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 Economics: Generate forecast of demand and submit an order to TEC • • • Spring selling season Nov Dec Jan Feb Mar Apr May Jun Receive order from TEC at the end of the month Each suit sells for p = $180 TEC charges c = $110/suit Discounted suits sell for v = $90 Jul Aug 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. utdallas.edu/~metin 4 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 Actual demand 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Forecast Forecasts and actual demand for surf wet-suits from the previous season utdallas.edu/~metin 7 How do we know the actual when the actual demand > forecast demand If we 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 8 Empirical distribution of forecast accuracy Order by A/F ratio utdallas.edu/~metin 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 0.57 100% 90% 80% 70% Probability 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 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. 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 Qm 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 Set Actual demand Forecast the mean of the normal distribution to Expected actual demand Expected A/F ratio Forecast Set the standard deviation of the normal distribution to Standard deviation of actual demand Standard deviation of A/F ratios Forecast utdallas.edu/~metin 11 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 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 1995 521 2817 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 Why not a mean of 3200? utdallas.edu/~metin Empirical vs normal demand distribution 1.00 0.90 Probability . 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0 1000 2000 3000 4000 5000 6000 Quantity 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 <= Q) The benefit of ordering one more unit is the reduction in the chance of underage: – Expected benefit on the Qth unit = Cu x (1-F(Q)) . 70 Expected gain or loss 80 60 Expected marginal benefit of understocking 50 40 30 Expected marginal loss of overstocking 20 10 As more units are ordered, the expected benefit from ordering one unit decreases while the expected loss of ordering one more unit increases. 0 0 800 utdallas.edu/~metin 1600 2400 3200 4000 4800 5600 6400 16 Expected profit maximizing order quantity To minimize the expected total cost of underage and overage, order Q units so that the expected marginal cost with the Qth unit equals the expected marginal benefit with the Qth unit: Co F (Q) Cu 1 F Q Cu C o Cu Rearrange terms in the above equation -> F (Q) 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 C o Cu 20 70 0.7778 Look up 0.7778 in the empirical distribution function graph Or, look up 0.7778 among the ratios: – If the critical ratio falls between two values in the table, choose the one that leads to the greater order quantity … 24 25 26 Percentile 72.7% 75.8% 78.8% … … 1.25 1.27 1.30 … … 212 635 1696 … … 170 500 1300 … … … HEATWAVE 3/2 HEAT 3/2 HAMMER 3/2 Forecast Actual demand A/F Ratio Rank … … Product description – 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 0.6 0.7 0.8 0.9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.6915 0.7257 0.7580 0.7881 0.8159 0.6950 0.7291 0.7611 0.7910 0.8186 0.6985 0.7324 0.7642 0.7939 0.8212 0.7019 0.7357 0.7673 0.7967 0.8238 0.7054 0.7389 0.7704 0.7995 0.8264 0.7088 0.7422 0.7734 0.8023 0.8289 0.7123 0.7454 0.7764 0.8051 0.8315 0.7157 0.7486 0.7794 0.8078 0.8340 0.7190 0.7517 0.7823 0.8106 0.8365 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 3192 0.77 1181 4101 Equivalently, Q = norminv(0.778,3192,1181) = 4096.003 utdallas.edu/~metin 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. Bean Cost 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 Co CSL= 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 Cumulative Probability 0.6 0.4 0.2 0 4 5 6 7 8 9 10 11 12 13 14 15 16 87 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<1100) utdallas.edu/~metin 25 Parkas at L.L. Bean Additional Expected Expected Expected Marginal 100s Marginal Benefit Marginal Cost Contribution 11th 5500.49 = 2695 500.51 = 255 2695-255 = 2440 12th 5500.29 = 1595 500.71 = 355 1595-355 = 1240 13th 5500.18 = 990 500.82 = 410 990-410 = 580 14th 5500.08 = 440 500.92 = 460 440-460 = -20 15th 5500.04 = 220 500.96 = 480 220-480 = -260 16th 5500.02 = 110 500.98 = 490 110-490 = -380 17th 5500.01 = 55 500.99 = 495 55-495 = -440 utdallas.edu/~metin 26 Revisit L.L. Bean as a Newsvendor Problem Total cost by ordering Q units: – C(Q) = overstocking cost + understocking cost Q C (Q) C o (Q x) f ( x)dx C u ( x Q) f ( x)dx 0 Q dC (Q) Co F (Q) Cu (1 F (Q)) 0 dQ Marginal cost of overage at Q* - Marginal cost of underage at Q* = 0 Cu F (Q ) P( D Q ) C o Cu * utdallas.edu/~metin * 27 Safety Stock Inventory held in addition to the expected demand is called the safety stock The expected demand is 1026 parkas but we order 1300 parkas. So the safety stock is 1300-1026=274 parka. utdallas.edu/~metin 28 The Newsvendor Model: Performance measures 29 utdallas.edu/~metin Newsvendor model performance measures For any order quantity we would like to evaluate the following performance measures: – Expected lost sales » The average number of units demand exceeds the order quantity – Expected sales » The average number of units sold. – Expected left over inventory » The average number of units left over at the end of the season. – Expected profit – Expected fill rate » The fraction of demand that is satisfied immediately – In-stock probability » Probability all demand is satisfied – Stockout probability » Probability some demand is lost utdallas.edu/~metin 30 Expected (lost sales=shortage) ESC is the expected shortage in a season ESC is not a percentage, it is the number of units, also see next page Demand - Q if Shortage 0 if Demand Q Demand Q ESC E(max{Dema nd in a season Q,0}) ESC (x - Q)f(x)dx x Q utdallas.edu/~metin 31 Inventory and Demand during a season 0 Q Inventory 0 Season utdallas.edu/~metin Upside down Q Inventory=Q-D D, Demand During A Season Demand During a Season 0 32 0 Q Upside down 0 Shortage =D-Q Q Shortage Season Demand utdallas.edu/~metin 0 D: Demand During s Season Shortage and Demand during a season 33 Expected shortage during a season First let us study shortage during the lead time Expected shortage E (max{ D Q,0}) ( D Q) f D ( D)dD where f D is pdf of D. D Q Ex: d1 9 with prob p1 1/4 Q 10, D d 2 10 with prob p2 2/4 , Expected Shortage? d 11 with prob p 1/4 3 3 3 Expected shortage max{0, (d i Q)} pi i 1 11 (d Q)}P( D d ) d 10 1 2 1 1 max{0, (9 - 10)} max{0, (10 - 10)} max{0, (11 - 10)} 4 4 4 4 utdallas.edu/~metin 34 Expected shortage during a season Ex: Q 10, D Uniform(6,12), Expected Shortage? D 12 1 10 2 1 1 D2 1 12 2 Expected shortage ( D 10) dD 10 D 10(12) 10(10) 6 6 2 6 2 6 2 D 10 D 10 2/6 12 utdallas.edu/~metin 35 Expected lost sales of Hammer 3/2s with Q = 3500 Normal demand with mean 3192, standard deviation=1181 – Step 1: normalize the order quantity to find its z-statistic. z Qm s 3500 3192 0.26 1181 – 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 see Appendix B table on p.380 of the textbook » or, in Excel L(z)=normdist(z,0,1,0)-z*(1-normdist(z,0,1,1)) see Appendix D on p.389 – Step 3: Evaluate lost sales for the actual normal distribution: Expected lost sales s L( z ) 1181 0.2824 334 Keep 334 units in mind, we shall repeatedly use it utdallas.edu/~metin 36 Measures that follow expected lost sales Demand=Sales+Lost Sales D=min(D,Q)+max{D-Q,0} or min(D,Q)=D- max{D-Q,0} Expected sales = m - Expected lost sales = 3192 – 334 = 2858 Inventory=Sales+Left Over Inventory Q=min(D,Q)+max{Q-D,0} or max{Q-D,0}=Q-min(D,Q) Expected Left Over Inventory = Q - Expected Sales = 3500 – 2858 = 642 utdallas.edu/~metin 37 Measures that follow expected lost sales Expected total underage and overage cost with (Q=3500) =70*334 + 20*642 Expected profit Price-Cost Expected sales Cost-Salvage value Expected left over inventory $70 2858 $20 642 $187, 221 What is the relevant objective? Minimize the cost or maximize the profit? Hint: What is profit + cost? It is 70*3192=Cu*μ, which is a constant. utdallas.edu/~metin 38 Type I service measure: Instock probability = CSL Instock probability: percentage of seasons without a stock out For example consider 10 seasons : 11 0 111 0 1 0 1 Instock Probabilit y 10 Write 0 if a season has stockout, 1 otherwise Instock Probabilit y 0.7 Instock Probabilit y 0.7 Probabilit y that a single season has sufficient inventory [Sufficien t inventory] [Demand during a season Q] InstockPro bability P(Demand Q) utdallas.edu/~metin 39 Instock Probability with Normal Demand N(μ,σ) denotes a normal demand with mean μ and standard deviation σ PN ( m , s ) Q Normdist(Q , m , s ,1) PN ( m , s ) m Q m Taking out the mean N (m ,s ) m Q m P Dividing by the StDev s s Qm P N (0,1) Obtaining standard normal distributi on s Qm Normdist ,0,1,1 s utdallas.edu/~metin 40 Example: Finding Instock probability for given Q μ = 2,500 /week; s= 500; Q = 3,000; Instock probability if demand is Normal? Instock probability = Normdist((3,000-2,500)/500,0,1,1) utdallas.edu/~metin 41 Example: Finding Q for given Instock probability μ = 2,500/week; s= 500; To achieve Instock Probability=0.95, what should be Q? Q = Norminv(0.95, 2500, 500) utdallas.edu/~metin 42 Type II Service measure: Fill rate Recall: Expected sales = m - Expected lost sales = 3192 – 334 = 2858 Expected sales Expected sales Expected demand m Expected lost sales 2858 1 m 3192 89.6% Expected fill rate Is this fill rate too low? Well, lost sales of 334 is with Q=3500, which is less than optimal. utdallas.edu/~metin 43 Service measures of performance 100% 90% 80% Expected fill rate 70% 60% 50% In-stock probability CSL 40% 30% 20% 10% 0% 0 1000 2000 3000 4000 5000 6000 7000 Order quantity utdallas.edu/~metin 44 Service measures: CSL and fill rate are different inventory CSL is 0%, fill rate is almost 100% 0 time inventory 0 utdallas.edu/~metin CSL is 0%, fill rate is almost 0% time 45 Summary Determine the optimal level of product availability – Demand forecasting – Profit maximization / Cost minimization Other measures – – – – – – – Expected shortages = lost sales Expected left over inventory Expected sales Expected cost Expected profit Type I service measure: Instock probability = CSL Type II service measure: Fill rate utdallas.edu/~metin 46 Homework Question: A newsvendor can price a call option Q1: Suppose that you own a simple call option for a stock with a strike price of Q. Suppose that the price of the underlying stock is D at the expiration time of the option. If D <Q, the call option has no value. Otherwise its value is D-Q. What is the expected value of a call option, whose strike price is $50, written for a stock whose price at the expiration is normally distributed with mean $51 and standard deviation $10? utdallas.edu/~metin 47