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To cite this article:Marshall Fisher, Kumar Rajaram, Ananth Raman, (2001) Optimizing Inventory Replenishment of Retail Fashion Products.Manufacturing & Service Operations Management 3(3):230-241. http://dx.doi.org/10.1287/msom.3.3.230.9889
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Marshall Fisher • Kumar Rajaram • Ananth RamanThe Wharton School, University of Pennsylvania, 3620 Locust Walk, 3207 SH-DH,
Philadelphia, Pennsylvania 19104-6366University of California, Los Angeles, Anderson School of Management, Los Angeles, California 90095-1481
Harvard Business School, Soldiers Field, Boston, Massachusetts 02163
W e consider the problem of determining (for a short lifecycle) retail product initial andreplenishment order quantities that minimize the cost of lost sales, back orders, and
obsolete inventory. We model this problem as a two-stage stochastic dynamic program,propose a heuristic, establish conditions under which the heuristic finds an optimal solution,and report results of the application of our procedure at a catalog retailer. Our procedureimproves on the existing method by enough to double profits. In addition, our method canbe used to choose the optimal reorder time, to quantify the benefit of leadtime reduction,and to choose the best replenishment contract.(Retailing; Inventory Replenishment; Stochastic Dynamic Programming; Heuristics)
1. IntroductionRetail inventory management is concerned with deter-mining the amount and timing of receipts to inventoryof a particular product at a retail location. Retail in-ventory-management problems can be usefully seg-mented based on the ratio of the product’s lifecycle Tto the replenishment leadtime L. If T/L � 1, then onlya single receipt to inventory is possible at the start ofthe sales season. This is the case considered in thewell-known newsvendor problem. At the other ex-treme, if T/L k 1, then it’s possible to assemble suf-ficient demand history to estimate the probability den-sity function of demand and to apply one of severalwell-known approaches such as the Q, R model.
The middle case, where T/L � 1 but is sufficientlysmall to allow only a single replenishment or a smallnumber of replenishments, has received much less at-tention both in the research literature and in retailpractice. As we describe in § 2, there is a small butgrowing literature on limited-replenishment inven-tory problems. Perhaps because of the lack of pub-lished analysis tools, we have found that retailers of-ten ignore the opportunity to replenish when T/L is
close to one and treat this case as though it were anewsvendor problem. This is unfortunate, because, aswe show with the numeric computations in this pa-per, planning for even a single replenishment, can, inthis case, dramatically increase profitability.
In this paper, we consider limited lifecycle retailproducts in which only a single replenishment is pos-sible. We model the problem of determining the initialand replenishment order quantities (to minimize thecost of lost sales, backorders, and obsolete inventoryat the end of the product’s life) as a two-stage sto-chastic dynamic program. We show that the second-stage cost function of this program may not be con-vex or concave in the inventory position after thereorder is placed, which means that simulation-basedoptimization techniques (Ermoliv and Wetts 1998)typically used to solve problems of this type are notguaranteed to find an optimal solution. For this rea-son, and also for computational efficiency, we for-mulate a heuristic for this problem. We show that thisheuristic finds an optimal solution if demand subse-quent to the time a reorder is placed is perfectly cor-related with demand prior to this time. While perfect
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001 231
correlation between early and late demand is unlike-ly, we believe this result indicates that our heuristicwill work well if this correlation is high. Thus, inpractice, it seems reasonable to expect good perfor-mance from this heuristic because the logical basis ofimplementing replenishment based on early sales isthat demand during the later season is highly corre-lated with early demand. In our application, the cor-relation between early and late demand was 0.95. Wealso apply simulation-based optimization techniques(Ermoliv and Wetts 1998) and find that our heuristicis much faster and finds solutions within 1% of theoptimization procedure if the correlation betweenearly and late demand is at least 60%. For lower cor-relations, the solutions are within 1% to 5%.
We have applied this process at a catalog retailerand find that it improved over their current processfor determining initial and replenishment quantitiesby enough to essentially double profits. Remarkably,compared to no replenishment, a single-optimized re-plenishment improves profit by a factor of five. A keychallenge in implementing short lifecycle replenish-ment is estimating a probability density function fordemand with no demand history. To circumvent thisproblem in our application, we applied the commit-tee-forecast process in Fisher and Raman (1996) andfound that it worked well.
The most important difference between catalog andtraditional retail management is that a catalog cus-tomer will generally accept a backorder if an item isstocked out. Because our application was at a catalogretailer, our model and heuristic are given for thisversion of the problem, but it is straightforward tomodify the model, heuristic, and proof of optimalityfor a case where backorders are not allowed.
In § 2 of this paper, we review the literature on shortlifecycle inventory replenishment. In § 3, we formulatethe problem; in § 4, we state our heuristic and establishoptimality conditions; in § 5, we show how to modifythe process when customers may return merchandise,and in § 6, report results of our application.
2. Literature ReviewAnalytical models for managing inventory for shortlifecycle products share many common features.
First, all are stochastic models, because they considerdemand uncertainty explicitly. Second, they considera finite selling period at the end of which unsold in-ventory is marked down in price and sold at a loss.In this sense, these models are similar to the classicnewsvendor model. Third, they model multiple pro-duction commitments such that sales information isobtained and used to update demand forecasts be-tween planning periods. The last two characteristics,finite-selling periods and multiple production com-mitments, differentiate style goods inventory modelsfrom other stochastic inventory models. Examples ofpapers that consider style goods inventory problemsinclude Murray and Silver (1966), Hausman and Pe-terson (1972), Bitran et al. (1986), Matsuo (1990), andFisher and Raman (1996). A detailed review of thesepapers can be found in Raman (1999).
Recent work that deals specifically with the retail-er’s inventory-management problem for short lifecy-cle products includes Bradford and Sugrue (1990),Eppen and Iyer (1997a), and Eppen and Iyer (1997b).Bradford and Sugrue model a decision that is similarto the one we study, but they do not consider theimpact of replenishment leadtimes. In addition, theirsolution procedure consists of complete enumeration,which works efficiently for smaller problems butcould be difficult to implement in larger, practical-sized problems. Eppen and Iyer (1997a) consider aproblem that is substantially different from ours.Even though their model allows the retailer to ‘‘buy’’and ‘‘dump’’ at the beginning of each period, the so-lution method they propose applies only when no‘‘buy’’ decisions are permitted after the first period.
Eppen and Iyer (1997b) model a backup agreementin place at a catalog retailer. A backup agreement isone of the mechanisms by which a retailer achievesreplenishment of branded merchandise supplied by amanufacturer to several retailers. In a backup agree-ment, a retailer places an initial order before the startof the sales season and commits to reorder a certainquantity during the season. After assessing sales dur-ing the early part of the season, if the retailer choosesto reorder less than this commitment, there is a pen-alty cost assessed for each unit not ordered. In thismodel, replenishment leadtimes are assumed to be
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001232
zero, which is reasonable because the manufacturerwould typically have produced the product and heldit in inventory for this and other retailers. Because thereplenishment leadtime is zero, it is not necessary forthe retailer to accept backorders from consumers.
In this paper, we consider the case where replen-ishment is achieved without backup agreements. Af-ter receiving an updated order, the manufacturer pro-duces and delivers products to the retailer after asignificant leadtime. The retailer is not required tocommit to any of the reorders. To compensate for thelong leadtime, consumer backorders are accepted bythe retailer. This case occurs when the manufacturersare either captive suppliers or wholly owned by theretailer and the retailer sources from several suchmanufacturers. Thus, in this case, it is crucial to mod-el the impact of leadtimes and backorders, althoughthis significantly complicates the analysis leading toa nonconvex optimization model. In addition to in-corporating replenishment leadtimes and backorders,our work differs from all these papers in the processthat we use to estimate demand densities and to com-pare our method to actual practice.
3. ModelWe model the supply decisions faced by a catalog re-tailer for a product with random demand over a salesseason of fixed length. The retailer must determinean initial order Q1 available at the start of the salesseason. At a fixed time t during the season, the re-tailer updates the demand forecast, based on ob-served sales, and places a reorder quantity Q2 thatarrives after a fixed leadtime L at time t � L.
Price is fixed throughout the season. Inventory leftover at the end of the season is sold at a salvage pricebelow cost. Customers who encounter a stockout willbackorder if there will be sufficient supply at somepoint in the future to satisfy the backorder. Specifi-cally, the opportunity to backorder is not offered to acustomer once the total supply quantity (Q1 � Q2)has been committed through sales or prior backor-ders. A lost sale is incurred when an item requestedby a customer is not in stock or not backordered.
We first model this problem and formulate a solu-
tion heuristic assuming that the reorder time t isfixed. Then, we determine an optimal reorder time tempirically for a given data set by parametricallysolving this problem with varying t. We are given:
Cu � Cost per unit of lost sale. This is set to thedifference between the per-unit sales priceand cost of the product.
Co � Cost per unit of leftover inventory. This isset to the difference between the per unitcost and the salvage price of the product.
Cb � Cost per unit of backorders. This is set tothe additional costs incurred in procure-ment and distribution when an order isbacklogged, plus an estimate of the cost ofcustomer ill will.
L � Length of replenishment leadtime.
Define the following variables:
X � Random variable representing total de-mand until the reorder is placed.
Y � Random variable representing total de-mand during the replenishment leadtime.
W � Random variable representing total de-mand after the reorder arrives until the endof the season.
R � Random variable representing total de-mand after the reorder is placed until theend of the season, where R � Y � W.
Q1 � Initial-order quantity.
Q2 � Reorder quantity.I � Inventory position after reorder is placed.
I � Q1 � Q2 � x.
For the given reorder time t, the decision processinvolves choosing Q1, observing x, and then deter-mining the inventory position I for the remainder ofthe season to minimize total backorder, understock,and overstock costs. This sequence of decisions isshown in Figure 1.
We consider random variables � and � with jointdensity function f (�, ). Let g(�) be the marginal den-sity on � defined by f (�, ) and h( � �) be the con-ditional density on given � defined by f (�, ). Wedefine E�/�(()) � # ()h( � �) � and E�((�)) ��
0
# (�)g(�) ��, where ( ) is any real-valued scalar�0
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001 233
Figure 1 The Replenishment Planning Process
function. Let (a)� � max(a, 0). We model this problemby the two-stage stochastic dynamic program (P1).
Z(t) � min C(Q ) � E [C (Q , x) � C (Q , x)] (P1)1 x 1 1 2 1Q �01
where
�C (Q , x) � C (x � Q )1 1 b 1
C (Q , x) � min C (I, x)2 1 2I
� �� E {C min((y � (Q � x) ) ,Y/X b 1
�I � (Q � x) )}1
�� E E {C (y � w � I)Y/X W/X u
�� C (I � y � w) }o
I � Q � x, I � 0 (P2)1
C1(Q1, x) represents the backorder costs Cb(x � Q1)�
during the period before the reorder is placed. C2(I, x)represents expected cost as a function of the inven-tory position I after the reorder is placed and consistsof two terms. The first term, EY/X{Cbmin((y � (Q1 �x)�)�, I � (Q1 � x)�)}, represents the expected costsof backorder during the replenishment leadtime. Be-cause backorders are accepted only if they can befilled from replenishment, it is important to recognizethat backorders during the replenishment leadtimecan never exceed the effective inventory position afterthe first period backlog is cleared (i.e., I � (Q1 � x)�).This condition is enforced by the operator min((y �(Q1 � x)�)�, I � (Q1 � x)�). The second term ofC2(I, x) is EY/XEW/X{Cu(y � w � I)� � Co(I � y � w)�},represents the expected overstock and understockcosts in the periods after the reorder is placed untilthe end of the season.
It is important to recognize that C2(I, x) is neither
convex nor concave in I. We illustrate this propertyusing the following example.
EXAMPLE 1. Let Q1 � 50 and x � 10. Let the con-ditional probability distribution for Y/(X � 10) beP(Y/(X � 10) � 100) � 0.5 and P(Y/(X � 10) � 200)� 0.5, while the conditional probability distributionfor W/(X � 10) is P(W/(X � 10) � 100) � 0.5 andP(W/(X � 10) � 200) � 0.5.
Let Cb � 15, Co � 20, Cu � 40, � 0.9, I1 � 80,and I2 � 110, I � I1 � (1 � )I2 � 83. By substi-tuting these values, using the values of Q1 and x, anddistributions Y/X and W/X to calculate expectations,it is easy to verify that:
� �C (I , x) � E [C min((y � (Q � x) ) ,2 Y/X b 1
�I � (Q � x) )]1
�� E E [C (y � w � I )Y/X W/X u
�� C (I � y � w) ]o
1 2� 9325 � C (I , x) � (1 � )C (I , x)2 2
� 9317.5.
This shows that C2(I, x) is not convex in I.Next, let I1 � 80 and I2 � 300, so that I � I1 �
(1 � )I2 � 107, while all the other values remainunchanged. Now,
� �C (I , x) � E [C min((y � (Q � x) ) ,2 Y/X b 1
�I � (Q � x) )]1
�� E E [C (y � w � I )Y/X W/X u
�� C (I � y � w) ]o
1 2� 8672.5 � C (I , x) � (1 � )C (I , x)2 2
� 8775.
This shows that C2(I, x) is not concave in I.In view of this characteristic of C2(I, x), simulation-
based optimization techniques (Ermoliv and Wetts1998), typically used to compute the solution to thisclass of problems, are not guaranteed to solve Prob-lem P2 and subsequently Problem P1 to optimality.Consequently, for this reason and run-time consider-ations, we elected to develop a heuristic. This heuris-tic is described in the next section.
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001234
Once we have developed a scheme to solve thisproblem, to find the optimal reorder time, we wouldperform a line search on t to solve (P):
Z* � min Z(t) (P)0�t�T�L
4. The Two-Period NewsvendorHeuristic
The purpose of this heuristic is to set Q1. In this re-gard, it is useful to understand the costs affected bythe choice of Q1. Firstly, a portion of Q1 may remainunsold at the end of the season, generating an over-stock. Secondly, during the interval 0 to t � L, if Q1
is too small, one may incur backorder costs. Duringthe interval t to t � L, one may also incur stockoutsif satisfied and backordered demand exceeds Q1 �Q2, but it seems more natural to think of this cost asresulting from the choice of Q2, not the choice of Q1.Given this, we let S � X � Y, U � X � Y � W, andchoose Q1 to solve:
¯Z (t) � min C(Q )h 1Q �01
� �� E C (s � Q ) � E C (Q � u) (PH)S b 1 U o 1
To solve this problem, let F1(s) and F2(u) be the dis-tribution functions of random variables S and U, re-spectively. The first-order condition for problem (PH)is:
�Z (t)h � �C (1 � F (Q )) � C F (Q ) � 0b 1 1 o 2 1�Q1
We set the heuristic order quantity Q to the valueh1
of Q1 that satisfies this condition. Rearranging terms,this is calculated as the solution to the followingequation:
CoF (Q ) � F (Q ) � 11 1 2 1Cb
Let f 1(s) and f2(u) be the density functions of ran-dom variables S and U, respectively. Because
2� Z (t)h � �C f (Q ) � C f (Q ) � 0,b 1 1 o 2 12�Q1
the first-order conditions are sufficient to establish
the optimality of Zh(t) at Q . Note that our choice ofh1
Q minimizes expected backordering costs during theh1
period before replenishment and minimizes expectedoverstock cost at the end of the season because ofQ . The following result establishes conditions underh
1
which this heuristic finds an optimal solution.
PROPOSITION 1. Suppose Z(t) is the optimal solution toProblem (P1) when random variables X, Y, and W are per-fectly correlated. Then, Zh(t) � Z(t).
PROOF. If random variables X, Y, and W are perfectlycorrelated, then Y � �X and W � �X, where �, � arepositive constants. Thus, E(Y/X � x) � �x, V(Y/X �x) � 0, E(W/X � x) � �x, and V(W/X � x) � 0. Whenall customers backorder, the optimal reorder quantityis Q � [x(1 � � � �) � Q1]�. If Q � 0, then one* *2 2
incurs no overstock and understock costs in the thirdperiod after the reorder arrives. The only costs in-curred will be possible backorder costs during the firsttwo periods represented by Cb[x(1 � �) � Q1]�. IfQ � 0, in addition to the backorder costs in the first*2two periods, one could incur an overstock of [Q1 � x(1� � � �)]� because of the initial order with associatedcosts Co[Q1 � x(1 � � � �)]�. Consequently, total ex-pected costs in the season when one has perfectly cor-related demand can be expressed as
�C(Q ) � E {C [x(1 � �) � Q ]1 X b 1
�� C [Q � x(1 � � � �)] }.o 1
Because by definition, S � X � Y � X(1 � �) and U� X � Y � W � X(1 � � � �), C(Q1) � ES{Cb(s �Q1)�} � EU{Co(Q1 � u)�} � C̄(Q1). Thus, Z(t) �
C(Q1) � C̄(Q1) � Zh(t). Q.E.D.minQ �Q1
It can be shown that Proposition 1 also holds underthe assumption of no customer backorders. In lightof this proposition, it is reasonable to expect goodperformance from this heuristic because the logicalbasis of implementing replenishment based on earlysales is that demand during the later season is highlycorrelated with early demand. In our application, wefound across all the products the correlation betweenX and Y to be around 0.96 and between X and W tobe about 0.95. This suggests that this heuristic couldprovide a simple and efficient basis to model the re-quired decisions in this application.
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001 235
Once one uses the two-period newsvendor heuris-tic to determine Q1 and observes demand x duringthe first period, the optimal solution to the minimi-zation problem P2 can be approximated by setting I� max(I*, Q1 � x), where I* is the newsvendor quan-tity defined on HR/X, the cumulative distribution of Rupdated by X � x (i.e., I* � H [(Cu � Cb)/(Cu � Cb
�1R/X
� Co)]. The quality of this approximation is also as-sessed in the application while evaluating the perfor-mance of the heuristic.
5. Modifications to Account forReturns of Merchandise
In catalog retailing, because customers place ordersbased on photographs displayed in catalogs, pur-chased merchandise is often returned if the actualproduct differs from what the customer expectedfrom the catalog. In this section, we describe how toextend our model to include merchandise returns.
Returned items can be resold if they are receivedbefore the season ends. This means that backordersin the interval (0, t � L) and stockouts during the in-terval (t � L, T ) may be reduced by the availability ofreturns, but returns that are received too late to beresold can contribute to overstock. Based on the prac-tice followed by the catalog retailer described in theapplication, we assume that a known fraction � ofcustomers return products, where 0 � � � 1. Thesereturns are immediately reusable if necessary to sat-isfy either a backorder or demand. We also assumethat recycled returns (i.e., returns on returns and soon) are not reusable during the sales season. Theseassumptions ensure that we make an unbiased com-parison with existing practice at this retailer.
Consequently, to adapt the heuristic to include re-turns of merchandise, we first consider the period un-til the reorder arrives. If Q1 is the initial-order quan-tity and s is the demand during this period, the totalnumber of reusable returns is � min(s, Q1). If s � Q1,the total backorders that occur during this period are[s � (Q1 � � min(s, Q1))]� � [s � Q1(1 � �)]�. Sim-ilarly, if u is the demand during the entire season,total reusable returns because of the initial-order
quantity are � min(u, Q1). If Q1 � u, the total over-stock that occurs during the entire season is
We use the procedure outlined in the previous sec-tion to determine Q as the solution to the followingh
1
equation.
C Qo 1F [Q (1 � �)] � F � 11 1 2� �(1 � �)C (1 � �)b
At the end of the first period, we observe realizeddemand x and use it to set the reorder quantity Q2
� I � (Q � x)� � � min(Q , x), where I � max(I*,h h1 1
(Q � x)� � � min(Q , x)), and I* is the newsvendorh h1 1
quantity defined on HR�/X, the cumulative distribu-tion of R’ � R(1 � �) updated by X � x (i.e., I* �H [(Cu � Cb)/(Cu � Cb � Co)]).�1
R�/X
6. ApplicationWe have tested the ideas presented in this paper at alarge catalog retailer. We applied the model and thetwo-period newsvendor heuristic to make purchasedecisions for 120 styles/colors from the women’sdress department appearing in a particular catalog.We chose this division because it represented a sig-nificant portion of the business. The sales season forthese products is T � 22 weeks, and the replenish-ment leadtime is L � 12 weeks. Because these prod-ucts are sold through mailorder catalogs, the priceduring the season is fixed. Around 35% of sales arereturned, i.e., � � 0.35.
In the process currently in place at this retailer, ini-tial-order quantities are set to forecast demand for the22-week season adjusted for anticipated merchandisereturns. Forecasts for each style/color are updated af-ter two weeks by dividing observed sales by the his-torical fraction of total-season sales for the depart-ment, which have been observed in the past to
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001236
Figure 2 Comparison of Early and Updated Forecasts
normally occur in the first two weeks. Reorders areplaced to make up the difference from an updatedforecast adjusted for returns. Specifically for a givenstyle/color, if f is the total forecast sales and � is theanticipated fraction of returns, then Q1 � (1 � �) f .Letting x2 be the actual sales observed at the end oftwo weeks and k2 be the fraction of total demand his-torically observed at this point for a group of similarproducts, then we set Q2 � ((1 � �)x2/k2 � Q1)�.Note that this procedure sets Q1 to the forecast salesnet of anticipated returns during the entire seasonand, hence, reorders are used as a reaction to larger-than-anticipated sales rather than something that isplanned for in advance.
Figure 2 shows the improvement in forecast accu-racy because of updating at the retailer. Each pointshows forecast and actual demand for a particularstyle/color combination. The left graph compares de-mand forecasts with actual demand for the averageof forecasts made by four expert buyers prior to thebeginning of the season. In the right graph, the fore-casts equal actual sales after two weeks into the sea-son divided by a factor representing the fraction oftotal sales historically observed after two weeks.
Application of our model requires a method to es-timate demand-probability distributions. This is par-ticularly challenging because there was no sales his-tory for any of the new dresses. However, we wereable to calculate forecast errors, defined as the differ-ence between buyer forecast and actual sales for sim-ilar products appearing in the same catalog from thepast two years. We used this information to conclude
that the distribution of forecast errors was normallydistributed with a large degree of confidence (�2 testholds at � � 0.01 level). We assumed that forecasterrors would follow a similar distribution in past andfuture seasons. This seemed reasonable because thesame individuals who forecasted product demand inthe past were also forecasting current season de-mand.
Because the demand for any given product is equalto its forecast plus the associated forecast error, thisimplies that the demand distribution for U for a givenproduct during the entire season is normally distrib-uted. While probability distributions for retail prod-ucts seem to have long tails, these result from plot-ting actual demand for products that seemindistinguishable (or at least similar) ex ante. How-ever, in contrast, U represents the demand distribu-tion for a given product.
To estimate normal parameters � and � of this dis-tribution, we implemented the procedure developedby Fisher and Raman (1996). In this method, themembers of a committee (comprised of four buyersin our case) independently provide a forecast of salesfor each product. The mean � is set to the average ofthese forecasts. The standard deviation of demand �
is set to ��c, where �c is the standard deviation of theindividual committee member’s forecasts and the fac-tor � is chosen so that the average standard deviationof historical forecast errors equals the average stan-dard deviation assigned to new products. In our ap-plication, we found � to be 1.4.
To estimate the parameters of distribution X andR, where U � X � R, we assume that (X, R) followsa nondegenerate bivariate normal distribution. Forthis distribution, it is well known (Bickel and Dock-sum 1977) that the marginal distribution of X is anunivariate normal distribution with mean �x andstandard deviation �x, while the marginal distribu-tion of R is also normally distributed with mean �R
and standard deviation �R. Let kt represent the pro-portion of total sales until reorder point t, �t the cor-relation between X and R, and �t the correlation be-tween X and U. We estimate kt, �t, and �t fromhistorical data and use the formulas developed inFisher and Raman (1996) to calculate
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001 237
Figure 3 Committee Standard Deviation Versus Forecast Error
2(1 � � )t� � k �, � � � � � � ,x t X t t 2[ ]�(1 � � )t
2(1 � � )t� � k (1 � �), and � � � .R t R 2�(1 � � )t
For the bivariate normal distribution (X, R), note thatthe updated distribution R/X � x is also normallydistributed with mean �R/X � �R � �t(x � �X)�R/�X
and standard deviation �R/X � �R�1 � � . Because2t
0 � �t � 1, this implies that �R/X � �R. Thus forecastupdating based on actual sales x reduces variance inthe distribution of demand during the remaining sea-son and permits a more accurate forecast. By usingreplenishment, the retailer can take advantage of thisimproved forecast by placing a more precise reorderthat directly contributes to higher expected profitsduring the remaining season.
To better understand the nature of forecast errors,we compared the standard deviation of the commit-tee forecast for individual products at the beginningof the season (i.e., �c) with its corresponding forecasterror. These results, shown in Figure 3, suggest thatwhen the committee agrees, they tend to be accurate,and that the committee process is a useful way todetermine what you can and cannot predict.
With the exception of the backorder penalty Cb, allthe cost parameters required for our analysis werereadily available. Estimating the backorder penalty ischallenging in practice because, in addition to the $1per-unit extra-transaction cost for procurement anddistribution associated with a backorder, there is an
intangible cost because of customer ill will. The com-pany was uncertain as to the exact value of the ill-will cost, but felt a value of Cb in the range $5 to $15was reasonable. We applied our analysis to three cas-es using $5, $10, and $15 per unit as values of Cb. Wealso analyzed the case Cb � 1 to insure that our heu-ristic did not outperform the current rules because weassessed an ill-will cost that was not used in the cur-rent rule.
Note that although ill-will costs can also be addedto Cu, we did not add them because an ill-will costin this application was charged only because man-agement was mainly interested in insuring that flex-ibility to backorder was not abused. Given the valuesof Cu and Co, if Cb � 0, then it is optimal to set Q1 �
0 and backorder all first-period demand. But, theseexcessive backorders would likely reduce marketshare in the long term. The omission of ill-will costsin Cu does not affect the analysis because, dependingon the product, Cu was two to four times greater thanbackorder costs, and consequently, it would never beoptimal to not satisfy demand to avoid a backorder.
As a practical matter, we found that historicallyaround 5% of customers chose not to accept the offerto backorder at this retailer. Consequently, we adjust-ed the backorder cost to account for this fraction oflost sales by defining an effective backorder cost, C�b� 0.95·Cb � 0.05·Cu, representing the costs of a bac-korder and stockout weighted by the expected frac-tion of customers who would choose either option.We replaced Cb with C in the definition of problem�b(PHr).
The first step in our methodology is to determinethe reorder time. It is important to accurately choosethis time because, if chosen too early in the season,actual sales will not be sufficient to provide an ac-curate revision of the second-period demand forecast.On the other hand, if the reorder time is too far intothe season, the benefit of replenishment is diminishedbecause it is then used to service only a small portionof the season. The specific choice of reorder point de-pends on the proportion of total sales observed dur-ing the initial weeks and the length of the replenish-ment leadtime. For instance, if this proportion is high,there is a long leadtime or both, one would choose
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001238
Figure 4 Reorder Time and Expected Coststhe reorder point early in the season to ensure that areasonable proportion of total sales is serviced by thereorder.
To determine the best reorder time, we used MonteCarlo simulation with the estimated distributions ofdemand to calculate Z(t) for Week t. In this procedure,for a given reorder time, we estimate the initial-orderquantity using our heuristic. We simulate x (as a re-alization of X), the distribution of total demand untilthe reorder is placed. We use x to calculate the back-order costs before the reorder is placed, update R/X� x, and calculate the expected costs during the re-maining season. We repeat this procedure for severalsimulated realizations of X and calculate the expectedcosts during the entire season associated with a reor-der time by averaging the costs associated with eachrealization. As discussed previously, using the bivari-ate normal distribution to model demand (X, R) en-sures that both X and R are univariate normal distri-butions and the variance of updated second-perioddemand R/X � x is also univariate normal whose var-iance is now reduced from �R to �R/X � �R�1 � � ).2
t
We repeat the simulation for several choices of re-order time. The results of this simulation are sum-marized in Figure 4. Because Z(t) attains its mini-mum at t � t* � 2, the reorder time is chosen to beat the end of Week 2. Note that the length of the re-plenishment leadtime assumed in this analysis is 12weeks. As the season lasts only 22 weeks, we cannotreorder after Week 10. Consequently, the value of Z(t)for t � 10 is set equal to the expected costs incurredfor a single period buy if we set Q1 to Q , the news-s
1
vendor quantity defined on the total distribution ofdemand (i.e., Q � F [Cu/(Cu � Co)].s �1
1 2
A key factor that influences the level of profitsgained by replenishment is the proportion of total de-mand over time observed during the early part of theseason and the time until the reorder arrives. Clearly,if this proportion is very high, then the benefits ofreplenishment are limited, as the reorder would onlyserve a small proportion of total-season demand. Inour application, we found that historically, for similarproduct lines, 10% of total demand is observed whenthe reorder is placed after two weeks, and 50% of
demand is observed at Week 10 when the reorder ar-rives. These values confirmed that replenishmentbased on actual sales was a viable strategy for thechosen product line and motivated us to apply ourmethod to determine initial- and replenishment-orderquantities.
For the 120 styles/colors in this department, wedetermined the initial-order quantity by solvingproblem (PHr) using the heuristic modified to includereturns. We then observed x, the sales until the sec-ond week, and set the reorder quantity Q2 using theprocedure developed in § 5. Because at the end of theseason we knew total sales and actual sales per weekfor each dress, we were able to calculate the stock-outs, overstock, backorders, dollar sales, and profitsthat would have resulted from our ordering policy.To compare our method with the current orderingrules, we also calculated these values for the currentpolicy. The results consolidated across all the 120dresses are tabulated in Table 1.
Observe that although the total orders placed byour method and existing practice are similar, thecomposition of these orders across the two periods isdifferent. Our heuristic reduces overstocks, stockouts,
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001 239
Table 1 Comparison of the Two-Period Newsvendor Heuristic withCurrent Practice
Profit Increase By Model(As % Current Sales) 4.92 3.52 2.23
Figure 5 Replenishment Costs and Leadtimes
and backorders enough to increase profits comparedwith the current rule from 2.23% to 4.92% of currentsales, depending on the value of Cb. Profit before taxfor this retailer is around 3% of sales. Consequently,our heuristic offers the potential to approximatelydouble profit.
Our results also show the impact of Cb on the so-lution. Order quantities for the current rule do notchange because the current rule does not consider Cb
in determining order quantities. When Cb � $5, weorder a smaller initial quantity because backordersare now relatively less expensive. This in turn increas-es backorders and stockouts but reduces overstocks,which increases profit improvement from 3.52% to4.92%. On the other hand, when Cb � $15, we ordera larger initial quantity because backorders are rela-tively more expensive. This reduces backorders andstockouts but increases overstocks, which reduces theprofit improvement from 3.52% to 2.23%.
To insure that our heuristic did not outperform thecurrent rules because we assessed an ill-will cost thatwas not used in the current rules (possibly because ill-will costs were not recorded in the books), we alsoconsidered the case when Cb � 1. Here, Cb only con-sists of the additional transaction cost per unit of pro-curement and distribution associated with a backorder.As expected, our heuristic ordered a substantiallysmaller amount initially than the cases with larger val-ues of Cb. This in turn increased backorders and stock-outs, but reduced overstocks. Over all, the profit im-provement over the current rules increased to 5.52%.
This is consistent with the general pattern in Table 1,which shows profit improvement increasing as Cb de-creases.
This analysis assumes a replenishment leadtime of12 weeks. It is easy to understand that reducing thistime could potentially increase the benefits of replen-ishment, because a greater portion of the season canbe serviced from the more accurate reorder. However,it is important to precisely calculate this benefit tojustify the costs of leadtime reduction. Our method-ology provides a framework to analyze these benefitsboth before and after sales are realized. To performthis analysis before actual sales are realized, we usea simulation to calculate expected costs for differentvalues of leadtimes. Using actual sales, and for thecase in which Cb � 10, we performed an analysisidentical to the one used to obtain the results report-ed in Table 1, but using the new choices of the lead-time. These results are summarized in Figure 5 andindicate that the length of the replenishment lead-times significantly influences the benefits of replen-ishments. This type of analysis could be used to de-cide between a domestic supplier with typically
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001240
Table 2 Percentage Performance Gap Between the Two-Period News-vendor Heuristic and a Simulation-Based Optimization Methodfor Different Levels of Demand Correlations
CorrelationPerformance Gap
(%) CorrelationPerformance Gap
(%)
0�0.1�0.2�0.3�0.4�0.5�0.6�0.7�0.8�0.9�0.99
5.24.153.11.81.51.21.10.80.50.30.1
00.10.20.30.40.50.60.70.80.90.99
5.2431.71.41.30.90.70.60.20.07
higher costs but shorter leadtimes and a foreign sup-plier with relatively low costs but long leadtimes.
To further evaluate the quality of this heuristic, wesolved (P1) using a simulation-based optimizationmethod.1 In this technique, we use simulation to nu-merically compute C(Q1) for selected values of Q1 inthe range [0, Q*], where Q* � F [Cu/(Cu � Co)] is�1
2
the newsvendor quantity2 defined across the wholedistribution of demand. Finally, we set Z(t) �
C(Q1). In the case where Cb � 10, this tech-*min0�Q�Q1
nique improved profit relative to the current rule byaround 3% of current sales, a lower improvementthan was achieved with our heuristic. In the caseswhere Cb equaled 5 or 15, the profit improvement wasalso marginally less than achieved with our heuristic.
In addition to resulting in a smaller profit gainthan the two-period newsvendor heuristic, we foundthat the solution time for this technique was around40 hours on a Dell Pentium II PC, as compared withless than a minute for the two-period newsvendorheuristic. These results provide strong justification forusing this heuristic in this application.
We also considered the impact on performance ofcorrelation between early and later demand. In ourapplication, across all products, we found the corre-lation between X and Y to be around 0.96 and be-tween X and W to be about 0.95. In view of Propo-sition 1, such high correlation suggests that theheuristic solution is very close to the optimal solutionfor this problem.
We performed a computational study to evaluatethe performance of the heuristic for different levels ofcorrelation between early and later season demand.Define �1 as the correlation between X and Y and �2
as the correlation between X and W. For simplicity,we set �1 � �2 � � and vary � from 0 to 0.99 in stepsof 0.1. For a given value of �, and using the committee
1Please refer to Ermoliv and Wetts (1998), ‘‘Numerical Techniquesfor Stochastic Optimization,’’ Springler Verlag, New York, for a the-oretical justification and a detailed description of this technique. Thesame simulation-based technique is used to numerically estimateC2(Q1, x) required in the computation of C(Q1). Here we vary I overthe range [0, 10Q*] and set C2(Q1, x) � min0�I�10Q* C2(I).2We choose this value because it is highly unlikely that an initial-order quantity greater than this quantity would not be sufficient tocover sales during the periods before the replenishment arrives.
estimates of � and � for each product, we calculatedQ1 using the heuristic and using the simulation-basedoptimization method. We then used these values ofthe initial-order quantities and Monte Carlo simula-tion with the estimated distribution of U for eachproduct to calculate expected costs for each tech-nique.
For a given value of �, let Ch represent the totalexpected cost of the heuristic across all products, andlet Cs represent the corresponding total cost of thesimulation-based optimization procedure. The per-centage-performance gap of the heuristic is definedas (Ch � Cs)/Cs � 100%. We report the percentage-performance gap across a range of values for demandcorrelation in Table 2. For each level of demand cor-relation, the run times for the heuristic across all theproducts was less than a minute, while the equivalentrun time for the optimization approach was aroundforty hours.
The results in Table 2 show that this gap variesfrom 0.06% to 5.2%, with the highest gaps occurringat the lowest levels of correlation. These results sug-gest that the heuristic provides an efficient basis toaddress this problem, even for cases that have modestlevels of correlation (e.g., ��� � 0.3) between early andlater sales. Because fashion replenishment makesmost sense for products with some degree of corre-lation between early and later sales, this heuristic
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FISHER, RAJARAM, AND RAMANOptimizing Inventory Replenishment of Retail Fashion Products
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT
Vol. 3, No. 3, Summer 2001 241
seems to be an accurate, simple, and intuitive way fora retailer to implement this strategy.
In conclusion, we believe that the method describedhere provides a useful framework to improve the ac-curacy and analyze several crucial aspects of replen-ishment-based planning.
AcknowledgmentsThe authors would like to thank Professor Leroy B. Schwarz, a Se-nior Editor, and two anonymous referees for several excellent sug-gestions during the review process.
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The consulting Senior Editor for this manuscript was Gary Eppen. This manuscript was received on November 30, 1999, and was with the authors 488days for 2 revisions. The average review cycle time was 45 days.
