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1 Optimizing Inventory Replenisment of Retail Fasion Products Marsall Fiser Kumar Rajaram Anant Raman Te Warton Scool, University of Pennsylvania, 3620 Locust Walk, 3207 SH-DH, Piladelpia, Pennsylvania University of California, Los Angeles, Anderson Scool of Management, Los Angeles, California Harvard Business Scool, Soldiers Field, Boston, Massacusetts We consider te problem of determining (for a sort lifecycle) retail product initial and replenisment order quantities tat minimize te cost of lost sales, back orders, and obsolete inventory. We model tis problem as a two-stage stocastic dynamic program, propose a euristic, establis conditions under wic te euristic finds an optimal solution, and report results of te application of our procedure at a catalog retailer. Our procedure improves on te existing metod by enoug to double profits. In addition, our metod can be used to coose te optimal reorder time, to quantify te benefit of leadtime reduction, and to coose te best replenisment contract. (Retailing; Inventory Replenisment; Stocastic Dynamic Programming; Heuristics) 1. Introduction Retail inventory management is concerned wit determining te amount and timing of receipts to inventory of a particular product at a retail location. Retail inventory-management problems can be usefully segmented based on te ratio of te product s lifecycle T to te replenisment leadtime L. If T/L 1, ten only a single receipt to inventory is possible at te start of te sales season. Tis is te case considered in te well-known newsvendor problem. At te oter extreme, if T/L k 1, ten it s possible to assemble sufficient demand istory to estimate te probability density function of demand and to apply one of several well-known approaces suc as te Q, R model. Te middle case, were T/L 1 but is sufficiently small to allow only a single replenisment or a small number of replenisments, as received muc less attention bot in te researc literature and in retail practice. As we describe in 2, tere is a small but growing literature on limited-replenisment inventory problems. Peraps because of te lack of publised analysis tools, we ave found tat retailers often ignore te opportunity to replenis wen T/L is close to one and treat tis case as toug it were a newsvendor problem. Tis is unfortunate, because, as we sow wit te numeric computations in tis paper, planning for even a single replenisment, can, in tis case, dramatically increase profitability. In tis paper, we consider limited lifecycle retail products in wic only a single replenisment is possible. We model te problem of determining te initial and replenisment order quantities (to minimize te cost of lost sales, backorders, and obsolete inventory at te end of te product s life) as a two-stage stocastic dynamic program. We sow tat te secondstage cost function of tis program may not be convex or concave in te inventory position after te reorder is placed, wic means tat simulation-based optimization tecniques (Ermoliv and Wetts 1998) typically used to solve problems of tis type are not guaranteed to find an optimal solution. For tis reason, and also for computational efficiency, we formulate a euristic for tis problem. We sow tat tis euristic finds an optimal solution if demand subsequent to te time a reorder is placed is perfectly correlated wit demand prior to tis time. Wile perfect 2001 INFORMS Vol. 3, No. 3, Summer 2001, pp /01/0303/0230$ electronic ISSN

2 correlation between early and late demand is unlikely, we believe tis result indicates tat our euristic will work well if tis correlation is ig. Tus, in practice, it seems reasonable to expect good performance from tis euristic because te logical basis of implementing replenisment based on early sales is tat demand during te later season is igly correlated wit early demand. In our application, te correlation between early and late demand was We also apply simulation-based optimization tecniques (Ermoliv and Wetts 1998) and find tat our euristic is muc faster and finds solutions witin 1% of te optimization procedure if te correlation between early and late demand is at least 60%. For lower correlations, te solutions are witin 1% to 5%. We ave applied tis process at a catalog retailer and find tat it improved over teir current process for determining initial and replenisment quantities by enoug to essentially double profits. Remarkably, compared to no replenisment, a single-optimized replenisment improves profit by a factor of five. A key callenge in implementing sort lifecycle replenisment is estimating a probability density function for demand wit no demand istory. To circumvent tis problem in our application, we applied te committee-forecast process in Fiser and Raman (1996) and found tat it worked well. Te most important difference between catalog and traditional retail management is tat a catalog customer will generally accept a backorder if an item is stocked out. Because our application was at a catalog retailer, our model and euristic are given for tis version of te problem, but it is straigtforward to modify te model, euristic, and proof of optimality for a case were backorders are not allowed. In 2 of tis paper, we review te literature on sort lifecycle inventory replenisment. In 3, we formulate te problem; in 4, we state our euristic and establis optimality conditions; in 5, we sow ow to modify te process wen customers may return mercandise, and in 6, report results of our application. 2. Literature Review Analytical models for managing inventory for sort lifecycle products sare many common features. First, all are stocastic models, because tey consider demand uncertainty explicitly. Second, tey consider a finite selling period at te end of wic unsold inventory is marked down in price and sold at a loss. In tis sense, tese models are similar to te classic newsvendor model. Tird, tey model multiple production commitments suc tat sales information is obtained and used to update demand forecasts between planning periods. Te last two caracteristics, finite-selling periods and multiple production commitments, differentiate style goods inventory models from oter stocastic inventory models. Examples of papers tat consider style goods inventory problems include Murray and Silver (1966), Hausman and Peterson (1972), Bitran et al. (1986), Matsuo (1990), and Fiser and Raman (1996). A detailed review of tese papers can be found in Raman (1999). Recent work tat deals specifically wit te retailer s inventory-management problem for sort lifecycle products includes Bradford and Sugrue (1990), Eppen and Iyer (1997a), and Eppen and Iyer (1997b). Bradford and Sugrue model a decision tat is similar to te one we study, but tey do not consider te impact of replenisment leadtimes. In addition, teir solution procedure consists of complete enumeration, wic works efficiently for smaller problems but could be difficult to implement in larger, practicalsized problems. Eppen and Iyer (1997a) consider a problem tat is substantially different from ours. Even toug teir model allows te retailer to buy and dump at te beginning of eac period, te solution metod tey propose applies only wen no buy decisions are permitted after te first period. Eppen and Iyer (1997b) model a backup agreement in place at a catalog retailer. A backup agreement is one of te mecanisms by wic a retailer acieves replenisment of branded mercandise supplied by a manufacturer to several retailers. In a backup agreement, a retailer places an initial order before te start of te sales season and commits to reorder a certain quantity during te season. After assessing sales during te early part of te season, if te retailer cooses to reorder less tan tis commitment, tere is a penalty cost assessed for eac unit not ordered. In tis model, replenisment leadtimes are assumed to be Vol. 3, No. 3, Summer

3 zero, wic is reasonable because te manufacturer would typically ave produced te product and eld it in inventory for tis and oter retailers. Because te replenisment leadtime is zero, it is not necessary for te retailer to accept backorders from consumers. In tis paper, we consider te case were replenisment is acieved witout backup agreements. After receiving an updated order, te manufacturer produces and delivers products to te retailer after a significant leadtime. Te retailer is not required to commit to any of te reorders. To compensate for te long leadtime, consumer backorders are accepted by te retailer. Tis case occurs wen te manufacturers are eiter captive suppliers or wolly owned by te retailer and te retailer sources from several suc manufacturers. Tus, in tis case, it is crucial to model te impact of leadtimes and backorders, altoug tis significantly complicates te analysis leading to a nonconvex optimization model. In addition to incorporating replenisment leadtimes and backorders, our work differs from all tese papers in te process tat we use to estimate demand densities and to compare our metod to actual practice. 3. Model We model te supply decisions faced by a catalog retailer for a product wit random demand over a sales season of fixed lengt. Te retailer must determine an initial order Q 1 available at te start of te sales season. At a fixed time t during te season, te retailer updates te demand forecast, based on observed sales, and places a reorder quantity Q 2 tat arrives after a fixed leadtime L at time t L. Price is fixed trougout te season. Inventory left over at te end of te season is sold at a salvage price below cost. Customers wo encounter a stockout will backorder if tere will be sufficient supply at some point in te future to satisfy te backorder. Specifically, te opportunity to backorder is not offered to a customer once te total supply quantity (Q 1 Q 2 ) as been committed troug sales or prior backorders. A lost sale is incurred wen an item requested by a customer is not in stock or not backordered. We first model tis problem and formulate a solution euristic assuming tat te reorder time t is fixed. Ten, we determine an optimal reorder time t empirically for a given data set by parametrically solving tis problem wit varying t. We are given: C u Cost per unit of lost sale. Tis is set to te difference between te per-unit sales price and cost of te product. Cost per unit of leftover inventory. Tis is set to te difference between te per unit cost and te salvage price of te product. Cost per unit of backorders. Tis is set to te additional costs incurred in procurement and distribution wen an order is backlogged, plus an estimate of te cost of customer ill will. L Lengt of replenisment leadtime. C o C b Define te following variables: X Random variable representing total demand until te reorder is placed. Y Random variable representing total demand during te replenisment leadtime. W Random variable representing total demand after te reorder arrives until te end of te season. R Random variable representing total demand after te reorder is placed until te end of te season, were R Y W. Initial-order quantity. Q 1 Q 2 Reorder quantity. I Inventory position after reorder is placed. I Q 1 Q 2 x. For te given reorder time t, te decision process involves coosing Q 1, observing x, and ten determining te inventory position I for te remainder of te season to minimize total backorder, understock, and overstock costs. Tis sequence of decisions is sown in Figure 1. We consider random variables and wit joint density function f (, ). Let g() be te marginal density on defined by f (, ) and( ) be te conditional density on given defined by f (, ). We define E / (()) # 0 ()( ) and E (()) # ()g(), were ( ) is any real-valued scalar Vol. 3, No. 3, Summer 2001

4 Figure 1 Te Replenisment Planning Process function. Let (a) max(a, 0). We model tis problem by te two-stage stocastic dynamic program (P1). Z(t) min C(Q ) E [C (Q, x) C (Q, x)] were Q 0 1 C (Q, x) C (x Q ) 1 1 b 1 C (Q, x) min C (I, x) I 1 x E Y/X{Cbmin((y (Q1 x) ), I (Q1 x) )} E E {C (y w I) Y/X W/X u C o(i y w) } (P1) I Q x, I 0 (P2) 1 C 1 (Q 1, x) represents te backorder costs C b (x Q 1 ) during te period before te reorder is placed. C 2 (I, x) represents expected cost as a function of te inventory position I after te reorder is placed and consists of two terms. Te first term, E Y/X {C b min((y (Q 1 x) ), I (Q 1 x) )}, represents te expected costs of backorder during te replenisment leadtime. Because backorders are accepted only if tey can be filled from replenisment, it is important to recognize tat backorders during te replenisment leadtime can never exceed te effective inventory position after te first period backlog is cleared (i.e., I (Q 1 x) ). Tis condition is enforced by te operator min((y (Q 1 x) ), I (Q 1 x) ). Te second term of C 2 (I, x) ise Y/X E W/X {C u (y w I) C o (I y w) }, represents te expected overstock and understock costs in te periods after te reorder is placed until te end of te season. It is important to recognize tat C 2 (I, x) is neiter convex nor concave in I. We illustrate tis property using te following example. EXAMPLE 1. Let Q 1 50 and x 10. Let te conditional probability distribution for Y/(X 10) be P(Y/(X 10) 100) 0.5 and P(Y/(X 10) 200) 0.5, wile te conditional probability distribution for W/(X 10) is P(W/(X 10) 100) 0.5 and P(W/(X 10) 200) 0.5. Let C b 15, C o 20, C u 40, 0.9, I 1 80, and I 2 110, I I 1 (1 )I By substituting tese values, using te values of Q 1 and x, and distributions Y/X and W/X to calculate expectations, it is easy to verify tat: C 2(I, x) E Y/X[Cbmin((y (Q1 x) ), I (Q1 x) )] EY/XE W/X[C u(y w I ) C o(i y w) ] C 2(I, x) (1 )C 2(I, x) Tis sows tat C 2 (I, x) is not convex in I. Next, let I 1 80 and I 2 300, so tat I I 1 (1 )I 2 107, wile all te oter values remain uncanged. Now, C 2(I, x) E Y/X[Cbmin((y (Q1 x) ), I (Q1 x) )] EY/XE W/X[C u(y w I ) C o(i y w) ] C 2(I, x) (1 )C 2(I, x) Tis sows tat C 2 (I, x) is not concave in I. In view of tis caracteristic of C 2 (I, x), simulationbased optimization tecniques (Ermoliv and Wetts 1998), typically used to compute te solution to tis class of problems, are not guaranteed to solve Problem P2 and subsequently Problem P1 to optimality. Consequently, for tis reason and run-time considerations, we elected to develop a euristic. Tis euristic is described in te next section. Vol. 3, No. 3, Summer

5 Once we ave developed a sceme to solve tis problem, to find te optimal reorder time, we would perform a line searc on t to solve (P): Z* min Z(t) (P) 0tTL 4. Te Two-Period Newsvendor Heuristic Te purpose of tis euristic is to set Q 1. In tis regard, it is useful to understand te costs affected by te coice of Q 1. Firstly, a portion of Q 1 may remain unsold at te end of te season, generating an overstock. Secondly, during te interval 0 to t L, if Q 1 is too small, one may incur backorder costs. During te interval t to t L, one may also incur stockouts if satisfied and backordered demand exceeds Q 1 Q 2, but it seems more natural to tink of tis cost as resulting from te coice of Q 2, not te coice of Q 1. Given tis, we let S X Y, U X Y W, and coose Q 1 to solve: Z (t) min C(Q ) 1 Q10 EC(s S b Q 1) EUC o(q1 u) (PH) To solve tis problem, let F 1 (s) andf 2 (u) be te distribution functions of random variables S and U, respectively. Te first-order condition for problem (PH) is: Z (t) Q 1 C (1 F (Q )) CF(Q ) 0 b 1 1 o 2 1 We set te euristic order quantity Q1 to te value of Q 1 tat satisfies tis condition. Rearranging terms, tis is calculated as te solution to te following equation: C o 1 1 C b 2 1 F (Q ) F (Q ) 1 Let f 1 (s) and f 2 (u) be te density functions of random variables S and U, respectively. Because 2Z (t) 2 Q 1 C f (Q ) C f (Q ) 0, b 1 1 o 2 1 te first-order conditions are sufficient to establis te optimality of Z (t) atq 1. Note tat our coice of Q 1 minimizes expected backordering costs during te period before replenisment and minimizes expected overstock cost at te end of te season because of Q 1. Te following result establises conditions under wic tis euristic finds an optimal solution. PROPOSITION 1. Suppose Z(t) is te optimal solution to Problem (P1) wen random variables X, Y, and W are perfectly correlated. Ten, Z (t) Z(t). PROOF. If random variables X, Y, and W are perfectly correlated, ten Y X and W X, were, are positive constants. Tus, E(Y/X x) x, V(Y/X x) 0, E(W/X x) x, and V(W/X x) 0. Wen all customers backorder, te optimal reorder quantity is Q2* [x(1 ) Q 1 ].IfQ2* 0, ten one incurs no overstock and understock costs in te tird period after te reorder arrives. Te only costs incurred will be possible backorder costs during te first two periods represented by C b [x(1 ) Q 1 ].If Q* 2 0, in addition to te backorder costs in te first two periods, one could incur an overstock of [Q 1 x(1 )] because of te initial order wit associated costs C o [Q 1 x(1 )]. Consequently, total expected costs in te season wen one as perfectly correlated demand can be expressed as C(Q ) E {C [x(1 ) Q ] 1 X b 1 C o[q1 x(1 )] }. Because by definition, S X Y X(1 )andu X Y W X(1 ), C(Q 1 ) E S {C b (s Q 1 ) } E U {C o (Q 1 u) } C (Q 1 ). Tus, Z(t) min Q Q C(Q 1 ) C (Q 1 ) Z (t). Q.E.D. 1 It can be sown tat Proposition 1 also olds under te assumption of no customer backorders. In ligt of tis proposition, it is reasonable to expect good performance from tis euristic because te logical basis of implementing replenisment based on early sales is tat demand during te later season is igly correlated wit early demand. In our application, we found across all te products te correlation between X and Y to be around 0.96 and between X and W to be about Tis suggests tat tis euristic could provide a simple and efficient basis to model te required decisions in tis application. 234 Vol. 3, No. 3, Summer 2001

6 Once one uses te two-period newsvendor euristic to determine Q 1 and observes demand x during te first period, te optimal solution to te minimization problem P2 can be approximated by setting I max(i*, Q 1 x), were I* is te newsvendor quantity defined on H R/X, te cumulative distribution of R updated by X x (i.e., I* H 1 R/X[(C u C b )/(C u C b C o )]. Te quality of tis approximation is also assessed in te application wile evaluating te performance of te euristic. 5. Modifications to Account for Returns of Mercandise In catalog retailing, because customers place orders based on potograps displayed in catalogs, purcased mercandise is often returned if te actual product differs from wat te customer expected from te catalog. In tis section, we describe ow to extend our model to include mercandise returns. Returned items can be resold if tey are received before te season ends. Tis means tat backorders in te interval (0, t L) and stockouts during te interval (t L, T ) may be reduced by te availability of returns, but returns tat are received too late to be resold can contribute to overstock. Based on te practice followed by te catalog retailer described in te application, we assume tat a known fraction of customers return products, were 0 1. Tese returns are immediately reusable if necessary to satisfy eiter a backorder or demand. We also assume tat recycled returns (i.e., returns on returns and so on) are not reusable during te sales season. Tese assumptions ensure tat we make an unbiased comparison wit existing practice at tis retailer. Consequently, to adapt te euristic to include returns of mercandise, we first consider te period until te reorder arrives. If Q 1 is te initial-order quantity and s is te demand during tis period, te total number of reusable returns is min(s, Q 1 ). If s Q 1, te total backorders tat occur during tis period are [s (Q 1 min(s, Q 1 ))] [s Q 1 (1 )]. Similarly, if u is te demand during te entire season, total reusable returns because of te initial-order quantity are min(u, Q 1 ). If Q 1 u, te total overstock tat occurs during te entire season is [Q1 (u min(u, Q 1))] [Q1 u(1 )]. Using tese results, we redefine (PH) to: Z (t) min C (Q ) r r 1 Q10 EC(s Q (1 )) S b 1 EUC o(q1 u(1 )). (PH r ) We use te procedure outlined in te previous section to determine Q 1 as te solution to te following equation. Co Q1 F [Q (1 )] F 1 (1 )C (1 ) At te end of te first period, we observe realized demand x and use it to set te reorder quantity Q 2 I (Q x) min(q 1 1, x), were I max(i*, (Q x) min(q 1 1, x)), and I* is te newsvendor quantity defined on H R/X, te cumulative distribution of R R(1 ) updated by X x (i.e., I* H 1 R/X[(C u C b )/(C u C b C o )]). 6. Application We ave tested te ideas presented in tis paper at a large catalog retailer. We applied te model and te two-period newsvendor euristic to make purcase decisions for 120 styles/colors from te women s dress department appearing in a particular catalog. We cose tis division because it represented a significant portion of te business. Te sales season for tese products is T 22 weeks, and te replenisment leadtime is L 12 weeks. Because tese products are sold troug mailorder catalogs, te price during te season is fixed. Around 35% of sales are returned, i.e., In te process currently in place at tis retailer, initial-order quantities are set to forecast demand for te 22-week season adjusted for anticipated mercandise returns. Forecasts for eac style/color are updated after two weeks by dividing observed sales by te istorical fraction of total-season sales for te department, wic ave been observed in te past to b Vol. 3, No. 3, Summer

7 Figure 2 Comparison of Early and Updated Forecasts normally occur in te first two weeks. Reorders are placed to make up te difference from an updated forecast adjusted for returns. Specifically for a given style/color, if f is te total forecast sales and is te anticipated fraction of returns, ten Q 1 (1 ) f. Letting x 2 be te actual sales observed at te end of two weeks and k 2 be te fraction of total demand istorically observed at tis point for a group of similar products, ten we set Q 2 ((1 )x 2 /k 2 Q 1 ). Note tat tis procedure sets Q 1 to te forecast sales net of anticipated returns during te entire season and, ence, reorders are used as a reaction to largertan-anticipated sales rater tan someting tat is planned for in advance. Figure 2 sows te improvement in forecast accuracy because of updating at te retailer. Eac point sows forecast and actual demand for a particular style/color combination. Te left grap compares demand forecasts wit actual demand for te average of forecasts made by four expert buyers prior to te beginning of te season. In te rigt grap, te forecasts equal actual sales after two weeks into te season divided by a factor representing te fraction of total sales istorically observed after two weeks. Application of our model requires a metod to estimate demand-probability distributions. Tis is particularly callenging because tere was no sales istory for any of te new dresses. However, we were able to calculate forecast errors, defined as te difference between buyer forecast and actual sales for similar products appearing in te same catalog from te past two years. We used tis information to conclude tat te distribution of forecast errors was normally distributed wit a large degree of confidence ( 2 test olds at 0.01 level). We assumed tat forecast errors would follow a similar distribution in past and future seasons. Tis seemed reasonable because te same individuals wo forecasted product demand in te past were also forecasting current season demand. Because te demand for any given product is equal to its forecast plus te associated forecast error, tis implies tat te demand distribution for U for a given product during te entire season is normally distributed. Wile probability distributions for retail products seem to ave long tails, tese result from plotting actual demand for products tat seem indistinguisable (or at least similar) ex ante. However, in contrast, U represents te demand distribution for a given product. To estimate normal parameters and of tis distribution, we implemented te procedure developed by Fiser and Raman (1996). In tis metod, te members of a committee (comprised of four buyers in our case) independently provide a forecast of sales for eac product. Te mean is set to te average of tese forecasts. Te standard deviation of demand is set to c, were c is te standard deviation of te individual committee member s forecasts and te factor is cosen so tat te average standard deviation of istorical forecast errors equals te average standard deviation assigned to new products. In our application, we found to be 1.4. To estimate te parameters of distribution X and R, were U X R, we assume tat (X, R) follows a nondegenerate bivariate normal distribution. For tis distribution, it is well known (Bickel and Docksum 1977) tat te marginal distribution of X is an univariate normal distribution wit mean x and standard deviation x, wile te marginal distribution of R is also normally distributed wit mean R and standard deviation R. Let k t represent te proportion of total sales until reorder point t, t te correlation between X and R, and t te correlation between X and U. We estimate k t, t, and t from istorical data and use te formulas developed in Fiser and Raman (1996) to calculate 236 Vol. 3, No. 3, Summer 2001

8 Figure 3 Committee Standard Deviation Versus Forecast Error [ ] (1 ) 2 t x t X t t 2 (1 ) t k,, (1 ) 2 t R t R 2 (1 ) t k (1 ), and. For te bivariate normal distribution (X, R), note tat te updated distribution R/X x is also normally distributed wit mean R/X R t (x X ) R / X and standard deviation R/X R 1. 2 t Because 0 t 1, tis implies tat R/X R. Tus forecast updating based on actual sales x reduces variance in te distribution of demand during te remaining season and permits a more accurate forecast. By using replenisment, te retailer can take advantage of tis improved forecast by placing a more precise reorder tat directly contributes to iger expected profits during te remaining season. To better understand te nature of forecast errors, we compared te standard deviation of te committee forecast for individual products at te beginning of te season (i.e., c ) wit its corresponding forecast error. Tese results, sown in Figure 3, suggest tat wen te committee agrees, tey tend to be accurate, and tat te committee process is a useful way to determine wat you can and cannot predict. Wit te exception of te backorder penalty C b, all te cost parameters required for our analysis were readily available. Estimating te backorder penalty is callenging in practice because, in addition to te $1 per-unit extra-transaction cost for procurement and distribution associated wit a backorder, tere is an intangible cost because of customer ill will. Te company was uncertain as to te exact value of te illwill cost, but felt a value of C b in te range $5 to $15 was reasonable. We applied our analysis to tree cases using $5, $10, and $15 per unit as values of C b.we also analyzed te case C b 1 to insure tat our euristic did not outperform te current rules because we assessed an ill-will cost tat was not used in te current rule. Note tat altoug ill-will costs can also be added to C u, we did not add tem because an ill-will cost in tis application was carged only because management was mainly interested in insuring tat flexibility to backorder was not abused. Given te values of C u and C o,ifc b 0, ten it is optimal to set Q 1 0 and backorder all first-period demand. But, tese excessive backorders would likely reduce market sare in te long term. Te omission of ill-will costs in C u does not affect te analysis because, depending on te product, C u was two to four times greater tan backorder costs, and consequently, it would never be optimal to not satisfy demand to avoid a backorder. As a practical matter, we found tat istorically around 5% of customers cose not to accept te offer to backorder at tis retailer. Consequently, we adjusted te backorder cost to account for tis fraction of lost sales by defining an effective backorder cost, C b 0.95 C b 0.05 C u, representing te costs of a backorder and stockout weigted by te expected fraction of customers wo would coose eiter option. We replaced C b wit Cb in te definition of problem (PH r ). Te first step in our metodology is to determine te reorder time. It is important to accurately coose tis time because, if cosen too early in te season, actual sales will not be sufficient to provide an accurate revision of te second-period demand forecast. On te oter and, if te reorder time is too far into te season, te benefit of replenisment is diminised because it is ten used to service only a small portion of te season. Te specific coice of reorder point depends on te proportion of total sales observed during te initial weeks and te lengt of te replenisment leadtime. For instance, if tis proportion is ig, tere is a long leadtime or bot, one would coose Vol. 3, No. 3, Summer

9 te reorder point early in te season to ensure tat a reasonable proportion of total sales is serviced by te reorder. To determine te best reorder time, we used Monte Carlo simulation wit te estimated distributions of demand to calculate Z(t) for Week t. In tis procedure, for a given reorder time, we estimate te initial-order quantity using our euristic. We simulate x (as a realization of X), te distribution of total demand until te reorder is placed. We use x to calculate te backorder costs before te reorder is placed, update R/X x, and calculate te expected costs during te remaining season. We repeat tis procedure for several simulated realizations of X and calculate te expected costs during te entire season associated wit a reorder time by averaging te costs associated wit eac realization. As discussed previously, using te bivariate normal distribution to model demand (X, R) ensures tat bot X and R are univariate normal distributions and te variance of updated second-period demand R/X x is also univariate normal wose variance is now reduced from R to R/X R 1 ). 2 t We repeat te simulation for several coices of reorder time. Te results of tis simulation are summarized in Figure 4. Because Z(t) attains its minimum at t t* 2, te reorder time is cosen to be at te end of Week 2. Note tat te lengt of te replenisment leadtime assumed in tis analysis is 12 weeks. As te season lasts only 22 weeks, we cannot reorder after Week 10. Consequently, te value of Z(t) for t 10 is set equal to te expected costs incurred for a single period buy if we set Q 1 to Q s 1, te news- vendor quantity defined on te total distribution of demand (i.e., Q s F [C u /(C u C o )]. A key factor tat influences te level of profits gained by replenisment is te proportion of total demand over time observed during te early part of te season and te time until te reorder arrives. Clearly, if tis proportion is very ig, ten te benefits of replenisment are limited, as te reorder would only serve a small proportion of total-season demand. In our application, we found tat istorically, for similar product lines, 10% of total demand is observed wen te reorder is placed after two weeks, and 50% of Figure 4 Reorder Time and Expected Costs demand is observed at Week 10 wen te reorder arrives. Tese values confirmed tat replenisment based on actual sales was a viable strategy for te cosen product line and motivated us to apply our metod to determine initial- and replenisment-order quantities. For te 120 styles/colors in tis department, we determined te initial-order quantity by solving problem (PH r ) using te euristic modified to include returns. We ten observed x, te sales until te second week, and set te reorder quantity Q 2 using te procedure developed in 5. Because at te end of te season we knew total sales and actual sales per week for eac dress, we were able to calculate te stockouts, overstock, backorders, dollar sales, and profits tat would ave resulted from our ordering policy. To compare our metod wit te current ordering rules, we also calculated tese values for te current policy. Te results consolidated across all te 120 dresses are tabulated in Table 1. Observe tat altoug te total orders placed by our metod and existing practice are similar, te composition of tese orders across te two periods is different. Our euristic reduces overstocks, stockouts, 238 Vol. 3, No. 3, Summer 2001

10 Table 1 Comparison of te Two-Period Newsvendor Heuristic wit Current Practice Figure 5 Replenisment Costs and Leadtimes Current Rule Model C ib $5 Model C ib $10 Model C ib $15 Initial Order Reorder Total Buy Overstock Omits Backorders Profit ($) Sales ($) Profit Increase By Model (As % Current Sales) and backorders enoug to increase profits compared wit te current rule from 2.23% to 4.92% of current sales, depending on te value of C b. Profit before tax for tis retailer is around 3% of sales. Consequently, our euristic offers te potential to approximately double profit. Our results also sow te impact of C b on te solution. Order quantities for te current rule do not cange because te current rule does not consider C b in determining order quantities. Wen C b $5, we order a smaller initial quantity because backorders are now relatively less expensive. Tis in turn increases backorders and stockouts but reduces overstocks, wic increases profit improvement from 3.52% to 4.92%. On te oter and, wen C b $15, we order a larger initial quantity because backorders are relatively more expensive. Tis reduces backorders and stockouts but increases overstocks, wic reduces te profit improvement from 3.52% to 2.23%. To insure tat our euristic did not outperform te current rules because we assessed an ill-will cost tat was not used in te current rules (possibly because illwill costs were not recorded in te books), we also considered te case wen C b 1. Here, C b only consists of te additional transaction cost per unit of procurement and distribution associated wit a backorder. As expected, our euristic ordered a substantially smaller amount initially tan te cases wit larger values of C b. Tis in turn increased backorders and stockouts, but reduced overstocks. Over all, te profit improvement over te current rules increased to 5.52%. Tis is consistent wit te general pattern in Table 1, wic sows profit improvement increasing as C b decreases. Tis analysis assumes a replenisment leadtime of 12 weeks. It is easy to understand tat reducing tis time could potentially increase te benefits of replenisment, because a greater portion of te season can be serviced from te more accurate reorder. However, it is important to precisely calculate tis benefit to justify te costs of leadtime reduction. Our metodology provides a framework to analyze tese benefits bot before and after sales are realized. To perform tis analysis before actual sales are realized, we use a simulation to calculate expected costs for different values of leadtimes. Using actual sales, and for te case in wic C b 10, we performed an analysis identical to te one used to obtain te results reported in Table 1, but using te new coices of te leadtime. Tese results are summarized in Figure 5 and indicate tat te lengt of te replenisment leadtimes significantly influences te benefits of replenisments. Tis type of analysis could be used to decide between a domestic supplier wit typically Vol. 3, No. 3, Summer

11 iger costs but sorter leadtimes and a foreign supplier wit relatively low costs but long leadtimes. To furter evaluate te quality of tis euristic, we solved (P1) using a simulation-based optimization metod. 1 In tis tecnique, we use simulation to numerically compute C(Q 1 ) for selected values of Q 1 in te range [0, Q*], were Q* F 1 2 [C u /(C u C o )] is te newsvendor quantity 2 defined across te wole distribution of demand. Finally, we set Z(t) min 0QQ1 * C(Q 1 ). In te case were C b 10, tis tecnique improved profit relative to te current rule by around 3% of current sales, a lower improvement tan was acieved wit our euristic. In te cases were C b equaled 5 or 15, te profit improvement was also marginally less tan acieved wit our euristic. In addition to resulting in a smaller profit gain tan te two-period newsvendor euristic, we found tat te solution time for tis tecnique was around 40 ours on a Dell Pentium II PC, as compared wit less tan a minute for te two-period newsvendor euristic. Tese results provide strong justification for using tis euristic in tis application. We also considered te impact on performance of correlation between early and later demand. In our application, across all products, we found te correlation between X and Y to be around 0.96 and between X and W to be about In view of Proposition 1, suc ig correlation suggests tat te euristic solution is very close to te optimal solution for tis problem. We performed a computational study to evaluate te performance of te euristic for different levels of correlation between early and later season demand. Define 1 as te correlation between X and Y and 2 as te correlation between X and W. For simplicity, we set 1 2 and vary from 0 to 0.99 in steps of 0.1. For a given value of, and using te committee 1 Please refer to Ermoliv and Wetts (1998), Numerical Tecniques for Stocastic Optimization, Springler Verlag, New York, for a teoretical justification and a detailed description of tis tecnique. Te same simulation-based tecnique is used to numerically estimate C 2 (Q 1, x) required in te computation of C(Q 1 ). Here we vary I over te range [0, 10Q*] and set C 2 (Q 1, x) min 0I10Q* C 2 (I). 2 We coose tis value because it is igly unlikely tat an initialorder quantity greater tan tis quantity would not be sufficient to cover sales during te periods before te replenisment arrives. Table 2 Correlation Percentage Performance Gap Between te Two-Period Newsvendor Heuristic and a Simulation-Based Optimization Metod for Different Levels of Demand Correlations Performance Gap (%) Correlation Performance Gap (%) estimates of and for eac product, we calculated Q 1 using te euristic and using te simulation-based optimization metod. We ten used tese values of te initial-order quantities and Monte Carlo simulation wit te estimated distribution of U for eac product to calculate expected costs for eac tecnique. Foragivenvalueof, letc represent te total expected cost of te euristic across all products, and let C s represent te corresponding total cost of te simulation-based optimization procedure. Te percentage-performance gap of te euristic is defined as (C C s )/C s 100%. We report te percentageperformance gap across a range of values for demand correlation in Table 2. For eac level of demand correlation, te run times for te euristic across all te products was less tan a minute, wile te equivalent run time for te optimization approac was around forty ours. Te results in Table 2 sow tat tis gap varies from 0.06% to 5.2%, wit te igest gaps occurring at te lowest levels of correlation. Tese results suggest tat te euristic provides an efficient basis to address tis problem, even for cases tat ave modest levels of correlation (e.g., 0.3) between early and later sales. Because fasion replenisment makes most sense for products wit some degree of correlation between early and later sales, tis euristic 240 Vol. 3, No. 3, Summer 2001

12 seems to be an accurate, simple, and intuitive way for a retailer to implement tis strategy. In conclusion, we believe tat te metod described ere provides a useful framework to improve te accuracy and analyze several crucial aspects of replenisment-based planning. Acknowledgments Te autors would like to tank Professor Leroy B. Scwarz, a Senior Editor, and two anonymous referees for several excellent suggestions during te review process. References Bickel, P., K. Doksum Matematical Statistics. Holden Day Publisers, San Francisco, CA. Bitran, G. R., E. Haas, H. Matsuo Production planning of style goods wit ig setup costs and forecast revisions. Oper. Res. 34(2) Bradford, J. W., P. K. Sugrue A Bayesian approac to te twoperiod style-goods inventory problem wit single replenisment and eterogeneous poisson demands. J. Oper. Res. Soc. 43(3) Eppen, G. D., A. V. Iyer. 1997a. Improved fasion buying using Bayesian updates. Oper. Res. 45(6) ,. 1997b. Backup agreements in fasion buying Te value of upstream flexibilty. Management Sci. 43(11) Ermoliv, Y., R.J.B. Wetts Numerical Tecniques for Stocastic Optimization. Springler Verlag, New York. Fiser, M., A. Raman Reducing te cost of demand uncertainty troug accurate response to early sales. Oper. Res. 44(4) Hausman, W. H., R. Peterson Multiproduct production sceduling for style goods wit limited capacity, forecast revisions and terminal delivery. Management Sci. 18(7) Matsuo, H A stocastic sequencing problem for style goods wit forecast revisions and ierarcical structure. Management Sci. 36(3) Murray, G. R., Jr., E. A. Silver A Bayesian analysis of te style goods inventory problem. Management Sci. (11) Raman, A Managing inventory for fasion products. Quantitative Models for Supply Cain Management. S. Tayur, R. Ganesan, and M. Magazine, eds. Kluwer Academic Publisers, Norwell, MA. Te consulting Senior Editor for tis manuscript was Gary Eppen. Tis manuscript was received on November 30, 1999, and was wit te autors 488 days for 2 revisions. Te average review cycle time was 45 days. Vol. 3, No. 3, Summer

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