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Newsvendor ModelChapter 11These slides are based in part on slides that come with Cachon & Terwieschbook Matching Supply with Demand http://cachon-terwiesch.net/3e/. If youwant to use these in your course, you may have to adopt the book as a textbookor obtain permission from the authors Cachon & Terwiesch.utdallas.edu/ metin1

Learning Goals Determine the optimal level of product availability– Demand forecasting– Profit maximization Service measures such as a fill rateutdallas.edu/ metin2

Motivation Determining optimal levels (purchase orders)– Single order (purchase) in a season– Short lifecycle items 1 month: Printed Calendars, Rediform6 months: Seasonal Camera, Panasonic18 months, Cell phone, NokiaMotivating Newspaper Article for toy manufacturer MattelMattel [who introduced Barbie in 1959 and run a stock out for several years then on] washurt last year by inventory cutbacks at Toys “R” Us, and officials are also eagerto avoid a repeat of the 1998 Thanksgiving weekend. Mattel had expected to shipa lot of merchandise after the weekend, but retailers, wary of excess inventory,stopped ordering from Mattel. That led the company to report a 500 millionsales shortfall in the last weeks of the year . For the crucial holiday sellingseason this year, Mattel said it will require retailers to place their full ordersbefore Thanksgiving. And, for the first time, the company will no longer takereorders in December, Ms. Barad said. This will enable Mattel to tailorproduction more closely to demand and avoid building inventory for orders thatdon't come.- Wall Street Journal, Feb. 18, 1999 utdallas.edu/ metinFor tax (in accounting), option pricing (in finance) and revenue managementapplications see newsvendorEx.pdf, basestcokEx.pdf and revenueEx.pdf.3

O’Neill’s Hammer 3/2 wetsuitutdallas.edu/ metin4

Hammer 3/2 timeline and economicsGenerate forecastof demand andsubmit an orderto TEC, supplierSpring selling seasonNov Dec JanFebMar Apr MayReceive orderfrom TEC at theend of themonthEconomics:JunJulAugLeftoverunits arediscountedEach suit sells for p 180TEC charges c 110/suitDiscounted suits sell for v 90The “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.utdallas.edu/ metin– Marketing’s forecast for sales is 3200 units.5

Newsvendor model implementation steps1.Gather economic inputs:a) selling price,b) production/procurement cost,c) salvage value of inventory2.Generate a demand model to represent demanda) Use empirical demand distributionb) Choose a standard distribution function: the normal distributionand the Poisson distribution – for discrete items3.Choose an aim:a) maximize the objective of expected profitb) satisfy a fill rate constraint.4.utdallas.edu/ metinChoose a quantity to order.6

The Newsvendor Model:Develop a Forecast7utdallas.edu/ metin

Historical forecast performance at O’Neill.70006000Actual 007000ForecastForecasts and actual demand for surf wet-suits from the previous seasonutdallas.edu/ metin8

How do we know the actualwhen the actual demand forecast demandAre the number of stockout units ( unmet demand demand-stock)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.– A vicious cycleUnderestimate the demand Stock less than necessary.Stocking less than the demand Stockouts and lower sales.Lower sales Underestimate the demand.– Demand filtering: Demand known exactly only when below thestock.– Shall we order more than optimal to learn about demand? Yes and no, if some customers complain about a stockout; see next page.utdallas.edu/ metin9

Observing a Portion of Unmet Demand Unmet demand are reported by partners (sales associates)Reported lost sales are based on customer complaintsNot everybody complains of a stock out,Not every sales associate records complaints,Not every complaint is reported,Only a portion of complaints are observed by IM?utdallas.edu/ metin

Empirical distribution of forecast accuracyOrder by A/F ratioutdallas.edu/ metinError* A/F 30-3421.23-13141.6019950.375210.8628170.5733 products, so increment probability by 3%.100%90%80%70%ProbabilityActual demandProduct descriptionForecastJR ZEN FL 3/290140EPIC 5/3 W/HD12083JR ZEN 3/2140143WMS ZEN-ZIP 4/3170163HEATWAVE 3/2170212JR EPIC 3/2180175WMS ZEN 3/2180195ZEN-ZIP 5/4/3 W/HOOD270317WMS EPIC 5/3 W/HD320369EVO 3/2380587JR EPIC 4/3380571WMS EPIC 2MM FULL390311HEATWAVE 4/3430274ZEN 4/3430239EVO 4/3440623ZEN FL 3/2450365HEAT 4/3460450ZEN-ZIP 2MM FULL470116HEAT 3/2500635WMS EPIC 3/2610830WMS ELITE 3/2650364ZEN-ZIP 3/2660788ZEN 2MM S/S FULL680453EPIC 2MM S/S FULL740607EPIC 4/31020732WMS EPIC 4/310601552JR HAMMER 3/21220721HAMMER 3/213001696HAMMER S/S FULL14901832EPIC 3/221903504ZEN 3/231901195ZEN-ZIP 4/338103289WMS HAMMER 3/2 FULL64903673* Error Forecast - Actual demand** A/F Ratio Actual demand divided by 251.501.75A/F ratioEmpirical distribution function for the historical A/F ratios.11

Normal distribution tutorial All normal distributions are specified by 2 parameters, mean m and st dev s.Each normal distribution is related to the standard normal that has mean 0 andst dev 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},wherez Q msor Q m z s– The above are two ways to write the same equation, the first allows you to calculate zfrom Q and the second lets you calculate Q from z.– Look up Prob{the outcome of a standard normal is z or lower} in theStandard Normal Distribution Function Table.utdallas.edu/ metin12

Using historical A/F ratios to choose aNormal 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 demandForecastSet the mean of the normal distribution toExpected actual demand Expected A/F ratio Forecast Set the standard deviation of the normal distribution toStandard deviation of actual demand Standard deviation of A/F ratios Forecastutdallas.edu/ metin13

O’Neill’s Hammer 3/2 normal distribution forecastProduct descriptionJR ZEN FL 3/2EPIC 5/3 W/HDJR ZEN 3/2WMS ZEN-ZIP 4/3Forecast Actual demand9014012083140143170156Error-5037-314A/F Ratio1.55560.69171.02140.9176 ZEN 3/2ZEN-ZIP 4/3WMS HAMMER 3/2 FULLAverageStandard 0.86330.56590.99750.3690Expected actual demand 0.9975 3200 3192Standard deviation of actual demand 0.369 3200 1181 Choose a normal distribution with mean 3192 and st dev 1181 to representdemand for the Hammer 3/2 during the Spring season.Why not a mean of 3200?utdallas.edu/ metin14

Fitting Demand Distributions:Empirical vs normal demand yEmpirical distribution function (diamonds) and normal distribution function with15mean 3192 and standard deviation 1181 (solid line)utdallas.edu/ metin

An Example of Empirical Demand:Demand for Candy (in the Office Candy Jar) An OPRE 6302 instructor believes that passing out candies (candies, chocolate,cookies) in a late evening class builds morale and spirit.This belief is shared by office workers as well. For example, secretaries keepoffice candy jars, which are irresistible:“ 4-week study involved the chocolate candy consumption of 40 adult secretaries.The study utilized a 2x2 within-subject design where candy proximity was crossed with visibility.Proximity was manipulated by placing the chocolates on the desk of the participant or 2 m fromthe desk. Visibility was manipulated by placing the chocolates in covered bowls that were eitherclear or opaque. Chocolates were replenished each evening. “People ate an average of 2.2 more candies each day when they were visible, and 1.8candies more when they were proximately placed on their desk vs 2 m away.” They ate 3.1candies/day when candies were in an opaque container. Candy demand is fueled by the proximity and visibility.What fuels the candy demand in the OPRE 6302 class? What undercuts the demand? Hint: The aforementioned study is titled “The office candydish: proximity's influence on estimated and actual consumption” and published in InternationalJournal of Obesity (2006) 30: 871–875.utdallas.edu/ metin16

The Newsvendor Model:The order quantity that maximizesexpected profit17utdallas.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 youknown demand.– In other words, suppose you had left over inventory (i.e., you over ordered). Cois the increase in profit you would have enjoyed had you ordered one fewerunit.– For the Hammer 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 youknown demand.– In other words, suppose you had lost sales (i.e., you under ordered). Cu is theincrease in profit you would have enjoyed had you ordered one more unit.– For the Hammer Cu Price – Cost p – c 180 – 110 70utdallas.edu/ metin18

Balancing the risk and benefit of ordering a unit Ordering one more unit increases the chance of overage– Probability of overage F(Q) Prob{Demand Q)– Expected loss on the Qth unit Co x F(Q) “Marginal cost of overstocking” The benefit of ordering one more unit is the reduction in the chance of underage– Probability of underage 1-F(Q)– Expected benefit on the Qth unit Cu x (1-F(Q)) “Marginal benefit of understocking”.70Expected gain or loss8060Expected marginal benefitof an extra unit inreducing understockingAs more units are ordered, the expected marginal benefit fromordering 1 more unit decreases while the expected marginal costof ordering 1 more unit increases.5040Expected marginaloverstocking costof an extra unit30201000800utdallas.edu/ metin160024003200400048005600640019

Expected profit maximizing order quantity To minimize the expected total cost of underage and overage, orderQ units so that the expected marginal cost with the Qth unit equalsthe expected marginal benefit with the Qth unit:Co F (Q) Cu 1 F Q CuCo 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 andoverage, choose Q such that we do not have lost sales (i.e., demandis Q or lower) with a probability that equals to the critical ratio utdallas.edu/ metin20

Expected cost minimizing order quantity withthe empirical distribution function Inputs: Empirical distribution function table; p 180; c 110; v 90; Cu 180-110 70; Co 110-90 20Evaluate the critical ratio:Cu70Co Cu 20 70 0.7778Look up 0.7778 in the empirical distribution function graphOr, look up 0.7778 among the ratios:– If the critical ratio falls between two values in the table, choose the one thatleads to the greater order quantity 242526Percentile72.7%75.8%78.8% 1.251.271.30 2126351696 1705001300 HEATWAVE 3/2HEAT 3/2HAMMER 3/2Forecast Actual demand A/F Ratio Rank Product description– Convert A/F ratio into the order quantityutdallas.edu/ metinQ Forecast * A / F 3200 *1.3 4160.21

Expected cost minimizing order quantity withthe normal distribution Inputs: p 180; c 110; v 90; Cu 180-110 70; Co 110-90 20; criticalratio 0.7778; mean m 3192; standard deviation s 1181Look up critical ratio in the Standard Normal Distribution Function 1330.8389– 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 Or, Q norminv(0.778,3192,1181) 3192 1181norminv(0.778,0,1) 4096utdallas.edu/ metin22

Another Example: Apparel IndustryHow many L.L. Bean Parkas to order?Demand data / ity ofdemandbeingthis lative ProbabilityProbabilityof demandof demandgreaterbeing this sizethan thisor less, F(.)size, 1.29.82.18.92.08.96.04.98.02.99.011.00.00Expected demand is 1,026 parkas,order 1026 parkas regardless of costs?utdallas.edu/ metinCost/Profit dataCost per parka c 45Sale price per parka p 100Discount price per parka 50Holding and transportation cost 10Salvage value per parka v 50-10 40Profit from selling parka p-c 100-45 55Cost of overstocking c-v 45-40 5Had the costs and demand been symmetric,we would have ordered the average demand.Cost of understocking 55Cost of overstocking 5Costs are almost always antisymmetric.Demand is sometimes antisymmetric23

Optimal Order Q*p sale price; v outlet or salvage price; c purchase priceF(Q) CSL In-stock probability Cycle Service Level Probability that demand will be at or below reorder pointRaising the order size if the order size is already optimalExpected Marginal Benefit P(Demand is above stock)*(Profit from sales) (1-CSL)(p - c)Expected Marginal Cost P(Demand is below stock)*(Loss from discounting) CSL(c - v)Define Co c-v overstocking cost; Cu p-c understocking cost(1-CSL)Cu CSL CoCSL Cu / (Cu Co)Cu55CSL F (Q ) P(Demand Q ) 0.917Cu Co 55 5*utdallas.edu/ metin*24

Optimal Order 04 5 6 7 8 9 10 11 12 13 14 15 16 87Optimal Order Quantity 13(‘00)utdallas.edu/ metin25

Marginal Profits at L.L. BeanApproximate additional (marginal) expected profit from ordering 1(‘00) extra parkasif 10(’00) are already ordered (55.P(D 1000) - 5.P(D 1000)) 100 (55.(0.49) - 5.(0.51)) 100 2440Approximate additional (marginal) expected profit from ordering 1(‘00) extra parkasif 11(’00) are already ordered (55.P(D 1100) - 5.P(D 1100)) 100 (55.(0.29) - 5.(0.71)) 100 1240AdditionalExpectedExpectedExpected Marginal100sMarginal Benefit Marginal CostContribution10 11 5500 .49 2695 500 .51 255 2695-255 2440utdallas.edu/ metin11 125500 .29 1595 500 .71 3551595-355 124012 135500 .18 990 500 .82 410990-410 58013 145500 .08 440 500 .92 460440-460 -2014 155500 .04 220 500 .96 480220-480 -26015 165500 .02 110 500 .98 490110-490 -38016 175500 .01 55 500 .99 49555-495 -44026

Revisit Newsvendor Problem with Calculus Total cost by ordering Q units:C(Q) overstocking cost understocking costQ 0QC (Q) Co (Q x) f ( x)dx Cu ( x Q) f ( x)dxdC (Q) Co F (Q) Cu (1 F (Q)) F (Q)(Co Cu ) Cu 0dQMarginal cost of raising Q* - Marginal cost of decreasing Q* 0CuF (Q ) P( D Q ) Co Cu*utdallas.edu/ metin*27

Safety StockInventory held in addition to the expecteddemand is called the safety stockThe expected demand is 1026 parkas butwe order 1300 parkas.So the safety stock is 1300-1026 274 parkas.utdallas.edu/ metin28

The Newsvendor Model:Performance measures29utdallas.edu/ metin

Newsvendor model performance measures For any order quantity we would like to evaluate the followingperformance measures:– Expected lost sales» The average number of demand units that exceed the order quantity– Expected sales» The average number of units sold.– Expected left over inventory» The average number of inventory units that exceed the demand– Expected profit– Fill rate» The fraction of demand that is satisfied immediately from the stock (no backorder)– In-stock probability» Probability all demand is satisfied– Stockout probability» Probability that some demand is lost (unmet)utdallas.edu/ metin30

Expected (lost sales shortage) ESC is the expected shortage in a season (cycle) ESC is not a percentage, it is the number of units, also see next page Demand - Q ifShortage 0if Demand QDemand QESC E(max{Dema nd in a season Q,0}) ESC (x - Q)f(x)dx ,f is the probability density of demand.x Qutdallas.edu/ metin31

Inventory and Demand during a seasonLeftover inventory0QLeftoverInventory0Seasonutdallas.edu/ metinDemandDuring aSeasonUpsidedownQLeftoverInventory Q-DD, DemandDuringA Season032

0QUpsidedown0Shortage D-QQShortageSeasonDemandutdallas.edu/ metin0D: Demand During a SeasonInventory and Demand during a seasonShortage33

Expected shortage during a seasonExpected shortage E (max{ D Q,0}) ( D Q) P( D d )d Q 1 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 3Expected shortage max{0, (d i Q)} pi i 111 (d 10)}P( D d )d 11121 1 max{0, (9 - 10)} max{0, (10 - 10)} max{0, (11 - 10)} 444 4utdallas.edu/ metin34

Expected shortage during a seasonExpected shortage E (max{ D Q,0}) ( D Q) f ( D)dDwhere f is pdf of Demand.D Q Ex:Q 10, D Uniform(6,12), Expected Shortage?D 12 1 10 2 11 D1 12 2Expected shortage ( D 10) dD 10 D 10(12) 10(10) 66 26 2 6 2 D 10D 10122 2/6utdallas.edu/ metin35

Expected lost sales of Hammer 3/2s with Q 3500Normal demand with mean 3192, standard deviation 1181– Step 1: normalize the order quantity to find its z-statistic.z Q ms 3500 3192 0.261181– Step 2: Look up in the Standard Normal Loss Function Table the expected lostsales for a standard normal distribution with that z-statistic: L(0.26) 0.2824 seetextbook Appendix B Loss Function Table.» or, in Excel L(z) normdist(z,0,1,0)-z*(1-normdist(z,0,1,1)) see textbook Appendix D.– Step 3: Evaluate lost sales for the actual normal distribution:Expected lost sales s L( z) 1181 0.2824 334Keep 334 units in mind, we shall repeatedly use itutdallas.edu/ metin36

The Newsvendor Model:Cycle service level and fill rate37utdallas.edu/ metin

Type I service measure: Instock probability CSLCycle service levelInstock probability: percentage of seasons without a stock outFor example consider 10 seasons :Instock Probabilit y 1 1 0 1 1 1 0 1 0 110Write 0 if a season has stockout,1 otherwiseInstock Probabilit y 0.7Instock Probabilit y 0.7 Probabilit y that a single season has sufficient inventory[Sufficien t inventory] [Demand during a season Q]InstockProbability P(Demand Q)utdallas.edu/ metin38

Instock Probability with Normal DemandN(μ,σ) denotes a normal demand with mean μ and standard deviation σP N ( m , s ) Q Normdist(Q, m , s ,1) P N ( m , s ) m Q m Taking out the mean N (m ,s ) m Q m P Dividing by theStDevss Q m P N (0,1) Obtaining standard normal distributions Q m Normdist ,0,1,1 s utdallas.edu/ metin39

Example: Finding Instock probability for given Qμ 2,500; s 500; Q 3,000;Instock probability if demand is Normal?Instock probability Normdist((3,000-2,500)/500,0,1,1)utdallas.edu/ metin40

Example: Finding Q for given Instock probabilityμ 2,500/week; s 500;To achieve Instock Probability 0.95, what should Q be?Q Norminv(0.95, 2500, 500)utdallas.edu/ metin41

Type II Service measureFill rateRecall:Expected sales m - Expected lost sales 3192 – 334 2858Expected salesExpected sales Expected demandmExpected lost sales 2858 1 m3192 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/ metin42

Service measures of performance100%90%80%Expected fillrate70%60%50%In-stock 0007000Order quantityutdallas.edu/ metin43

Service measures: CSL and fill rate are differentinventoryCSL is 0%, fill rate is almost 100%0timeinventory0utdallas.edu/ metinCSL is 0%, fill rate is almost 0%time44

The Newsvendor Model:Measures that follow from lost sales45utdallas.edu/ metin

Measures that follow from expected lost sales Demand Sales Lost SalesD 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 Leftover InventoryQ min{D,Q} max{Q-D,0} or max{Q-D,0} Q-min{D,Q}Expected Leftover Inventory Q - Expected Sales 3500 – 2858 642utdallas.edu/ metin46

Measures that follow from expected lost salesEconomics: Each suit sells for p 180TEC charges c 110/suitDiscounted suits sell for v 90Expected total underage and overage cost with (Q 3500) 70*334 20*642Expected profit Price-Cost Expected sales Cost-Salvage value Expected left over inventory 70 2858 20 642 187, 221What is the relevant objective? Minimize the cost or maximize the profit?Hint: What is profit cost? It is 70*(3192 334 2858) Cu*μ, which is a constant.utdallas.edu/ metin47

Profit or [Underage Overage] Cost;Does it matter?(p : price; v : salvage value; c : cost) per unit.D : demand; Q : order quantity. (p - c)D - (c - v)(Q - D) if [D Q] Overage Profit(D,Q) (pc)Qif[D Q] Underage (c - v)(Q - D) if [D Q] Overage Cost(D,Q) (pc)(DQ)if[D Q] Underage (p - c)D if D Q Profit(D,Q) Cost(D,Q) (p c)D(pc)DifD Q E[Profit(D, Q)] E[Cost(D,Q)] (p - c)E(Demand ) Constant in QM ax E[Profit(D, Q)] and M in E[Cost(D,Q)] are equivalent ;QQBecause they yield the same optimal order quantity.utdallas.edu/ metin48

Computing the Expected Profit with Normal Demands Expected Profit Profit(D, Q) f(D) dD Suppose that the demand is Normal with mean μ and standard deviation σExpected Profit (p - v) μ normdist(Q, μ, σ,1) - (p - v) σ normdist(Q, μ, σ,0)- (c - v) Q normdist(Q, μ, σ,1) (p c) Q (1 normdist(Q, μ, σ,1))Example: Follett Higher Education Group (FHEG) won the contract to operate the UTDbookstore. On average, the bookstore buys textbooks at 100, sells them at 150 andunsold books are salvaged at 50. Suppose that the annual demand for textbooks hasmean 8000 and standard deviation 2000. What is the annual expected profit of FHEG fromordering 10000 books? What happens to the profit when standard deviation drops to 20 andorder drops to 8000?Expected Profit is 331,706 with order of 10,000 and standard deviation of 2000: (150-50) 10000,8000,2000,1) ected Profit is 399,960 with order of 8000 and standard deviation of 20: * 00,8000,20,1) s.edu/ metin49

Summary Determine the optimal level of product availability– Demand forecasting– Profit maximization / Cost minimization Other measures––––––Expected shortages lost salesExpected left over inventoryExpected salesType I service measure: Instock probability CSLType II service measure: Fill rateExpected cost is equivalent to expected profitutdallas.edu/ metin50

– In other words, suppose you had left over inventory (i.e., you over ordered). C o is the increase in profit you would have enjoyed had you ordered one fewer unit. – For the Hammer C o Cost –Salvage value c –v 110 –90 20 C u underage cost – The cost of ordering one fewe