Omega 37 (2009) 106–115www.elsevier.com/locate/omega

Analysis of a two-echelon inventory system with returns�

Subrata Mitra∗

Indian Institute of Management Calcutta, Joka, Diamond Harbour Road, Kolkata 700 104, India

Received 17 April 2006; accepted 3 October 2006Available online 17 November 2006

Abstract

Product take-back and recovery activities have grown in recent times as a consequence of stringent government regulationsand increased customer awareness of environmental pollution. Inventory management in the context of product returns has drawnthe attention of many researchers. However, the inherent complexity of the system with uncertain returns makes the analysis ofthe system extremely difficult. So far, the literature on this type of system is mostly limited to single echelons. The few papersavailable in literature on multi-echelon systems with returns base their analyses on simplified assumptions such as non-existenceor non-relevance of set-up and holding costs at different levels. In this paper, we relax these assumptions and consider a two-echelon system with returns under more generalized conditions. We develop a deterministic model as well as a stochastic modelunder continuous review for the system, and provide numerical examples for illustration.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Two-echelon inventory; Product recovery; Reverse logistics; Continuous review

1. Introduction

Product take-back after use for disposal or recov-ery is recently receiving growing attention for severalreasons. First of all, government legislations in manydeveloped countries hold organizations responsible forhandling their products and packaging after being usedand discarded by customers. Secondly, customers alsohave become more aware of the environmental pollutioncaused by the landfill and incineration of used productsand packaging, and would like the manufacturers to takeresponsibility of recycling them. Customers would evenprefer to buy environment-friendly products (Carter andEllram [1] estimated the market to be over $200 bil-lion), which would put pressure on the manufacturers

� This manuscript was processed by Associate Editor RuudH. Teunter.

∗ Tel.: +91 33 24678300; fax: +91 33 24678307.E-mail address: subrata@iimcal.ac.in.

0305-0483/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.omega.2006.10.002

to initiate returns of their used products and recover theeconomic value as much as possible. Adoption of envi-ronmentally friendly practices not only helps the man-ufacturers comply with the government regulations andcustomers’ demand, but also enhances their corporateimage. Finally, organizations are now taking a proactiveapproach to product recycling instead of a passive ap-proach in the past from an economic point of view. Forexample, the cost of remanufacturing a product is gen-erally much lower (40–60%) than that of manufacturingor procuring a new product [2], and a remanufacturedproduct is considered to be “as good as new” and issold in the primary market along with the new productat the same price and with the same warranty [3].

Inventory management in the context of product re-turns has drawn a lot of attention from researchers. Thefact that returns are more uncertain than demands interms of quantity, quality, and timing makes inventorycontrol more difficult than that without returns. In an

S. Mitra / Omega 37 (2009) 106–115 107

inventory system with returns, there are three types ofinventory—returned units, recovered units and manu-factured or procured units. It is difficult to ascertain theappropriate holding cost rates for returned and recov-ered units (Teunter et al. [4] compared the performancesof different methods for setting the holding cost ratesin an average cost inventory model with returns). If theholding cost rates are different for recovered and manu-factured/procured units, then a stock depletion rule hasto be put in place in case the recovered units and themanufactured/procured units exist simultaneously in theserviceable stock.

Literature on reparable item inventory managementexists since 1960s. In these systems, return of every unitgenerates demand for one unit. So, there is a perfect cor-relation between demand and return. On the contrary,in inventory systems with returns, it is usually assumedthat demand and return are mutually independent.Assumption of independence of demand and return isespecially reasonable if the mean time-to-return is largecompared to the time between placing orders. Examplesof these systems include return of a photocopier afterthe expiry of the lease/rent period, which might notnecessarily generate demand for a new machine. How-ever, it may also be possible to have correlated demandand return for products with short life cycles, suchas reusable containers [5] and single-use cameras [6].Reviews of literature on inventory systems with returnsare available in Fleischmann et al. [7] and Guide et al.[8]. Recent references include Teunter [9,10] for deter-ministic models, Mahadevan et al. [11] for a periodicreview model and Fleischmann and Kuik [12] for acontinuous review model.

The references cited so far consider a single loca-tion inventory system. The inherent complexity of thesystem makes it difficult to extend to a multi-echelonsystem. In a single location inventory system, customerdemand occurs directly at the location where both recov-ered units and manufactured/procured units are stocked.But in multi-echelon systems, the higher echelons haveto satisfy demand from the lower echelons, which mightcome in batches. This batch demand coupled with theexistence of set-up costs for both recovery and manufac-turing/procurement at the higher echelon complicatesthe system to a great extent. Literature on multi-echeloninventory systems with returns is few and far between.Those available in literature are based on simplified as-sumptions for tractability. Muckstadt and Isaac [13] de-veloped a model for a one-warehouse, N -retailer distri-bution system with returns where the retailers did nothave set-up costs and they followed an (S−1, S) contin-uous review policy, S being the order-up-to level. Since

customer demands faced by the retailers were Poissondistributed, the demand process at the warehouse couldbe described by compound Poisson. Moreover, the re-covery process at the warehouse was described by aqueuing system, which was independent of the choiceof the policy parameters. Thus the set-up cost and hold-ing cost at the level of recoverable units had no rele-vance in the model. The warehouse operated under an(R, Q) continuous review policy for its serviceable in-ventory where R was the reorder point and Q was theorder quantity. Korugan and Gupta [14], for a similardistribution system, made the same assumptions aboutthe retailers and the demand process at the warehouse.They even did not consider the set-up cost for the ser-viceable inventory, and developed a model based on anopen queuing network with finite buffers. For a reviewon multi-echelon inventory research with returns, read-ers are referred to Fleischmann and Minner [15] andDekker et al. [16].

The literature on multi-echelon inventory systemswithout returns is very rich. Efforts should be made toapply those results for systems with returns. Minner [17]extended his work on strategic safety stock placementin forward supply chains to supply chains with returns.His basic approach was based on Simpson’s model [18]for a serial supply chain under a base stock policy. Thevariables were the service times at different locations,and the objective was to minimize the investment insafety stocks. Based on the definition of the system,set-up costs and shortage costs had no relevance in themodel. The problem addressed in the current paper gen-eralizes the description of the system for a two-echelonserial supply chain considering set-up costs and holdingcosts for all the inventory levels and shortage costs forthe serviceable inventory levels. The item under con-sideration has a high demand justifying the increase inrecycling activities for high-demand, low-value itemsin recent times compared with high-value, low-volumeitems in the past [19]. We develop a deterministic modeland a stochastic model of the system. For the stochas-tic model, we assume that the system is under con-tinuous review. Numerical examples are provided forillustration.

The motivation for the problem is derived fromgrowing recycling of electronic products, such as tele-vision and photocopier, at the end of their useful lives.In many developing countries such as India, consumersare frequently bombarded with incentives/offers toexchange their used electronic products for new andtechnologically advanced models. Unlike in the de-veloped countries, there is no separate market forremanufactured or refurbished products in developing

108 S. Mitra / Omega 37 (2009) 106–115

countries such as India as yet. So, no matter whetherthe used products are returned for exchange or re-turned as they have reached their end-of-life, they arecannibalized and the parts and components that canbe reused (directly or maybe after some amount ofrefurbishing) are separated out for manufacturing newproducts. There are many common electronic compo-nents such as resistors, capacitors and basic transistorsthat are used in all models of the products no matterhow advanced they might be. Unlike in the developedcountries, in developing countries such as India manu-facturers are not obligated by law to declare whetherin a new product, used components have been incorpo-rated or not. These used/refurbished parts/componentsare invisible to the consumers and hence carry the samevalue as new parts/components to them. The number ofunits of electronic products sold represents the demandfor the specific parts/components. Also, since the usageperiod of these electronic products varies from personto person depending on various factors such as usagepattern, obsolescence of models, etc., sales and returnsmay not have any correlation at all.

The paper is organized as follows. Section 2 presentsthe problem description. The models are formulated inSection 3. Section 4 presents numerical examples, fol-lowed by a conclusion in Section 5.

2. Problem description

Consider a two-echelon inventory system consistingof a depot and a distributor. The distributor faces cus-tomer demand and places orders with the depot forreplenishment. Defective but recoverable units are re-turned to the distributor, which are immediately returnedto the depot for recovery. Hence we can assume, withoutloss of generality, that defective but recoverable unitsare directly returned to the depot. As the recovery ofreturned units takes place at the depot, and not at thedistributor, the recoverable inventory is relevant for thedepot, and not for the distributor. Since the return rateis usually less than the demand rate, it is not possiblefor the depot to meet the demand of the distributor withthe recovered units alone. The depot also has to procurefrom an outside supplier with infinite capacity. Thus thedepot has two inventories—recoverable inventory of de-fective units and serviceable inventory of recovered andprocured units. It is assumed that the recovered units areas good as new and have the same value as the procuredunits. It is also assumed that the time to recover a batchof defective units is much less than the lead time ofprocurement at the depot such that, without loss of gen-erality, the recovery time can be considered to be zero.

Recoverableinventory(Stage 3)

Serviceableinventory(Stage 2)

Outside supplier

Serviceable inventory(Stage 1)

Customer return

Depot

Distributor

Customer demand

Fig. 1. A one-depot, one-distributor inventory system with returns.

Independence is assumed between demand and return.The inventory system is represented by Fig. 1.The cost components of the system are set-up costs

and inventory holding costs for the distributor and thetwo inventories of the depot. In addition, there are short-age costs for the distributor and the serviceable inven-tory of the depot for the stochastic situation. The objec-tive is to determine the values of the policy variables atall the levels, which minimize the total cost (TC) of thesystem.

Hereafter, the serviceable inventory at the distributor,the serviceable inventory at the depot, and the recover-able inventory at the depot are referred to by Stages 1,2 and 3, respectively.

3. Model formulation

In this section, we develop mathematical models forthe problem. First we develop a deterministic model fordeterministic demand and return rates. Then we developa stochastic model for stochastic demand and returnrates. Before going into the models, let us introduce thefollowing symbols.

Ai set-up cost at Stage i (i = 1, 2, 3)

hi holding cost per unit per unit time at Stage i (i =1, 2, 3)

pi shortage cost per unit at Stage i (i = 1, 2)

li lead time at Stage i (i = 1, 2)

Ri reorder point at Stage i (i = 1, 2)

ki safety factor at Stage i (i = 1, 2)

� mean demand per unit time� standard deviation of demand per unit time

S. Mitra / Omega 37 (2009) 106–115 109

r fraction of demand returned per unit time (0 < r < 1)Q order quantity at Stage 1S2 order-up-to level at Stage 2n integer

3.1. Deterministic model

In this model, it is assumed that the demand rate (�)

and the return rate (r�) are deterministic, stationary anduniform through time. Stage 1 follows a stationary or-dering policy, i.e., in each cycle it places an order ofQ units with Stage 2. Hence for each cycle at Stage 2,there should be an integer (say, n) number of cycles atStage 1. In other words, each cycle at Stage 2 satisfiesnQ units of demand from Stage 1. Stage 2 procuresthese nQ units from two sources—inventory of returnedunits and an outside supplier. The cycle length at Stage2 is nQ/�. In each cycle, Stage 2 can recover rnQ

units of returned inventory. The remaining (1 − r)nQ

units have to be procured from the outside supplier.This procurement policy apparently places the prob-lem in the category of dual/multiple supplier (or supplymode) inventory problems that have been widely ad-dressed in literature. Dual/multiple suppliers (or supplymodes) are resorted to in practice essentially for threereasons. First, since different suppliers have differentlevels of uncertainties in terms of quantity, quality andtiming of their deliveries, the procurement order can besplit among multiple suppliers to reduce the effectivelead time and thereby the chance of stockout. The deci-sion variables in this case are the order quantities allo-cated to different suppliers. The literature that addressesthis problem includes Ramasesh et al. [20], Sedarageet al. [21], Fong et al. [22], Chen et al. [23] and Kelle andMiller [24]. Second, suppliers may be classified basedon unit price, lead time and flexibility, and a trade-offmay be approached between fast, flexible but expensivesuppliers and slow, inflexible but inexpensive suppliers.Besides reducing the chance of stockout, in this case theobjective also includes the minimization of transactioncosts. Agrawal et al. [25] and Yan et al. [26] studied thisproblem with periodic demand forecast updates avail-able while placing orders with the suppliers. Finally,suppliers may have capacity constraints that justify theexistence of multiple suppliers. Syam and Shetty [27]addressed such a situation with supplier capacity limi-tations. Readers interested in literature on dual/multiplesuppliers (or supply modes) may please refer to an ex-cellent review of this class of problems by Minner [28].Though it might appear that the present problem hassimilarities with the dual/multiple supplier (or supplymode) problems mentioned above, there is a subtle dif-

ference in that one of the supply modes in this problem,i.e., the stream of returned units, is beyond the controlof the decision-maker, who has to decide only on thenumber of units to be procured from the outside sup-plier. From literature on multiple-supplier problems, weknow that phasing deliveries from different supplierscan not only reduce the inventory holding cost but alsomake it possible to utilize the updated demand forecastinformation at the time of placing orders. In this case,too, there is potential of inventory cost saving at Stage2 if demand from Stage 1 could be met alternately fromthe returned inventory and with the units procured fromthe outside supplier. However, the problem with mod-elling this scheme is that the value of rn, and hence(1 − r)n, might not come out as an integer, and, there-fore, the assumption of stationary ordering policy atStage 1 would break down. To maintain the same as-sumption, the value of n may have to be set at an ex-orbitantly high level (for example, if r is 0.1, n has tobe a multiple of 10 to make rn, and hence (1 − r)n,an integer). This obviously would increase the cost ofthe system significantly if the inventory holding cost atStage 2 were reasonably high. To overcome this diffi-culty and bring in tractability in modelling, here we as-sume that the order is placed with the outside supplierand the recovery process is initiated in such a mannerthat the whole of nQ units are replenished simultane-ously at the beginning of each cycle at Stage 2. Hencewe only have to be concerned with making n an inte-ger in the modelling. It is no longer necessary to ensurethat rn, and hence (1 − r)n, is an integer. However, ifthe value of n is such that rn, and hence (1 − r)n, isan integer (e.g. r = 0.5, n = 2), then of course we canreduce the inventory holding cost at Stage 2 by phas-ing the recovery process and delivery from the outsidesupplier without affecting the cost at Stage 1.

Fig. 2 shows the on-hand inventory levels at Stages1, 2 (both installation and echelon) and 3 with time forn = 2. The lead time between Stages 1 and 2 (l1) is as-sumed to be zero. Had it been positive, the inventoryplot at Stage 1 would have to be shifted by the sameamount, and there would have been pipeline stock be-tween Stages 1 and 2. Since the pipeline stock cost isnot a relevant cost in the model, this does not influencethe analysis.

From Fig. 2, it is apparent that the number of set-upsper unit time at Stages 1, 2 and 3 are �/Q, �/nQ and�/nQ, respectively. The average echelon inventories atStages 1 and 2 are Q/2 and nQ/2, respectively, andthe echelon holding costs at Stages 1 and 2 are h1 − h2and h2, respectively. The average on-hand inventory atStage 3 is 1

2 (r�×nQ/�)=rnQ/2. Hence the expression

110 S. Mitra / Omega 37 (2009) 106–115

Q

2Q

2rQOn-hand stockat Stage 3

On-hand stockat Stage 1

echelon stockat Stage 2

Echelon stock

time

time

time

On-hand and

Fig. 2. On-hand stocks at Stages 1, 2 and 3 over time.

for the TC per unit time can be written as follows:

TC = A1�

Q+ A2�

nQ+ A3�

nQ+ Q

2(h1 − h2)

+ nQ

2h2 + rnQ

2h3 = �

Q

(A1 + A2

n+ A3

n

)

+ Q

2[h1 + (n − 1)h2 + rnh3]. (1)

The optimal values of Q and n can be obtained fromthe above expression following the procedure outlinedin Silver et al. [29, p. 480]. The expressions for optimalQ and n are given below:

Q∗ =√

2�(A1 + A2/n + A3/n)

h1 + (n − 1)h2 + rnh3,

n∗ =√

(A2 + A3)(h1 − h2)

A1(h2 + rh3). (2)

It can be seen from the above expressions that in theabsence of returns they would simply converge with theformulas derived in Silver et al. [29, p. 480] for a two-stage forward flow-only deterministic inventory system.In case n∗ is not an integer, one has to compute both TC(�n∗�) and TC (�n∗�), and take one of �n∗� and �n∗� forwhich TC(·) is lower. Since TC is a convex function inQ and n, this would ensure optimality of the solution.

3.2. Stochastic model

This is an extension of the deterministic model withstochastic, stationary and uniformly occurring demandand return rates. The item is considered to be fast mov-ing, and hence demand per unit time is assumed to be

Normal with mean � and variance �2. Likewise, returnper unit time is also assumed to be Normal with mean�r (=r�, say) and variance �2

r (=k�2, say). Demandper unit time and return per unit time are independentof each other. Without loss of generality, it is assumedthat k = r , which ensures that the coefficient of varia-tion ((1/

√r)�/�) for return per unit time is higher than

the same (�/�) for demand per unit time, representingthe fact that returns are relatively more uncertain thandemands. However, had the values of k and r been dif-ferent, the analysis presented in the paper would nothave changed. It is also assumed that �?�/

√r to en-

sure that the probabilities of demand rate and return ratebeing negative are negligible. Both Stages 1 and 2 areassumed to follow echelon-stock-based continuous re-view policies. The policy variables are the reorder point(R1) and the fixed order quantity (Q) for Stage 1, andthe reorder point (R2) based on echelon stock and theorder-up-to level (S2) for Stage 2. The system follows anested policy, i.e., when a stage reorders, all its down-stream stages also reorder. As mentioned in the contextof the deterministic model, there should be n cycles atStage 1 for each cycle at Stage 2. Out of these n cy-cles at Stage 1, the first (n − 1) will be termed normalreplenishments based on the reorder point R1, and thelast one will be termed joint replenishment by Stage 2based on the reorder point R2 (see [30]). To computethe echelon inventory position at Stage 2, the echelonon-hand stock at Stage 2 should comprise the cyclestock at Stage 2 and the on-hand stock at Stage 1. InDe Bodt and Graves’ model [30], safety stock was dis-tributed between both the stages, and it was assumedthat the entire stock at Stage 2 was available at Stage1 to satisfy end-item demand during the joint replen-ishment by Stage 2. Mitra and Chatterjee [31] analysedDe Bodt and Graves’ model from the implementationpoint of view, and suggested a modification of theirmodel allowing safety stock only at Stage 1 that notonly takes care of implementation, but also improvesthe performance of the system under certain conditions.In this paper, we follow Mitra and Chatterjee’s modelfor keeping safety stock only at Stage 1 as a protectionagainst uncertainty in end-item demand. Stage 2 doeshold safety stock, but that is to take care of the uncertainsupply of returned units. When the echelon inventoryposition at Stage 2 reaches or goes below R2, an orderquantity equal to the difference between S2 and safetystock, if any, at Stage 2 is placed with the outside sup-plier. The recovery process of returned units is also ini-tiated in such a way that deliveries from the supplier andthe recoverable inventory are realized simultaneously.Since returns are uncertain, it may so happen that after

S. Mitra / Omega 37 (2009) 106–115 111

replenishment the on-hand stock at Stage 2 is less thannQ, the cycle stock required at Stage 2 to meet the de-mand from Stage 1. In that case, the shortfall is madeup by expediting delivery from the supplier at someexpediting cost (p2) to ensure that the on-hand stockat Stage 2 at the beginning of each cycle is at leastnQ. Any shortfall in end-item demand at Stage 1 isbackordered.

The expressions for R1 and R2 are straightforwardand are given below:

R1 = �l1+k1�√

l1 and R2 = �(l1+l2)+k1�√

l1+l2

where k1 is the safety factor at Stage 1.

To derive the expression for S2, we need to first derivethe distribution of recoverable inventory at Stage 3 be-tween two successive replenishments by Stage 2. Thetime between two successive replenishments by Stage 2is given by nQ/demand rate, which is a random variablesince demand rate itself is random. Since return rate isalso random, the distribution of recoverable inventoryat Stage 3 between two successive replenishments byStage 2 is a convolution of two random variables, whichis difficult to derive. To simplify the analysis, as it is al-ready assumed that the mean demand rate is very highcompared to the standard deviation of demand rate, thetime between two successive replenishments by Stage2 can be approximately taken as constant and equal tonQ/�. Hence the distribution of recoverable inventoryat Stage 3 during nQ/� is also Normal with mean rnQ

and variance rnQ�2/�. The extent of error introduceddue to this approximation in distribution is examinedthrough a set of numerical experiments in Section 4. Thecomponents of S2 would be the quantity to be orderedfrom the outside supplier had there been no uncertaintyin returns and an amount of safety stock as a protectionagainst uncertainty in returns. Hence the expression forS2 can be written as follows:

S2 = (1 − r)nQ + k2�

√rnQ

�

where k2 is the safety factor at Stage 2.

To derive the expression for the expected total cost(ETC) per unit time, we need to derive the expressionsfor the expected total set-up cost, expected total aver-age cycle stock cost, expected total safety stock costand expected total shortage cost, all of them per unit oftime. The expressions for the expected total set-up costand expected total average cycle stock cost, per unitof time, will be the same as given by TC in (1). Theexpressions for safety stock and expected shortage per

replenishment cycle (ESPRC) at Stages 1 and 2 aregiven below.

Safety stock at Stage 1=n−1

nk1�

√l1+1

nk1�

√l1+l2,

Safety stock at Stage 2 = k2�

√rnQ

�,

ESPRC at Stage 1

= n − 1

n�√

l1{�(k1) − k1 + k1�(k1)}

+ 1

n�√

l1 + l2{�(k1) − k1 + k1�(k1)},

ESPRC at Stage 2 = �

√rnQ

�{�(k2) − k2 + k2�(k2)}.

Here, �(·) and �(·) denote the pdf and cdf of the stan-dard normal distribution, respectively. While the deriva-tions of the expressions for safety stock and ESPRC atStage 1 are available in Mitra and Chatterjee [31], theexpressions for safety stock and ESPRC at Stage 2 arederived in the Appendix. Hence the expression for ETCcan be written as follows:

ETC = A1�

Q+ A2�

nQ+ A3�

nQ+ Q

2(h1 − h2)

+ nQ

2h2 + rnQ

2h3 +

(n − 1

nk1�

√l1

+1

nk1�

√l1 + l2

)h1 + k2�

√rnQ

�h2

+ p1�

Q

[n − 1

n�√

l1{�(k1) − k1 + k1�(k1)}

+1

n�√

l1 + l2{�(k1) − k1 + k1�(k1)}]

+ p2�

nQ�

√rnQ

�{�(k2) − k2 + k2�(k2)}.

(3)

Solving for the optimal values of Q, n, k1 and k2 fromthe above expression will require iterative solving offour simultaneous equations. To make matters simple,we may alternatively use the optimal values of Q andn obtained from the deterministic model (2), and solvefor optimal k1 and k2 from (3). Using the optimal valuesof Q and n from the deterministic models is the com-mon approach in models with forward logistics only,and is known to work well there. The validity of thisapproximation will be tested in the next section on nu-merical experimentation. The difference between thismodel and the models available in related literature is

112 S. Mitra / Omega 37 (2009) 106–115

that this model is more general, and the approximationonly concerns the derivation of (nearly) optimal policyparameters.

The optimal values of k1 and k2 can be obtained fromthe following expressions:

1 − �(k1) = Qh1

p1�, 1 − �(k2) = nQh2

p2�. (4)

4. Numerical examples and discussions

We illustrate the model developed in this paper bythe following numerical example:

� = 100, � = 5, r = 0.5,

A1 = 25, A2 = 100, A3 = 50,

h1 = 2, h2 = 1, h3 = 0.3,

p1 = 20, p2 = 10,

l1 = 0.25, l2 = 0.5.

Solving for Q and n using (2), we get Q=77.8 and n=2.Solving for k1 and k2 using (4), we get k1 = 1.42 andk2=1.01. Hence R1=28.55, R2=81.15 and S2=82.25.The replenishment policy for the system would be asfollows. At Stage 1, whenever the inventory positionfalls at or below 28.55 units, an order quantity of 77.8units will be placed with Stage 2, and at Stage 2, anorder is placed with the outside supplier to bring itsinventory position to 82.25 units whenever its echeloninventory position falls at or below 81.15 units. As soonas replenishments arrive at Stage 2, inventory at Stage 3is recovered and made available at Stage 2 as serviceableinventory.

Next we conduct an experiment to examine the extentof error introduced in the modelling due to the assump-tions and approximations made through the followingset of numerical examples:

� = 100, � = 1, 5, 10, r = 0.1, 0.3, 0.5, 0.7, 0.9,

A1 = 25, A2 = 100, A3 = 50,

h1 = 2, h2 = 1, h3 = 0.1, 0.3, 0.5,

p1 = 20, p2 = 10,

l1 = 0.25, l2 = 0.25, 0.5, 1.

The maximum coefficients of variation for demand rateand return rate in the numerical examples are 0.1 and0.32, respectively. This is to ensure that the probabili-ties of demand rate and return rate being negative arenegligible, as mentioned before.

A full factorial design gives 135 problem instancesfor each of which we first determine the values ofthe policy variables based on the model developed inthis paper, and then using those values simulate thesystem in SLAM II (simulation language for alterna-tive modelling) for 1000 time units (data was collectedafter the first 100 time units to allow the system to sta-bilize and eliminate initial bias) to compute the ETCper unit time. To simulate demand and return occurringuniformly through time, we divide the base time unitinto a number of periods to avoid lumpiness in demandand return. The number of periods is decided based onthe data to ensure maintaining negligible probabilitiesof negative demand and return. Different numbers ofperiods may of course produce differences in simulationoutputs. However, the objective is to make a trade-offbetween the magnitudes of mean demand and return toensure uniformity of occurrence, and the probabilitiesof negative demand and return. Should negative demandor return be generated, the observation is excluded fromthe final analysis.

After having an idea about the ranges of optimalvalues of the policy variables from the model, wemake a search through simulation using SLAM IIwithin these ranges to determine the combination ofvalues of the policy variables that gives the least ETC(ETCleast) per unit time. This may not give the op-timal solution to the problem, but does represent the“best” solution obtained through simulation. We thencompute the per cent error for each problem instancefrom ((ETC − ETCleast)/ETCleast) × 100. Table 1below shows the per cent errors for all the 135 probleminstances.

It is seen from Table 1 that the error ranges from0.41% to 5.27%, and the average error is 1.84%. Theaverage per cent errors for different values of � and thesame for different values of r are plotted in Figs. 3 and4, respectively.

It is seen from Fig. 3 that the average per cent er-ror increases with �. This is expected because the ap-proximation of the distribution of recoverable inventoryat Stage 3 between successive replenishments by Stage2 was based on the assumption that �?�/

√r . When

the gap between � and � closes down, the average percent error is expected to increase. Similarly from Fig. 4we see that the average per cent error increases with r .This is also expected because if r is increased, it willincrease the variability of returns, and as such the safetystock and ESPRC at Stage 2 will also increase. Sincein reality the recoverable inventory is a convolution oftwo random variables, the average per cent error is thusexpected to increase with r .

S. Mitra / Omega 37 (2009) 106–115 113

Table 1Per cent errors for different values of �, r, l2 and h3

r h3 � = 1 � = 5 � = 10

l2 = 0.25 l2 = 0.5 l2 = 1 l2 = 0.25 l2 = 0.5 l2 = 1 l2 = 0.25 l2 = 0.5 l2 = 1

0.1 0.1 0.41 0.41 0.43 0.72 0.92 0.92 1.02 1.61 1.860.3 0.58 0.57 0.55 0.56 0.70 0.89 1.21 1.79 2.170.5 0.63 0.58 0.64 0.75 1.15 1.23 1.06 1.63 1.91

0.3 0.1 0.69 0.68 0.64 0.83 0.98 1.17 1.76 2.27 2.580.3 0.70 0.68 0.62 1.10 1.08 1.06 2.36 2.30 2.810.5 0.64 0.55 0.62 1.30 1.38 1.33 2.13 2.21 2.66

0.5 0.1 0.82 0.78 0.83 1.57 1.93 1.97 2.32 2.98 3.140.3 0.75 0.64 0.73 1.82 1.87 1.78 2.97 2.94 3.330.5 0.61 0.63 0.67 1.61 1.61 1.87 2.34 2.60 3.50

0.7 0.1 0.88 0.85 0.91 2.46 2.40 2.66 3.36 3.88 3.700.3 0.90 1.02 0.92 2.13 2.21 2.00 3.29 3.76 3.600.5 0.89 0.87 0.97 2.27 2.30 2.10 3.40 3.60 4.10

0.9 0.1 1.12 1.10 1.04 2.94 2.84 2.66 5.08 4.88 5.270.3 1.03 0.98 1.02 2.77 2.75 2.93 4.23 3.97 4.130.5 0.96 1.01 0.94 2.62 2.49 2.75 3.97 4.31 4.55

0.77

1.76

2.99

0

1

2

3

4

1 10

Standard deviation of demand

Ave

rag

e %

err

or

5

Fig. 3. Average % errors for different values of �.

We have also computed the average per cent errorsfor different values of l2 and h3. The average errorsare 1.72%, 1.84% and 1.96% for l2 = 0.25, 0.5 and 1,respectively. The average errors are 1.87%, 1.83% and1.82% for h3 = 0.1, 0.3 and 0.5, respectively. We seethat the average per cent errors are not very sensitive tol2 and h3.

We conclude from the numerical experiment that themodel developed performs very well with respect to theleast-cost solution obtained through simulation with anoverall average error of 1.84%. The performance of themodel is better when the values of � and r are lower.Even for high values of � and r , the maximum errorobtained is 5.27%. The model is found to be insensitiveto the values of l2 and h3.

Like any other theoretical model, the model pre-sented in this paper is not without its limitations. Thedemand and return processes have been assumed to beNormally distributed, which is acceptable for fast mov-

0.99 1.38

1.82.28

2.75

00.51

1.52

2.53

0 0.2 0.4 0.6 0.8 1

r

Ave

rag

e %

err

or

Fig. 4. Average % errors for different values of r .

ing items with small coefficients of variation to en-sure low probabilities of the negative tail. However, foritems having high coefficients of variation for demandsand/or returns, the model is not applicable; probablyLog-Normal, Gamma or any other suitable distributionwith no negative tail would be appropriate in these sit-uations. Also, it has been assumed that the demand andreturn processes are independent of each other. Whenthis is not the situation (for example, for reparable iteminventory systems, there is a perfect positive correlationbetween demands and returns), the correlation betweendemands and returns has to be taken into account, andthe model has to be modified accordingly.

5. Conclusion

Inventory management in the context of productreturns has drawn attention of the researchers. Butthe inherent complexity of the system with uncertain

114 S. Mitra / Omega 37 (2009) 106–115

returns makes the analysis of the system extremely dif-ficult. Most of the papers available in literature ad-dress the problem for a single echelon. The literatureon multi-echelon systems with returns is very limited.Those available base their analyses on simplified as-sumptions. In this paper, we considered a two-echeloninventory system with returns under generalized con-ditions, and developed a deterministic and a stochasticmodel for the system. To analyse the stochastic model,we made certain approximations that were tested in ex-tensive numerical experiments. We found that the modelwe developed performed very well with respect to thebest solution obtained through simulation. We intend toapply the results obtained in this paper for more generalmulti-echelon inventory systems with returns.

Appendix A

A.1. Derivation of the expressions for safety stock(SS2) and ESPRC (ESPRC2) at Stage 2

Let S denote the recoverable inventory at Stage 3 be-tween two successive replenishments by Stage 2. HenceS ∼ N(rnQ, rnQ�2/�) with density function f (·). Letd denote the recoverable inventory at Stage 3 had therebeen no uncertainty in returns. Hence d = rnQ. Thenthe expression for SS2 can be written as follows:

SS2 =∫ ∞

d−k2�√

rnQ/�

(s + k2�

√rnQ

�− d

)f (s) ds.

Assuming that the probability of shortage at Stage 2 isnegligible, it can be written that

SS2 ≈∫ ∞

0

(s + k2�

√rnQ

�− d

)f (s) ds

≈ E(S) + k2�

√rnQ

�− d .

Since E(S) = d = rnQ, SS2 = k2�

√rnQ

�.

The expression for ESPRC2 can be written as follows:

ESPRC2=∫ d−k2�

√rnQ/�

0

(d−k2�

√rnQ

�−s

)f (s) ds.

Let s = rnQ + z�√

rnQ/� where Z is N(0, 1).

Then

ESPRC2 = − �

√rnQ

�

∫ −k2

−√�rnQ/�

(z + k2)�(z) dz

= �

√rnQ

�

[�(k2) − �

(√�rnQ

�

)

+k2

{�(k2) − �

(√�rnQ

�

)}]

≈ �

√rnQ

�{�(k2)−k2+k2�(k2)} as �?

�√r

.

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