demand seasonality in retail inventory management

European Journal of Operational Research xxx (2014) xxx–xxx

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Demand seasonality in retail inventory management

http://dx.doi.org/10.1016/j.ejor.2014.03.0300377-2217/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +1 972 883 5947.E-mail address: [email protected] (D. Honhon).

Please cite this article in press as: Ehrenthal, J. C. F., et al. Demand seasonality in retail inventory management. European Journal of Operational R(2014), http://dx.doi.org/10.1016/j.ejor.2014.03.030

J.C.F. Ehrenthal a, D. Honhon b,⇑, T. Van Woensel c

a LOG CONNECT AG, Embrach, Switzerlandb Jindal School of Management, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USAc School of Industrial Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

a r t i c l e i n f o

Article history:Received 15 February 2013Accepted 20 March 2014Available online xxxx

Keywords:RetailingInventory controlLost salesNon-stationary demandSeasonality

a b s t r a c t

We investigate the value of accounting for demand seasonality in inventory control. Our problem is moti-vated by discussions with retailers who admitted to not taking perceived seasonality patterns intoaccount in their replenishment systems. We consider a single-location, single-item periodic review lostsales inventory problem with seasonal demand in a retail environment. Customer demand has seasonal-ity with a known season length, the lead time is shorter than the review period and orders are placed asmultiples of a fixed batch size. The cost structure comprises of a fixed cost per order, a cost per batch, anda unit variable cost to model retail handling costs. We consider four different settings which differ in thedegree of demand seasonality that is incorporated in the model: with or without within-review periodvariations and with or without across-review periods variations. In each case, we calculate the policywhich minimizes the long-run average cost and compute the optimality gaps of the policies which ignorepart or all demand seasonality. We find that not accounting for demand seasonality can lead to substan-tial optimality gaps, yet incorporating only some form of demand seasonality does not always lead to costsavings. We apply the problem to a real life setting, using Point-of-Sales data from a European retailer.We show that a simple distinction between weekday and weekend sales can lead to major costreductions without greatly increasing the complexity of the retailer’s automatic store ordering system.Our analysis provides valuable insights on the tradeoff between the complexity of the automatic storeordering system and the benefits of incorporating demand seasonality.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The main challenge in managing retail inventory is to matchreplenishment and demand, that is providing items on the shelfjustified by an upcoming shopper demand. Economies of scale insupply, inadequate store execution and demand variation oftenlead to out-of-stocks and excess inventory. While store executionand retail out-of-stocks have received considerable attention inacademia and business practice (see Aastrup & Kotzab, 2010), theimpact of demand variation has largely been overlooked (Bijvanc& Vis, 2011).

Demand in retailing is known to vary depending on the day ofthe week and time of year, around important holidays such asChristmas, and the seasons. For example, ice cream is in higherdemand in the summer months. Demand is also generally notevenly distributed within the day. For instance, in businessdistricts more customers shop just after working hours on week-days. Retailers can, to some degree, reduce demand variation, for

instance by reducing promotions or offering ‘‘everyday low prices’’.However, customer buying habits, like shopping on weekends,limit a retailer’s ability to completely smooth demand variations.This creates the need for retailers to account for seasonality in theirinventory control and shelf inventories, in (partial) synchroniza-tion with the demand pattern (Aviv & Federgruen, 1997).

Not accounting for demand seasonality leads to systematic mis-matches in demand and supply at the item-store level, resulting inhigher than needed costs. For example, Gruen and Corsten (2008)show that out-of-sync replenishment and demand lead to recur-ring out-of-stocks in retail stores on specific days of the week.Yet, from our conversation with retailers, it appears that many ofthem lack the capabilities or skills to incorporate demand season-ality into their store ordering and shelf replenishment systems. Sixmedium-sized European retailers reported to us that their auto-mated store ordering (ASO) systems do not have the technicalcapabilities to account for demand seasonality within the week.Similarly, two of Europe’s largest retailers told us that even thoughtheir ASO system allows for different yearly seasonality patternsfor each item, they currently refrain from using refined seasonalitypattern analysis because of the added level of complexity. We also

esearch

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spoke to a company which sells forecasting and ERP-type softwarefor retailers. They reported that their program has the technicalcapability of monitoring sales patterns to a very fine granularity,yet so far none of their clients has ever asked for this informationto be incorporated into replenishment decisions; it was only usedfor planning checkout staffing. All the retailers we spoke to wereaware of the possible inefficiencies of not exploiting seasonalitybut were unsure of the return on investment for updating theirsystems. One retailer we met was in the process of investigatingthe mismatch between intra-day sales patterns and the replenish-ment times and building its own tool to address the problem.

We study a setting where demand has a known seasonality pat-tern with two types of variations: within-review period andacross-review periods (the review period is defined as the timebetween when two replenishment orders are placed). We applyour findings to real-life examples where demand follows a season-ality cycle over the days of the week and the times of the day. Fig. 1illustrates the type of seasonality pattern under investigation. Itshows the sales of a specific cigarette product (global brand, singlebox of 10 cigarettes) at a European retail store for two months in2010. Demand varies depending on the day of the week: weeklysales peak on Saturday and Sunday. Demand is also not evenly dis-tributed within a day: on Saturday, most of the sales occur before5 pm while on Sunday most of the sales occurs after 5 pm. Inessence, the demand rate varies across days of the week and withinthe day, but displays the same periodic pattern every week.

Given seasonal customer demand, we research the followingquestions: (1) What is the cost of neglecting demand seasonalityin retail inventory control? (2) For which type of products is thiscost most significant? (3) How much can be gained by partiallyincorporating demand seasonality into an ASO system and whatis the best way to do so? We answer these questions in a settingwhich is suitable for retail environments: we consider a singlelocation, single item, lost sales inventory problem with handlingcosts, fractional lead time (i.e., a lead time shorter than the reviewperiod) and batch ordering. Further, we assume that demand isnon-stationary, in particular the distribution of demand is differentbefore and after the order is received within a review period andalso exhibits a seasonal pattern. This means there are two typesof demand variations: within-review period variations andacross-review periods variations.

We consider four different settings in which the retaileraccounts for varying degrees of seasonality. In the first setting,the retailer accounts for both types of demand variations. In thesecond setting, the retailer only accounts for across-review periods

Fig. 1. Sales seasonality in cigarette sales

Please cite this article in press as: Ehrenthal, J. C. F., et al. Demand seasonality(2014), http://dx.doi.org/10.1016/j.ejor.2014.03.030

variations. In the third setting, the retailer only accounts forwithin-review periods variations. In the fourth setting, the retailerignores all form of seasonality, which often corresponds to currentretail practice.

In each setting, we calculate the inventory policy which mini-mizes the long-run average cost criterion by solving a Markov deci-sion process and show that it does not have a simple structure.Then, the performance of the inventory policies in last three set-tings is evaluated using the true distribution of demand. Thisallows us to discuss when it is most important to incorporatewithin- and across-review periods variations into the ASO systemand which type of variation is the most crucial. Third, we applythe problem to a real-life setting, using Point-of-Sales (POS) datafrom a European retailer and replenishment information. We alsoexplore simplified policies of only distinguishing between week-day and weekend sales.

We find that incorporating seasonal variations can lead to asubstantial decrease in costs – on average 4.91% in our applicationusing POS data. Not surprisingly, we find that product categorieswith more variable demands lead to the highest optimality gapsbut there is more to the story. First, we see that optimality gapsincrease with average demand and decrease with case pack size,and the costs associated with ordering, handling and purchasing.Second, we see that from our POS data analysis that across-daysvariations have a greater impact than intra-day variations. In fact,taking intra-day variations into account without acknowledgingacross-days variations can lead to an increase in costs, i.e., moreinformation is not always better. Finally we see that, in many cases,a very significant portion of the cost savings can be achieved with asimple distinction between weekday and weekend sales, providedthe weekend is defined appropriately for each product category.

The closest paper to ours is a recent study by Tunc, Kilic, Tarim,and Eksioglu (2011), who investigate the best stationary policygiven that demand is non-stationary. They find that a stationarypolicy may be a good approximation only if demand uncertaintyis high, setup costs are high and penalty costs are low. Our papershave fundamentally different approaches. Tunc et al. (2011) searchfor the best stationary ðs; SÞ policy given non-stationary demand:the retailer knows that demand is non-stationary but restricts him-self to a stationary policy because it is easier to implement. In con-trast, this paper considers the best non-stationary inventory policygiven a demand pattern that leaves out some seasonality, forexample because the retailer uses weekly demand estimates. Fur-ther, Tunc et al. (2011) use a model with backorders, no lead time,a batch size of one, no handling costs and finite horizon, which

during a two-month period in 2010.

in retail inventory management. European Journal of Operational Research


J.C.F. Ehrenthal et al. / European Journal of Operational Research xxx (2014) xxx–xxx 3

means their work does not apply to the fast-moving consumergoods retail industry. In contrast, we consider an infinite-horizonaverage cost, lost-sales inventory model, with batch sizes and han-dling costs.

Our study contributes to the literature in several ways. Our firstand main contribution is that we give insights on the tradeoffbetween the complexity of an ASO system and the cost benefitsof incorporating demand seasonality to various degrees. Takingdemand seasonality into account in an ASO system generallymeans making it more complex and opaque, which can decreasethe store managers’ trust in the system. Interestingly we show thatincorporating more of demand seasonality is not always beneficial.It is therefore important that retailers carefully analyze thedemand patterns of their products before considering an upgradeof their ASO system. Our analysis can be used to provide motiva-tion for improving an ASO system and provides suggestions onhow to proceed. Our second contribution is theoretical: we are,to the best of our knowledge, the first authors to consider theimpact of ignoring demand seasonality in a model tailored forretailing, i.e., a model with lost sales, lead times, batch sizes, andhandling costs. In our analysis we calculate the optimal policy foreach possible demand setting, using dynamic programming andshow that it does not have a simple structure. Our third contribu-tion is empirical: we include a study based on real-life sales dataand illustrate the magnitude of the potential cost savings fromincorporating demand seasonality.

The remainder of this paper is organized as follows. We providea review of relevant papers in Section 2. Our model is presented inSection 3. In Section 4, we study the problem in four different set-tings wherein the retailer accounts for seasonality to variousdegrees. Section 5 applies the model to a real-life situation. Sec-tion 6 presents the conclusions and points towards furtherresearch.

1 modðx; yÞ is the remainder of the division of x by y. For exampled9 ¼ modð9� 1;7Þ þ 1 ¼ 2.

2. Literature review

In contrast to inventory models with backordering, whereunmet demand is satisfied at a later time, lost-sales inventorymodels assume that customers are unwilling to wait for theirorder. Such behavior is frequent for retail out-of-stocks (Campo,Gijsbrechts, & Nisol, 2000). Lost-sales models penalize unmetdemand to capture the negative effects of out-of-stocks. Theseinclude diminished customer loyalty and increased in-store logis-tics costs through staff attending shoppers and looking for out-of-stock items in back rooms (Gruen, Corsten, & Bharadwaj, 2002).

Because lost sales models are more difficult to analyze mathe-matically than backorder models, they have received much lessattention in the inventory literature. With the exception of the caseof zero lead time, for which it can be shown that (state-dependent)ðR; s; SÞ policies are optimal (see e.g., Cheng & Sethi, 1999; Veinott &Wagner, 1965), there is no simple optimal replenishment policy inlost sales inventory systems. As Bijvanc and Vis (2011) write intheir comprehensive review on lost sales inventory modeling,‘‘not much is known about an optimal replenishment policy whenexcess demand is lost’’. Therefore, the focus of most papers is onstudying the performance of practically motivated policies (seee.g., Hill & Johansen, 2006; Johansen & Hill, 2000), obtaining struc-tural properties of the optimal policy (see e.g., Huh & Janakiraman,2010; Zipkin, 2008) or deriving bounds (e.g., Karlin & Scarf, 1958,chap. 10; Morton, 1969). Most of the methods employed in thesepapers also apply to the case of non-stationary demand.

A recent development related to our research is the consider-ation of fractional lead time, i.e., when the lead time is shorter thanthe review period. Kapalka, Katircioglu, and Puterman (1999) studythe steady-state behavior of the on-hand inventory when the


objective is to minimize the long-run average cost subject to a fillrate constraint. Janakiraman and Muckstadt (2004) analyze prop-erties of the optimal policy and compare it to the best order-up-to policy. Chiang (2006) uses dynamic programming to analyzesuch a model with no fixed costs. Sezen (2006) uses simulationprogramming to study the impact of review period length on theaverage on hand inventory level and fill rate.

Also recently, handling costs have been added to lost-salesinventory modeling. Curs�eu, Van Woensel, Fransoo, and Erkip(2011) include store handling cost into the single-item inventoryproblem in a retail setting. This reflects Van Zelst, van Donselaar,van Woensel, Broekmeulen, and Fransoo (2009)’s finding that itemhandling (e.g. receipt of goods, shelf replenishment) accounts for75% of store logistics costs, while inventory accounts for theremaining 25% (see also Saghir & Jönson, 2001). Curs�eu et al.(2011) examine the optimal policy under the long-run average costcriterion using parameter values typical for grocery retailing. Theyshow that retailers can achieve substantial cost reductions byincluding handling costs in their inventory decisions.

Studies which look specifically at the issue of non-stationarityin demand are scarce (Tunc et al., 2011) and most focus on the casewhen unsatisfied demand is backordered. Karlin (1960a, 1960b)studies the discounted version of the non-stationary demand prob-lem and gives an algorithm to calculate the optimal policy. Zipkin(1989) proposes an alternative policy for the average-cost problem.Johnson and Thompson (1975) proves the optimality of myopicinventory policies for certain dependent demand processes.Morton (1978) obtains bounds on the infinite horizon problemby allowing disposal in each period. Song and Zipkin (1993) obtainsome monotonicity properties for the optimal policy in a settingwhere demand rates vary with an underlying state-of-the-worldvariables. Tarlim and Kingsman (2006) propose heuristic methodsfor calculating near optimal ðs; SÞ policies with non-stationarydemand while Silver (1981) and Morton and Pentico (1995) focuson alternative inventory control policies.

3. Model

We consider a periodic review inventory replenishment systemwith lost sales, batch ordering and inventory handling costs, simi-lar to that of Curs�eu et al. (2011). The review cycle length R, i.e. thetime between when two successive orders are placed, is exogenousto the model and set as the time unit. The lead time L is fixed andshorter than the review period length (i.e., we have fractional leadtimes). Hence, each review period is divided into two parts: a per-iod of length L before order is received, and a period of length R� Lafter the order is received and before the next order is placed.Demands in these two parts are independent discrete random vari-ables which exhibit a seasonal pattern with a fixed period of �d, thatis, the demand during the lead time in review period t has the samedistribution as demand during the lead time in review period t þ �d.Similarly, demand after the order is received in review period t hasthe same distribution as demand after the order is received inreview period t þ �d. Let Dd;L and Dd;R�L denote the demand in thed-th review period within a season respectively before and afterthe order is received, for d ¼ 1; . . . ; �d. Without loss of generality,we assume that review period 1 is the start of a new season so thatthe season index associated with review period t is a value between1 and �d that indicates which period within the season review per-iod t corresponds to: dt ¼ modðt � 1; �dÞ þ 1.1 Table 1 gives the nota-tion used in this paper.



Table 1Notation.

R Review period

L Lead time (0 6 L 6 R)t Period indexdt Season index for period t�d period (i.e., length of a season)It Inventory on hand at the beginning of period tat Quantity ordered in period tDd;L Demand during lead time in d-th review period within a seasonDd;R�L Demand after the receipt of the order in d-th review period within a

seasonkd;L expected value of Dd;L

kd;R�L expected value of Dd;R�L

q Fixed batch sizeK Fixed cost per orderK1 Handling cost per batchK2 Handling cost per unith Holding cost per unit of inventoryp Penalty cost for each unit of sales lost during a perioddA Across-review periods demand slopedW Within-review period demand slope

2 We use the following notation: W when the retailer ignores within-review periodvariations and W when the retailer accounts for within-review period variations.Similarly A when the retailer ignores across-review periods variations and A when theretailer accounts for across-review periods variations.


The timing of events within a review period is as follows. (i) Atthe start of each review period, the on-hand inventory It and sea-son index dt are observed and an order at is placed. The order at

is a non-negative integer multiple of a fixed batch size q, i.e.at 2 f0; q;2q; . . .g. The replenishment cost incurred by the retailerincludes a fixed component (ordering cost), a variable componentwhich is proportional to the number of batches (handling cost asexplained in Curs�eu, Van Woensel, Fransoo, van Donselaar, &Broekmeulen (2009)) and a variable component proportional tothe number of units (purchasing cost). (ii) The lead time demandDdt ;L is realized and is satisfied, if possible, from on-hand inventory.All demand not immediately satisfied is lost and a penalty cost isincurred. (iii) The order placed at the start of the period arrives Ltime units into the same review cycle. (iv) The demand Ddt ;R�L

occurs and, again, all demand not directly satisfied from on-handinventory is lost and a penalty cost is incurred. (v) The on-handinventory level at the end of the review period is therefore equalto Itþ1 ¼ ððIt � Ddt ;LÞ

þ þ at � Ddt ;R�LÞþ

where ðxÞþ ¼maxf0; xg forany x 2 R. If this quantity is positive, a holding cost is incurred.

The retailer’s objective is to find the ordering policy which min-imizes the long-run average cost. We formulate this problem as aMarkov decision process. The season index and the inventory on-hand at the beginning of a review period, i.e., ðdt ; ItÞ, characterizethe system state so the state space is S ¼ fð1;0Þ; ð1;1Þ; . . . ;

ð2; 0Þ; ð2;1Þ; . . . ; ð�d;0Þ; ð�d;1Þ; . . .g. The actions correspond to theorder quantities so the action space A ¼ f0; q;2q; . . .g, for every

state in S. Let pd;i;jðaÞ ¼ P j ¼ ðði� Dd;LÞþ þ a� Dd;R�LÞþ� �

denote

the probability of going from state ðd; iÞ 2 S to stateðmodðd; �dÞ þ 1; jÞ 2 S, when the order quantity is a 2 A. Theper-period expected cost in state ðd; iÞ 2 S given action a ¼ nq, isdefined as:

cd;iðaÞ ¼ K1fa>0g þ K1nþ K2nqþ hE ðði� Dd;LÞþ þ a� Dd;R�LÞþh i

þ p E ðDd;L � iÞþ� �

þ E ðDd;R�L � a� ði� Dd;LÞþÞþh in o

where 1fEg is the indicator function for event E;K is the orderingcost, K1 is the handling cost, K2 is the purchasing cost, h is the hold-ing cost and p is the penalty cost for unmet demand.

The inventory policy which minimizes the long-run averagecost can be obtained using value iteration (see for examplePuterman, 2005).


4. Accounting for demand seasonality

For notational convenience, we assume from now on that Dd;L

and Dd;R�L have a Poisson distribution with mean kd;L and kd;R�L

respectively, for d ¼ 1; . . . ; �d. Our model can also be implementedwith any discrete distribution.

If a retailer has a sophisticated ASO which allows him to createa separate bookkeeping of demand for each period within the sea-son before and after the order is received, we assume that thisretailer will be able to calculate the true distribution of Dd;L andDd;R�L for d ¼ 1; . . . ; �d. This retailer is taking into account both thewithin-review-period variations and the across-review-periodsvariations and calculates 2�d inputs variables in his ASO. We referto this setting as the WA setting, where W stands for Within andA for Across. Let kWA be the vector that summarizes the demand dis-tribution parameters used by the retailer when calculating itsinventory policy:

kWA ¼k1;L k2;L . . . k�d;L

k1;R�L k2;R�L . . . k�d;R�L

!ð1Þ

Now assume the retailer keeps a record of the demand heobserves for each period of a season but, in its bookkeeping, doesnot distinguish between before and after the order is received.The retailer uses only �d input variables in its ASO system: one foreach review period in a season. This retailer (correctly) finds thatdemand in the d-th review period within a season has a Poissondistribution with mean kd;L þ kd;R�L for d ¼ 1; . . . ; �d. He then (incor-rectly) assumes that demand is evenly distributed during thereview period, that is, he finds that Dd;L has a Poisson distributionwith mean L

R ðkd;L þ kd;R�LÞ and Dd;R�L has a Poisson distribution withmean R�L

R ðkd;L þ kd;R�LÞ for d ¼ 1; . . . ; �d. We refer to this case as theWA setting since it ignores within-review period variations butstill acknowledges across-review periods variations.2 Let kWA bethe vector that summarizes the demand distribution parametersused by the retailer when calculating its inventory policy:

kWA ¼LR ðk1;L þ k1;R�LÞ L

R ðk2;L þ k2;R�LÞ . . . LR ðk�d;L þ k�d;R�LÞ

R�LR ðk1;L þ k1;R�LÞ R�L

R ðk2;L þ k2;R�LÞ . . . R�LR ðk�d;L þ k�d;R�LÞ

!

Next assume the retailer acknowledges that the demand distri-bution before and after receiving the order in a review period is dif-ferent, but does not recognize the seasonality across-reviewperiods. In this case, the retailer uses only two input variables inits ASO system: one for each part of the review period (beforeand after the order is shelved). This retailer (incorrectly) finds that

Dd;L has a Poisson distribution with mean 1�d

P�dd¼1kd;L and Dd;R�L has a

Poisson distribution with mean 1�d

P�dd¼1kd;R�L for d ¼ 1; . . . ; �d. We

refer to this case as the WA setting, since the retailer seeswithin-review-period variations but does not account for across-review-periods variations. Let kWA be the vector that summarizesthe demand distribution parameters used by the retailer when cal-culating its inventory policy:

kWA ¼

1�d

X�d

d¼1

kd;L1�d

X�d

d¼1

kd;L . . . 1�d

X�d

d¼1

kd;L

1�d

X�d

d¼1

kd;R�L1�d

X�d

d¼1

kd;R�L . . . 1�d

X�d

d¼1

kd;R�L

0BBBBB@

1CCCCCA



Fig. 2. Illustration of the k vector in all four settings.


Finally assume the retailer neither accounts for within-review-period nor across-review-periods variations. In this case, the retai-ler uses only one input variable in its ASO system: the totaldemand over one review period. This retailer (incorrectly) findsthat this demand has a Poisson distribution with mean1�d

P�dd¼1ðkd;L þ kd;R�LÞ. He then assumes that demand is evenly dis-

tributed within each review period, that is, he finds that Dd;L has

a Poisson distribution with mean LR

1�d

P�dd¼1ðkd;L þ kd;R�LÞ and Dd;R�L

has a Poisson distribution with mean R�LR

1�d

P�dd¼1ðkd;L þ kd;R�LÞ. We

refer to this case as the WA setting since the retailer ignores bothwithin-review-period and across-review-periods variations. LetkWA be the vector that summarizes the demand distributionparameters used by the retailer when calculating its inventorypolicy:

kWA ¼

LR

1�d

X�d

d¼1

ðkd;L þ kd;R�LÞ LR

1�d

X�d

d¼1

ðkd;L þ kd;R�LÞ . . . LR

1�d

X�d

d¼1

ðkd;L þ kd;RÞ

R�LR

1�d

X�d

d¼1

ðkd;L þ kd;R�LÞ R�LR

1�d

X�d

d¼1

ðkd;L þ kd;R�LÞ . . . R�LR

1�d

X�d

d¼1

ðkd;L þ kd;R�LÞ

0BBBBB@

1CCCCCA

Example 1 illustrates the calculation of the k vectors.

Example 1. Let �d ¼ 7 and

3 The dynamic program can be simplified in the WA and WA cases because theretailer does not account for across-review-periods variations so it suffices to use It asa state variable.

kWA ¼0:8 1:4 1:8 2 2 1:8 1:40:2 0:6 1:3 2 3 4:2 5:6

� �

Fig. 2 illustrates the demand vectors as used in the four settingsWA;WA;WA, and WA when R ¼ 1 and L ¼ 0:5. By looking at theWA setting, we see that the true demand pattern is such thataverage demand rate and the proportion of sales which take placeduring the lead time both increase during the season. In the WA


setting, the retailer recognizes the differences between reviewperiods but assumes that demand is evenly spread during eachreview period. In the WA and WA settings, the average demandrate is the same for each review period, however, in the WA set-ting, the retailer understands that there is on average moredemand after the order is received, while in the WA setting heassumes that the demand rate is the same before and after theorder is received.

In all four settings, we solve for the optimal inventory policygiven the corresponding k vector using the method described inSection 3.3 Let UWA;UWA;UWA and UWA denote the inventory policy

in the WA;WA;WA and WA settings respectively.We calculate the long-run average cost of each policy using the

probability transition matrix and cost vector calculated based onkWA in (1), which embraces within- and across-periods demand

variations. Then we calculate the performance gap of the three pol-icies which ignore part of the seasonality with respect to the long-run average cost in the WA policy:

100� CðUÞ � CðUWAÞCðUWAÞ

where CðUÞ denotes the long-run average cost of policy U, forU ¼ UWA;UWA and UWA.



Fig. 3. Example of the optimal policy in all four settings.


4.1. Comparison of the policies

In this section, we compare the structure of the UWA;UWA;UWA

and UWA inventory policies. Fig. 3 shows the optimal order quanti-ties as a function of on-hand inventory for all four settings whenK ¼ K1 ¼ K2 ¼ 0; p ¼ 50; �k ¼ 5, and �d ¼ 7 and the following kWA:

kWA ¼6:79 1:21 2:14 3:07 4:00 4:93 5:860:30 0:54 0:77 1:00 1:23 1:46 1:70

� �

Note that in the kWA vector above, demand increases across-review periods within a season. In particular, more demand needsto be satisfied with the order placed in review period dþ 1 com-pared to d for d 2 f1; . . . ;6g.

From Fig. 3, we see that the structure of ordering policies differfrom an ðs; S;nqÞ policy as for low levels of inventory on-hand, theadvice is to order the same quantity. We also see that the orderquantity seems to be monotone in the inventory on-hand in allfour settings. This is true in most cases but not always, as also


noted by Curs�eu et al. (2011), Hill and Johansen (2006), andJansen (1998).

Further, we see that the inventory policy in the WA and WAsettings is the same for each period within the season while ittends to increase with the season index in the other two settings.This is because the retailer ignores across-review-periods varia-tions in the WA and WA settings and therefore treats every reviewperiod as identical. This means to the retailer in the WA and WAorders too much at the start of the season when demand is lowand too little at the end of the season when demand is high. Fur-ther, a pairwise comparison of the policies in the WA and WA set-tings and the policies in the WA and WA settings reveals thatignoring within review-period variations leads to lower-than-opti-mal order quantities. This is because most of the demand occursduring the lead time, therefore the retailer tends to underestimatedemand uncertainty when it assumes that demand is evenly dis-tributed during the review period as is the case in the WA andWA settings.



Fig. 4. Results of the sensitivity analysis.


4.2. The value of accounting for demand seasonality

In this section, we study the ordering policies in the WA;WAand WA settings by varying the model parameters. For the costparameters, we use K ¼ f0;10;20g;K1 ¼ f0;5;10g;K2 ¼ f0;1;2g;p ¼ f25;50;75g and h ¼ 1. The value of the cost parameters arethus set relative to the unit holding cost. For the case pack size,we use q 2 f1;3;6;12g. We vary the kWA by setting �d ¼ 7 and usingthe following formulas for d 2 f1; . . . ;7g:

kd;L ¼ð1� dWÞ kþ 3dA

� �for d ¼ 1

ð1� dWÞ kþ ðd� 5ÞdA� �

for d ¼ 2; . . . ;7

(ð2Þ

kd;R�L ¼ dW kþ ðd� 4ÞdA� �

ð3Þ

In these formulas k is the average demand per review period, dW

is the within-review period demand slope, i.e., a parameter thatallows for the proportion of demand within a review period to varyin a systematic way and dA is the across-review periods demandslope, i.e., a parameter that allows the demand in each periodwithin a season to vary in a systematic way. Increasing dW leadsto more demand after the lead time (within each review period)compared to during the lead time. Increasing dA leads to moredemand at the end rather than the start of the season. Our choiceof values for kd;L and kd;R�L implies that more demand needs to besatisfied with the order placed in review period dþ 1 comparedto d for d 2 f1; . . . ;6g.4 Note that in order for all the mean demand

values to be positive, we need dA 6k3. We use

k ¼ f0:1;1;5;10g; dW 2 f0;0:1; . . . ;0:9;1g and dA 2 0; k30 ;

2k30 ; . . . ; k

3

n o.

4 The order placed at the start of review period d has to cover the R� L portion ofreview period d plus the L portion of review period dþ 1. Our choice of values for kd;L

and kd;R�L imply that kd;R�L þ kdþ1;L is increasing in d.


Based on these parameters, we generated 17,424 problem instancesin our sensitivity analysis, resulting in 69,696 policy and averagecost calculations. The base case we use has K ¼ K1 ¼ K2 ¼ 0;p ¼ 50; q ¼ 1; k ¼ 5; dW ¼ 0:5 and dA ¼ 0.

Fig. 4 shows how the average optimality gaps of the three pol-icies which ignore part or all demand seasonality vary with eachparameter. In most cases, the average optimality gap of the WApolicy is the largest, followed by the WA policy, and then by theWA policy. This means that using more input variables in theASO system generally leads to better decisions and lower averagecost. However, this is not always the case. When dA is low, so thereis little variation across periods in a season, the optimality gap ofthe WA policy can be higher than that of the WA policy. This isbecause there is more to gain by realizing that demand is notevenly spread across the review period. Also, when dW is high, sothat little demand occurs during the lead time, the retailer benefitsfrom ignoring both types of variations rather than accounting forwithin-review-period variations only. In other words, less informa-tion is better! This is because the retailer who ignores both types ofseasonality benefits from two effects running in opposite direc-tions. Not seeing that most of the demand occurs after the leadtime results in higher than optimal order quantities. Not seeingthat most of the demand occurs at the end of the season leads tolower than optimal order quantities in periods with higher seasonindices. Combined, the two effects partially offset each other. Incontrast, the retailer who only ignores across-review periods vari-ations but accounts for within-periods variations will order too lit-tle at the start of the season, resulting in higher stock-out penalties.Though this counter-intuitive result is a byproduct of our mathe-matical model, it nonetheless illustrates the fact that retailersshould use caution when incorporating demand seasonality intotheir ASO. It is important to do a careful analysis and understandthat some forms of seasonality are more important to incorporate




than others. The next section shows how to perform such an anal-ysis in practice.

Fig. 4 shows that the optimality gaps decrease in K;K1;K2 and q.This can be explained by the fact that larger fixed costs (per orderor per batch) and larger case pack sizes lead to larger optimal orderquantities. As a result, it becomes less important to observe across-and within-review periods variations because the inventoryordered is used to cover multiple periods of demand. Moreover,the optimality gaps of the three policies are increasing in the pen-alty cost p. This is because a higher penalty cost makes the under-stocking associated with ignoring seasonality more costly. Further,an increase in the average demand k leads to an increase in opti-mality gaps since more demand to satisfy means more opportunityfor errors when seasonality is ignored.

An increase in dA, which has the effect of shifting more of thedemand to the end of the season, leads to worse performance ofthe policies in the WA and WA settings. This is in line with expec-tations, since the retailer ignores across-review periods variationsin these settings. Varying dW away from 0.5 means that thedemand within a review period is distributed less evenly betweenthe two sub periods (before and after the order is received). As aresult, the WA policy, which assumes demand is evenly distributedwithin the review period, performs worse. The impact on the opti-mality gaps of the WA setting is more difficult to analyze becauseof the two opposite effects offsetting each other as discussedabove.

5 Specifically, we talked to the store head manager, replenishment and checkoutstaff, the corporate heads of logistics and sales as well as the forecasting and ASOteam in the retailer’s headquarters.

6 Utilitarian items are primarily instrumental and functional, whereas hedonicitems provide more experiential value, fun, excitement, and pleasure (see Sloot,Verhoef, & Franses, 2005).

5. Application to a real-life setting

In this section, we evaluate the value of accounting for demandseasonality using item-store level POS data collected at a Europeanretailer for a single store in the entire year of 2010. The store stocksabout 10.000 items (i.e. stock-keeping units) and is located at amajor intersection, with two competitors’ stores nearby. It is open7 days a week from 6 am to 10 pm.

The retailer records sales by the second and stores this data bythe hour. So it is possible to observe monthly, weekly, daily andintraday demand variation. Following Van Donselaar, Gaur, vanWoensel, Broekmeulen, and Fransoo (2010), we conducted severalobservations on replenishment (after shelving) and on-shelf avail-ability (before store closing) to ensure that tasks are executed asplanned and sales data are not distorted by out-of-stocks. We didnot find frequent or systematic out-of-stocks and therefore assumehistorical sales data to be a reasonable representation of shopperdemand. In what follows we use the terms sales and demandinterchangeably.

We analyze sales data at the item-level and find that demandduring the week follows specific patterns. These patterns varyamong items and are distinct from store sales patterns (see alsothe example of cigarette sales in Section 1, Fig. 1). We focus ouranalysis on a sample of 10 items from 10 different product catego-ries. All of these items are included in the ASO system.

For each of the items, shelf replenishment is structured as fol-lows. Every evening at 5 pm, the ASO system observes on-handinventory and calculates the order quantities. The store managereither approves or overwrites the orders and sends a confirmationto the distribution center. All orders are processed in the sameregional distribution center which the retailer owns and operates.There are no direct deliveries from suppliers. The order placedevery evening at 5 pm arrives by truck from the distribution centerthe next morning. Until 10am, store staff shelves all items accord-ing to the planograms. The replenishment staff is well trained inshelf replenishment. The store has a small backroom for organizingitems for shelf replenishment, e.g. to dispose of excess packaging,and to organize returns of roll cages and recycling materials. Items


left over from shelf replenishment (one jar of jam, two deodorants,etc.) are organized in backroom bins until the next scheduled shelfreplenishment.

Based on our discussion with the store management,5 it appearsthere is no systematic practice of overwriting the ASO’s order quan-tities. In particular, we believe that the store manager does notadvance orders to balance workload as discussed in Van Donselaaret al. (2010). Overwriting the ASO occurs on rare occasions whenstore manager expects demand to deviate from the order suggestionbased on her personal experience.

The ASO system generates orders in three steps. First, the sys-tem calculates an average of the past 52 weeks of sales data,excluding promotions and special events, for example the nationalday. Second, it divides the resulting weekly demand estimate byseven in order to obtain the daily demand estimate. Third, the sys-tem compares the demand estimate to on-hand inventory and cal-culates the daily order quantities according to a formulaaccounting for case pack size and shelf space. It follows that thecurrent ASO system neither accounts for intra-week sales varia-tions nor intra-day variations. So for example, it ignores if an itemtypically sells more on Saturday than on Monday. Also, the ASOsystem ignores if an item sells more at night (after the 5 pm orderplacement) than before lunch (after the 10am replenishment of thelast order).

The team in charge of ASO at this retailer is thinking of incorpo-rating intra-week sales variations to a certain degree by calculatingtwo daily demand estimates: one for weekday sales and one forweekend sales. However, they are unclear about whether to treatFriday sales as weekday or weekend.

5.1. Modeling

As mentioned earlier, the 10 items we study can be orderedevery day at 5 pm and the store is open every day from 6 am to10 pm. Hence, we use R ¼ 16 hours = 1 (business) day for thelength of the review period. Given that the new order is on theshelves by 10am, we use (10 � 5) + (10 � 6) = 9 hours = 9/16 dayfor our estimate of the lead time L. Therefore, in this section,within-review-period variations mean intra-day variations whileacross-review periods correspond to across-days variations. Notethat a review period overlaps with two days of the week but forsimplicity we call each review period by the day of the week wherethere is the most overlap. Also let day 1 in a week be Sunday, day 2be Monday, etc.

For each item, the sales data exhibit a clear seasonality patternwith a period of one week, i.e., �d ¼ 7 days. Fig. 1 in Section 1 givesan example of the observed seasonality patterns. As in Section 3 wemodel the demand using a Poisson distribution, which proved tobe a good fit for our data set.

The 10 items we selected cover a broad range of sales velocity,weekly and daily sales patterns, case-packs sizes, and other attri-butes, including global brands, national brands, regional brands,and private labels; hedonic and utilitarian items; items for imme-diate consumption and home storage; as well as varying degrees ofshelf life. This product selection approach is common in on-shelfavailability research (see, for instance, Campo et al., 2000; Gruen& Corsten, 2008).6 Weeks during which there was a promotion ora special event (specifically Easter, Christmas, and the nationalday) were ignored in our analysis.



Table 2Information on the 10 selected items.

Item Average demand Avg. wk. demand Case pack size VW VA

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Energy drink L 9.58 8.69 8.5 8.09 9.28 10.19 8.11 126.52 24 0.06 0.22(2.5 dl can) R–L 16.75 7.36 6.24 7.13 7.4 7.25 11.95Caffeinated soda L 6.8 1.83 2.78 4.36 4.2 3.27 4.82 58.83 4 0.08 0.42(1.5 l bottle) R–L 6.78 1.97 2.89 3.78 4.4 3.49 7.46Milk L 2.86 1.59 3.51 2.95 3.21 3.33 3.12 51.58 6 0.16 0.16(1 l tetra pack) R–L 3.57 4.19 4.33 4.41 3.84 4.26 6.41Croissants L 1.07 2.51 2.32 2.44 2.60 2.82 1.93 30.68 1 0.07 0.06(single bun) R–L 3.33 1.80 1.90 1.97 1.85 2.09 2.05Lettuce L 1.27 1.50 1.46 1.62 1.53 1.21 3.28 26.86 1 0.15 0.19(single head) R–L 3.11 1.93 1.87 2.05 2.01 2.03 2.01Cigarettes L 1.24 1.57 1.68 1.57 1.67 1.61 1.83 22.36 10 0.06 0.21(single box of 10) R–L 3.39 1.32 1.09 1.25 1.13 1.40 1.61Sausage L 0.73 0.32 0.64 0.71 0.77 0.73 1.08 14.67 1 0.20 0.45(double pack) R–L 0.67 0.54 1.19 1.44 1.48 1.58 2.79Potato chips L 1.71 0.64 0.56 0.54 0.59 0.7 1.02 11.17 4 0.06 0.58(300gr pack) R–L 1.6 0.5 0.25 0.54 0.46 0.62 1.44Eggs L 0.34 0.24 0.37 0.29 0.42 0.37 0.34 5.30 1 0.12 0.17(pack of six) R–L 0.57 0.34 0.33 0.37 0.29 0.47 0.56Orange juice L 0.4 0.22 0.26 0.24 0.23 0.19 0.26 4.82 1 0.18 0.29(4x1l tetra pack) R–L 0.59 0.17 0.44 0.52 0.44 0.3 0.56

7 For instance, Zipkin (2000) notes this difficulty as an obstacle for practicalimplementation. Only partial solution approaches to this problem exist in research,such as observational studies on store handling cost (e.g. Curs�eu et al., 2009) andestimations on stockout penalty costs (e.g. Anderson, Fitzsimons, & Simester, 2011;Olivares, Musalem, Bradlow, Terwiesch, & Corsten, 2011).


Table 2 gives the average sales values during the lead time (L),after the shelving of the new order (R� L) for each day of the weekas well as average weekly demand and the corresponding casepack size.

We also include two measures of sales variability: VW and VA

which respectively capture the extent of within-review periodsvariations and across-review periods variations. They are calcu-lated as follows:

VW ¼17

X7

d¼1

916� kd;L

kd;L þ kd;R�L

��

VA ¼ CV kd;L þ kd;R�L

d¼1;...;7

where 9=16 is the percentage of time that corresponds to the leadtime within a review period and CV stands for coefficient ofvariation.

Since the current replenishment policy in the retailer’s ASO sys-tem neither accounts for intra-day nor across-days variations, wechoose to model it using the WA policy introduced in Section 4.Note that this is not an exact representation of the retailer’s ASOsystem for two main reasons: (1) the policy used by the ASO sys-tem is a simpler policy which resembles an ðs; S;nqÞ policy (2)the order quantity chosen by the ASO system is not explicitly basedon a minimization of the costs involved. In terms of costs, the WApolicy is therefore a lower bound on the performance of the policyused by the ASO system, which implies that our results on theimpact of incorporating seasonality are on the conservative side.

We compare this policy to the WA policy which takes intoaccount both within-day and across-days variations, the WA policywhich incorporates across-days but not within-day variations andthe WA policy which considers within-day but not across-daysvariations. We also construct six new settings in which the retailergroups the days of the week into weekdays versus weekend daysand calculates a separate average daily demand for each group(as mentioned before, the retailer we spoke to considered imple-menting a similar idea). The six settings differ with respect to thedefinition of weekend (either Friday–Saturday, Saturday–Sundayor Friday–Saturday–Sunday) and whether or not they account forintra-day variations. In settings where the retailer does not accountfor intra-day variations, he assumes that the demand is evenly dis-tributed throughout the review period. In his book-keeping, he


keeps track of only two values of demand: the daily weekdaydemand and daily weekend demand. We refer to these cases asthe WFS;WSS and WFSS settings when the weekend is definedrespectively as Friday–Saturday, Saturday–Sunday and Friday–Sat-urday–Sunday. In settings where the retailer accounts for intra-dayvariations, he keeps track of four quantities: weekday demandbefore the leadtime, weekday demand after the leadtime, weekenddemand before the leadtime and weekend demand after the lead-time. We refer to these as the WFS;WSS and WFSS settings whenthe weekend is defined respectively as Friday–Saturday, Satur-day–Sunday and Friday–Saturday–Sunday. The calculation of thecorresponding k vectors is given in the Appendix A.

Calculating the minimum long-run average cost under eachsetting requires an estimate of the cost parameters, which inpractice is a very difficult task.7 For this reason, we decided to varythe cost parameters in a way similar to the approach in Section 4.2.We use p 2 f25;50;75g;K 2 f0;5;10;15;20;25g;K1 2 f0;5;10g;K2 2f0;1;2g;h ¼ 1, leading to 162 problem instances per item. This way,we are able to calculate the value of incorporating seasonality undera wide range of cost treatments. The value of the case pack size qwas set equal to the actual value observed in practice for each item(see Table 2, third column from right).

5.2. Results

Fig. 5 gives the results of our analysis. For each item, we reportthe minimum, average, and maximum optimality gaps for the 9sub-optimal policies compared to the performance of the WA pol-icy across all 162 problem instances.

First we see that the optimality gaps of the WA policy, which isa conservative estimate of current practice by the retailer, can besubstantial. The maximum value we encountered in our numericalanalysis was equal to 66.27%. This means that the retailer’s costscan be 66.27% higher than optimal when he ignores all forms ofdemand seasonality. The average optimality gaps across allproduct categories was 4.91%, with the greatest average values



Fig. 5. Average optimality gaps of the 9 sub-optimal policies for the 10 items.

Table 3Average optimality gaps and number of input variations in the ASO system for the 9 settings.

Setting WA WA WA WSS WFS WFSS WSS WFS WFSS

Avg opt. gap (%) 4.91 5.46 0.86 1.64 4.19 2.36 1.34 4.14 2.19Nb input var. 1 2 7 2 2 2 4 4 4


achieved for the caffeinated soda (16.10%), the potato chips(12.98%) and the energy drink (6.81%). Not surprisingly categorieswith more variable demands, as measured by VW and VA, lead thehighest optimality gaps but sales velocity and case pack size alsoseem to have an impact, which confirms our findings fromSection 4.2.

For all 10 items, switching from the WA to the WA policy, that isaccounting for across-days seasonality, would lead to a decrease inthe average optimality gaps. This is especially notable with theenergy drink (from 6.81% to 1.48%), the caffeinated soda (from16.10% to 0.87%), the sausage (from 4.83% to 1.12%) and the potatochips (from 12.98% to 0.29%). This means that the retailer has a lotto gain from incorporating across-days variations. In fact doing soalone brings the retailer within 1.85% of optimality for all itemsand within 1% for seven of them.

We also see that a significant reduction in costs can be achievedby simply distinguishing between weekdays and weekend days:for example from 12.98% to 0.62% for potato chips. In some caseshowever, it is important to use the appropriate definition for the


weekend. For example, the weekend should clearly be defined asSaturday–Sunday for the energy drink, caffeinated soda, croissantand potato chips items, as Friday–Saturday for sausage and asFriday–Saturday–Sunday for lettuce. Not defining the weekendproperly can lead to worse performance than the WA policy (e.g.,for the energy drink, croissants, cigarettes and orange juice). Incor-porating intra-day variations generally leads to a further reductionin costs but this improvement is not as important compared to thecost decrease that comes with the weekday-weekend distinction.

Interestingly, the connection between the number of input vari-ables included in the ASO system and the optimality gaps is notstraightforward: increasing the complexity of the ASO by addingmore variables does not necessarily mean greater cost savings. Thiscan be seen in Table 3. For example, switching from the WA policyto the WA policy does not always lead to an reduction in the aver-age optimality gaps. In particular, the retailer would be worse off ifit was to incorporate intra-day variations without taking intoaccount across-days variations for the energy drink, caffeinatedsoda, sausage, potato chips and cigarettes. This is another example




of ‘less information is better’ as discussed in Section 4.1. Also wesee that incorporating only two input variables in the ASO systemusing the appropriate definition of the weekend can do better thanusing four input variables and an inadequate definition of theweekend. This suggests that there is an important tradeoffbetween ASO complexity and the benefits of incorporating demandseasonality, which has not received research attention previously.

In summary, we found that ignoring seasonal variations canlead to an increase in costs (on average 4.91%). The greatest benefitcomes from incorporating across-days variations and much of thisbenefit can be captured with a distinction between weekday andweekend sales, provided the weekend is defined appropriatelyfor each product category. In some cases, capturing intra-day salesvariations without accounting for across-days differences can leadto an increase in costs, showing that the retailer should carefullyanalyze demand seasonality before deciding how to incorporateit into his ASO.

6. Conclusions and future research

The aim of our study was to provide insights on the impact ofincorporating demand seasonality in a retail replenishment sys-tem. Our motivation stems from the observation that many retail-ers do not (fully) consider seasonality in practice because of a lackof technical capabilities of their ASO system or a desire to reducecomplexity.

Our paper yields some important insights for retail manage-ment. By accounting for non-stationary demand in inventory man-agement, retailers can reduce inventory holding, handling andstock-out cost substantially. Cost savings are higher for fast-mov-ing items with high demand variability across-review periodsand low case pack size. Yet, in some cases, accounting for moredemand seasonality can lead to an increase in costs, which sug-gests that the upgrade in the ASO system should be consideredcarefully. Our numerical analysis with real life data indicatespotential cost savings that are considerable for the retailing indus-try with its tight margins and rigorous focus on cost-efficiency (anaverage reduction of 4.91% in optimality gaps). We show that it ismost important to incorporate across-days variations and thatlarge cost savings can be already achieved with a simpledistinction between weekday and weekend sales, provided thatthe weekend is defined appropriately for each item (the optimalitygap goes down to an average of 1.64%). Doing so improves thereturn on investment in automated store ordering systems thatare equipped with the capabilities of accounting for non-stationarydemand.

In our study, we did not focus on obtaining structural propertiesof the optimal policy. For the general case with fixed ordering andhandling and fractional lead times, we have shown that the order-ing policy in all four settings does not have a simple structure.However, it may be a special case of our model. We leave this pos-sibility for future research, as this question is of a different scopethan the current paper.

As our model is not specific to a particular demand pattern, itmay also be employed in ultra-high selling retail outlets with mul-tiple deliveries per day, for instance. The smaller the review period,the more accurate the information for shelf replenishment.

kWðFSÞ ¼deke ddkd ddkd ddkd ddkd

ð1� deÞke ð1� ddÞkd ð1� ddÞkd ð1� ddÞkd ð1� ddÞkd ð


Moreover, albeit the focus of this research was the retailingindustry, we believe the insights to be of interest to similar set-tings, where product values are low compared to handling costs,demand is non-stationary and non-availability is penalized bycustomers.

In order to limit the complexity of the inventory problem athand, we excluded minimum shelf inventory requirements, shelfspace restrictions (we assume shelf space to be sufficient fordemand during lead time), and perishability concerns in our anal-ysis. Incorporating minimum stock requirements and shelf spacerestrictions into the model may provide interesting insights undernon-stationary demand (see also Curs�eu et al., 2011). Also, weinvestigate a single-item, single-location problem. Future researchcould work towards building multi-item and/or multi-echelonmodels and investigate workload scheduling and balancing in theretail supply chain (such as adjustments to delivery times as wellas seasonal capacity constraints and cost) and at the store level.Lastly, we do not consider promotional events. Future researchcould work towards understanding promotional non-stationarity,e.g. the item-level intraday and across-days effects of various typesof promotions. Combining the insights of promotional demand var-iation with our non-stationary demand inventory model account-ing for ordering and handling costs may provide additionalinsights to retail marketers and logistics managers for increasingthe return on promotions. Finally, another possible avenue forfuture research is the study of easy-to-implement heuristic policieswhich incorporate demand seasonality. For example, it would beinteresting to study how well a ðsd; Sd;nqÞ policy, with d beingthe season index, would perform compared to the optimal policyused in this paper.

Acknowledgments

The authors would like to thank the retailer (who wishes toremain anonymous) who provided the data used in Section 5.

Appendix A

In this Appendix we show how to calculate the demand vector k

for the weekday/weekend settings in Section 5. In the six settingswhich distinguish between weekday and weekend sales, the k vec-tors are calculated as follows:

kWðFSÞ ¼9

16 kd 916 kd 9

16 kd 916 kd 9

16 kd 916 ke 9

16 ke

716 kd 7

16 kd 716 kd 7

16 kd 716 kd 7

16 ke 716 ke

!

where kd ¼ 15

P5d¼1 kd;L þ kd;R�L

and ke ¼ 12


.

kWðSSÞ ¼9

16 ke 916 kd 9

16 kd 916 kd 9

16 kd 916 kd 9

16 ke

716 ke 7

16 kd 716 kd 7

16 kd 716 kd 7

16 kd 716 ke

!

where kd ¼ 15


and ke ¼ 1

2

Pd¼1;7 kd;L þ kd;R�L

.

kWðFSSÞ ¼9

16 ke 916 kd 9

16 kd 916 kd 9

16 kd 916 ke 9

16 ke

716 ke 7

16 kd 716 kd 7

16 kd 716 kd 7

16 ke 716 ke

!

where kd ¼ 14


and ke ¼ 13

Pd¼1;6;7 kd;L þ kd;R�L

.

ddkd deke

1� ddÞkd ð1� deÞke

!




where kd ¼ 15


; ke ¼ 1

2


;

dd ¼P5

d¼1kd;LP5

d¼1kd;Lþkd;R�Lð Þ

and de ¼P7

d¼6kd;LP7


.

kWðSSÞ ¼ddkd ddkd ddkd ddkd ddkd deke deke

ð1� ddÞkd ð1� ddÞkd ð1� ddÞkd ð1� ddÞkd ð1� ddÞkd ð1� deÞke ð1� deÞke

!

where kd ¼ 15


; ke ¼ 12

Pd¼1;7 kd;L þ kd;R�L

;

dd ¼P6

d¼2kd;LP6


and de ¼P

d¼1;7kd;LP

d¼1;7kd;Lþkd;R�Lð Þ.

kWðFSSÞ ¼deke ddkd ddkd ddkd ddkd deke deke

ð1� deÞke ð1� ddÞkd ð1� ddÞkd ð1� ddÞkd ð1� ddÞkd ð1� deÞke ð1� deÞke

!

where kd ¼ 14


; ke ¼ 1

3

Pd¼1;6;7 kd;L þ kd;R�L

;

dd ¼P5

d¼2kd;LP5


and de ¼P

d¼1;6;7kd;LP

d¼1;6;7kd;Lþkd;R�Lð Þ.

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