THE NEWSVENDOR PROBLEM

Transcription

1 Clamshell Beach Press CBP WP THE NEWSVENDOR PROBLEM Arthr V. Hill The John and Nancy Lindahl Professor, Carlson School of Management, University of Minnesota, Spply Chain & Operations Department, th Avene Soth, Minneapolis, MN Voice Copyright 2016 Clamshell Beach Press Revised Febrary 12, Introdction Early each morning, the owner of a corner newspaper stand needs to order newspapers for that day. If the owner orders too many newspapers, some papers will have to be thrown away or sold as scrap paper at the end of the day. If the owner does not order enogh newspapers, some cstomers will be disappointed and sales and profit will be lost. The newsvendor problem is to find the best (optimal) nmber of newspapers to by that will maximize the expected (average) profit given that the demand distribtion and cost parameters are known. The newsvendor problem is a one-time bsiness decision that occrs in many different bsiness contexts sch as bying seasonal goods for a retailer, making the last by or last prodction rn decision, setting safety stock levels, setting target inventory levels, selecting the right capacity for a facility or machine, and overbooking cstomers. These contexts share a common mathematical strctre with the following for elements: a decision variable (), ncertain demand (D), nit overage cost (co), and nit nderage cost (c). This paper is intended to give readers both a mathematical and intitive nderstanding of the newsvendor model to solve the newsvendor problem. This model is one of the most celebrated models in all of operations management and operations research and has been in the literatre for over 100 years (Edgeworth, 1888; Arrow, Harris, & Marschak, 1951). This paper presents the newsvendor problem in the standard retail context. The reader is encoraged to explore the companion Excel workbook Newsvendor Model.xls available from Clamshell Beach Press. The remainder of this paper is organized as follows. Section 2 defines the contexts of the newsvendor problem and the for elements of the mathematical strctre. Sections 3 and 4 present the newsvendor problem with discrete (integer) demand and continos (non-integer) demand. Section 5 presents a simple example with graphs for the continos demand case sing both the trianglar and normal distribtion approaches. Section 6 presents a simple way to estimate the critical ratio that is needed for these two models. Section 7 then discsses behavioral isses related to the newsvendor problem, and Section 8 concldes the paper with a smmary of the main concepts. Appendices 1 and 2 derive the newsvendor model with discrete Copyright 2015 Clamshell Beach Press,

2 and continos demand. Appendix 3 derives the expected vales. Appendix 4 presents the VBA code for the inverse Poisson CDF and Appendix 5 presents the VBA code for the inverse trianglar CDF. 2. Contexts of the newsvendor problem The newsvendor problem is a one-time bsiness decision that occrs in many different bsiness contexts: Bying seasonal goods for a retailer Retailers have to by seasonal goods (sometimes called style goods ) once per season. (Note that a season can be a day, week, year, etc.) For example, most swimsits can only be prchased seasonally. If a byer orders too few swimsits for the selling season, the retailer will have lost sales and dissatisfied cstomers. If the byer orders too many swimsits, the retailer will have to sell them at a clearance price or throw some away. Gpta, Hill, and Bozdine-Chameeva (2006) extend the newsvendor model to handle mltiple seasons (periods), each with a different price elasticity of demand. Making the last by or last prodction rn decision Manfactrers have to make a last by (or last prodction rn) for a prodct (or component) that is near the end of its life cycle. If the order size is too small, the firm will have stockots and disappointed cstomers. If the order size is too large, the firm will only be able to sell the items for their salvage vale. Hill, Giard, and Mabert (1989) considered a similar problem within the context of selecting a keep qantity for an aging service parts inventory. Setting safety stock levels A distribtor has to set the safety stock level for an item. If the safety stock is too low, stockots will occr. If safety stock is too high, the firm has too mch carrying cost. Nearly all safety stock models are newsvendor problems with the selling season being one order cycle or one review period. Setting target inventory levels A salesperson carries inventory in the trnk of a vehicle. The inventory is controlled by a target inventory level. If the target is too low, stockots will occr. If the target is too high, the salesperson will have too mch carrying cost. Selecting the right capacity for a facility or machine If the capacity of a factory or a machine over the planning horizon is set too low, stockots will occr. If capacity is set too high, the capital costs will be too high. Overbooking cstomers If an airline overbooks too many passengers, it incrs the cost of giving away free tickets to inconvenienced passengers. If the airline does not overbook enogh seats, it incrs an opportnity cost of lost revene from flying with empty seats. All of these newsvendor problem contexts share a common mathematical strctre with the following for elements: A decision variable () The newsvendor problem is to find the optimal for a one-time decision, where is the decision qantity (order qantity, safety stock level, overbooking level, capacity, etc.). * denotes the optimal (best) vale for. Uncertain demand (D) Demand is a random variable defined by the demand distribtion (e.g., normal distribtion, Poisson distribtion, etc.) and estimates of the parameters of the Copyright 2015 Clamshell Beach Press, Page 2

3 demand distribtion (e.g., mean, standard deviation). Demand may be either discrete (integer) or continos. This paper develops the newsvendor models for both discrete and continos demand and for nearly all commonly sed demand distribtions. Unit overage cost (co) This is the cost of bying one nit more than the demand dring the one-period selling season. In the standard retail context, the overage cost is the nit cost (c) less the nit salvage vale (s), i.e., co = c s. The salvage vale is the salvage revene less the salvage cost reqired to dispose of the nsold prodct. Unit nderage cost (c) This is the cost of bying one nit less than the demand dring the one-period selling season. This is also known as the stockot (or shortage) cost. In the retail context, the nderage cost is compted as the lost contribtion to profit, which is the nit price (p) less the nit cost (c), i.e., c = p c. The lost cstomer goodwill (g) associated with a lost sale can also be inclded (i.e., c = p c + g). However, it is difficlt to estimate the g parameter in the retail context becase it is the net present vale of ftre lost profit from this cstomer and all other cstomers affected by this cstomer s negative word of moth. Since co and c are both cost parameters, taxes shold be considered for both or neither. Given that the newsvendor problem is in a single period, cash flows do not need to be disconted. 3. The newsvendor problem with discrete demand The model When demand only takes on integer (whole nmber) vales, it is said to be discrete. 1 With order qantity and specific demand D, the cost for the one-period selling season is: 2 co ( D) if D Cost(, D) c ( D ) if D (1) For discrete demand, the demand distribtion is defined by the probability mass fnction 3 p(d). The eqation for the expected cost, therefore, is given by: 1 (2) ECost( ) p( D) Cost(, D) c p( D)( D) c p( D)( D ) o D0 D0 D The first term in eqation (2) is the expected overage (scrap) cost and the second term is the expected nderage (shortage) cost. As is shown in Appendix 1, the optimal order qantity * can be fond at the vale where the expected cost fnction is flat. This is where the expected costs 1 A discrete random variable only takes on integer (whole nmber) vales. This cold be based on the Poisson distribtion, another theoretical discrete distribtion, or an empirical discrete distribtion (sing historical data). 2 A mathematically concise expression is C(, D) c ( D) c ( D ), where ( x) max( x, 0). o 3 The probability mass fnction p(d) is the probability that demand is exactly the integer D. The cmlative distribtion fnction P(D) is the probability that demand is less than or eqal to D. Copyright 2015 Clamshell Beach Press, Page 3

4 for and 1 nits are approximately eqal (i.e., ECost ( ) ECost ( 1) ). Therefore, * is the smallest vale of sch that the following relationship holds tre: * c P( *) p( D) (3) c c D0 o Appendix 1 derives eqation (3). The vale R c / ( c co ) is called the critical ratio or critical fractile and is always between zero and one. 4 The optimal is denoted as * and can be fond with a simple search procedre starting at = 1 and increasing ntil the above relationship is satisfied. When c = co, the critical ratio is R = 0.5, which is consistent with the intition that sggests that * shold be eqal to the median demand when the costs are eqal. Newsvendor example with the Poisson distribtion For example, a byer for a manfactrer mst decide how mch to make with the last manfactring rn before a prodct is discontined. The firm crrently has zero in stock and the forecast for the lifetime demand is 4 nits. (The forecast is the mean of the distribtion.) The demand over the lifetime of the prodct is assmed to be a Poisson distribted random variable (a reasonable assmption). The nderage cost (c) and overage cost (co) are estimated to be $1000 and $100 per nit, respectively. The c parameter is large becase a stockot will disappoint cstomers and becase the prodct will not be manfactred again. The critical ratio is R c / ( c co ) = 1000/1100 = Hill, Giard, and Mabert (1989) developed a decision spport system to help managers solve the newsvendor problem in this bsiness context. Figre 1 shows the Poisson probabilities p(d) and the cmlative Poisson probabilities P(D). The optimal (maximm expected profit) vale of can be fond by finding the smallest vale of sch that P() The optimal vale of for this problem, therefore, is * = 7. 4 The word fractile is a statistical term for the vale associated with a fraction of a distribtion. For example, the median of the distribtion is the 0.50 fractile of the distribtion. Copyright 2015 Clamshell Beach Press, Page 4

5 Figre 1. Poisson probabilities with mean λ = 4 Probability p (D ) Demand (D ) D p(d) P(D) * Implementing the model in Excel The cmlative Poisson distribtion can be implemented in Excel with the fnction POISSON(, λ, TRUE). While Excel does not provide a fnction for the inverse of the cmlative Poisson, it is easy to find the that satisfies eqation (3) with a simple search. Appendix 4 implements a simple VBA fnction for Excel for the Poisson inverse Cmlative Distribtion Fnction (CDF) where * = poisson_inverse(r, λ). Figre 2 shows the expected profit for this example, again showing the optimal vale at * = 7 nits. Notice that the expected profit does not change very mch with small deviations from * and that it is better to err on the high side than on the low side for this example. As derived in Appendix 3, the expected profit is ( p s g)( P( 1) P( )) ( p c g) g, which is $3,607 for this example. Copyright 2015 Clamshell Beach Press, Page 5

6 Figre 2. Expected profit verss order qantity for the example with * = 7 $4,000 $3,500 Expected Profit ($) $3,000 $2,500 $2,000 $1,500 $1,000 $500 $ Order antity ( ) 4. The newsvendor problem with continos demand The model As with the discrete demand case, the cost for order qantity and specific demand D is: co ( D) if D Cost(, D) c ( D ) if D (4) We assme that demand (D) is a continos random variable 5 with density fnction f( D ) and cmlative distribtion fnction FD. ( ) The expected cost fnction is given by: ECost( ) Cost( D, ) f ( D) c ( D) f ( D) c ( D ) f ( D) o D0 D0 D (5) This eqation is analogos to eqation (2) for the discrete demand problem. In order to find the optimal, we take the derivative of the expected cost fnction and set it to zero to find: 5 A continos random variable can take on any real vale, inclding fractional vales. Copyright 2015 Clamshell Beach Press, Page 6

7 decost( ) c o F ( ) c (1 F ( )) 0 d c F ( ) c c 1 * F c co c o (6) Testing the second derivative proves that * is a global optimm. As mentioned before, R c / ( c co ) is the critical ratio and is always between zero and one. In order to find *, the optimal vale of, it is necessary to find the associated with the cmlative probability distribtion so that F( *) c / ( c c ). Mathematicians write this as o 1 * F ( c / ( c co )), where F 1 (.) is the inverse of the cmlative distribtion fnction (also called the inverse distribtion fnction). 6 Appendix 2 presents the derivation for eqation (6). Appendix 3 derives expressions for the expectations for the nmber of nits sold, lost sales, nits salvaged, cost, and profit. This appendix also shows the relationship between the expected profit and expected cost. Implementing the continos demand model with the normal distribtion Microsoft Excel incldes the inverses for several cmlative distribtions, inclding the normal, lognormal, and gamma distribtions. For the normal distribtion, the Excel fnction for the optimal * is NORMINV(R, μ, σ). For example, a newsvendor problem has costs co = $100 and c = $1000 and a critical ratio of R The demand is normally distribted with 4 and 1 nits. The optimal order qantity is then * NORMINV(0.909, 4, 1) 5.34 nits. The Encyclopedia of Operations Management (Hill, 2012) incldes Excel fnctions for the inverse cmlative distribtions for all commonly sed continos probability distribtions. Implementing the continos demand model with the trianglar distribtion When little or no historical information abot demand is available and/or the demand distribtion is not symmetrical, the trianglar distribtion is a practical approach. An experienced person (or team) estimates three parameters: minimm demand (Dmin), most likely demand (Dml), and maximm demand (Dmax). It is best to start with Dmin and Dmax so that people do not anchor on the mode. Excel does not inclde the trianglar distribtion or its inverse, bt the inverse for the trianglar distribtion is easy to derive and implement (Hill & Sawaya, 2004). With the trianglar distribtion, the optimal order qantity for critical ratio R c / ( c c ) is: o 6 Mathematicians also write this as * arg min( ECost ( )). Copyright 2015 Clamshell Beach Press, Page 7

8 F R (7) 1 * ( ) Dmin for R 0 Dmin R( Dmax Dmin )( Dml Dmin ) for 0 R ( Dml Dmin ) / ( Dmax Dmin ) Dmax (1 R)( Dmax Dmin )( Dmax Dml ) for ( Dml Dmin ) / ( Dmax Dmin ) R 1 Dmax for R1 Appendix 5 presents the VBA code for this fnction. 5. The newsvendor example with continos demand The problem A retailing firm bys swimsits for the smmer season. The firm bys its swimsits from a low cost provider in Asia, bt is only able to make a single prchase per year. The estimated demand is 5000 nits, with a minimm of 2000 and a maximm of 8000 nits. The selling price is p = $20 per nit. The firm pays c = $5.00 per nit. The firm can sell excess inventory otside North America for a salvage vale of s = $2.00 per nit. The management believes that no significant goodwill is lost with a lost sale (i.e., g = 0). Therefore, the nderage cost (cost of a lost sale) is c p c g = $15 and the overage cost (cost of one nit of extra inventory) is co c s = $3. The critical ratio is then R c / ( c c ) 15 / %. o The soltion with the trianglar demand distribtion approach With the trianglar distribtion, we have (Dmin, Dml, Dmax) = (2000, 5000, 8000). Using the inverse cmlative distribtion fnction for the trianglar distribtion (eqation (7)), the optimal order qantity is * F 1 (0.833) 6,268 nits and the optimal expected profit is approximately $69,464. Figres 3 and 4 show the graphs for the demand and profit. Copyright 2015 Clamshell Beach Press, Page 8

9 Figre 3. Demand distribtion for the example problem with the trianglar distribtion Demand Distribtion Density fnction f (D ) Figre 4. Expected profit for the example problem with the trianglar distribtion Expected Profit ($) $80,000 $70,000 $60,000 $50,000 $40,000 $30,000 $20,000 $10,000 $- The soltion with the normally distribted demand approach 2,000 2,375 2,750 3,125 3,500 3,875 4,250 4,625 5,000 5,375 5,750 6,125 6,500 6,875 7,250 7,625 8,000 Demand (D ) Expected Profit verss Order antity 2,000 2,375 2,750 3,125 3,500 3,875 4,250 4,625 Order antity ( ) 5,000 5,375 5,750 6,125 6,500 6,875 7,250 7,625 With the trianglar distribtion, the estimated standard deviation is ˆ ( Dmax Dmin ) / 6 = ( )/6 = 1000 nits. Using the inverse cmlative normal distribtion, the optimal order qantity 1 is * F (0.833) 5967 nits and the optimal expected profit is abot $70,503. * can be fond in Excel with NORMINV(0.833, 5000, 1000). Figres 5 and 6 show the graphs. 8,000 Copyright 2015 Clamshell Beach Press, Page 9

10 Figre 5. Demand distribtion for the example problem with the normal distribtion Demand Distribtion Density fnction f (D ) ,000 2,480 2,960 3,440 3,920 4,400 4,880 5,360 5,840 6,320 6,800 7,280 7,760 8,240 8,720 Figre 6. Expected profit for the example problem with the normal distribtion Expected Profit ($) $80,000 $70,000 $60,000 $50,000 $40,000 $30,000 $20,000 $10,000 $- Demand (D ) Expected Profit verss Order antity 2,000 2,480 2,960 3,440 3,920 4,400 4,880 5,360 5,840 6,320 Order antity ( ) 6,800 7,280 7,760 8,240 8,720 In this example, the trianglar and normal distribtion approaches have practically the same optimal order qantity and optimal expected profit. However, when the most likely vale is not close to the midpoint of the minimm and maximm vales (i.e., the distribtion is skewed), the optimal soltions may be far apart. When this is tre, the trianglar distribtion approach will likely be a better method than the normal distribtion approach. Copyright 2015 Clamshell Beach Press, Page 10

11 6. Estimating the critical ratio The critical ratio is normally calclated sing the eqation R c / ( c co ), which reqires estimates of both the nderage and overage costs. As mentioned in the introdction, in the retail context, the nderage cost is price mins nit cost pls lost goodwill (i.e., c p c g ) and the overage cost is nit cost less salvage vale (i.e., co c s ). While price, cost, and salvage vale are sally fairly easy to estimate, the lost goodwill (g) is often very difficlt to estimate, which means that the nderage cost and the critical ratio are also hard to estimate. Goodwill is hard to estimate becase it is difficlt to predict how cstomers will react to a stockot sitation. Some disappointed cstomers might be satisfied with an alternative prodct from the retailer or might retrn at a later date, which means that g is close to zero. However, some cstomers might leave the store disappointed and never come back, which means that the retailer wold lose the net present vale of the lifetime stream of profit from those cstomers. Even worse, some cstomers might give a bad report to many others, which might frther damage the retailer s brand, sales, and profits. Goodwill is also hard to estimate in other newsvendor problem contexts sch as the overbooking context where it is difficlt to estimate the lost goodwill associated with trning away a passenger at the boarding gate for a departing flight. Another approach for calclating the critical ratio is to estimate the ratio of the cost parameters r c / c withot estimating either the nderage or overage cost. It is easy to show that once r is o known, the critical ratio (R) can be calclated sing R r / ( r 1). 7 To help the reader better nderstand the critical ratio R and the parameter ratio r, consider the following examples. When r c / co 1, the critical ratio for the newsvendor problem is R = 1/(1+1) = 0.50, which means that the optimal is at the median of the demand distribtion. When r is mch greater than one, the critical ratio R is close to 1, which leads to an optimal on the far right tail of the demand distribtion. When r is close to zero, the critical ratio R is close to zero, which leads to an optimal on the far left tail of the demand distribtion. For example, it might be difficlt to estimate the lost goodwill parameter g and the associated nderage cost c, bt the decision makers might be able to estimate that the cost of a lost sale is ten times higher than the cost of having too mch inventory at the end of the period. For this sitation, the ratio r c / co is 10 and the critical ratio is R r / ( r 1) = 10/(10+1) = 10/ Proof: c c (1/ c ) c / c r o o R c c c c (1/ c ) c / c 1 r 1, where r c / c. o o o o o Copyright 2015 Clamshell Beach Press, Page 11

12 7. Behavioral isses with the newsvendor problem Reward systems In this athor s experience, decision makers (byers, analysts, planners, and managers) who solve the newsvendor problem as a part of their job often make bad decisions becase their reward systems are not aligned with the economics. Decision makers are instrcted to optimize expected profit, bt are then sbjected to a measrement system that focses on the easier-to-measre metrics that do not reflect the proper economic balance of the relevant costs (e.g., cost of lost sales and cost of excess inventory). This misalignment leads decision makers to respond to the voice that is yelling the lodest at the moment and ignore (or at least discont) harder-to-measre balancing metrics. For example, in a project this athor did with a large msic retailer, the cost of excess inventory for new releases was low becase the retailer cold retrn CDs to the manfactrer for a small restocking fee (abot 15 percent of the cost). The cost of a stockot for the retailer was high de to high margins (abot $10 per CD at the time). Applying the newsvendor model, the retailer s byers shold have been aggressively overbying on a consistent basis. However, excess inventory was easy to measre and lost sales were hard to measre, which often led byers to give more weight to excess inventory and less weight to lost sales in their bying decisions. In other words, it appeared that byers were being driven by their reward system to nder-by even thogh the economics shold have led them to overby. This meant that the critical ratio for the byers was different from the critical ratio for the retailer, which reslted in a misalignment between the goals for the byers and the retail firm. The byers overage cost was c0 = c s + b, where b is the byer s reptational cost per nit remaining at the end of the selling season. The b and g parameters are both difficlt to estimate, bt can sometimes be impted (inferred) from historical byer behavior (Olivares, Terwiesch, & Cassorla, 2008). 8 Low margin (critical ratio) conditions Several research papers have sed hman experiments to stdy the behavioral isses with the repeated newsvendor problem (e.g., Schweitzer & Cachon, 2000; Bolton & Katok, 2008). Nearly all of this research has fond that sbjects tend to nder-by in high critical ratio (high margin) sitations and overby in low critical ratio (low margin) sitations. This pattern cannot be explained by risk aversion, risk-seeking preferences, loss avoidance, waste aversion, or nderstanding opportnity costs. Moritz and Hill (2010) fond that sbjects have cognitive dissonance in low margin sitations becase they are faced with the dilemma of meeting cstomer demand (which sggests a large order qantity) and optimizing expected profit (which sggests a small order qantity). They conclde that sbjects tend to estimate a positive goodwill parameter (e.g., g > 0), even when told that goodwill is not lost with a shortage (e.g., g = 0). 8 Do not confse the byer s reptational cost parameter (b), which is for overbying, and the retailer s lost goodwill parameter (g), which is for nder-bying. Copyright 2015 Clamshell Beach Press, Page 12

13 High margin (critical ratio) conditions Moritz, Hill, and Donohe (2010) fond that decision makers in a high margin condition tended to anchor on the previos period demand when making repeated newsvendor decisions over several periods. They also fond that the simple three-qestion cognitive reflection test (CRT) developed by Frederick (2005) predicted the degree to which decision makers anchored on the previos period demand, where high CRT people had significantly less anchoring. CRT measres the degree to which people allow their system 1 (atomatic, implsive) thinking to be moderated by their system 2 (analytical) thinking. In other words, the CRT measres the degree to which individals are more patient and less implsive when presented with a jdgmental task. 8. Conclsions The newsvendor logic is fndamental to solving many operations problems. The newsvendor model provides both sefl intition and a sefl tool. Explicitly defining the overbying and nder-bying costs, and calclating the critical ratio can often lead to better economic decisions than those made only on the basis of experience, intition, myopic reward systems, politics, power, or personalities. If managers are willing and able to make some assmptions abot the form of the demand distribtion and estimate the demand distribtion and cost parameters, they can imbed the newsvendor model in a decision spport system to help byers make better economic decisions. This type of decision spport system is particlarly valable for retail style goods where many decisions have to be made rotinely and where these decisions have a significant financial impact on the firm. The inpts to the newsvendor model inclde (a) the form of the demand distribtion (e.g., normal), (b) the parameters of the demand distribtion (e.g., mean), and (c) estimates of the overage and nderage cost parameters. The goodwill parameter, a component of the nderage cost, is often difficlt to estimate. This paper introdced the concept of the byer s reptational cost per nit remaining at the end of the selling season. This parameter is also difficlt to estimate. Decision makers (byers, analysts, managers) who solve the newsvendor problem as a part of their job often make bad decisions. The reward systems shold be designed to align byer behavior with the firm s economic objectives. Also, it appears that in low margin sitations, byers tend to be biased toward inferring a higher goodwill parameter; and in high margin sitations, byers with low CRT tend to pt too mch weight on last season s actal demand. The newsvendor economic logic appears in many different bsiness contexts sch as bying for a one-time selling season, making a final prodction rn, setting safety stocks, setting target inventory levels, and making capacity decisions. These contexts all have a single decision variable, random demand, and known overage and nderage costs. The newsvendor model provides a sefl tool for solving these problems and practical insights into how to think abot these problems. Copyright 2015 Clamshell Beach Press, Page 13

14 9. References Arrow, K., T. Harris, and J. Marschak (1951). Optimal Inventory Policy, Econometrica, 19 (3) Bolton, G.E. and E. Katok (2008). Learning by Doing in the Newsvendor Problem: A Laboratory Investigation of the Role of Experience and Feedback, Manfactring and Service Operations Management, 10(3), Edgeworth, F.Y. (1888). The Mathematical Theory of Banking, Jornal of the Royal Statistical Society, 53, Frederick, S. (2005). Cognitive Reflection and Decision Making, Jornal of Economic Perspectives, 19(4), Gpta, D., A.V. Hill, and T. Bozdine-Chameeva (2006). A pricing model for clearing end of season retail inventory, Eropean Jornal of Operational Research, 170 (2), Hadley, G., and T.M. Whitin (1963). Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ. Hill, A.V. (2012). The Encyclopedia of Operations Management, Financial Times Press, New York, New York. Hill, A.V., V. Giard, and V.A. Mabert (1989). A Decision Spport System for Determining Optimal Retention Stocks for Service Parts Inventories, IIE Transactions, 21 (3), Hill, A.V., and W.J. Sawaya III (2004). Prodction Planning for Medical Devices with an Uncertain Approval Date, IIE Transactions, 36 (4), Landsman, Z., and A.E. Valdez (2005). Tail conditional expectations for exponential dispersion models, ASTIN Blletin, 35 (1), Moritz, B.B., and A.V. Hill (2010). Asymmetric Ordering Behavior in Newsvendor Inventory Decisions: Cstomer Service and Cognitive Dissonance, University of Minnesota working paper. Moritz, B.B., A.V. Hill, and K.L. Donohe, Individal Differences in the Newsvendor Problem: Behavior and Cognitive Reflection, nder second review by Management Science (October 5, 2010). Olivares, M., C. Terwiesch, and L. Cassorla (2008). Strctral Estimation of the Newsvendor Model: An Application to Reserving Operating Room Time, Management Science, 54 (1), Schweitzer, M.E. and G.P. Cachon (2000). Decision bias in the newsvendor problem with known demand distribtion: experimental evidence, Management Science, 46, Silver, E.A., D.F. Pyke, and R. Peterson (1998). Inventory management and prodction planning and schedling, Third edition, John Wiley & Sons, New York. Winkler, R.L, G.M. Roodman, and R.R. Britney (1972). The Determination of Partial Moments, Management Science, 19 (3), Copyright 2015 Clamshell Beach Press, Page 14

15 Appendix 1: Derivation of the newsvendor model with discrete demand 9 The optimal expected cost will be where the expected costs for ordering nits is approximately the same as the expected cost for ordering + 1 nits. This is the point at the bottom of the total expected cost crve where the crve is flat, which is the global minimm expected cost. Setting ECost ( ) ECost ( 1) and applying eqation (2), we find: 1 (8) c p( D)( D) c p( D)( D ) c p( D)( 1 D) c p( D)( D 1) o o D0 D D0 D1 However, the left side can be rewritten as c p( D)( D) c p( D)( D ) o D0 D1 becase D 0 when D. Combining terms and defining the cmlative distribtion fnction as F( ) p( D) yields: D0 c p( D) c p( D) 0 o D0 D1 c F( ) c (1 F( )) 0 o c F( ) p( D) c c D0 o Therefore, the optimal order qantity for the discrete demand newsvendor problem can be fond by finding the smallest sch that D0 (9) F( ) p( D) c / ( c c ), where the qantity o R c / ( c co ) is the critical ratio (fractile). Note that this reslt reqires no assmptions abot the demand distribtion other than it mst be a discrete distribtion. The next section proves that expression (9) also holds for continos demand distribtions. Appendix 2: Derivation of the newsvendor model for continos demand For order qantity and specific demand D, the cost is: co ( D) for D Cost(, D) c ( D ) for D (10) For continos demand with density f( D ), the expected cost is: 9 The derivations in the appendices show many more intermediate mathematical steps than are normally shown in a research paper. This is done to help all readers nderstand the details of the derivations. Copyright 2015 Clamshell Beach Press, Page 15

16 o D0 ECost( ) Cost( D, ) f ( D) D0 c ( D) f ( D) c ( D ) f ( D) D c f ( D) c Df ( D) c Df ( D) c f ( D) o o D0 D0 D D c F( ) c H ( ) c ( H ( )) c (1 F( )) o o c F( ) c F( ) c H ( ) c H ( ) c c o o ( c c )( F( ) H ( )) c ( ) o (11) where F ( ) is the demand distribtion fnction evalated at and Df ( D) H ( ). Note: In eqation (11), the mean demand is D0 D Df ( D) and is called the complete expectation becase the range of integration is (0, ). The partial expectation of demand is ( ) ( ) H Df D with a range of integration (0, ). 10 Given that H( ), it is clear that D0 Df ( D) H ( ). D By the fndamental law of calcls, F '( ) f ( ) and according to Leibniz s rle 11 H '( ) f ( ). Therefore, the first derivative of ECost() with respect to is: decost( ) ( c c )( F '( ) F( ) H '( )) c d o ( c c )( f ( ) F( ) f ( )) c o ( c c ) F( ) c o (12) 10 Winkler, Roodman, and Britney (1972) se the term partial moment rather than partial expectation. We assme that demand is always non-negative (i.e., D 0). Winkler et al. se the notation E ( D ) for the partial expectation of 0 the random variable D in the range (0,). This paper will se the simpler notation H. ( ) 11 Leibniz s rle states that sitation, d xh( y ) xh( y ) r x y dh y dg y (, ) ( ) ( ) r( x, y) dx dx r( h( y), y) r( g( y), y). In this xg ( y ) xg ( y ) dy y dy dy d H '( ) dh ( ) / d Df ( D) f ( ) d r( x, y) r( D, ) Df ( D)., where y D 0, x D, h( ), g ( ) 0, and Copyright 2015 Clamshell Beach Press, Page 16

17 Setting this derivative to zero leads to: F c ( ) c c o (13) where the qantity R c / ( c c ) is the critical ratio (fractile). The second derivative of ECost ( ) is o 2 2 d ECost d co c f ( ) / ( ) ( ), which is non-negative for all vales of. 1 Therefore, ECost ( ) is a convex fnction and * F ( c / ( c co )) is the globally optimal order qantity. Note that this reslt does not reqire any assmptions abot the demand distribtion and therefore is tre for all continos probability distribtions. Appendix 3: Derivations of expected vales The derivations in this section are developed for continos demand. Expectations for discrete demand are identical when integrals are replaced by smmations, f( D ) is replaced by pd. ( ) These derivations show the details to make the mathematical reasoning as accessible as possible for the non-mathematical reader. Expected nmber of nits sold For order qantity and specific demand D, the nmber of nits sold is: D for D Sold(, D) for D (14) An alternative expression for the nmber of nits sold is min( D, ). For continos demand with density f( D ), the expected nmber of nits sold is then given by: ESold( ) Sold(, D) f ( D) D0 D0 Df ( D) f ( D) (15) D H ( ) (1 F( )) where H ( ) is the partial expectation of demand, which is defined as D H( ) Df ( D) D0, and f ( D) 1 F( ). Winkler, Roodman, and Britney (1972) prove that the partial expectation (partial first moment) for a normally distribted random variable is: Copyright 2015 Clamshell Beach Press, Page 17

18 H( ) Df ( D) F( z) f( z) (16) D0 where and are the mean and standard deviation of demand, F () z and f () z are the cmlative and probability density fnctions for the standard normal distribtion, and z( ) /. Note that F( ) F ( z), bt that f ( ) f( z). Therefore, for normally distribted demand, the expected nmber of nits sold is: ESold( ) H ( ) (1 F( )) F ( z) f ( z) (1 F( )) F ( z) f ( z) F( ) F( ) f ( z) F( ) ( ) F( ) f ( z) (17) Expected nmber of nits of lost sales For order qantity and specific demand D, the nmber of nits of lost sales is: 0 for D Lost(, D) D for D (18) Alternative expressions for lost sales inclde max( D,0) and ( D,0). For continos demand with density fnction f( D ), the expected nits of lost sales is: ELost( ) Lost(, D) f ( D) D D D0 ( D ) f ( D) Df ( D) f ( D) D H ( ) (1 F( )) (19) where Df ( D) Df ( D) D and f ( D) 1 F( ) D. D Copyright 2015 Clamshell Beach Press, Page 18

19 For normally distribted demand, H ( ) can be replaced with eqation (16) to find the expected nmber of nits of lost sales: ELost( ) H ( ) (1 F( )) F ( z) f ( z) (1 F( )) (1 F( )) (1 F( )) f ( z) ( )(1 F( )) f ( z) (20) Expected nmber of nits salvaged For order qantity and specific demand D, the nmber of nits salvaged is: D for D Salvage(, D) 0 for D (21) Alternative expressions for the nmber of nits salvaged inclde max( D,0) and ( D,0). For continos demand with density fnction f( D ), the expected nmber of nits salvaged is: ESalvage( ) Salvage(, D) f ( D) D0 D0 ( D) f ( D) f ( D) Df ( D) D0 D0 F( ) H ( ) (22) For normally distribted demand, E ( ) can be replaced with eqation (16) to find the expected nmber of nits salvaged: ESalvage( ) F( ) H ( ) F( ) F ( z) f ( z) F( ) F( ) f ( z) ( ) F( ) f ( z) (23) Expected cost As developed in eqation (11), the expected cost for continos demand is: Copyright 2015 Clamshell Beach Press, Page 19

20 ECost ( ) ( c c )( F( ) H( )) c ( ) (24) o For normally distribted demand, E ( ) can be replaced with eqation (16) to find the expected cost: ECost( ) ( c c )( F( ) F ( z) f ( z)) c ( ) o ( c c )(( ) F( ) f ( z)) c ( ) o (25) where z( ) / and F () z and f () z are the CDF and PDF for the standard normal. (Note that F ( z) F( ) and f ( z) f ( ).) Expected profit For order qantity and specific demand D, the profit is: pd s( D) c if D Profit(, D) p g( D ) c if D (26) The expected profit fnction, therefore, is: D0 EProfit( ) Profit(, D) f ( D) D0 [ pd s( D)] f ( D) [ p g( D )] f ( D) c D p Df ( D) s f ( D) s Df ( D) D0 D0 D0 p f ( D) g Df ( D) g f ( D) c D D D ph ( ) sf( ) sh ( ) p(1 F( )) g( H ( )) g(1 F( )) c ph ( ) sf( ) sh ( ) p pf( ) g gh ( ) g gf( ) c ph ( ) sh ( ) gh ( ) pf( ) sf( ) gf( ) p c g g ( p s g) H ( ) ( p s g) F( ) ( p c g) g ( p s g)( H ( ) F( )) ( p c g) g (27) For normally distribted demand, ( ) H can be replaced with eqation (16) to find the expected profit: Copyright 2015 Clamshell Beach Press, Page 20

21 EProfit( ) ( p s g)( H ( ) F( )) ( p c g) g ( p s g)( F( ) f ( z) F( )) ( p c g) g ( p s g)(( ) F( ) f ( z)) ( p c g) g (28) Relationships between expected vales All nits ordered at the beginning of the period mst be either sold or salvaged at the end of the period: Taking expectations of both sides leads to the relationship: Sold ( D, ) Salvage ( D, ) (29) ESold ( ) ESalvage ( ) (30) The demand in a period mst be converted into either a sale or a lost sale. In other words, the demand is always the sm of the nmber of nits sold and the nmber of nits of lost sales. For any order qantity and demand realization: D Sold ( D, ) Lost( D, ) (31) Taking expectations of both sides, it is clear that the average demand is the sm of the expected nmber of nits sold and the expected nmber of nits of lost sales: ESold ( ) ELost ( ) (32) The expected cost in eqation (24) can be related to expected profit as follows: ECost( ) ( c c )( F( ) H ( )) c ( ) o ( p c g c s)( F( ) H ( )) ( p c g)( ) ( p s g)( F( ) H ( )) ( p c g) ( p c g) ( p c g) ( p s g)( H ( ) F( )) ( p c g) ( p c) g ( p s g)( H ( ) F( )) ( p c g) ) g g ( p c) g [( p s g)( H ( ) F( )) ( p c g) g] g ECost( ) ( p c) EProfit( ) EProfit( ) ( p c) ECost( ) (33) In other words, for any order qantity, the expected profit is the profit for selling the average demand, ( p c), less the expected cost. The expected cost, therefore, can be interpreted as the cost of demand variability and the organization shold be willing to pay that amont to redce ncertainty to zero. In other words, when the demand has no variability, the expected cost is zero Copyright 2015 Clamshell Beach Press, Page 21

22 and the expected profit is ( p c). Given that the qantity ( p c) is a constant and is independent of, maximizing expected profit is eqivalent to minimizing expected cost. Smmary of the expected vale eqations Table 1 smmarizes the expectations and Table 2 smmarizes the relationships between expected vales for all demand distribtions (both continos and discrete). Tables 3 and 4 smmarize the expectations for normally distribted demand and Poisson distribted demand. Table 1. Smmary of expected vales for all demand distribtions 12 Expected nits sold ESold ( ) H( ) (1 F( )) Expected salvage ESalvage ( ) F( ) H( ) Expected lost sales ELost ( ) H( ) (1 F( )) Expected cost 13 ECost ( ) ( c co )( F( ) H( )) c ( ) Expected profit EProfit ( ) ( p s g)( H( ) F( )) ( p c g) g pesold ( ) s ESalvage( ) g ELost( ) c Table 2. Relationships between expectations for all demand distribtions All nits ordered will always be either sold or salvaged. ESold ( ) ESalvage ( ) The realized demand will always be eqal to the actal nmber of nits sold pls the lost sales. D Sold(, D) Lost(, D) min( D, ) max( D,0) Average demand is always the expected nits sold pls the expected nits of lost sales. ESold ( ) ELost ( ) Expected profit is always the profit for the average demand less the expected cost. EProfit ( ) ( p c) ECost ( ) Table 3. Smmary of expected vales for normally distribted demand 14 Expected nits sold ESold ( ) ( ) F( ) f ( z) Expected salvage ESalvage ( ) ( ) F( ) f ( z) Expected lost sales ELost ( ) ( )(1 F( )) f ( z) Expected cost ECost ( ) ( c c )(( ) F( ) f ( z)) c ( ) o Expected profit EProfit ( ) ( p s g)(( ) F( ) f ( z)) ( p c g) g 12 H ( ) Df ( D) for continos demand and D0 H ( ) Dp( D) for discrete demand. 13 The expected cost is the sm of the expected nderage cost pls the expected overage cost. 14 The expected vales for the normal distribtion are derived by replacing H ( ) with F ( z ) f ( z ), where z ( ) /. The proof for this relationship can be fond in Winkler, Roodman, and Britney (1972). D0 Copyright 2015 Clamshell Beach Press, Page 22

23 Table 4. Smmary of expected vales for Poisson 15 distribted demand with mean λ Expected nits sold ESold ( ) F( 1) (1 F( )) Expected salvage ESalvage ( ) F( ) F( 1) Expected lost sales ELost ( ) F( 1) (1 F( )) Expected cost ECost ( ) ( c co )( F( ) F( 1)) c ( ) Expected profit EProfit ( ) ( p s g)( F( 1) F( )) ( p c g) g The expected vales for other distribtions can be fond sing the partial expectation fnctions presented in Table 5. These eqations were derived by the athor from the tail conditional expectation fnctions presented in Landsman and Valdez (2005). Table 5. Partial expectation fnctions Continos distribtions Normal H( x) F ( z) f ( z), where z( x ) / Lognormal 2 /2 2 Normal H( x) e (1 F (( ln( x)) / )) Exponential H( x) ( x) F ( x) x Gamma H( x) FGamma ( x 1, ) Discrete distribtions Poisson H( x) F ( x 1) Poisson Exp Binomial H( x) F ( x 1 p, n 1) Binomial Negative binomial H( x) F ( x 1 p, 1) Appendix 4: VBA code for the inverse of the trianglar CDF Fnction trianglar_inverse(p, a, b, c) As Doble ' Compte the inverse of the trianglar distribtion at probability p ' given trianglarly distribted demand with parameters ' (minimm, most likely, maximm) = (a, b, c). If p <= (b - a) / (c - a) Then trianglar_inverse = a + Sqr(p * (c - a) * (b - a)) Else trianglar_inverse = c - Sqr((1 - p) * (c - a) * (c - b)) End If End Fnction NB 15 Hadley and Whitin (1963) prove that the partial expectation for the Poisson distribtion is H( ) P( 1). Copyright 2015 Clamshell Beach Press, Page 23

24 Appendix 5: VBA code for the inverse of the Poisson CDF Fnction poisson_inverse(p, lambda) ' p =cmlative probability and lambda = mean of the Poisson distribtion. ' This rotine trncates the reslt at xmax = 60. Dim x As Integer Const xmax = 60 For x = 1 To xmax poisson_inverse = x If Application.WorksheetFnction.Poisson(x, lambda, Tre) >= p Then Exit Fnction Next x MsgBox poisson_inverse( & Format(p, 0.00% ) & ) was trncated at _ & Val(xmax) &., vbexclamation End Fnction Related resorces from Clamshell Beach Press: The companion Excel workbook newsvendor model.xls is available for download from Other related Excel workbooks from Clamshell Beach Press inclde slowmove.xls and safety stock.xls. The Seasonal Bying paper applies the newsvendor logic to the retail bying context. The Trianglar Distribtion paper presents the details of the trianglar distribtion. Acknowledgements: The athor thanks Jonathan Hill for his helpfl edits on the mathematics in earlier versions of this paper. The athor also thanks Sheryl Holt (Writing Stdies Department, University of Minnesota) and Lindsay Conner (Word Ot Commnications, for their helpfl edits of earlier versions of this paper. Copyright 2016 Clamshell Beach Press. All rights reserved. Copying or distribting any part of this docment in any form withot prior written permission from Clamshell Beach Press is illegal. Written permission may be obtained by sending an to info@clamshellbeachpress.com. Copyright 2015 Clamshell Beach Press, Page 24