inventory control in multi-channel retail

European Journal of Operational Research 227 (2013) 101–111

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European Journal of Operational Research

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

Production, Manufacturing and Logistics

Inventory control in multi-channel retail

Frank Schneider a,⇑, Diego Klabjan b

a Supply Chain Management & Management Science, University of Cologne, Cologne, Germanyb Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA

a r t i c l e i n f o

Article history:Received 15 February 2012Accepted 1 December 2012Available online 8 December 2012

Keywords:Inventory controlMulti-channel retailPeriodic review lost salesPoisson demandNon-linear costMultiple demand streams

0377-2217/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2012.12.001

⇑ Corresponding author. Tel.: +49 178 807 8898; faE-mail addresses: [email protected]

(F. Schneider), [email protected] (D. Klabja

a b s t r a c t

In recent years multi-channel retail systems have received increasing interest. Partly due to growingonline business that serves as a second sales channel for many firms, offering channel specific priceshas become a common form of revenue management. We analyze conditions for known inventory controlpolicies to be optimal in presence of two different sales channels. We propose a single item lost salesmodel with a lead time of zero, periodic review and nonlinear non-stationary cost components withoutrationing to realistically represent a typical web-based retail scenario. We analyze three variants of themodel with different arrival processes: demand not following any particular distribution, Poisson distrib-uted demand and a batch arrival process where demand follows a Pòlya frequency type distribution. Weshow that without further assumptions on the arrival process, relatively strict conditions must beimposed on the penalty cost in order to achieve optimality of the base stock policy. We also show thatfor a Poisson arrival process with fixed ordering costs the model with two sales channels can be trans-formed into the well known model with a single channel where mild conditions yield optimality of an(s, S) policy. Conditions for optimality of the base stock and (s, S) policy for the batch arrival process withand without fixed ordering costs, respectively, are presented together with a proof that the batch arrivalprocess provides valid upper and lower bounds for the optimal value function.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

In recent years multi-channel retail systems have experiencedincreasing interest. Partly due to the growing online business thatserves as a second sales channel for many firms, offering a productfor different prices depending on the sales channel has become acommon form of revenue management. Many retailers haveopened an online channel in addition to brick-and-mortar. Anotherrecent example are orders placed online and merchandise pickedup in-store (e.g. Home Depot and Best Buy offer this service to cus-tomers). Additionally, many small businesses use platforms likeeBay and Amazon Marketplace on the one hand, and a self-managed online store on the other hand, as sales channels.

In all these cases different prices can accompany the same prod-uct. There are various additional scenarios in which customergroups pay different prices for the same product. Examples includerebates for certain ‘‘valuable’’ customers, added-value productslike cooled beverages that can be price-tagged using RFID technol-ogy and other price discrimination settings.

In general, we focus on the possibility of selling items managedunder the same stock keeping unit (SKU) for different prices.

ll rights reserved.

x: +49 221 470 7950., [email protected]).

Inventory control is typically performed at the SKU level and musttake this aspect into account. In this work we concentrate onfinding optimal ordering policies for the periodic review singleitem lost sales model with fixed setup costs, two different demandclasses, and a lead time of zero in absence of inventory rationing.

This models a typical retail selling scenario with multiple (inour case two) sales channels. A firm orders a single SKU under peri-odic review from a supplier (single echelon). For each orderingdecision a fixed setup and variable procurement cost is incurred.Items – managed under a single SKU – are then sold via twodifferent sales channels for different selling prices to customers.Demand streams are stochastic, independent and follow non-stationary distributions. If demand cannot be satisfied, it is lostand a penalty cost is incurred.

We assume distinct penalty costs for both sales channels. This isconsistent with the outlined setting of an online retailer that on theone hand sells through shopping platforms and, on the other hand,runs a self-managed online store. Stock-out penalties are typicallyhigher for self-managed online stores, due to higher sales marginsand loss of customer goodwill in case of low availability ofproducts.

We assume prices for the different sales channels are exoge-neous and we do not consider the pricing problem. Additionally,we do not allow rationing for any one of the sales channels. Thereason for both aspects is the high (price) transparency customers

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102 F. Schneider, D. Klabjan / European Journal of Operational Research 227 (2013) 101–111

typically have in online businesses. Availability of products in asingle channel and stock out in the other is discovered frequentlyand causes complaints about this practice as well as bad ratingsof stores. As a result many online retailers do not consider ration-ing to be a valid tool of revenue management. The assumption ofzero lead time and only two sales channels is a limitation of ourwork owed to the complexity of lost sales models.

The contributions of this paper are the analyses of lost salesinventory models with two sales channels, nonlinear, non-stationary cost functions and non-stationary demand in a profitmaximization scenario with fixed setup cost and no rationing.To the best of our knowledge, this is the first work addressing thissetting by means of a rigorous mathematical analysis. We provideconditions for optimality of the base stock policy for a generalarrival process and of the (s, S) policy for Poisson arrivals. We alsomodel consecutive batch arrivals of orders in both sales channelsand show that costs in this setting serve as valid bounds on thetrue cost.

The remainder of this paper is organized as follows. In Sec-tion 1.1 we provide a short summary of past literature in the fieldof lost sales models with various demand classes. This is followedby Section 1.2 which introduces background results and notation.Section 2 introduces the basic cost components and a simple arri-val model, where customer demand does not follow a particulardistribution and we present results of the model analysis in Sec-tion 3. In Section 4 we present and analyze the model includinga Poisson arrival process of customers. This section also providesa numerical study based on real world data. Section 5 includesthe model and analysis of the batch arrival process together withupper and lower bounds on the true value function. We concludethe article in Section 6.

1.1. Literature review

Lost sales models have received considerably less attentionthan models with backlogging of demand, since their analysis isin general more involved. Karlin and Scarf [17] were the first toaddress the lost sales problem with non-zero lead time. Theyshowed optimality of the (s, S) policy for the case of a lead timeof one period. Their results are however limited to a single de-mand class, linear cost functions and stationary demand distribu-tions. Morton [20] extended the work of Karlin and Scarf tomodels with arbitrary stochastic lead times, and derived variousproperties of an optimal policy without explicitly stating it. Again,only a single demand class was considered. Nahmias [21] andJohansen [11] introduced myopic heuristics for lost sales modelswith lead time, due to the absence of a provable optimal policy.Only one demand class with stationary demand is considered.Various papers (see Reiman [23], Hill [9]) consider continuous re-view models with demand following the Poisson distribution. Acomparison between policies that place an order based on a fixedfrequency and base stock policies is provided in Reiman [23].Again, no fixed setup costs and only a single demand class areconsidered.

Another stream of literature considering lost sales modelsexplicitely requires demand distribution to be Pòlya frequencyfunctions of order 2, as we do in some of our results. Rosling[24] considers an inventory model with nonlinear (fixed plus lin-ear) shortage cost. However, in this work backlogging of demandis assumed and all parameters are stationary. Huggins and Olsen[10] prove optimality of (s, S) policies in a lost sale model in anexpediting context with zero lead times for general cost func-tions, but assume stationary cost and demand parameters for asingle demand stream model. Li and Yu [19] show quasiconcavityfor a number of lost sales models with a single demand stream,but do only consider lost revenue as penalty cost for lost sales.

Additionally, holding costs are assumed to be linear. Also seePorteus [22] Section 9.1 for details on inventory control for Pòlyafrequency distributions.

Early work addressing the problem of different demand classeswas performed by Veinott [12,13]. He presented a critical levelpolicy for a periodic review model with several demand classes,a lead time of zero and partial backlogging of demand. Topkis[25] showed optimality of the policies by Veinott. Both papersconcentrate mainly on finding critical levels for rationing salesto low class demand in order to meet defined service levels. At-kins and Katircioglu [3] described a single item lost sales modelwith two demand classes in a periodic review setting, but withthe focus on meeting service levels. Cohen et al. [6] performeda comparable analysis in the case of inventory issued to two de-mand classes with different priority at the end of a period. Theperiod’s total demand is known when allocating inventory tothe demand classes. Nahmias and Demmy [21] introduced a sin-gle period inventory model with rationing of sales for differentdemand classes. Demand only occurs at the end of a period andis known before the decision whether to satisfy demand or notis made. Ha [8] analyzes a stationary make-to-stock system withseveral demand classes and lost sales. Here, rationing levels aredetermined from which on demand from certain classes shouldbe rejected. Downs et al. [7] used a linear programming approachto calculate order quantities in a finite horizon non-zero lead timelost sales model for various products. Their approach is based onhistoric data. Fixed ordering cost (which adds nonlinearity) is notconsidered. Zipkin [28] presents bounds on the optimal policy forthe periodic review single item lost sales model with lead timeand also discusses an extension to multiple demand classes,although no optimal policies are mentioned or proven. Arslanet al. [2] study a single-product inventory model for multiple de-mand classes, but assume backlogging of demand. Zhou et al. [26]discuss a capacitated single item lost sales make-to-stock modelwith multiple demand classes, but again no fixed set-up costsare considered. Cattani and Souza [5] perform a numerical com-parison of rationing policies with a pure first-come, first servepolicy as modeled in our paper in the particular case of directmarketing in two sales channels. A general treatment of inventorytheory is given in Zipkin [27] where fundamental results on lostsales models are summarized. An extensive review of work oninventory control and fulfillment in multi-channel distributionis given in Agatz et al. [1].

The presented past literature shares two aspects with ourmodels: lost sales and multiple demand classes. Our models areunique due to the following characteristics. We are the first toincorporate general non-stationary non-linear penalty and hold-ing cost functions in a lost sales setting with two demand streams.We concurrently consider fixed plus linear procurement cost aswell as holding and penalty costs for both streams. We explicitelydisallow inventory rationing for the two demand streams as thisreflects many common multi-channel retail scenarios weinvestigated.

1.2. Background results

Let d : R! f0;1g be defined as d(u) = 1 for u > 0 and d(u) = 0otherwise. Additionally, notation x+ = max(x,0) for x 2 R is used.We use the term discrete function to denote a function f : I! R

where I # Z. We refer to function g : I ! R defined by g(u) =f(u + 1) � f(u) as the first difference of function f. We refer to the firstdifference of function g as the second difference of function f.

Definition 1. Let v be an arbitrary finite sequence of real numbersv1, v2, . . . , vn with vi < vi+1 for all i = 1, 2, . . . , n � 1. Then the numberof sign changes of function f : I ! R is given by

F. Schneider, D. Klabjan / European Journal of Operational Research 227 (2013) 101–111 103

maxv

Xi

jsgnðf ðv iÞÞ � sgnðf ðv iþ1ÞÞj( )

where sgn : R! f�1;0;1g is defined as sgn (x) = �1 for x < 0, sgn(x) = 0 for x = 0 and sgn (x) = 1 otherwise.

In studying lost sales models with setup costs Pòlya frequencydistributions play a vital role. We say that discrete functionf : I! R; I # Z is PFr if f is a Pòlya frequency sequence of order rin the sense of Karlin [15, Chapter 8].

Many known distributions (e.g. normal, binomial, Poisson, gam-ma) are Pòlya frequency distributions of infinite order (see e.g.Karlin [15]). For a thorough discussion of Pòlya frequency distribu-tions and proofs of the presented propositions see Karlin [14] orKarlin and Proschan [16].

Definition 2 (Adopted from Keilson and Gerber [18]). A discretefunction is said to be strongly unimodal, if its convolution with anyunimodal discrete function is unimodal.

Proposition 1. Let f : Z! R be a discrete function. If there exists�1 < n <1;n 2 R such that sgn (f(u)) = �1 for all u < n and sgn(f(u)) = 1 for all u > n, then sequence g defined by gðuÞ ¼

Puk¼�1f ðkÞ

is strongly unimodal. The first difference of any unimodal sequencehas at most one sign change.

The proof of Proposition 1 is straight forward.

Proposition 2 (Adopted from Barlow and Proschan [4] and Karlin[15]). Discrete functions that are at least PF2 are strongly unimodal.The Poisson distribution is a Pòlya frequency function.

Proposition 3. Let f : Z! R be a discrete function with a single signchange as in Proposition 1. Then the convolution of f with a discretedistribution h defined by gðuÞ ¼

P1k¼�1f ðu� kÞhðkÞ, where h P 0

and h is at least PF2, also has a single sign change. Additionally, thesign changes order in the same direction as f.

Proof. The first part directly follows from Definition 2 and Propo-sitions 1 and 2. For the second part assume the sign of f changesorder from � to + with increasing u. Then from f(u) < 0 for u < n,where n is defined as in Proposition 1, we get limu?�1g(u) < 0and conversely we have f(u) > 0 for u > n and therefore limu?1g(u) > 0. As a result the sign must change from � to + with increas-ing u. The same can be shown if the order of the sign change isreversed. h

In some of the proofs we require convexity of the value functionor one period cost functions. However, since we are analyzing asetting with discrete quantities, cost functions have domains I # Z.When we deal with convexity of a function f : I! R; I # Z, we referto convexity of the linear interpolation function �f : J ! R, with J # R,defined by �f ðxÞ ¼ f ðuÞ þ ðx� uÞðf ðuþ 1Þ � f ðuÞÞ, where u 2 I is suchthat u 6 x < u + 1.

2. Modeling

In this section we present our modeling framework. In the gen-eral case customers purchase a product through two different saleschannels. Customers place orders without a specific order andaccording to an arbitrary distribution of the order sequence. Wecall this scenario the general order process. All subsequent modelsare special cases of this setting. In all models we analyze a finitehorizon setting with terminal costs assumed to be zero. Through-out the paper we denote the inventory on hand by xt 2 N0 andthe order up to level by yt 2 N0. The actual order quantity, i.e.,

the decision taken in our model, is given by yt � xt and always re-stricted to yt P xt.

The model considers two different sales channels of a singleproduct. We refer to the channels as the high and low class channel(based on the corresponding high or low selling price). The inven-tory level on hand is reviewed periodically. The sequence of eventsis as follows. A certain amount of inventory xt is observed at thebeginning of time period t. Then a replenishment order is placed.After the replenishment order arrives with zero lead time nonneg-ative demand is realized and the cost is accounted for. Note thatthis is equivalent to placing a replenishment order at the end ofa business day and receiving an overnight shipment on the nextmorning. This setting especially fits a conventional in-store retailbusiness with fixed opening hours, but only approximates thesetting of an online store or continuous 24/7 shopping. Typically,customers also place orders during the lead time of a shipment,which of course is non-zero in reality. However, depending onthe distribution of the demand over time it can be the case thatvery low volumes are demanded between the time of order andshipment, which renders the model a good approximation.

2.1. Basic cost components

In all models demand exceeding inventory on hand is lost andincurs a penalty cost in addition to lost sales revenue. Inventoryexceeding demand in a time period is carried over to the next timeperiod and incurs a holding cost. The cost components consideredin all models are

1. the procurement cost,2. the penalty cost for lost sales,3. the holding cost, and4. sales revenue (modeled as negative cost).

As a result, the one period cost function has six components.Since multiple (in our case two) sales channels are considered,we have one penalty cost component for each channel, whereasonly a single holding cost function is considered. For all time peri-ods t of the planning horizon let hN

t : N0 ! Rþ be the inventoryholding cost and let hH

t : N0 ! Rþ; hLt : N0 ! Rþ be the penalty cost

functions for high and low class sales channels, respectively. Thehigh (low) class channel is the sales channel associated with thehigher (lower) sales price of the product. Herein we refer to thesum of the negative sales revenue and penalty cost as the shortagecost.

A few general standard assumptions with respect to the costfunctions are made. Holding and penalty costs are assumed to befinite, nonnegative and are only defined on domain N0. Addition-ally, we assume hN

t ð0Þ ¼ hLt ð0Þ ¼ hH

t ð0Þ ¼ 0.We introduce procurement cost in time period t as ct(yt � xt) +

Ktd(yt � xt). The procurement cost component consists of twoparts: a linear part, which includes the variable per item procure-ment cost ct, and a fixed cost Kt, which is incurred once every timean order is placed irrespective of its quantity. We also considersales revenue explicitely and assume it to be linear in the numberof items sold through each sales channel. The expression for reve-nue (defined as a cost) is given by �pH

t AHt � pL

t ALt , where AH

t and ALt

denote the quantities sold through high and low class channel inperiod t, respectively. Similarly, pH

t and pLt are the per unit selling

prices of the two channels.

2.2. General order process model

In general orders are assumed to be placed in an arbitrary unor-dered sequence in both channels. Random vector Wt denotes thearrival sequence of orders during time period t. Each coordinate


of Wt encodes a binary random variable and the ith coordinateencodes the channel in which order number i was placed. Thenumber of coordinates of Wt is itself random. An example of a real-ization is given by

wt ¼ fLHLLHHLHLLHLHL . . . Lg

which encodes that the first order is placed in the low class saleschannel, the second one is placed in the high class sales channel,and so forth (see Fig. 1). We denote the ith order of period t by wti

(or Wti when we refer to random variables). Quantity jwtj denotesthe number of coordinates in wt, which is the number of orders intime period t. The total number of orders for each channel in periodt is given by

DHt ðWtÞ ¼ jfijWti ¼ Hgj

and

DLt ðWtÞ ¼ jfijWti ¼ Lgj

We also define

AHt ðyt ;WtÞ ¼ jfijWti ¼ H; i 6 ytgj

and

ALt ðyt ;WtÞ ¼ jfijWti ¼ L; i 6 ytgj

as the amount of the goods sold to high and low class channel,respectively, given inventory level yt. Note that AL

t ðyt;wtÞþAH

t ðyt;wtÞ 6 yt and DLt P AL

t , DHt P AH

t for yt P 0 and any realizationwt of Wt.

Using the above definitions we can write the one period costfunction Q t : N0 ! R, which is defined by

Q tðytÞ ¼ ctyt þX

wt

�pLt AL

t ðyt;wtÞ � pHt AH

t ðyt;wtÞ�

þ hNt yt � AL

t ðyt ;wtÞ � AHt ðyt ;wtÞ

� �þ hL

t DLt ðwtÞ � AL

t ðyt ;wtÞ� �

þ hHt DH

t ðwtÞ � AHt ðyt;wtÞ

� ��PðWt ¼ wtÞ

and denotes the controllable part of expected variable cost in periodt for given order up to level yt. Value function Gt : N0 ! R repre-sents expected cost-to-go until the end of the time horizon forcurrent inventory level xt and is defined by the optimality equation

GtðxtÞ ¼minytPxt

(�ctxt þ Ktdðyt � xtÞ þ Q tðytÞ

þ cX

wt

Gtþ1 yt � AHt ðyt;wtÞ � AL

t ðyt;wtÞ� �

PðWt ¼ wtÞ)ð1Þ

where 0 6 c 6 1 is a discount factor.

Fig. 1. Timeline for one period o

3. Inventory control for the general order process model

The presented model is very general and thus – without furtherassumptions – does not exhibit structured optimal policies (evenin the abscence of fixed setup costs). This is mainly due to the factthat selling quantities cannot be written in a closed form expres-sion, which makes analysis of the value function tedious.

We note that even the standard single item lost sales modelwith a single demand stream requires additional assumptions(e.g. monotone cost functions and the sum of penalty and holdingcost must exceed the discounted procurement cost, see Zipkin [27,Section 9.4.6]) in order to prove optimality of base stock policies inthe abscence of fixed costs.

In the next theorem we give conditions for optimality of basestock policies under the abscence of fixed costs, but for the generalarrival sequence.

Theorem 1. If in each period t

1. fixed ordering cost Kt = 0,2. penalty cost functions are linear and their slopes bH

t and bLt

obey

f the ge

cctþ1 � hNt ð1Þ 6 bL

t þ pLt ¼ bH

t þ pHt

3. and holding costs are convex,

then the base stock policy is optimal for the general arrival process.

The theorem’s conditions can be expressed as follows. Threeconditions must be met: (a) penalty costs must be linear in theamount of lost sales, (b) shortage cost per item must be equal forboth sales channels, and (c) this quantity must not be below nexttime period’s discounted per unit procurement cost minus holdingcost for the first unit. If this is the case, then – in the abscence offixed ordering costs – the base stock policy is optimal.

Condition 2 of the theorem might not always hold in practice. Itrequires the sum of the per unit penalty cost and selling price to beequal in both channels, whereas it is expected that lost salespenalties are higher for customers willing to pay a higher priceper unit. However, in special cases (e.g. a sales channel for keyaccounts with rebates) the condition is likely to hold.

The detailed proof is provided in Appendix A. Here, we outlinethe main ideas at a high level. Optimality of base stock policies isshown by induction. A sufficient condition is convexity of the valuefunction in each time period. We first rewrite (1) in terms ofGtðxtÞ ¼ GtðxtÞ þ ctxt . It then suffices to show that for any givenwt expression

QtðytÞ � cctþ1ðyt � jWt jÞþ ð2Þ

is convex. To show this we distinguish case yt < jwtj, where the pen-alty and procurement costs are incurred (i.e., demand is higher thaninventory), case yt > jwtj, where the holding and procurement costs

neral order process.


are incurred, and case yt = jwtj, where only the procurement costsare incurred. The procurement costs are linear and can be omittedin the analysis. In the first case all possible arrival sequences oftwo subsequent orders (HH, HL, LH, LL) are analyzed. Here, theequality in condition 2 of the theorem ensures nondecreasing firstdifferences of (2). The inequality of condition 2 provides this inthe remaining two cases.

Note that the conditions of Theorem 1 do not restrict the natureof the underlying arrival sequence, i.e., stochastic process Wt. It fol-lows from the proof that under convexity of Gt our assumptions aresufficient and necessary. However, base stock policies do not re-quire convexity of Gt (but convexity of Gt implies optimality of basestock policies) and therefore in the abscence of this convexity ourconditions might be too strict or not sufficient.

Example. To illustrate the conditions of our theorems we presentnumerical examples based on the following setting. As in ouroriginal motivation, we consider an online retailer with two saleschannels. According to our model a single product is sold throughthese channels (e.g. websites) and can be purchased through eitherchannel as long as at least one item of this SKU is in stock.

For illustration of Theorem 1 let us assume that no fixed order-ing cost is incurred as required by condition 1, and thus Kt = 0 forall t. Daily or at least weekly shipments are common in manyonline retail segments and we therefore assume a discount factorc = 1. Furthermore, let us assume stationary cost parameters forease of exposition. We assume a unit retail price of pL

t ¼ 0:75 forthe first channel and pH

t ¼ 1:00 for the second channel. The equal-ity in condition 2 then requires the unit penalty cost for the firstchannel to be 0.25 below the unit penalty cost for items soldthrough the second channel, e.g. bH

t ¼ 0:10 and bLt ¼ 0:35. Assum-

ing a procurement cost of ct = 0.20, any (nonnegative) convex hold-ing cost function satisfies the inequality in condition 2, renderingbase stock policy optimal for the outlined arrival process.

4. Poisson arrivals

An interesting special case of the general arrival process is thecase of Poisson arrivals, which is discussed in this section.

4.1. Model

For the Poisson arrival process order placement in both saleschannels is assumed to be Poisson distributed. Let s be the lengthof a time period and kt be the arrival rate for orders in period t. Thetotal number of orders per period then follows

PðjWtj ¼ kÞ ¼ e�kts ðktsÞk

k!

where jWtj is the total number of orders in period t and kts is meanand variance of the underlying Poisson distribution. For every orderthere exists certain probability 0 6 v 6 1 that it was placed throughthe high class channel, and probability 1 � v that it comes from thelow class channel.

This process is equivalent to the following process. Let highclass orders be placed based on rate kH

t and low class orders beplaced based on rate kL

t . If the two processes are independent, thenthis process is identical to the above process with kt ¼ kH

t þ kLt and

v ¼ kHt

kHt þkL

t.

Expected orders through high and low class channel for a par-ticular realization wt of Wt are given by vjwtj and (1 � v)jwtj,respectively.

The expected number of goods sold is composed of two terms.The first term captures all cases where total demand exceeds yt,i.e., jwtjP yt, while the second treats the case where jwtj < yt. Thetotal number of goods sold through high class channel is givenby

ytvX1k¼yt

PðjWt j ¼ kÞ þ vXyt�1

k¼0

kPðjWtj ¼ kÞ ð3Þ

The corresponding expression for low class channel can be ob-tained by replacing v by 1 � v in (3).

Incurred penalty costs in a time period depend on the differencebetween realized demand and the number of goods sold. Since de-mand is a random variable specified by the arrival sequence Wt, theamount of unsatisfied demand is also stochastic. Let us denotethese quantities for high and low class channel by

NHt ðyt ;WtÞ ¼ DH

t ðWtÞ � AHt ðyt;WtÞ

and

NLt ðyt;WtÞ ¼ DL

t ðWtÞ � ALt ðyt ;WtÞ

Note that NLt P 0 and NH

t P 0.The expected penalty costs for the high class channel in period t

is given by

X1k¼0

hHt ðkÞP NH

t ðyt;WtÞ ¼ k� �

Conditioning PðNHt ðyt;WtÞ ¼ kÞ on the total number of orders

yields

PðNHt ðyt ;WtÞ ¼ kÞ ¼

X1i¼0

PðNHt ðyt ;WtÞ ¼ kjjwtj ¼ iÞPðjwt j ¼ iÞ

In the next step term PðNHt ðWtÞ ¼ kjjwtj ¼ iÞ is developed. Two cases

are distinguished.

1. Let first yt P i.In this case inventory exceeds i. Therefore, all demand is satis-fied and thus

P NHt ðyt ;WtÞ ¼ kjjwt j ¼ i

� �¼

1 if k ¼ 0;0 if k > 0:

�ð4Þ

2. Let now yt < i.In this case inventory is not sufficient to satisfy all demand. Theprobability of not fulfilling exactly k high class orders is equal tothe probability of k orders being placed in the high class channelafter depletion of inventory. If jwtj exceeds inventory yt by lessthan k, the probability of not fulfilling k orders is 0. If jwtj exceedsthe inventory by k or more units, the probability follows theBinomial distribution as indicated earlier. To summarize,

P NHt ðyt ;WtÞ¼kjjwt j¼ i

� �¼

0 if i<kþyt;

i�yt

k

� �vkð1�vÞi�yt�k if iPkþyt

8><>:

Probability PðNHt ðyt;WtÞ ¼ kÞ can be written as the combination of

the two cases discussed above and it reads

P NHt ðyt;WtÞ¼k

� �¼Xyt�1

i¼0

P NHt ðyt ;WtÞ¼kjjwt j¼ i

� �Pðjwt j¼ iÞ;

þX1i¼yt

P NHt ðyt;WtÞ¼kjjwtj¼ i

� �Pðjwt j¼ iÞ


The expected value for the penalty costs is then given by

E hHt ðyt ;WtÞ

� �¼X1k¼0

hHt ðkÞ

Xyt�1

i¼0

P�

NHt ðyt;WtÞ

#¼ kjjwtj ¼ i

�Pðjwt j ¼ iÞ

"

þX1

i¼kþyt

P NHt ðyt ;WtÞ¼ kjjwt j ¼ i

� �Pðjwtj ¼ iÞ

The expression for expected penalty costs in the low class channelcan be derived accordingly.

Because of (4) and hHt ð0;WtÞ ¼ 0, the first summation term in

the brackets is always equal to zero. The final expression for theexpected value of the penalty costs is then given by

E hHt ðyt;WtÞ

� �¼X1k¼0

hHt ðkÞ

X1i¼kþyt

i� yt

k

� �vkð1� vÞi�yt�kPðjwtj ¼ iÞ

Equivalently, the expected penalty costs for the low class chan-nel can be written by exchanging v and 1 � v in the aboveformulae.

Holding costs are determined by the amount of inventory heldat the end of the period after satisfying realized demand. If demandexceeds inventory on hand, clearly, no holding costs are incurred.Expected holding costs are given by

E hNt ðyt ;WtÞ

� �¼Xyt

k¼0

Pðjwt j ¼ kÞhNt ðyt � kÞ

Sales revenue is determined by the total number of items soldthrough high and low class channels. Rewriting (3) for low classdemand and multiplying by the selling price yields expectedrevenue (defined as a cost)

Eðrtðyt ;WtÞÞ¼�pHt ytv

X1k¼yt

PðjWt j¼kÞþvXyt�1

k¼0

kPðjWt j¼kÞ

0@

1A

�pLt ytð1�vÞ

X1k¼yt

PðjWt j¼kÞþXyt�1

k¼0

kð1�vÞPðjWt j¼kÞ

0@

1A

¼� vpHt þð1�vÞpL

t

� �yt

X1k¼yt

PðjWt j¼kÞþXyt�1

k¼0

kPðjWt j¼kÞ

0@

1A

Having all cost components specified, the optimality equationGt : N0 ! R for the model can now be written as

GtðxtÞ ¼minytPxtf�ctxt þ Ktdðyt � xtÞ þ Q tðytÞ þ cE½Gtþ1ððyt � jWt jÞþÞ�g

with

Q tðytÞ ¼ E hHt ðyt;WtÞ

� �þ E hL

t ðyt ;WtÞ� �

þ E hNt ðyt;WtÞ

� �þ E�

rtðyt ;WtÞ�þ ctyt ð5Þ

4.2. Analysis

For the Poisson model (as well as the model with batch arrivals,which is discussed later in Section 4), two sets of sufficient condi-tions for optimality of (st, St)-type policies are established. Wepresent both sets, since their conditions are not subsets of eachother. The first set of conditions provides convexity of the one per-iod cost function and preserves Kt-convexity of the value function.It is given in Theorem 2.

Theorem 2. If for all t

1. penalty and holding cost functions are convex and nondecreasing,2. Kt P cKt+1, and

3. pHt v þ pL

t ð1� vÞP � hHt ð1Þv þ hL

t ð1Þð1� vÞ� �

,

then the (st, St) policy is optimal for the Poisson model.Condition 1 requires cost functions for penalty and holding

costs to be convex and nondecreasing, which is a common assump-tion in inventory control. Condition 2 is fulfilled if discounted setupcosts for placing an order do not increase over time. Assuming thatthe process of placing and processing an order is not subject tochange, we expect this condition to be fulfilled in practice. The lastcondition is always satisfied in case of nonnegative sales revenue.We can conclude that Theorem 2 holds under very mild conditions.

We return to our example from Section 3 to illustrate that thetheorem’s conditions can commonly be satisfied. We choose e.g.Kt = 25 for all t and a discount factor of 0 6 c 6 1, which satisfiescondition 2. All other parameters can take arbitrary values, e.g.,those given in our example of Section 3, as long as cost functionsare convex and take a common value for zero items.

The proof of Theorem 2 is provided in Appendix B. It uses thefact that expected sales revenue as well as penalty costs in bothsales channels can be expressed as simple functions, which arethen shown to be convex if hL

t and hHt are convex functions. The

model then reduces to a single item lost sales model for whichoptimality of the (st, St) policy is known for a lead time of zeroand convex cost functions (see e.g. Porteus [22, Section 7.1]).

We provide a second set of optimality conditions for (st, St)policy relying on unimodality of the value function for each timeperiod, which is another sufficient condition for optimality of(st, St) type policies. It is presented in Theorem 3.

Theorem 3. Let M be the binomially distributed random variablewith parameters k 2 N and 0 6 v 6 1, i.e., M � B(k, v). Also defineaL

t ðuÞ ¼ hLt ðuþ 1Þ � hL

t ðuÞ and aHt ðuÞ ¼ hH

t ðuþ 1Þ � hHt ðuÞ. If for all t

1. pHt v þ pL

t ð1� vÞ > ct ,

2. ct�1 þminuP0 hNt�1ðuþ 1Þ�

�hN

t�1ðuÞ�

P c pHt vþ

�pL

t ð1� vÞþmaxkEM vaH

t ðk�MÞ þ ð1� vÞaLt ðMÞ

�and

3. penalty costs are nondecreasing,

then the (st, St) policy is optimal for the Poisson model.The theorem’s conditions can be illustrated as follows. First, the

average per item sales price in both channels must exceed the peritem procurement cost. In practice this is commonly the case;otherwise the retailer is better off by not selling the product inone or both of the two channels. The second condition of Theorem 3is more involved. Its interpretation is that the maximum dis-counted expected per unit shortage cost in the next period mustbe bounded from above by the minimum per item holding costplus the procurement cost in the current time period. This can eas-ily be checked in certain cases (e.g., for convex holding costs andconcave penalty costs by inspecting incremental costs for the firstitem). In practice condition 2 is satisfied in a number of scenarios.Examples include high holding costs (e.g. for perishable goods),products with low profit margins and low penalty costs, or in thecase of rapidly dropping sales prices. The last condition simplyrequires penalty costs to be nondecreasing, which is a trivialrequirement. Note that shortage costs need not be equal in bothchannels as was the case for the general order process.

To provide evidence that the conditions are satisfied in certaincases, we state a numerical example. We assume stationary param-eters and omit time indices. Consider the example from Section 3and assume retail prices of pL = 0.75 and pH = 1.00 for the two saleschannels, respectively. For v = 0.3 and c = 0.70 the first condition isclearly satisfied. For a perishable good relatively high holding costsare a reasonable assumption. We assume minimum per item hold-ing costs to be 0.25 and linear penalty costs. The second term on the

Table 1Optimality conditions imposed by Theorem 2 and 3 for (st, St) policy.

Cost type Theorem 2 Theorem 3

Procurement(linear)

Free Below averagerevenue

Procurement(fixed)

Nondecreasing from t to t + 1 Free

Penalty Convex, nondecreasing and Freea

Average for firstitem P average revenue

Holding Convex, nondecreasing Nondecreasinga

a Additionally, condition 2 of Theorem 3 must hold.


right hand side of condition 2 then reduces to c(aL(1 � v) + aHv),where aL and aH are the slopes of the penalty cost functions. Penaltycosts typically represent loss of goodwill induced by stock outs. Weassume values of aL = 0.1 and aH = 0.2 and constant retail prices forour good. condition 2 then requires a discount factor of c 6 0.99 toprovide optimality of the (st, St) policy.

The proof of Theorem 3 is provided in Appendix C. Here we out-line the proof at a high level. We prove optimality of the (st, St)policy by showing unimodality of Lt = Qt(yt) + cE[Gt+1((yt �Wt)+)].This is achieved by proving that precisely one sign change of thefirst difference of Lt exists and it occurs from � to +. We rewritethe first difference of Lt(yt) as a convolution of a piecewise definedfunction g(k) with the Poisson distribution. After showing that g(k)must have precisely one sign change that occurs from � to +, weuse Definition 2 and Proposition 2 to show that Lt(yt + 1) � Lt(yt)must also have exactly one sign change from � to +. This providesunimodality of Lt and standard arguments show optimality of the(st, St) policy.

Remark 1. Note that both theorems of this section do not requirethe underlying distribution of total demand in a period, i.e.,random variable jWtj, to be Poisson as their proofs do not rely onunique characteristics of this distribution. For Theorem 2 it issufficient that for a given total demand the distribution betweenthe two sales channels is binomial, whereas jWtj can follow anarbitrary distribution. Theorem 3 additionally requires the distri-bution of jWtj to be at least PF2. However, we formulated thetheorems in a Poisson based framework, since it is commonlyassumed that demands follow a Poisson distribution in ouroutlined setting.

4.3. Discussion

The two sets of conditions from the previous section cover dif-ferent cost configurations. While the first set requires convex costfunctions and nonincreasing discounted setup costs, the second setof conditions does not impose any restrictions on the setup costs.Instead, the minimum cost incurred by ordering a unit exceedingdemand must be higher than the maximum discounted shortagecost in the next time period. Condition 2 of Theorem 3 ensuresunimodality of the value function. If it does not hold, there canbe cases in which buying an extra unit results in higher expectedcosts, as the holding costs exceed potential savings in penalty cost,but buying two extra units reduces total expected cost. This resultsin a policy struture that is more complex than (st, St). Table 1 pro-vides an overview of cost configurations that provide optimality ofthe (st, St) policy.

4.4. Numerical study

To assess the value of following an (st, St) type policy in a prac-tical setting, we performed a number of computational experi-ments with cost parameters not satisfying the conditions ofTheorems 1–3. In order to compare an optimal solution againstan optimal (st, St) type policy, we first solved the dynamic programoptimally by backwards recursion. We also obtained values for st

and St in each period by following these steps.

1. Set St ¼ arg minytfQ tðytÞ þ cE Gtþ1 ðyt � jWt jÞþ

� � g.

2. Set st ¼ max �yt with

�yt 2 fyt jyt 6 St ;Kt þ Q tðStÞ þ cE Gtþ1 ðSt � jWtjÞþ� �

6 Q tðytÞ þ cE Gtþ1 ðyt � jWtjÞþ� �

g

3. If such an st does not exist, set st = 0.

In these steps Gt+1 is the function in time period t + 1, when fol-lowing the (st, St) policy as described above in each period.

There are two main reasons rendering (st, St) type policies non-optimal: the demand distribution is not log-concave (as requiredby Theorem 3) or cost functions are not convex (as required byTheorem 2).

While there is a number of well-known non-log-concave distri-butions (e.g. Student’s t, Pareto, Cauchy, F), we focus on the morepractical setting of non-convex cost functions. Commonly observedare non-convex holding or procurement costs, e.g. for renting stor-age in a warehouse or bulk purchasing, where holding and pro-curement cost is a staircase function.

We performed a number of experiments with selling prices,variable procurement cost, holding and penalty cost held constant.We allowed expected demand to vary over time and used non-sta-tionary fixed setup cost in some of the instances. Cost parametersfrom a local retailer for pet supplies who sells through eBay as wellas a self-managed online store were used. Sales through the self-managed online store yield higher sales margins. The self-managedonline store is therefore considered as the high class channel. Salesthrough eBay generate higher absolute revenue, but margins arelower due to fees incurred and better comparability of competingoffers.

For the experiments we chose a single SKU, in our case a trans-port box for pets. We set sales prices to pH = 6.05, pL = 5.25 and ahigh class to low class ratio of v = 0.25 as observed from historicsales data for the boxes. Procurement cost for the product is c = 3and penalty costs in both channels were estimated by the salesmargin of an average shopping cart including our product andset to hL(k) = 3.7k and hH(k) = 4.5k. We set the discount factorc = 0.99995, representing an annual inflation rate of 2%, as we as-sumed period length of one day.

We performed experiments with fixed ordering costK = [0, 5, 10, 15] and holding cost as a fraction of procurement costof h = [0.1, 0.2, 0.3, 0.4], as these were parameters for which the re-tailer was not able to provide specific estimates. We investigated aproblem with T = 30 periods. For each time period we randomlygenerated a mean value k 2 [10, 300] for the Poisson distributionwith equal probability. We generated and solved 40 instancesand report the average and maximum cost increase of followingan (st, St) policy over an optimal policy as a percentage of total costfor each parameter combination.

Table 2 shows results of our experiments. We observed subop-timal performance in all instances with staircase procurement cost.Higher fixed setup cost reduces the cost difference in all instances,since the fixed setup cost essentially smoothes out ripples in thevalue function introduced by the staircase shaped cost functions.Higher holding cost also decreases the cost difference in most in-stances, mainly due to the higher total cost. For linear procurementcost and stationary fixed ordering cost, (st, St) was optimal for allinstances, including staircase holding cost.

Table 2Cost increase (st, St) vs. optimal policy for 30 periods in percent.

h K

0 5 10 15 20

(a) Linear holding cost, staircase procurement cost0.1 Avg 1.25 1.04 0.87 0.73 0.62

Max 1.73 1.42 1.14 0.94 0.81

0.2 Avg 0.74 0.62 0.53 0.46 0.41Max 1.15 0.98 0.81 0.68 0.66

0.3 Avg 0.56 0.48 0.43 0.39 0.35Max 0.99 0.82 0.69 0.60 0.59

0.4 Avg 0.52 0.44 0.38 0.34 0.30Max 1.32 1.09 0.88 0.70 0.56

(b) Staircase holding and procurement cost0.1 Avg 1.31 1.09 0.92 0.77 0.66

Max 1.73 1.44 1.20 1.01 0.93

0.2 Avg 0.89 0.74 0.64 0.57 0.53Max 1.19 0.97 0.86 0.86 0.85

0.3 Avg 0.81 0.70 0.63 0.58 0.54Max 1.40 1.19 1.03 0.89 0.80

0.4 Avg 0.92 0.80 0.70 0.63 0.57Max 2.41 2.26 2.06 1.83 1.59

Table 3Cost increase of (st, St) vs. optimal policy for 30 periods and non-stationary fixed costin percent.

Procurement cost Holding cost Average Maximum

Linear Linear 0.25 0.81Linear Staircase 0.56 1.23Staircase Linear 1.63 1.89Staircase Staircase 1.79 1.97


We also performed a number of experiments with non-station-ary fixed cost and randomly set Kt to values between 0 and 150. Innone of the investigated instances (st, St) was optimal and we re-port the maximum and average deviation for different shapes ofthe cost functions in Table 3 for the holding cost fraction of 0.1.In the case of non-stationary fixed cost, even with all costs linear,(st, St) is not optimal in all instances. This is caused by the violationof condition 2 of Theorem 3 by our parameter choice in all exper-iments, which enables varying Kt to cause sub-optimality of (st, St)as indicated by condition 2 of Theorem 2; otherwise all conditionsof Theorem 3 would be met, and Kt had no influence on optimality.In all instances the deviation with non-stationary fixed cost washigher than in the case of stationary fixed cost.

In all our real-world instances we observed that the cost in-curred when following an (st, St) policy is within 3% of the mini-mum cost. The additional effort to compute the optimal policy ismost likely not justifiable in many practical cases. However, notethat it is possible to choose parameters that result in an arbitrarilybad performance of the (st, St) policy.

Fig. 2. Timeline for one period of the HL

5. Demand channel batch arrivals

The main difficulty of the previously described models is thefact that they rely on the actual arrival sequence (for the generalorder arrival model) or on a particular distribution function (as isthe case in the Poisson arrival model). To circumvent this, we studyin this section the full batch cases where all orders through onechannel are placed at once (a valid interpretation is that demandfrom one channel has priority and must be satisfied first). This isclearly not completely realistic in our outlined setting and there-fore only approximates the previous models. In this section westudy policies of this approximate model and provide a relation-ship to the general order process model. Fig. 2 illustrates the batcharrival process.

We consider two cases and assume that either

1. all orders through high class channel arrive and are fulfilledfirst; left over inventory is then used to fulfill ordersthrough low class channel (this model and the underlyingquantities are denoted by superscript HL) or

2. this order is reversed and all orders through the low classchannel must be satisfied first, before left over inventoryis assigned to satisfy orders from the high class channel(this model and the underlying quantities are denoted bysuperscript LH).

Total demand in each channel is distributed according to anarbitrary discrete distribution. Distributions of high and low classdemand are assumed to be independent of each other. LetPðDH

t ¼ hÞ and P DLt ¼ l

� �be the probability of having exactly h

and l orders from the high and low class channel, respectively.We first state the two models. The one period cost function

Qt : N30 ! R for the HL case reads

QHLt yt ;D

Ht ;D

Lt

� �¼ �pH

t min yt ;DHt

� �� pL

t min yt �min yt;DHt

� �;DL

t

� �� þ hN

t yt � DHt � DL

t

� �þ� �

þ hLt DL

t �min yt �min yt ;DHt

� �;DL

t

� �� þ hH

t DHt �min yt;D

Ht

� �� þ ctyt ð6Þ

and the optimality equation GHLt : N0 ! R is stated as

GHLt ðxtÞ ¼min

ytPxt

X1h¼0

X1l¼0

�ctxt þ Ktdðyt � xtÞ þ Q HLt ðyt ;h; lÞ

h(

þcGHLtþ1 ðyt � h� lÞþ� �i

P DLt ¼ l

� �P DH

t ¼ h� �)

ð7Þ

On the other hand, the LH setting is

variant of the batch arrival process.


Q LHt yt;D

Lt ;D

Ht

� �¼ �pL

t min yt ;DLt

� �� pH

t min yt �min yt ;DLt

� �;DH

t

� �� þ hN

t yt � DHt � DL

t

� �þ� �

þ hHt DH

t �min yt �min yt ;DLt

� �;DH

t

� �� þ hL

t DLt �min yt;D

Lt

� �� þ ctyt

with the optimality equation

GLHt ðxtÞ ¼min

ytPxt

X1l¼0

X1h¼0

�ctxt þ Ktdðyt � xtÞ þ Q LHt ðyt ; l;hÞ

h(

þc GLHtþ1 ðyt � h� lÞþ� �i

P DHt ¼ h

� �P DL

t ¼ l� �)

In this analysis we consider model HL, i.e., demand from thehigh class channel demand is satisfied first. The results can easilybe adapted for model LH. We first study optimal policies withoutthe fixed ordering cost.


1. pHt þ hH

t ð1Þ � hHt ð0Þ

� �P pL

t þ limu!1 hLt ðuþ 1Þ � hL

t ðuÞ� �

,2. Kt = 0 and3. holding and penalty costs are convex and nondecreasing,

then the base stock policy is optimal for model HL.

Theorem 4 has a simple interpretation. The expression on theleft-hand side in condition 1 represents the shortage cost for thefirst lost sale in the high class channel. The expression on the rightdepicts the maximum cost of a lost sale in the low class channel.Condition 1 simply requires the per item shortage cost in the highclass channel to be always equal to or higher than in the low classchannel.

The result from Theorem 4 is not surprising. The analyzed sce-nario can be transformed into a single item, single demand streammodel by serially joining the penalty cost and sales revenue func-tions at quantity DH

t . Conditions 1 and 3 of the theorem then as-sures convexity of the shortage cost at this breakpoint.

The detailed proof is provided in Appendix D. It is based onshowing convexity of the value function in (7) by induction. First,(6) is inspected for an arbitrary fixed realization of the random de-mand variables. As mentioned, the one period cost function thenreduces to a simple piecewise defined function with two breakpoints. Convexity between breakpoints follows from the generalassumption that the holding and penalty costs are convex. Condi-tion 1 of the theorem provides increasing first differences at thebreakpoints. As a result, (6) – which is a weighted sum over fixedrealizations of the random variables – is convex. Together with theinduction assumption that Gt+1 is convex and monotone, it can beestablished that (7) is convex and the base stock policy is optimal.

We again give a numerical example of these conditions. Condi-tion 1 requires sales revenue plus the penalty cost for the first unitfor the high class channel to be greater than or equal to the salesrevenue plus the maximum per unit penalty cost for the low classchannel. For ease of exposition we assume the penalty cost func-tions to be linear and choose pH

t ¼ 1:00; hHt ðkÞ ¼ 0:2k; pL

t ¼ 0:75and hL

t ðkÞ ¼ 0:1k as in the example of Section 4.2. Additionally,we assume no setup costs are incurred. This satisfies all of the con-ditions of the theorem and renders the base stock policy optimal.

In presence of fixed ordering costs, we can state two sets of con-ditions, under which (st, St)-type policies are optimal. The sets – as

in Section 4.2 – rely on different properties of the value functionand are not just subsets of each other. The first is a generalizationof Theorem 4 that allows for Kt > 0. It is based on convexity of theone period cost function and Kt-convexity of the value function.

Proposition 4. If for all t

1. holding and penalty cost functions are convex andnondecreasing,

2. pHt þ hH

t ð1Þ � hHt ð0ÞP pL

t þ limu!1 hLt ðuþ 1Þ � hL

t ðuÞ� �

and3. Kt P cKt+1,

then the optimal policy for model HL is of (st, St) type.

Proof. We know from Theorem 4 and its corresponding proofthat given condition 1 of Theorem 4, GHL

t and Q HLt are convex in

all periods in the abscence of fixed costs, i.e., Kt = 0 for all t.For Kt > 0, let

ftðytÞ ¼ E Q HLt ðyt ;h; lÞ þ cGHL

tþ1 yt � DHt � DL

t

� �þ� ��

The optimality equation now reads

GHLt ðxtÞ ¼ �ctxt þmin ftðxtÞ; min

ytPxt½Kt þ ftðytÞ�

�

Following the proof of Theorem 2, we note that assuming GHLtþ1 is

Kt+1-convex, ft must be cKt+1-convex. Assuming GHLTþ1 ¼ 0, it is then

easy to show by backwards induction that an (st, St)-type policy isoptimal for the batch arrival model with fixed ordering costs, ifKt P cKt+1. h

The second set of conditions provides unimodality of the valuefunction and is in the same spirit as Theorem 3.


1. ct�1 þminu hNt�1ðuþ 1Þ � hN

t�1ðuÞ� �

> c max pHt þmaxu hH

t ðuþ 1Þ � hHt ðuÞ

� �;

�pL

t þmaxu hLt ðuþ 1Þ � hL

t ðuÞ� �

Þ,

2. ct < pLt þ hL

t ðuþ 1Þ � hLt ðuÞ and ct < pH

t þ hHt ðuþ 1Þ � hH

t ðuÞfor all u and

3. DHt and DH

t are random variables with a distribution functionthat is at least PF2,

then the optimal policy for model HL is of (st, St) type.

The crucial requirement of Theorem 5 lies in condition 1. It re-quires the procurement cost plus the minimum per item holdingcost in the current period to be higher than the next period’s dis-counted maximum per item shortage cost (sum of lost revenueand penalty cost). This can be the case for rapidly dropping salesprices or high holding costs (e.g. for perishable goods). Low dis-count factors have a similar interpretation and can also lead to thiscondition to hold.

Additionally, we need the per item shortage cost to be greaterthan the per item procurement cost. Setting retail prices abovethe procurement cost and assuming nonnegative penalty costssuffices to fulfill this condition. The last condition is a technicalcondition requiring the distribution functions of the demandstreams to be at least PF2, which is the case for many commondistributions (e.g. Poisson, binomial, gamma, normal, etc.). Notethat Theorem 5 does not formulate any conditions including Kt,which is typical for theorems based on unimodality of the valuefunction.

Table 4Maximum gap in expected cost between upper and lower bound for T = 30 in percent.

h K

(a) Varying K and h for v = 0.25, D = 1.8

0 5 10 15 20

0.1 0.18 0.19 0.21 0.24 0.260.2 0.22 0.22 0.23 0.23 0.230.3 0.04 0.03 0.03 0.02 0.010.4 0.01 0.00 0.00 0.00 0.00

D v

(b) Varying v and D for K = 10, h = 0.10.1 0.3 0.5 0.7 0.9

1 0.24 0.29 0.27 0.25 0.172 0.41 0.43 0.38 0.34 0.213 0.54 0.55 0.46 0.39 0.244 0.66 0.60 0.49 0.41 0.24


The proof for this theorem is provided in Appendix E. Again, weoutline it at a high level. The proof is carried out by induction. Weinspect the expression

RtðytÞ¼X1h¼0

X1l¼0

Q HLt ðyt ;h;lÞþcGHL

tþ1 yt�DHt �DL

t

� �þ� �� P Dl

t¼ l� �

P DHt ¼h

� �

It can be shown that conditions 2 and 3 of Theorem 5 assert thatthe first difference of Rt(yt) is the convolution of a function with asingle sign change from – to + with a strongly unimodal discretefunction. According to Proposition 3 the first difference of Rt(yt)must also have a single sign change from� to +. It then follows thatRt(yt) is unimodal and standard arguments show that the (st, St)policy is optimal.

For a numerical example we use the parameters from thecorresponding example of Theorem 3. With pL = 0.75, pH = 1.00,aL = 0.1 and aH = 0.2, condition 3 of the theorem is satisfied.Condition 2 requires 0.7 + minuP0(hN(u + 1) � hN(u)) P 1.2c. Set-ting c = 0.99 as before requires minimum per item holding costsof 0.488, whereas in the Poisson arrival case a value of 0.25 wassufficient.

As for the Poisson arrival process the presented sets of optimal-ity conditions rely on different fundamental properties of the costfunctions. The first set requires nonincreasing setup costs and con-vex holding and shortage costs. It uses the fact that shortage costsare simple to define in a closed form expression in the case of batcharrival, as there is a single breakpoint at which the functionchanges its form.

The second set does not constrain values for setup costs, butneeds shortage costs to be nondecreasing and limited by the min-imum holding plus procurement cost in the previous time period.The reason for the last part is that otherwise there could be inven-tory levels at which it would be cost optimal to order up to aninventory level greater than St.

5.1. Upper and lower bounds

Intuitively it is clear that the case in which high class orders areplaced and satisfied first represents a lower bound on the real cost ina time period, if we have higher penalty costs and sales margins inthe high class channel. In the same manner, case LH where all lowclass orders are placed and satisfied first intuitively represents anupper bound. Theorem 6 provides a formal proof of necessary con-ditions on selling prices and penalty cost functions for the batchmodel to be valid lower and upper bounds on the true cost function.


1. hLt ðu1 þ 1Þ � hL

t ðu1Þ 6 hHt ðu2 þ 1Þ � hH

t ðu2Þ for all u1;u2 2 N0

and2. pL

t 6 pHt

hold, then GHLt and GLH

t are lower and upper bounds for Gt, i.e.,

GHLt ðxtÞ 6 GtðxtÞ 6 GLH

t ðxtÞ ð8Þ

holds.

The theorem’s conditions are in line with the intuitive idea thata firm could maximize its profit, if it was able to always satisfy de-mand associated with the higher shortage cost first. However, it issurprising that we cannot simply require shortage cost to be higherin one channel than in the other. We explicitely need the conditionthat sales prices in the high channel must be equal to or exceedsales prices in the low channel. Otherwise, in the case of sufficientinventory on hand, we could achieve lower total cost by selling tothe low class channel first.

The proof is provided in Appendix F. The proof is based on thefact that given QHL

t 6 Qt 6 QLHt for all t; GHL

t 6 Gt 6 GLHt does hold.

This can easily be seen by induction. Given an arbitrary demandrealization in a time period, we can omit procurement cost inour analysis, since it is simply a linear term in Qt(yt) and identicalfor GHL

t ; GLHt and Gt. We then fix yt and inspect possible realizations

of demand for a period. Two cases are inspected. First we assumeall demand can be satisfied. Then no penalty costs are incurredand condition 2 of the theorem asserts that Q HL

t 6 Q t 6 Q LHt . In

the case when demand exceeds supply, condition 1 of the theoremasserts Q HL

t 6 Q LHt for any arbitrary realization of demand. It can

then be argued that Qt must always lie within the given bounds,since total demand is always equal for GHL

t ; GLHt and Gt and the min-

imum and maximum penalty costs that can be incurred given afixed total demand are given by the assumed batch arrivals. Thiscompletes the proof.

5.2. Numerical study

To assess the quality of the upper and lower bounds, we per-formed a number of numerical experiments based on the real-world setting from Section 4.4, which satisfies the conditions ofTheorem 6. We performed experiments with fixed ordering costK = [0, 5, 10, 15] and holding cost as a fraction of procurement costof h = [0.1, 0.2, 0.3, 0.4]. We investigated a problem withT = 30 periods and k = 30. In our first study we used a high classto low class ratio of v = 0.25 and all costs were assumed to be lin-ear. The difference in shortage cost D = ph + hH � pL � hL = 1.8 waschosen to match our real-world example. In the second study weperformed a number of experiments for h = 0.1 and K = 10 andvaried v as well as D. Table 4 reports the maximum relative gapin expected cost between upper and lower bound, i.e.,

maxx0P0GLH

0 ðx0Þ�GHL0 ðx0Þ

GLH0 ðx0Þ

.

Table 4 shows that the gap is below 1% for all computed in-stances. Table 4a indicates that for increasing holding cost the rel-ative gap decreases, as a result of higher total cost incurred. Forholding cost rates of 0.1 and 0.2 the gap increases in K, but for hold-ing cost rates of 0.3 and 0.4 the gap decreases in K. This is causedby the fact that for high holding cost the product is ordered inevery period in both models, whereas for low holding cost theordering decisions are different in both models. If high class de-mand is satisfies first, the policy is to always order. If low class de-mand is satisfied first, the product is not ordered in every periodfor high values of K.

Table 4b shows that increasing the difference in sales prices andpenalty cost increases the relative gap for given v. This is intui-tively clear, because the difference in revenue and cost in both


models must be higher if the difference in cost of a lost saleincreases. Increasing the fraction of high class demand of total de-mand induces two competing effects. On the one hand, intuitivelythe gap takes its maximum at v = 0.5 if the same ordering decisionsare taken in both models. On the other hand, total profit increaseswhen the fraction of high class demand is increased such that forhigh D the relative gap decreases monotonely for increasing v.

6. Concluding remarks

Although the model with only two sales channels analyzed inthis work is very basic, the conditions listed for optimality of basicinventory control policies are relatively restrictive for general or-der processes. This is a property inherent to lost sales models,which are complex due to their special nonlinear structure insystem dynamics. However, we succeeded to analyze models thatclosely parallel true circumstances in a real world retail scenariowith two sales channels. As expected, there is a trade-off betweenrestrictiveness of assumptions regarding the arrival process on theone hand and restrictiveness of optimality conditions on the otherhand. While the absence of assumptions on the arrival processleads to requesting the penalty costs to be linear and to have thedifference in slopes equal to the difference in selling prices, model-ing order arrivals by a Poisson process mitigates this requirementsand allows general convex penalty cost functions. We also havebeen able to provide two independent sets of optimality conditionsfor the (st, St)-policy under Poisson arrival and batch arrival pro-cesses; one based on providing convexity of the one period costfunction and the other one based on unimodality of the valuefunction.

In short, we have provided useful theory and insights into apractical retail setting with multiple sales channels or, equvialent-ly, price discrimination. We hope that our results can be of value topractitioners in inventory control managing multiple sales chan-nels. Our results shed light on structure of optimal inventory con-trol policies and the corresponding restrictions on modelparameters. To the best of our knowledge, there has been no workincluding nonlinear non-stationary cost components, two saleschannels and lost sales in a revenue maximization setting withoutinventory rationing.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.ejor.2012.12.001.

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