvendor model implementation steps ? Gather economic inputs: – selling price, – production/procurement cost, – salvage value of inventory ? Generate a demand model to represent demand – Use empirical demand distribution – Choose a standard distribution function » the normal distribution, » the Poisson distribution. Choose an objective: – maximize expected profit – satisfy a fill rate constraint.

utdallas. edu/~metin ? Choose a quantity to order. 5 The Newsvendor Model: Develop a Forecast 6 utdallas. edu/~metin Historical forecast performance at O? Neill 7000 6000 5000 Actual demand . 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Forecast utdallas. edu/~metin Forecasts and actual demand for surf wet-suits from the previous season 7 How do we know the actual when the actual demand > forecast demand ? If e underestimate the demand, we stock less than necessary.

? The stock is less than the demand, the stockout occurs. ? Are the number of stockout units (= unmet demand) observable, i. e. , known to the store manager? – Yes, if the store manager issues rain checks to customers.

– No, if the stockout demand disappears silently. ? No implies demand filtering. That is, demand is known exactly only when it is below the stock. ? Shall we order more than optimal to learn about demand when the demand is filtered? utdallas. edu/~metin 8Empirical distribution of forecast accuracy Order by A/F ratio Actual demand Product description Forecast JR ZEN FL 3/2 90 140 EPIC 5/3 W/HD 120 83 JR ZEN 3/2 140 143 WMS ZEN-ZIP 4/3 170 163 HEATWAVE 3/2 170 212 JR EPIC 3/2 180 175 WMS ZEN 3/2 180 195 ZEN-ZIP 5/4/3 W/HOOD 270 317 WMS EPIC 5/3 W/HD 320 369 EVO 3/2 380 587 JR EPIC 4/3 380 571 WMS EPIC 2MM FULL 390 311 HEATWAVE 4/3 430 274 ZEN 4/3 430 239 EVO 4/3 440 623 ZEN FL 3/2 450 365 HEAT 4/3 460 450 ZEN-ZIP 2MM FULL 470 116 HEAT 3/2 500 635 WMS EPIC 3/2 610 830 WMS ELITE 3/2 650 364 ZEN-ZIP 3/2 660 788 ZEN 2MM S/S FULL 680 453 EPIC 2MM S/S FULL 740 607 EPIC 4/3 1020 732 WMS EPIC 4/3 1060 1552 JR HAMMER 3/2 1220 721 HAMMER 3/2 1300 1696 HAMMER S/S FULL 1490 1832 EPIC 3/2 2190 3504 ZEN 3/2 3190 1195 ZEN-ZIP 4/3 3810 3289 WMS HAMMER 3/2 FULL 6490 3673 * Error = Forecast – Actual demand ** A/F Ratio = Actual demand divided by Forecast Error* A/F Ratio** -50 1. 56 37 0. 69 -3 1.

02 7 0. 96 -42 1. 25 5 0. 97 -15 1. 08 -47 1. 17 -49 1.

15 -207 1. 54 -191 1. 50 79 0.

80 156 0. 64 191 0. 56 -183 1. 42 85 0. 81 10 0. 98 354 0. 25 -135 1.

27 -220 1. 36 286 0. 56 -128 1. 19 227 0. 67 133 0.

82 288 0. 72 -492 1. 46 499 0. 59 -396 1. 30 -342 1.

23 -1314 1. 60 1995 0. 37 521 0. 86 2817 0. 57 00% 90% 80% 70% Probability 60% 50% 40% 30% 20% 10% 0% 0. 00 0. 25 0.

50 0. 75 1. 00 1. 25 1. 50 1. 75 A/F ratio Empirical distribution function for the historical A/F ratios. utdallas.

edu/~metin 9 Normal distribution tutorial ? ? ? All normal distributions are characterized 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. – Prob{demand is Q or lower} = Prob{the outcome of a standard normal is z or lower}, where z? Q? m s or Q ? m ? z ? s (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. utdallas. edu/~metin 10 Using historical A/F ratios to choose a Normal distribution for the demand forecast ? Start with an initial forecast generated from hunches, guesses, etc. – O? Neill? s initial forecast for the Hammer 3/2 = 3200 units.

? Evaluate the A/F ratios of the historical data: A/F ratio ? Actual demand Forecast ? Set the mean of the normal distribution to Expected actual demand ? Expected A/F ratio ? Forecast ? Set he standard deviation of the normal distribution to Standard deviation of actual demand ? Standard deviation of A/F ratios ? Forecast 11 utdallas. edu/~metin O? Neill? s Hammer 3/2 normal distribution forecast Product description JR ZEN FL 3/2 EPIC 5/3 W/HD JR ZEN 3/2 WMS ZEN-ZIP 4/3 Forecast Actual demand 90 140 120 83 140 143 170 156 Error -50 37 -3 14 1995 521 2817 A/F Ratio 1. 5556 0. 6917 1. 0214 0. 9176 … ZEN 3/2 ZEN-ZIP 4/3 WMS HAMMER 3/2 FULL Average Standard deviation … 3190 3810 6490 … … 1195 3289 3673 … 0.

3746 0. 8633 0. 5659 0. 9975 0. 3690 Expected actual demand ? 0. 9975 ? 3200 ? 3192 Standard deviation of actual demand ? 0. 369 ? 3200 ? 1181 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.

12 ? utdallas. edu/~metin Why not a mean of 3200? ? Empirical vs normal demand distribution 1. 00 0.

90 0. 80 0. 70 0.

60 0. 50 0. 40 0. 30 0. 20 0. 10 0.

00 0 1000 2000 3000 Quantity 4000 5000 6000 . Probability Empirical distribution function (diamonds) and normal distribution function with 13 mean 3192 and standard deviation 1181 (solid line) utdallas. edu/~metin The Newsvendor Model: The order quantity that maximizes expected profit 14 utdallas. edu/~metin “Too much” and “too little” costs ? 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 ? 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 utdallas.

edu/~metin 15 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), where F(Q) = Prob{Demand F (Q) ? Cu Co ? Cu ? ? The ratio Cu / (Co + Cu) is called the critical ratio. Hence, to minimize the expected total cost of underage and overage, 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 utdallas. edu/~metin 17 Expected cost minimizing order quantity with the empirical distribution function ? ? Inputs: Empirical distribution function table; p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20 Evaluate the critical ratio: Cu 70 Co ? Cu ? 20 ? 70 ? 0. 7778 ? ? Look up 0. 7778 in the empirical distribution function graph Or, look up 0.

778 among the ratios: – If the critical ratio falls between two values in the table, choose the one that leads to the greater order quantity … … … … … … … Product description HEATWAVE 3/2 HEAT 3/2 HAMMER 3/2 … Forecast Actual demand A/F Ratio Rank 170 500 1300 … 212 635 1696 … 1. 25 1. 27 1. 30 … 24 25 26 Percentile 72.

7% 75. 8% 78. 8% – Convert A/F ratio into the order quantity utdallas. edu/~metin Q ? Forecast * A / F ? 3200 *1. 3 ? 4160. … 18 Hammer 3/2? s expected cost minimizing 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: z 0. 5 0. 0.

7 0. 8 0. 9 0 0. 6915 0. 7257 0. 7580 0.

7881 0. 8159 0. 01 0. 6950 0. 7291 0. 7611 0. 7910 0.

8186 0. 02 0. 6985 0. 7324 0. 7642 0.

7939 0. 8212 0. 03 0. 7019 0. 7357 0. 7673 0.

7967 0. 8238 0. 04 0.

7054 0. 7389 0. 7704 0. 7995 0. 8264 0. 05 0. 7088 0.

7422 0. 7734 0. 8023 0. 8289 0.

06 0. 7123 0. 7454 0. 7764 0. 8051 0.

8315 0. 07 0. 7157 0. 7486 0. 7794 0. 8078 0. 8340 0.

08 0. 7190 0. 7517 0.

7823 0. 8106 0. 8365 0. 09 0. 7224 0. 7549 0. 7852 0.

8133 0. 8389 – 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: Q ? m ? z ? s ? 192 ? 0. 77 ? 1181 ? 4101 utdallas. edu/~metin ? Equivalently, Q = norminv(0.

778,3192,1181) = 4096. 003 19 Another Example: Apparel Industry How much to order? Parkas at L. L. Bean Demand D_i 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Probabability p_i . 01 . 02 . 04 .

08 . 09 . 11 . 16 . 20 . 11 . 10 .

04 . 02 . 01 . 01 Cumulative Probability of demand Probability of demand being this size or less, F() greater than this size, 1-F() .

01 . 99 . 03 .

97 . 07 . 93 . 15 .

85 . 24 . 76 . 35 . 65 . 51 .

49 . 71 . 29 . 82 . 18 .

92 . 08 . 96 . 04 . 98 . 02 .

99 . 01 1. 00 . 00 Expected demand is 1,026 parkas. utdallas. edu/~metin 20 Parkas at L. L. BeanCost per parka = c = $45 Sale price per parka = p = $100 Discount price per parka = $50 Holding and transportation cost = $10 Salvage value per parka = v = 50-10=$40 Profit from selling parka = p-c = 100-45 = $55 Cost of understocking = $55/unit Cost of overstocking = c-v = 45-40 = $5/unit utdallas. edu/~metin 21 Optimal level of product availability p = sale price; v = outlet or salvage price; c = purchase price CSL = Probability that demand will be at or below order quantity CSL later called in-stock probability Raising the order size if the order size is already optimal Expected Marginal Benefit of increasing Q= Expected Marginal Cost of Underage =P(Demand is above stock)*(Profit from sales)=(1-CSL)(p – c) Expected Marginal Cost of increasing Q = Expected marginal cost of overage =P(Demand is below stock)*(Loss from discounting)=CSL(c – v) Define Co= c-v; Cu=p-c (1-CSL)Cu = CSL CoCSL= Cu / (Cu + Co) utdallas. edu/~metin 22 Order Quantity for a Single Order Co = Cost of overstocking = $5 Cu = Cost of understocking = $55 Q* = Optimal order size Cu 55 CSL ? P( Demand ? Q ) ? ? ? 0. 917 Cu ? Co 55 ? 5 * utdallas. edu/~metin 23 Optimal Order Quantity 1. 2 0. 917 1 0. 8 0. 6 0. 4 0. 2 0 4 5 6 7 8 9 10 11 12 13 14 15 16 87 Cumulative Probability Optimal Order Quantity = 13(‘00) utdallas. edu/~metin 24 Parkas at L. L. Bean ? Expected demand = 10 („00) parkas ? Expected profit from ordering 10 („00) parkas = $499 ? Approximate Expected profit from ordering 1(„00) extra parkas if 10(? 00) are already ordered = 100. 55. P(D>=1100) – 100. 5. P(D

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