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International Journal of Production ResearchVol. 50, No. 5, 1 March 2012, 1243–1264

Reverse Logistics: a stochastic EOQ-based inventory control model for mixed

manufacturing/remanufacturing systems with return policies

Alberto Alinoviy, Eleonora Bottani* and Roberto Montanari

Department of Industrial Engineering, University of Parma, Parma, Italy

(Final version received February 2011)

This paper focuses on mixed manufacturing/remanufacturing systems, where manufacturing or purchase ofnew items integrates product reuse or remanufacturing, with the purpose to achieve a complete and timelydemand satisfaction. We formulate a stochastic Economic Order Quantity (EOQ)-based inventory controlmodel for a mixed manufacturing/remanufacturing system. The model is intended to identify the need ofplacing a manufacturing/purchasing order, to avoid the occurrence of stock-out situations. We then formulatea total cost minimisation problem, to derive the optimal return policy, this latter being a financial incentivepaid to customers to increase the flow of returned items. The model developed is investigated throughsimulations, in order to assess the effect of stochasticity (of demand, return fraction and return delay) on theoptimal return policy of the system; then, it is validated through a case study, to derive indications concerningits practical application in real cases. Our study ultimately provides a framework for practitioners to establishEOQ policies in reverse logistics contexts and to evaluate the opportunity of establishing a return policy inthose contexts.

Keywords: reverse logistics; EOQ inventory control; optimal return policy; uncertainty management;asset management

1. Introduction

Reverse Logistics (RL) is the process of planning, implementing and controlling the efficient and effective flow ofmaterials, products and information from the points of consumption to the point of origin, for the purpose ofrecapturing value or proper disposal (Rogers and Tibben-Lembke 1999). RL practices, including product recall,recycling and disposal, are currently observed in a lot of economic and manufacturing environments, and theReverse Logistics Executive Council (Rogers and Tibben-Lembke 1999) estimates in billions of dollars theAmerican annual RL costs, resulting in a non-irrelevant fraction of the Gross Domestic Product.

There are several reasons why RL practices are increasingly applied in manufacturing and logisticsenvironments. First, RL is an answer to ecological and environmental problems. Long-term business sustainabilityobjectives and the increasing cost of traditional disposal force manufacturers to make significant efforts toreintegrate used product into industrial production processes. Hence, more and more companies should improveservices provided to customers as regards replacing defective goods, repairing used products, refurbishing returnedproducts, recalling harmful goods and disposing product waste. All the above activities generate reverse flows in thesupply chain, and require their proper management for a company to remain competitive and differentiated.Legislation and directives, consumer awareness and social responsibilities toward the environment are furtherdrivers for RL (Castell et al. 2004, Bloemhof and van Nunen 2005, Ravi and Shankar 2005). In the context ofEurope, for instance, German manufacturers are generally responsible for the final destination of their products;moreover, the Packaging Ordinance (VerpackV 1998) mandates that German industries organise the reclamation ofreusable packaging waste. Procedures for collection, recycling and recovery targets for all types of electrical goodsare also set by the Waste Electrical and Electronic Equipment Directive (European Commission 2003), whichimposes responsibility for the disposal of electric and electronic equipment on manufacturers. A further importantact is the End of Life Vehicles Directive 2000/53/EC (European Commission 2000), which sets clear quantifiedtargets for reuse, recycling and recovery of vehicles and components.

*Corresponding author. Email: [email protected] address. Continuous Improving Department, Heinz Italia S.p.A., Milan, Italy.

ISSN 0020–7543 print/ISSN 1366–588X online

� 2012 Taylor & Francis

http://dx.doi.org/10.1080/00207543.2011.571921

http://www.tandfonline.com

Virtually any market field should deal with the problem of retrieving products from the market and defining

their proper disposition. Besides the case of vehicles and electrical equipment, logistics assets (or returnable

transport items) are a further example of products commonly involved in forward flows towards supply chain

partners and reverse flows. The proper management of forward and return flows of assets is a crucial question, as

their failure to enter the return channels, due to loss, theft or damage, involves relevant costs for companies (Breen

2006).When approaching the analysis of RL systems, it is important to remark that reverse flows may be regarded by

firms under two different perspectives. Under some circumstances, product returns are only considered as a logistic

cost; thus, the typical goal of a company is to minimise their extent. However, more recently, companies have

realised that RL can become a leverage for gaining competitive advantage (Marien 1998); hence, economic

profitability from remanufacturing, reuse or recycling may also be the reason for favouring return flows. In the case

of logistics assets, for instance, remanufacturing/refurbishing and reuse are of particular interest, as assets reused do

not add to the amount of items to be recycled or destroyed, thus reducing the total logistics cost of a company

(European Commission 2007). Nonetheless, to work correctly, remanufacturing environments require a continuous

supply of market-used products (Klausner and Hendrickson 2000) and, consequently, a typical business goal is

maximising return flows, instead of reducing them. To this extent, a company could decide to establish a ‘return

policy’, i.e. a financial incentive paid to their customers to favour product or asset returns. In the current paper, we

refer to this latter situation.Those contexts where assets can be reused, or returned products can be remanufactured and sold in the finished-

goods market, are known as mixed manufacturing/remanufacturing systems (MRSs). In such systems, some items

are remanufactured after their use by the customer, whereas manufacturing or purchasing of new items are required

when the return flow is insufficient to fulfil customers’ demand. MRSs are relevant to the RL context; however, their

analysis is complicated, because of:

(1) the complexity of material flows. In particular, a crucial question is represented by the necessity of a joint

coordination of the forward and reverse flows (Ketzenberg 2009);(2) the uncertainty of demand, lead time and returns, which can significantly undermine the synchronisation of

forward and reverse flows (Chanintrakul et al. 2009).

According to this premise, in the current paper we formulate a stochastic analytic model representing a useful

framework for the establishment of an EOQ policy for a mixed MRS, and analytically derive the optimal return

policy. Moreover, we investigate the effects of stochasticity on the optimal return policy for an MRS operating

under the proposed EOQ policy.The paper is organised as follows. In the next section, we review the relevant literature concerning models for RL

systems, with particular attention to inventory control and optimal return policy. In Section 3, we present the EOQ

model for an MRS, and assess the effects of stochasticity on the optimal return policy for the MRS analysed. In

Section 4, a case study is presented to show the application of the model to a real situation, and to derive insights for

practical implementations. The paper ends by a discussion of findings, managerial issues and suggestions for further

research.

2. Literature review

In accordance with the industrial interest raised by RL issues in recent years, literature on the topic is huge and

addresses different aspects related to MRSs. Owing to the focus of this paper, we analysed papers dealing with: (1)

inventory control models for MRSs; and (2) optimisation of the return policy and of the whole RL system.With regard to the first point, inventory management in RL systems has received much attention in recent years,

and numerous EOQ-based inventory control models for MRSs have been developed so far. It is commonly accepted

that the first contribution to this area was given by Schrady (1967), who formulated a deterministic EOQ-based

model, assuming constant demand and return rates and fixed lead times for external orders and recovery. More

recent studies include Richter (1996), who considered an EOQ model with constant disposal and used-product

collection rates. Koh et al. (2002) developed a joint EOQ/EPQ (Economic Production Quantity) model, assuming

that demand could be satisfied both by remanufacturing used products and purchasing new items. Oh and Hwang

(2006) proposed an optimal inventory control policy for a recycling system, assuming deterministic demand and

1244 A. Alinovi et al.

return fraction and null production lead times. Other works on the same topic include Teunter (2004), Dobos andRichter (2004), Inderfurth et al. (2005) and Konstantaras and Skouri (2009).

Most of the proposed models focus on the calculation of optimal ordering and production lot sizes underdeterministic conditions, and assume the feasibility of infinite production and remanufacturing rates, withconsequent instantaneous flows of newly remanufactured or recovered goods. A typical critique to those modelsthus consists of the scarce applicability of the formulation (Jaber and Rosen 2008). Specifically, in practical cases, itis nearly prohibitive to punctually estimate parameters such as order cost, inventory holding cost and shortage cost,which are basic inputs for EOQ lot sizing formulae, and whose wrong estimation may produce misleading results.A further limitation of those approaches is that the problem is formulated as deterministic, according to thetraditional EOQ model. However, it is plainly acknowledged by researchers that a main challenge of supply chainmanagement is making decisions when facing uncertainty or incomplete information. Hence, deterministic modelsare often unrealistic and of limited practical usefulness. Stochastic models for RL systems are proposed in a limitedgroup of papers. For instance, Fleischmann et al. (2002) proposed a basic inventory model, assuming Poisson-distributed demand and returns, and derived the optimal control policy of the system. A limitation of this work isthat the authors assumed independent demand and return rates, whereas establishing a return policy could involvedependence between demand and returns. More recently, Alinovi et al. (2009) simulated an RL system withinstantaneous integrative manufacturing flows; in this work, however, the authors assumed the delivery lead time tobe deterministically zero.

We already mentioned that firms operating in MRSs need a continuous supply of used products, and that returnpolicies are a possible means to ensure product return. Hence, a further topic investigated in the literature is theoptimisation of the return policy and of the RL system as a whole. In the case RL systems (with or without a returnpolicy) are examined for optimisation, the problem is often addressed by means of linear programming models,intended to identify the minimum total cost or maximum profit of the system (e.g. Thierry et al. 1995,Veerakamolmal and Gupta 1999, Hu et al. 2002, Beamon and Fernandes 2004, Savaskan et al. 2004, Lieckens andVandaele 2007). This is consistent with the fact that the total cost is one of the main performance measures of RLsystems, and of supply chains in general, which allows efficiency of the channel (Pochampally et al. 2009).Minimising the total cost, while assuring a defined service level to the customer, is also the strategy used in thisstudy. Focusing on works which consider return policies, Linton et al. (2005) studied the available managementoptions for inventory control in environments collecting large waste flows of durable goods to supportremanufacturing activities, with particular reference to the case of cathode ray tube (CRT) televisions. The authorsconsider the opportunity of additional expenditures to improve public awareness and interest, in order to increasethe supply of raw-material waste in the short and medium term. The opportunity to motivate consumers to sell theirused products with financial incentives has also been addressed by Lee (2001), Tsay (2001), Lau and Lau (1999) andEmmons and Gilbert (1998). Setoputro and Mukhopadhyay (2004) proposed a model for determining the optimalreturn policies as a function of given market reaction parameters, in an e-business environment. Liang et al. (2009)also studied the formulation of a financial incentive for consumers to assure continuous supply of used products, inanalogy to financial option pricing. However, these contributions only studied uncertainties affecting finishedproduct demand; conversely, stochastic returns also significantly affect the return policy.

In comparison with previous research, our work aims at being more representative of real contexts; with thispurpose in mind, we have paid particular attention to:

. formulating the model to include stochasticity of demand, return fraction and lead times, at the same timewithout focusing on specific probabilistic distributions;

. considering non-zero ordering and manufacturing lead time.

Conversely, we do not concentrate on lot sizing, assuming lot sizes to be given parameters, derived from externalconstraints (e.g. capacity of transport vehicles) or a firm’s estimations.

3. A stochastic EOQ-based inventory control model for RL systems with return policies

3.1 Notation, introduction and definitions

Throughout the paper, we use the notation proposed in Table 1.We consider a simple MRS system, made up of two echelons, i.e. a manufacturer and the end customers (see

Figure 1). This system integrates manufacturing activities with reuse of returned market-used items. We assume that

International Journal of Production Research 1245

returned items are of the same quality as the newly manufactured ones or, alternatively, that all returned itemshaving a low quality level can be instantaneously refurbished to the standard quality required for sale. The firstcircumstance may suite more properly for MRSs dealing with assets, which commonly undergo lots of reuse cyclesbefore needing substantial maintenance interventions; conversely, the second hypothesis may hold for properproduct remanufacturing systems.

Table 1. Notation used in the paper.

Symbol Description Measurement unit

� time domain –tmax length of the time horizon [days]t, k generic days of the planning horizon, t, k 2 Nj1 � t, k � tmaxf g –d tð Þ daily demand [items/day]� tð Þ daily demand expected value [items/day]�d demand standard deviation [items/day]~d tð Þ historical demand (as the outcome of the random variable d tð Þ on the historical day t) [items/day]r tð Þ daily return flow [items/day] tð Þ return fraction –� return fraction expected value –� return fraction standard deviation –RLTðtÞ return lead time [days]’RLT return lead time probability mass function –�RLT return lead time expected value [days]�RLT return lead time standard deviation [days]IH tð Þ on-hand inventory [items]I tð Þ inventory position [items]p tð Þ items supplied daily [items/day]s tð Þ items ordered daily [items/day]nDLT delivery lead time [days]EOQ order lot size [items]�� maximum tolerated stock-out probability –q forecasting day –i generic day in the forecasting horizon, i 2 qþ 1, qþ 2, . . . , , qþ nDLT

� �–

Yqi random return amount on day i, based on information available on day q [items/day]

Dq total random demand during the forecasting horizon, based on information availableon day q

[items]

Yq total random returns during the forecasting horizon, based on information availableon day q

[items]

Uq random net demand, i.e. demand minus returns, during the forecasting horizon,based on information available on day q

[items]

�q stock-out probability during the forecasting horizon –Zki Bernoulli random variable (with k� i) –’Zki

probability mass function of Zki –pki Bernoulli parameter of Zki –T demand period [days]e demand error [items/day]� return policy [%]�0 RL base cost [E/item]� unitary cost of returned items [E/item]p1 inferior asymptotic limit of the curve � �ð Þ –p2 superior asymptotic limit of the curve � �ð Þ –� flex point of the curve � �ð Þ –� ‘standard deviation’ of the curve � �ð Þ –�1, �2 beta random variable parameters –�max maximum feasible standard deviation for the beta variable –IH0 on-hand inventory at t¼ 0 [items]TC total cost function [E]cp unitary purchasing/manufacturing cost [E/item]hI daily inventory holding cost [E/item/day]


As Figure 1 shows, the system faces the external customer demand d tð Þ, and receives reverse flows of market-returned items r tð Þ. We consider a discrete time domain, represented by t 2 �, where � ¼ t 2 N : 1 � t � tmaxf g. d tð Þis a non-negative continuous random variable, with � tð Þ and �2d parameters. Note that time dependence for � tð Þallows modelling seasonal behaviour of demand. Return flows are strictly related to the historical demand, as thewhole quantity demanded at day t is supposed to result in a ‘return fraction’ tð Þ after RLTðtÞ days. tð Þ is thus acontinuous random variable defined over the interval 0, 1½ �, with � and �2 parameters. RLTðtÞ is a discrete non-negative random variable, with probability mass function ’RLT, and �RLT and �2RLT parameters.

Demand is to be satisfied through on-hand inventory, IH tð Þ, which can either be replenished through marketreturns or manufacturing/external purchase. We assume that the supplied quantity p tð Þ is the result of a previouscorresponding order s tð Þ of the same size. This latter was either placed to the supplier (in case of purchase) or to themanufacturing plant (in case of internal production) nDLT days earlier. Hence, we may define the inventory positionI tð Þ as the sum of IH tð Þ and the quantities ordered, but not yet delivered, which are expected to be physicallyavailable within nDLT days. Orders are placed for a fixed amount of items, defined a priori and correspondingto EOQ.

3.2 Analytical derivation of the stock-out probability

The inventory control model is aimed at determining whether the manufacturer should place a manufacturing/purchasing order, to prevent the occurrence of out-of-stock situations, due to insufficient returns or excessivedemand.

Let q be a generic day in the time domain V, on which we want to evaluate the opportunity to place an order (cf.Figure 2). At each day q, the decision to place an order is based on the computation of the probability of stock-outoccurrence in the following days; specifically, the manufacturer should estimate the stock-out probability �q duringthe forecasting horizon (i.e., from qþ 1 to qþ nDLT). We assume that qþ nDLT � tmax, remarking that tmax shouldbe taken instead of qþ nDLT when the above inequality does not hold. We consider the day i in the forecastinghorizon i 2 qþ 1, qþ 2, . . . , qþ nDLT

� �� and the amount of returned product on that day, Y

qi . The distribution of

Yqi is expected to depend on the information on historical demand available at day q, as it is reasonable that the

returned quantity depends upon the demanded quantities and on their probability to be returned on a defined day.

Figure 1. A simple integrated RL system.


One of the underlying assumptions of the model, in fact, is that the manufacturer does not have historical data of

return flows available, so that it has to estimate the amount of future return flows only based on the historical

deliveries. Such circumstance can be observed in the case the manufacturer did not establish a proper tracking

system, which may allow to precisely link returns with past deliveries.We hereafter use the symbol ‘�’ to denote historical (known) data, which reflect the known outcomes of the

corresponding random variables; for instance, ~d tð Þ denotes the known outcome of the random variable d tð Þ at day t.

According to the definitions of �q, Dq, Yq and Yqi in Table 1, the following equations hold:

Dq ¼XqþnDLT

i¼qþ1

d ið Þ ð1Þ

Yq ¼XqþnDLT

i¼qþ1

Yqi ð2Þ

Uq ¼ Dq � Yq ¼XqþnDLT

i¼qþ1

d ið Þ � Yqi ð3Þ

�q ¼ P Uq 4 I qð Þð Þ ð4Þ

where we denote as P �ð Þ the probability of the event in brackets.We now define Zki as follows:

Zki ¼1, if kþ RLT kð Þ ¼ i

0, else

�ð5Þ

By denoting as pki the probability of the quantity d kð Þ � kð Þ to return on day i, ’Zkijð Þ can be written as follows:

’Zkijð Þ ¼ P Zki ¼ jð Þ ¼

pki, if j ¼ 1

1� pki, if j ¼ 0

�ð6Þ

1 2 … q … q+nDLT–1 …q+nDLT tmax

Forecasting horizon (i=q+1,…q+nDLT)

Time domain W

q+1

Day on which we have to decide whether an order

should be placed

Historical demand ( )td

~

q+1 q+2 i … q+nDLT–1 q+nDLTq+nDLT–2…q

qiY , d(i) for

i =q+1,…q+nDLTDq, Yq, Uq, Iq ⇒ λq

Figure 2. Scheme for the computation of �q.


The calculation of pki may be easily obtained from the probability mass function of RLT, as follows:

pki ¼ P RLT kð Þ ¼ i� kð Þ ¼ ’RLT i� kð Þ ð7Þ

It is important to remark that Equation (7) is valid regardless of the specific distribution of RLT(t), meaning that,

for developing the theoretical model, no assumptions are required in this regard. Because we assume that any

delivery of a given past day results in a return on only one future day, the following constraint holds:

X1i¼k

Zki ¼ 1, 8k ¼ 1, . . . , tmax ð8Þ

Having hypothesised that no historical data on return flows are available, to derive an analytic expression of �q,some probabilistic parameters should be estimated, namely:

(1) the expected value and variance of return quantity at day i (propositions 1 and 2);(2) the correlation between returned quantities at two different days (proposition 3). This parameter is required

to derive the variance of the total return flow over the forecasting horizon;(3) the expected value and variance of the total return flow over the forecasting horizon (proposition 4);(4) the correlation between the total demand over the forecasting horizon and the total returned flow

(proposition 5);(5) the distribution of the net demand over the forecasting horizon (proposition 6).

For simplicity, we will henceforth introduce a notation change for time-dependent variables, moving the indication

of the day t from the argument in brackets to a newly introduced subscript (e.g., d tð Þ will be rewritten as dt).

Proposition 1: According to the historical data available on the forecasting day q, the expected value of the returned

quantity on day i can be expressed as follows:

E Yqi

� �¼ � �

Xqk¼1

~dk � pki þXi

k¼qþ1

�k � pki

!ð9Þ

See the Appendix for the proof of Equation (9).

Proposition 2: According to the historical data available on the forecasting day q, the variance of the return quantity

on day i has the following expression:

Var Yqi

� �¼Xqk¼1

~d2k � �2 þ �2

� pki � �

2 p

2ki

h in o

þXi

k¼qþ1

�2d þ �2k

� �� 2 þ �

2

� pki � �

2k � �

2 � p

2ki

h i: ð10Þ


Proposition 3: Let Yqi and Y

qj be the random variables representing the quantities to be returned on days i and j of the

forecasting horizon (i, j 2 qþ 1, qþ 2, . . . , qþ nDLT

� �, i 6¼ j), according to the information available on day q. Y

qi and

Yqj are not independent and their covariance has the following expression:

Cov Yqi ,Y

qj

h i¼ ��2

�Xqk¼1

~d2k � pki � pkj

þ

Xmin i,jf g

k¼qþ1

�2k � pki � pkj

� �24

35: ð11Þ


It is not surprising that the expression for covariance in Equation (11) is negative. We actually expected Yqi and

Yqj to be negatively correlated, according to their ‘competition’ as to receiving returns of past deliveries. As an

additional remark, we can note that no contribution to covariance is given by deliveries occurring between min i, j� �

and max i, j� �

. The reason is that there is no chance for items delivered in that period to return on min i, j� �

; hence, no

‘competition’ between the two variables arises under this circumstance. Finally, we remark that covariance tends to


be lower in the absolute value and it approaches zero when i and j are relatively (compared to �RLT) distant from

each other. The mathematical reason for this result is that for any day k � i, j, at least one between pki and pkj shouldbe close to zero, so that all addends in Equation (11) give little contribution to the whole sum. Once again, this canbe interpreted as a result of little ‘competition’ between days in ‘attracting’ past deliveries for return.

Proposition 4: The expected value and the variance of Yq have the following expressions:

E Yq½ � ¼ � �XqþnDLT

i¼qþ1

Xqk¼1

~dk � pki þXi

k¼qþ1

�k � pki

!: ð12Þ

Var Yq½ � ¼XqþnDLT

i¼qþ1

Xqk¼1

~d2k � �2 þ �2

� pki � �

2 p

2ki

h in o(

þXi

k¼qþ1

�2d þ �2k

� �� 2 þ �

2

� pki � �

2k � �

2 � p

2ki

h i

�2 � �2 �Xj5i

Xqk¼1

~d2k � pki � pkj

þXj

k¼qþ1

�2k � pki � pkj

� �" #): ð13Þ

See the Appendix for the proof of Equations (12) and (13).

Proposition 5: Covariance between Dq and Yq has the following expression:

Cov Dq,Yq½ � ¼ � � �2d �

XqþnDLT

i¼qþ1

Xik¼qþ1

pki ð14Þ


Proposition 6: The distribution of the net demand during the forecasting horizon (Uq) may find normal approximationwith the following parameters:

E Uq½ � ¼XqþnDLT

i¼qþ1

�i � E ½Yqi �

� �ð15Þ

Var½Uq� ¼ nDLT � �2d þ Var½Yq� � 2 � Cov½Dq,Yq� ð16Þ

See the Appendix for the proof of Equations (15) and (16).

The stock-out probability �q over the forecasting horizon reflects the probability of the net demand to exceed thecurrent inventory position; on the basis of the propositions above, and through a normal approximation, it can becomputed as per Equation (17):

�q ¼ P Uq 4 Iq� �

� 1��Iq � E Uq½ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var Uq½ �p

� ð17Þ

where we denoted the cumulative distribution function of the normal random variable by �. In the case �q exceeds adefined threshold (�q4 ��), a manufacturing/purchasing order should be placed, to prevent shortage.

3.3 Model simplifications

To improve its ease of application, the model described above can be simplified, without compromising thecorrectness of the results provided. More specifically, we have hypothesised that Y

qi represents the returns on day i,

forecasted at day q; hence, the returns distribution of a given day i in the planning horizon should change accordingto the position of q within the forecasting horizon and to its proximity to i. A possible simplification consists of onlyconsidering Yi variables, assuming to know historical data until i, so that the distribution of returns for each day i


will not change for all forecasting days including i in their respective forecasting horizon. This assumption is

unrealistic, because historical data are known only until the forecasting day and not until each day of the forecasting

horizon. Notwithstanding, the effects of such an unrealistic framework may be negligible if we assume that

�RLT nDLT: in such a context, the demanded quantities on days immediately preceding any day i in the

forecasting horizon and following day q, have very low probability to generate a return on day i. Hence, historical

demand on those days (which would not be known in real contexts) does not significantly affect the resulting stock-

out probability. On the basis of this premise, the simplified definition of Yi is as follows:

Yi ¼Xik¼1

~dk � k � Zki ð18Þ

The definitions of k and Zki remain unchanged. For brevity, we omit the proofs for the following equations, which

can be easily obtained following the achievements of Section 3.2. Where necessary, we use a left-hand side

superscript (�) to avoid ambiguity with the notation previously used.

E Yi½ � ¼ � �Xik¼1

~dk � pki ð19Þ

Var Yi½ � ¼Xik¼1

~d2k � �2 þ �2

� pki � �

2 � p

2ki

h in o, ð20Þ

Cov Yi,Yj

� �¼ ��2

�Xmin i,jf g

k¼1

~d2k � pki � pkj, ð21Þ

�Yq ¼XqþnDLT

i¼qþ1

Yi, ð22Þ

E �Yq½ � ¼ � �XqþnDLT

i¼qþ1

Xik¼1

~dk � pki, ð23Þ

Var �Yq½ � ¼XqþnDLT

i¼qþ1

Xik¼1

~d2k � �2 þ �2

� pki � �

2 p

2ki

h in o� 2�2

Xj5i

Xjk¼1

~d2kpkipkj

( )ð24Þ

�Uq ¼ Dq � �Yq ¼XqþnDLT

i¼qþ1

di � Yið Þ, ð25Þ

E �Uq½ � ¼XqþnDLT

i¼qþ1

�i � E ½Yi�ð Þ, ð26Þ

Var½�Uq� ¼ nDLT � �2d þ Var½�Yq�, ð27Þ

��q ¼ P �Uq 4 Iq� �

� 1��Iq � E �Uq½ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var �Uq½ �p

� : ð28Þ

The above simplified formulae will be used in the remainder of the paper.


3.4 Derivation of the optimal return policy

We now consider the opportunity for the manufacturer to offer a return policy to increase the volume of returneditems. On the basis of the definitions provided in the previous sections, we can rewrite Equation (18) to expressreturn flows for each t 2 �, as a function of demand and return fraction, i.e.:

r tð Þ ¼Xtk¼1

d kð Þ � kð Þ � Zkt ð29Þ

Let � represent the return policy, expressed as a percentage of the logistics cost �0 of the item. In the case a returnpolicy is applied, the unitary cost of returned items increases to � ¼ �0 � 1þ �ð Þ. At the same time, the return policy isexpected to enhance the return volume, thus reducing the cost of manufacturing/purchasing new items. Hence, theopportunity of introducing a return policy is modelled inside a total cost-minimisation problem, which allowsderiving the optimal value of the return policy. As proposed by Thierry et al. (1995) for returns in function of theintensity of environmental policies, we model customer compliance for returning items with an ‘S’ curve, as in thefollowing equation:

� ¼ p1 þ p2 � p1� �

�1ffiffiffiffiffiffi2�p

�

Z �

�1

e�12

x��ð Þ

2

dx, ð30Þ

According to the inventory control model described, the manufacturer places an order, of quantity EOQ, if thestock-out probability exceeds a defined threshold, i.e.:

s tð Þ ¼EOQ, if ��t ��

0, else

�ð31Þ

and:

p tð Þ ¼0, if t � nDLT

s t� nDLTð Þ, if t4 nDLT

�: ð32Þ

If we define IH0 as the starting on-hand inventory level (cf. Table 1), the transition equations for inventories can beexpressed as follows:

IH tð Þ ¼ IH t� 1ð Þ þ r tð Þ � d tð Þ þ p tð Þ, ð33Þ

I tð Þ ¼ I t� 1ð Þ þ r tð Þ � d tð Þ þ s tð Þ, ð34Þ

where IH 0ð Þ ¼ I 0ð Þ ¼ IH0 . The formulation of the optimal return policy for the MRS can be finally obtained asfollows:

min�

TC ¼Xtmax

t¼1

cp � p tð Þ þ � � r tð Þ þ hI � IH tð Þ ð35Þ

3.5 Effects of stochasticity on the optimal return policy

Our next purpose is to investigate the behaviour of a simulated system operating under the EOQ model described inSections 3.2–3.4; in particular, we focus on calculating the optimal return policy under different stochasticconditions. Since the analysis performed here is based on simulation, some assumptions should be made with regardto the demand behaviour. Without loss of generality, we consider the case of seasonal demand, with:

� tð Þ ¼ aþ b � sin !tð Þ ð36Þ

where ! ¼ 2�=T. Demand stochasticity is then modelled as:

d tð Þ ¼ � tð Þ þ e ð37Þ

where e�N ð0, �2dÞ.


We have previously mentioned that the inventory control model is valid regardless of the distribution of RLT(t).

Nonetheless, to apply Equation (7), either for simulation purpose or in a real scenario, we have to assume (or know)

the distribution of RLT tð Þ. We suppose here that RLT(t) is the integer rounding of a normal random variable, with

parameters �RLT and �2RLT, so pkt is calculated as:

pkt ¼ �t� kþ 0:5ð Þ � �RLT

�RLT

� ��

t� k� 0:5ð Þ � �RLT

�RLT

� : ð38Þ

We model as a beta random variable over the interval 0, 1½ � with parameters �1 and �2, which can be calculated

from � and � as follows:

�1 ¼ � �� 1� �

� ��2

� 1

" #

�2 ¼ 1� � � �

�� 1� �

� ��2

� 1

" # :8>>>>><>>>>>:

ð39Þ

The positivity constraint for �1 and �2 requires �2 5� � 1� � � �

; thus, we define a maximum allowed value

of � as:

�max :¼ k �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� � 1� �

� �qð40Þ

with k 2 0, 1� ½.We report in Table 2 the parameters used for the simulated scenario. Note that the hypothesis �RLT nDLT is

adequately respected, since the maximum value of �RLT used in simulation (cf. Table 3) is 12 and the probability for

a normal distribution with expected value 50 and standard deviation 12 to be lower than 20 (nDLT) is negligible

(� 0:62 %).The first simulation investigates the effect of demand stochasticity (�d) on the optimal �; �d was varied from 0 to

20 (step 5). The optimal value of � was derived by comparing the experimental mean values of the total cost obtained

with different �, ranging from 0–1 (step 0.05), with 50 replications for each �. As Table 3 shows, a (slight) decreasing

trend for the optimal return policy as a function of �d is observed, whereas the costs calculated in correspondence of

an optimal return policy showed an increasing trend.

Table 2. Simulation parameters for the basic scenario.

AcronymNumerical

value

a [–] 50b [–] 10T [days] 250tmax [days] 750EOQ [items] 544nDLT [days] 20�0 [E/item] 3.2cp [E/item] 10hI [E/item/day] 0.1p1 [–] 0.3p2 [–] 0.95� [–] 0.5� [–] 0.2�RLT [days] 50�RLT [days] 1�d [items/day] 0� [–] 0�� [–] 0.01


The sensitivity analysis for �RLT (Table 4) shows a similar behaviour. Specifically, we found a decreasing trendfor the optimal return policy and an increasing trend for the optimal cost, as a function of �RLT. Conversely, theoptimal return policy is almost insensitive to � (Table 5), although the optimal cost still displays an increasingtrend.

4. Case study

In this section, we exploit the inventory control model developed before to evaluate the usefulness of establishing areturn policy in a case study, referring to the problem of asset (i.e., pallet) management between a company and itscustomers. The company considered is headquarted in Milan (Italy), where it operates as a confectionery andcroissant manufacturer, offering more than 400 bakery and ice cream products. Such products are primarily sold inhotels, restaurants and catering (HO.RE.CA.) and retail markets, where the company serves approximately 50customers. To deliver products to customers, the company handles about 300,000 pallets, 13% of which should beannually replaced, due to damage, theft and loss; theft and loss are responsible for about half the amount of palletspurchased annually. Although the current return fraction for assets is about 87% (i.e., 1–39,000/300,000), thecompany incurs significant costs for the purchase of new pallets, accounting for about 350,000 E/year, havingestimated in 9 E/asset the cost of a new asset.

Thanks to some contacts with the company, we got the data related to forward and reverse flows of assets forabout 660 working days (from day 220 to day 880). A partial list of flows is provided in Figure 3. It can be seen fromFigure 3 that, in the current situation, the company knows the amount of assets shipped and returned each day, aswell as the stock of assets available. According to the company’s asset management policy, an order for purchasingnew pallets is placed anytime the current stock is lower than a defined threshold. The quantity purchased is always550 pallets, corresponding to a full-truck-load shipment. The amount of pallets shipped and received each day, inturn, is generated by fulfilment of customers’ orders and returns from customers respectively. As an example,Figure 3 shows the details of quantity shipped and returned at day 225. It can be seen from the figure that eachshipment is coded as ‘Zxx-yyyy’, where ‘Z’ indicates the product, ‘xx’ is the customer’s identifier and ‘yyyy’ isthe shipping number. The company sends products (and pallets) in response to the customer’s demand, by means offull-truck-load shipments; hence, each shipment includes 29–33 pallets. Once an order is shipped, it arrives at the

Table 5. Sensitivity analysis for the effect of �RLTon the optimal return policy.

�RLT [days]Optimal return

policy [%]Optimalcosts [E]

1 0.7 362.255 0.6 375.258 0.65 383.7710 0.6 390.9212 0.6 393.10

Table 3. Sensitivity analysis for the effect of �d on theoptimal return policy.

�d[items/day]

Optimal returnpolicy [%]

Optimalcosts [E]

0 0.7 362.255 0.7 363.1710 0.7 365.6715 0.65 378.4920 0.65 393.84

Table 4. Sensitivity analysis for the effect of � on theoptimal return policy.

� [–]Optimal return

policy [%]Optimalcosts [E]

0 0.7 362.250.05 0.7 362.200.1 0.65 363.930.15 0.7 367.250.20 0.7 371.11


Figure

3.Partiallist

offorw

ard

andreverse

flowsforthecase

studycompany.


customer’s site within two working days, and the pallets received are included in the stock of assets available at thecustomer’s site. The procedure adopted by the company for managing reverse flows of assets is the deferredexchange (European Commission 2007). This means that customers return pallets to the company when they cansend back a full-truck-load shipment, including approximately 520–550 empty pallets. Once the return flow isshipped, it will reach the company within two working days. The time required to collect 520–550 pallets is 50 dayson average (cf. Table 6); as an example, it can be seen from Figure 3 that pallets which return to the company at day225 were shipped to the customer starting from day 189.

The case study company considered is a good example of a mixed MRS, where pallets can be either sent back bycustomers or purchased externally; hence, the issue of avoiding the occurrence of out-of-stock situations is relevant,as well as the potential application of a return policy for enhancing the return flows. To our knowledge, financialincentives are not currently exploited by the company.

Historical data concerning forward and reverse flows were used to derive the statistical parameters of demandand returns, to allow their correct simulation. Table 6 shows a list of input data derived from the analysis of thehistorical flows of the company, as well as from direct contacts with it. Then, we have carried out an economicanalysis by comparing:

. the ‘AS IS’ scenario, which reflects the current asset management policy of the company, as it results fromthe forward and reverse flows in Figure 3; and

. several ‘TO BE’ scenarios, where we apply the inventory control model proposed in Section 3 to decidewhether an order should be placed, and evaluate the usefulness of exploiting a return policy, with differentvalues.

Results of the analysis are proposed in Table 7 and graphically shown in Figure 4. As a first outcome, it can be seenfrom Table 7 that, to increase the current return rate of 87%, the company should apply a relatively high returnpolicy, ranging from �¼ 0.79 in TO BE-1 to �¼ 1.20 in TO BE-10. This is due to the fact that the return rate is ratherhigh in the AS IS scenario. For the same reason, we observe that, despite the return policy decreasing the cost ofexternal purchase, there are no TO BE scenarios which score a lower total cost than the AS IS situation.Consequently, the percentage variation of costs when moving from the AS IS to the TO BE scenario is alwaysgreater than zero. Looking at Figure 4, one can also see that the total cost as a function of � first decreases thenincreases, with two scenarios (i.e., TO BE-6 and TO BE-7) scoring a total cost which is close to the AS IS situation.Besides the high � of the AS IS situation, we also conjecture that TO BE scenarios are not profitable in terms of thetotal cost because of the relatively low value of assets cp compared with the logistic cost �0. The cost of new assetsbeing low, it is not advantageous for the company to pay a high logistic cost for increasing the return flows;conversely, for assets having a higher economic value, increasing the return flows can be more profitable to thecompany. Hence, we have investigated the sensitivity of the percentage variation of the total cost as a function of cp,

Table 6. Input data for the case study company.

Acronym Numerical value

tmax 880 days�(t) 621.11 pallet/day�d 6.17 pallet/day� 0.87� 0EOQ 550 pallets�RLT 50 days�RLT 9.82 dayscp 9E/pallethI 0.04E/day/palletnDLT 4 days�0 0.8E/palletp1 0.3p2 0.95� 0.5� 0.2


ranging from 9–19E/asset. The results, proposed in Table 8 and Figure 5, support our conjecture: as a matter of fact,TO BE scenarios generate savings (i.e. a negative percentage variation of the total cost) when cp exceeds 13E/asset.It can also be observed that, the higher the cost of assets, the more profitable the application of a return policy: forinstance, for cp¼ 19E/asset, some TO BE scenarios generate savings which exceed 13.9% (e.g. �¼ 0.932 or�¼ 0.934).

5. Managerial implications, conclusions and future research directions

The aim of our paper was to propose a stochastic EOQ-based analytic model for inventory management in RLenvironments, with a particular focus on the case of MRS applying return policies. The model developed has somestrong points. First, it is more realistic than several works available in the literature, as it incorporates stochasticityof demand, return fraction and lead times, which commonly affect real scenarios. Hence, we think that theformulation proposed may serve practitioners as a theoretical framework for the implementation of EOQ policies inRL and MRS. At the same time, the model is general, as its formulation does not require specific probabilitydistributions for demand, return fraction and lead times. Second, our approach is based on the estimation of astock-out probability to define whether an order should be placed; ultimately, it may constitute a useful instrumentto increase supply chain efficiency and control inventory levels without compromising customer satisfaction. Third,

TO BE-1

TO BE-2 TO BE-3

TO BE-4

TO BE-5

TO BE-6 TO BE-7

TO BE-8

TO BE-9

TO BE-10

1.380,00

1.400,00

1.420,00

1.440,00

1.460,00

1.480,00

1.500,00

1.520,00

1.540,00

1.560,00

Figure 4. Total cost as a function of the TO BE scenario for the case study company.

Table 7. Results of the case study.

Scenario � � �

Cost ofreturn flows

[E/day]

Cost ofpurchasing[E/day]

Inventorycost [E/day]

Totalcost [E/day]

Percentagevariation

AS IS!TO BE

AS IS 87.00% 0.800 – 432.13 730.07 109.39 1271.60 –TO BE-1 88.19% 1.430 0.787 783.12 654.80 112.84 1550.76 21.95%TO BE-2 90.11% 1.440 0.800 805.95 556.96 125.03 1487.94 17.01%TO BE-3 90.76% 1.480 0.850 834.33 526.85 125.54 1486.72 16.92%TO BE-4 92.23% 1.519 0.899 870.34 459.11 131.28 1460.74 14.87%TO BE-5 90.03% 1.531 0.914 884.74 383.85 138.32 1406.91 10.64%TO BE-6 93.15% 1.546 0.932 894.30 353.74 149.57 1397.61 9.91%TO BE-7 93.15% 1.547 0.934 895.02 353.74 149.57 1398.33 9.97%TO BE-8 93.25% 1.600 1.000 926.74 353.74 148.39 1428.87 12.37%TO BE-9 93.25% 1.620 1.025 938.26 353.74 152.38 1444.38 13.59%TO BE-10 93.25% 1.760 1.200 1019.41 353.74 152.38 1525.53 19.97%


the estimation of the stock-out probability is only based on the knowledge of historical deliveries, whereas it doesnot exploit return flows as input data; hence, the model is suitable to be adopted also by a company which did notestablish a proper tracking system for return flows. As regards the limitations of the model, the main one is itscomplexity, which is a consequence of the need for including stochasticity in the mathematical formulation. Thiscould prevent its adoption in practical cases. Moreover, we should remark that our model may not be adequate forcontexts where the delivery lead time cannot be approximated as deterministic.

The model developed was investigated through simulations, and then tested by means of a case study. Results ofthe simulation lead to the following considerations. First, we observed a negative effect of demand and return leadtime stochasticity on the optimal return policy, confirming that the calculation of an optimal return policy withdeterministic models would not be adequate. We may additionally deduce that stochasticity seems to make returnsless appealing for firms, although no apparent effect of return fraction stochasticity was observed. The application

Figure 5. Percentage variation of the total cost as a function of � and cp for the case study company.

Table 8. Percentage variation of the total cost as a function of � and cp for the case study company.

cp [E/asset]

Scenario � � 9 10 11 12 13 14 15 16 17 18 19

TO BE-1 88.19% 0.787 21.95% 20.03% 18.32% 16.79% 15.42% 14.18% 13.06% 12.03% 11.09% 10.22% 9.43%TO BE-2 90.11% 0.800 17.01% 14.58% 12.43% 10.50% 8.77% 7.20% 5.78% 4.49% 3.30% 2.21% 1.20%TO BE-3 90.76% 0.850 16.92% 14.25% 11.88% 9.76% 7.86% 6.14% 4.58% 3.15% 1.85% 0.65% –0.46%TO BE-4 92.23% 0.899 14.87% 11.77% 9.02% 6.56% 4.35% 2.35% 0.54% –1.12% –2.63% –4.03% –5.31%TO BE-5 90.03% 0.914 10.64% 7.18% 4.10% 1.35% –1.12% –3.35% –5.37% –7.22% –8.91% –10.47% –11.90%TO BE-6 93.15% 0.932 9.91% 6.24% 2.99% 0.08% –2.53% –4.90% –7.04% –9.00% –10.79% –12.43% –13.95%TO BE-7 93.15% 0.934 9.97% 6.30% 3.04% 0.13% –2.49% –4.85% –7.00% –8.96% –10.75% –12.40% –13.92%TO BE-8 93.25% 1.000 12.37% 8.55% 5.17% 2.14% –0.57% –3.03% –5.26% –7.29% –9.16% –10.87% –12.45%TO BE-9 93.25% 1.025 13.59% 9.70% 6.25% 3.17% 0.40% –2.10% –4.38% –6.45% –8.35% –10.09% –11.70%TO BE-10 93.25% 1.200 19.97% 15.70% 11.91% 8.53% 5.49% 2.74% 0.25% –2.03% –4.11% –6.03% –7.80%


of the model to a case study, referring to the problem of assets management between a company and its customers,highlighted some further points. First, we noted that the application of a return policy for the company examined isnot profitable compared to the existing asset management policy. This is due to the fact that the current return rateis sufficiently high (i.e. about 87%); thus, increasing the return flows does not involve further benefits for thecompany. At the same time, we found that the return policy could be profitable for assets of higher economicalvalue (from 13–19E/asset), leading to a significant decrease of the total cost (up to 13.9%) compared with thecurrent scenario. The above results suggest that the application of a return policy should be carefully evaluatedaccording to the specific operating condition considered; the model we developed was proven to be a useful tool forsupporting such an evaluation.

Starting from the current paper, an interesting future research direction could be to develop a model consideringthe availability of additional historical information, namely connections between return flows and historicaldeliveries, which may apply to contexts where specific tracking systems have been implemented.

Acknowledgements

The authors wish to express their gratitude to the anonymous reviewers and to the guest editor, whose constructive suggestionsled to a significant improvement of the earlier version of the manuscript.

References

Alinovi, A., et al., 2009. Reverse Logistics: an optimal return policy based on a simulation approach. In: M. Macchi and

R. Pinto, eds., Proceedings of the 11th international conference on the modern information technology in the innovation

processes of the industrial enterprises, Vol. 1, 15–16 October 2009, Bergamo, Italy, 531–539.Beamon, M.B. and Fernandes, C., 2004. Supply-chain network configuration for product recovery. Production Planning and

Control, 15 (3), 270–281.Bloemhof, J. and van Nunen, J., 2005. Integration of environmental management and SCM. ERIM Report Series Research in

Management; ERS-2005-030-LIS.Breen, L., 2006. Give me back my empties or else! A preliminary ananlysis of customer compliance in reverse logistics pratices

(UK). Management Research News, 29 (9), 532–551.Castell, A., Clift, R., and France, C., 2004. Extended producer responsibility policy in the European Union – a horse or a camel?

Journal of Industrial Ecology, 8 (1/2), 4–7.Chanintrakul, P., et al., 2009. Reverse logistics network design: a state-of-the-art literature review. International Journal of

Business Performance and Supply Chain Modelling, 1 (1), 61–81.

Dobos, I. and Richter, K., 2004. An extended production/recycling model with stationary demand and return rates. InternationalJournal of Production Economics, 90 (3), 311–323.

Emmons, H. and Gilbert, S.M., 1998. Note: the role of returns policies in pricing and inventory decisions for catalogue goods.Management Science, 44 (2), 276–283.

European Commission, 2000. Directive 2000/53/ec of the European Parliament and of the Council of 18 September 2000on end-of-life vehicles. Official Journal, L 269, 34. Available from: http://eur-lex.europa.eu/LexUriServ/

LexUriServ.do?uri=CONSLEG:2000L0053:20050701:EN:PDF [Accessed 13 August 2009].European Commission, 2003. Directive 2002/96/EC of the European Parliament and of the Council of 27 January 2003 on waste

electrical and electronic equipment (WEEE). Official Journal, L 37, 24–38. Available from: http://eur-lex.europa.eu/

LexUriServ/LexUriServ.do?uri=OJ:L:2003:037:0024:0038:EN:PDF [Accessed 13 August 2009].European Commission, 2007. BRIDGE – building radio frequency identification for the global environment – WP9: returnable

transport items: the market for EPCglobal applications. Available from: www.bridge-project.eu [Accessed 11 July 2008].Fleischmann, M., Kuik, R., and Dekker, R., 2002. Controlling inventories with stochastic returns: a basic model. European

Journal of Operational Research, 138 (1), 63–75.Hu, T., Sheu, J., and Huan, K., 2002. A reverse logistics cost minimization model for the treatment of hazardous wastes.

Transportation Research Part E, 38 (6), 457–473.Inderfurth, K., Lindner, G., and Rachaniotis, N.P., 2005. Lot sizing in a production system with rework and product

deterioration. International Journal of Production Research, 42 (24), 5167–5184.Jaber, M.Y. and Rosen, M.A., 2008. The economic order quantity repair and waste disposal model with entropy cost. European

Journal of Operational Research, 188 (1), 109–120.Ketzenberg, M., 2009. The value of information in a capacitated closed loop supply chain. European Journal of Operational

Research, 198 (2), 491–503.Klausner, M. and Hendrickson, C.T., 2000. Reverse Logistics strategy for product take-back. Interfaces, 30 (3), 156–165.


Koh, S.G., et al., 2002. An optimal ordering and recovery policy for reusable items. Computers and Industrial Engineering, 43 (1),

59–73.Konstantaras, I. and Skouri, K., 2009. Lot sizing for a single product recovery system with variable setup numbers. European

Journal of Operational Research, 203 (2), 326–335.Lau, H.S. and Lau, A.H.L., 1999. Manufacturer’s pricing strategy and return policy for a single period commodity. European

Journal of Operational Research, 116 (2), 291–304.Lee, C.H., 2001. Coordinated stocking, clearance sales, and return policies for a supply chain. European Journal of Operational

Research, 131 (3), 491–513.Liang, Y., Pokharel, S., and Lim, G.H., 2009. Pricing used products for remanufacturing. European Journal of Operational

Research, 193 (2), 390–395.Lieckens, K. and Vandaele, N., 2007. Reverse logistics network design with stochastic lead times. Computers and Operations

Research, 34 (2), 395–416.Linton, J.D., Yeomans, J.S., and Yoogalingam, R., 2005. Recovery and reclamation of durable goods: a study of television

CRTs. Resources, Conservation and Recycling, 43 (4), 337–352.Marien, E.J., 1998. Reverse logistics as competitive strategy. Supply Chain Management Review, 34 (2), 43–52.Oh, Y.H. and Hwang, H., 2006. Deterministic inventory model for recycling system. Journal of Intelligent Manufacturing, 17 (4),

423–428.Pochampally, K.K., Gupta, S.M., and Govindan, K., 2009. Metrics for performance measurement of a reverse/closed-loop

supply chain. International Journal of Business Performance and Supply Chain Modelling, 1 (1), 8–32.Ravi, V. and Shankar, R., 2005. Analysis of interactions among the barriers of reverse logistics. Technological Forecasting and

Social Change, 72 (8), 1011–1029.Richter, K., 1996. The EOQ and waste disposal model with variable setup numbers. European Journal of Operational Research,

95 (2), 313–324.Rogers, D.S. and Tibben-Lembke, R.S., 1999. Going backwards: Reverse Logistics trends and practices. Pittsburg, PA: RLEC

Press.Savaskan, R.C., Bhattacharya, S., and Van Wassenhove Luk, N., 2004. Closed-loop supply chain models with product

remanufacturing. Management Science, 50 (2), 239–252.Schrady, D.A., 1967. A deterministic inventory model for repairable items. Naval Research Logistics Quarterly, 14 (3), 391–398.

Setoputro, R. and Mukhopadhyay, S.K., 2004. Reverse logistics in e-business – optimal price and return policy. International

Journal of Physical Distribution & Logistics Management, 34 (1), 70–88.

Teunter, R.H., 2004. Lot-sizing for inventory systems with product recovery. Computer and Industrial Engineering, 46 (3),

431–441.

Thierry, M., et al., 1995. Strategic issues in product recovery management. California Management Review, 37 (2), 114–135.Tsay, A.A., 2001. Managing retail channel overstock: markdown money and return policies. Journal of Retailing, 71 (4), 457–492.Veerakamolmal, P. and Gupta, S.M., 1999. Analysis of design efficiency for the disassembly of modular electronic products.

Journal of Electronics Manufacturing, 9 (1), 79–95.VerpackV, 1998. Verordnung uber die Vermeidung und Verwertung von Verpackungsabfallen – VerpackV (Packaging

Ordinance). BGBl, Teil I, Bonn.

Appendix

Proof of proposition 1

To prove Equation (9), we first introduce two new random variables, Yq�i and Yqþ

i , representing the return amounts on day i, dueto historical deliveries until day q and to future deliveries until day i, respectively. Hence, we can state that:

Yq�i ¼

Xqk¼1

~dk � k � Zki ðA1Þ

Yqþi ¼

Xik¼qþ1

dk � k � Zki ðA2Þ

Yqi ¼ Y

q�i þ Y

qþi ðA3Þ


Owing to the linearity properties of the expected value of random variables, we deduce:

E Yqi

� �¼ E Y

q�i þ Y

qþi

� �¼ E Y

q�i

� �þ E Y

qþi

� �¼ E

Xqk¼1

~dk � k � Zki

" #

þ EXi

k¼qþ1

dk � k � Zki

" #¼Xqk¼1

E ~dk � k � Zki

h iþXi

k¼qþ1

E dk � k � Zki½ �: ðA4Þ

Further simplifications for Equation (A4) can be obtained observing that ~dk are deterministic and known parameters, which canbe extracted from the argument of the expected value on account of the linearity properties. Moreover, k and Zki areindependent random variables (for any k), so that the expected value of their product can be transformed into the product oftheir expected values. Similar considerations can be drawn from the independence of the three random variables dk, k and Zki.According to the definitions above, and having recognised that pki is the expected value of Zki, we obtain:

E Yqi

� �¼Xqk¼1

~dk � E k½ � � E Zki½ � þXi

k¼qþ1

E dk½ � � E k½ � � E Zki½ �

¼Xqk¼1

~dk � � � pki þXi

k¼qþ1

�k � � � pki ¼ � �Xqk¼1

~dk � pki þXi

k¼qþ1

�k � pki

!

which proves Proposition 1.


In order to calculate the variance of Yqi , we first use Equation (A3) and observe that Yq�

i and Yqþi are independent, as they

represent returns on day i due to previous deliveries which occurred in disjoint time horizons (i.e. the days until and after q,respectively). This observation allows writing the variance of the sum of the two random variables as the sum of the variances, asfollows:

Var½Yqi � ¼ Var½Y

q�i þ Y

qþi � ¼ Var½Y

q�i � þ Var½Y

qþi � ðA5Þ

We shall continue by calculating Var½Yq�i �, which is the variance of a sum of products, according to Equation (A1). All addends

are independent of one another, since deliveries fulfilled on different days will have independent future destination, and thisobservation again allows writing the sum variance as a sum of variances. In addition, we can extract ~dk from the variancearguments, owing to its deterministic nature; as a result, we obtain:

Var Yq�i

� �¼ Var

Xqk¼1

~dk � k � Zki

" #¼Xqk¼1

Var ~dk � k � Zki

h i¼Xqk¼1

~d2k � Var k � Zki½ � ðA6Þ

We now calculate the variance of the product k � Zki, applying the general variance property, according to whichVar A½ � ¼ E ½A2� � E A½ �ð Þ

2, for any random variable A. We will then use the independence between k and Zki to convert theexpected value of their product into the product of their expected values: note that the independence between k and Zki implies 2k and Z2

ki to be independent too, and hence similar transformations can be operated, as follows:

Var k � Zki½ � ¼ E ½ k � Zkið Þ2� � E k � Zki½ �ð Þ

2¼ E ½ 2

k � Z2ki� � E k½ � � E Zki½ �ð Þ

2

¼ E ½ 2k� � E ½Z

2ki� � E k½ �ð Þ

2� E Zki½ �ð Þ

2ðA7Þ

By exploiting again the aforementioned variance property (this time as E ½A2� ¼ Var A½ � þ E A½ �ð Þ2), Equation (A7) can be

expressed as a function of the model parameters:

E ½ 2k� � E ½Z

2ki� � E k½ �ð Þ


2

¼ Var k½ � þ E k½ �ð Þ2

� �� Var Zki½ � þ E Zki½ �ð Þ

2� �

� E k½ �ð Þ2� E Zki½ �ð Þ

2

¼ �2 þ �2

� ½ pki � 1� pkið Þ þ p2ki� � �

2 p

2ki

¼ �2 þ �2

� pki � �

2 p

2ki: ðA8Þ

Hence, we obtain the variance of Yq�i as:

Var Yq�i

� �¼Xqk¼1

~d2k � �2 þ �2

� pki � �

2 p

2ki

h iðA9Þ


On the basis of a similar approach, we obtain an expression for the variance of Yqþi as follows:

Var Yqþi

� �¼ Var

Xik¼qþ1

dk � k � Zki

" #¼

Xik¼qþ1

Var dk � k � Zki½ �

¼Xi

k¼qþ1

E dk � k � Zkið Þ2

� �� E dk � k � Zki½ �ð Þ

2� �

¼Xi

k¼qþ1

E dkð Þ2� kð Þ

2� Zkið Þ

2� �

� E dk½ � � E k½ � � E Zki½ �ð Þ2

� �

¼Xi

k¼qþ1

E dkð Þ2

� �� E kð Þ

2� �

� E Zkið Þ2

� �� E dk½ �ð Þ

2� E k½ �ð Þ


2� �

¼Xi

k¼qþ1

Var dk½ � þ E dk½ �ð Þ2

� �� Var k½ � þ E k½ �ð Þ

2� �

� Var Zki½ � þ E Zki½ �ð Þ2

� �� E dk½ �ð Þ

2� E k½ �ð Þ


2�

¼Xi

k¼qþ1

�2d þ �2k

� �� 2 þ �

2

� pki � �

2k�

2 p

2ki

h i: ðA10Þ

Proposition 2 can be finally obtained by substituting the results of Equations (A9)–(A10) in Equation (A5), which completesproof 2.


In order to prove Proposition 3, some covariance properties have to be applied. We first apply the bilinearity property, accordingto which Cov½

Pni Ai,

Pmj Bj� ¼

Pni

Pmj Cov½Ai,Bj�, given any random variables Ai, i ¼ 1, . . . , , n and Bj, j ¼ 1, . . . , ,m. Hence, we

obtain:

Cov½Yqi ,Y

qj � ¼ Cov½Y

q�i þ Y

qþi ,Y

q�j þ Y

qþj �

¼ Cov½Yq�i ,Yq�

j � þ Cov½Yq�i ,Yqþ

j � þ Cov½Yqþi ,Yq�

j � þ Cov½Yqþi ,Yqþ

j � ðA11Þ

It is immediate to observe that Cov½Yq�i ,Y

qþj � ¼ Cov½Y

qþi ,Y

q�j � ¼ 0, on the basis of the independence between the variables in

brackets. In fact, Yq�i and Y

qþj are independent because they represent returns on days i and j due to previous deliveries which

occurred in disjoint time horizons. Analogous considerations apply for Yqþi and Y

q�j .

We now calculate Cov½Yq�i ,Yq�

j �, being aware that, in this case, no independence assumption holds. We will firstly apply thebilinearity property; then, we will use the property Cov A,B½ � ¼ E A � B½ � � E A½ � � E B½ �, given any random variables A and B.

Cov½Yq�i ,Y

q�j � ¼ Cov

Xqk¼1

~dk � k � Zki,Xqh¼1

~dh � h � Zhj

" #¼Xqk¼1

Xqh¼1

Cov ~dk � k � Zki, ~dh � h � Zhj

h i

¼Xqk¼1

Xqh¼1

E ~dk � k � Zki �~dh � h � Zhj

h i� E ~dk � k � Zki

h i� E ~dh � h � Zhj

h i : ðA12Þ

On the basis of the deterministic nature of ~dk and ~dh and of the independence between k and Zki, we can easily obtain:

E ½ ~dk � k � Zki� ¼~dk � E ½ k � Zki� ¼

~dk � E ½ k� � E ½Zki� ¼~dk � � � pki

E ½ ~dh � h � Zhj� ¼~dh � E ½ h � Zhj� ¼

~dh � E ½ h� � E ½Zhj� ¼~dh � � � phj

In order to develop E ½ ~dk � k � Zki � ~dh � h � Zhj�, two distinct cases have to be considered.

. Case k ¼ h: in this case, we have

E ½ ~dk � k � Zki �~dh � h � Zhj� ¼ E ½ ~dk � k � Zki �

~dk � k � Zkj�

¼ E ½ ~d2k � 2k � Zki � Zkj� ¼

~d2k � E ½ 2k� � E ½Zki � Zkj�

¼ ~d2k � �2 þ �

2

� E ½Zki � Zkj�

We should remark that Zki and Zkj are not independent, according to constraint (8) implying their sum to be lower than 1.Hence, one between Zki and Zkj should score zero, as they are binary variables and we assumed i 6¼ j. We therefore deduce that


E ½Zki � Zkj� ¼ 0 and obtain:

E ½ ~dk � k � Zki �~dh � h � Zhj�

��k¼h¼ 0:

. Case k 6¼ h: in this case, we have

E ½ ~dk � k � Zki � ~dh � h � Zhj� ¼ ~dk � ~dh � E ½ k� � E ½ h� � E ½Zki� � E ½Zhj� ¼ ~dk � ~dh � �2 � pki � phj:

Note that Zki and Zhj are independent when k 6¼ h. Finally, we can write the expression of Cov½Yq�i ,Yq�

j � as:

Cov½Yq�i ,Y

q�j � ¼

Xqk¼1

Xqh¼1

kh � ~dk � ~dh � �2 � pki � phj �

~dk � ~dh � �2 � pki � phj

ðA13Þ

where kh is defined as binary parameter scoring 1 if k 6¼ h and zero otherwise. Omitting null addends, Equation (A12) can bemore simply rewritten as:

Cov½Yq�i ,Y

q�j � ¼ ��

2 �Xqk¼1

~d2k � pki � pkj ðA14Þ

In a similar way, looking back at Equation (A11), we obtain an expression for Cov½Yqþi ,Yqþ

j �, as follows:

Cov½Yqþi ,Y

qþj � ¼ Cov

Xik¼qþ1

dk � k � Zki,Xj

h¼qþ1

dh � h � Zhj

" #

¼Xi

k¼qþ1

Xjh¼qþ1

Cov dk � k � Zki, dh � h � Zhj

� �

¼Xi

k¼qþ1

Xjh¼qþ1

E dk � k � Zki � dh � h � Zhj

� �� E dk � k � Zki½ � � E dh � h � Zhj

� �� Following the same approach described before, we have to consider two further situations.

. Case k ¼ h: in this case, we have

E ½dk � k � Zki � dh � h � Zhj� ¼ E ½d2k� � E ½ 2k� � E ½Zki � Zkj� ¼ 0,

because E ½Zki � Zkj� ¼ 0.

. Case k 6¼ h: in this case, we have

E ½dk � k � Zki � dh � h � Zhj� ¼ E ½dk� � E ½dh� � E ½ k� � E ½ h� � E ½Zki� � E ½Zhj�

¼ �k � �h � �2 � pki � phj:

Combining the results described above, we obtain:

Cov½Yqþi ,Yqþ

j � ¼ ��2 �

Xmin i,jf g

k¼qþ1

�2k � pki � pkj ðA15Þ

Proposition 3 can be finally derived, by substituting Equations (A14–A15) into Equation (A11) and omitting two null addends,thus completing proof 3.


Proof for Equation (12) directly follows from Equation (2), by substituting E ½Yqi � with its expression derived in Equation (9) and

applying the linearity property of the expected value of random variables. Proof for Equation (13) can be obtained applying thevariance property, according to which Var

Pni Ai

� �¼Pn

i Var Ai½ � þ 2Pn

i

Pj5i Cov Ai,Aj

� �, for any random variables Ai. In this

case, we have:

Var Yq½ � ¼ VarXqþnDLT

i¼qþ1

Yqi

" #¼

XqþnDLT

i¼qþ1

Var Yqi

� �þ 2

XqþnDLT

i¼qþ1

Xj5i

Cov Yqi ,Y

qj

h iðA16Þ

Equation (13) can now be easily obtained by substituting the expressions of Equations (10 and 11) for Var½Yqi � and Cov½Y

qi ,Y

qj �

respectively, into Equation (A16), thus completing proof 4.



To prove Proposition 5, we apply covariance bilinearity properties as follows:

Cov Dq,Yq½ � ¼ CovXqþnDLT

i¼qþ1

di,XqþnDLT

j¼qþ1

Yqj

" #¼

XqþnDLT

i¼qþ1

XqþnDLT

j¼qþ1

Cov di,Yqj

h i

¼XqþnDLT

i¼qþ1

XqþnDLT

j¼qþ1

Cov di,Xqk¼1

~dk � k � Zkj þXj

k¼qþ1

dk � k � Zkj

" #

¼ 0þXqþnDLT

i¼qþ1

XqþnDLT

j¼qþ1

Cov di,Xj

k¼qþ1

dk � k � Zkj

" #

¼XqþnDLT

i¼qþ1

XqþnDLT

j¼qþ1

Xjk¼qþ1

Cov di, dk � k � Zkj

� �ðA17Þ

If i 6¼ k, we have Cov½di, dk � k � Zkj� ¼ 0, due to independence between the argument variables. Conversely, if i ¼ k, thefollowing calculation must be performed:

Cov½di, dk � k � Zkj� ¼ Cov½dk, dk � k � Zkj� ¼ E ½d2k � k � Zkj�

� E ½dk� � E ½dk � k � Zkj� ¼ �2d þ �2k

� �� pkj � �

2k � � � pkj

¼ � � pkj � �2d þ �

2k � �

2k

� �¼ � � pkj � �

2d

Hence, we obtain:

Cov Dq,Yq½ � ¼XqþnDLT

i¼qþ1

Xi

k¼qþ1

� � pki � �2d

which proves Proposition 5.


Proof for Equation (15) directly follows from Equation (3), applying the linearity properties of the expected value of randomvariables. Equation (16) can be proved as follows:

Var Uq½ � ¼ VarXqþnDLT

i¼qþ1

di � Yqi

" #¼ Var

XqþnDLT

i¼qþ1

di

" #þ Var½Yq� � 2 � Cov½Dq,Yq�

¼ nDLT � �2d þ Var½Yq� � 2 � Cov½Dq,Yq�

where we used independence among demanded quantities on different days to convert the variance of their sum into the sum oftheir variances.

With regard to the distribution of the random variable Uq describing the net demand, we assert that a normal approximationcan be adopted. As a matter of fact, Uq is calculated as a linear combination, with additions and subtractions, of randomvariables, some of which (i.e. demanded quantities during delivery lead time) are independent from one another. Theapproximation should get better with the increasing length of the delivery lead time, as a consequence of an increase in thenumber of summed addends. Moreover, we can say that the addends representing quantities to be returned, i.e. Yq

i , arethemselves sums of other random variables as from their definition in Equations (A1–A3), so that the number of addends resultsto be quite larger than what one could firstly figure out. Finally, in various contexts, the daily demand itself can be correctlyapproximated with a normal distribution, since demand is the result of a sum of random variables, which are the daily customers’purchase decisions and this should contribute to improve the approximation. This completes proof 6.


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