European Journal of Operational Research 247 (2015) 804–819

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Supply chain design for unlocking the value of remanufacturing under

uncertainty

Wenyi Chen a, Beste Kucukyazici a, Vedat Verter a,∗, María Jesús Sáenz b

a Desautels Faculty of Management, McGill University, Canadab MIT-Zaragoza International Logistics Program, Zaragoza Logistics Center, Spain

a r t i c l e i n f o

Article history:

Received 14 October 2014

Accepted 25 June 2015

Available online 6 July 2015

Keywords:

Uncertainty

Product recovery

Closed-loop supply chains

Remanufacturing

WEEE

a b s t r a c t

Owing to the technological innovations and the changing consumer perceptions, remanufacturing has gained

vast economic potential in the past decade. Nevertheless, major OEMs, in a variety of sectors, remain reluctant

about establishing their own remanufacturing capability and use recycling as a means to satisfy the extended

producer responsibility. Their main concerns seem to be the potential for the cannibalization of their pri-

mary market by remanufactured products and the uncertainty in the return stream in terms of its volume

and quality. This paper aims at assisting OEMs in the development of their remanufacturing strategy, with

an outlook of pursuing the opportunities presented by the inherent uncertainties. We present a two-stage

stochastic closed-loop supply chain design model that incorporates the uncertainties in the market size, the

return volume as well as the quality of the returns. The proposed framework also explicitly represents the

difference in customer valuations of the new and the remanufactured products. The arising stochastic mixed-

integer quadratic program is not amenable to solution via commercial software. Therefore, we develop a so-

lution procedure by integrating sample average approximation with the integer L-shaped method. In order

to gather solid managerial insights, we present a case study based on BSH, a leading producer of home appli-

ances headquartered in Germany. Our analysis reveals that, while the reverse network configuration is rather

robust, the extent of the firm’s involvement in remanufacturing is quite sensitive to the costs associated with

each product recovery option as well as the relative valuation of the remanufactured products by the cus-

tomers. In the context of the BSH case, we find that among the sources of uncertainty, the market size has the

most profound effect on the overall profitability, and it is desirable to build sufficient expansion flexibility in

the forward network configuration.

© 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the

International Federation of Operational Research Societies (IFORS). All rights reserved.

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1. Introduction

Despite the increasing awareness concerning the benefits of re-

covering the remaining economic value in the end-of-use and end-of-

life products, recycling the material content of the returns continues

to be a more prevalent form of product recovery. Remanufacturing

the returned products so that they perform as good as their new ver-

sions constitutes a higher form of recovery that is not used to the

extent desired by the policy makers in many cases (Atasu,

Van Wassenhove, & Sarvary, 2009). In Europe, for example, the Waste

Electrical and Electronic Equipment (WEEE) Directive of 2002 has

been widely criticized as a “recycling law”, and consequently its re-

cast in 2012 aims at increasing the remanufacturing levels, among

∗ Corresponding author. Tel.: +1 514 398 4661.

E-mail addresses: Wenyi.chen@mail.mcgill.ca (W. Chen), Beste.kucukyazici@

mcgill.ca (B. Kucukyazici), vedat.verter@mcgill.ca (V. Verter), mjsaenz@zlc.edu.es

(M. Jesús Sáenz).

c

http://dx.doi.org/10.1016/j.ejor.2015.06.062

0377-2217/© 2015 Elsevier B.V. and Association of European Operational Research Societies (

All rights reserved.

ther improvements.1 The firm has an option of using the third-party

r developing in-house capability for remanufacturing. The extent of

EMs’ voluntary involvement in remanufacturing, however, often de-

ends on the economic benefits they expect directly (or indirectly)

ut of this activity (Guide, Teunter, & Wassenhove, 2003b). An im-

ortant factor that clouds the OEMs’ capability to assess the potential

enefits of engaging in remanufacturing is the uncertainty in the vol-

me, quality and timing of the returns. Particularly, for the economic

iability of in-house remanufacturing, which we study in this paper,

he firm needs to ascertain that there would be sufficient volume of

eturns eligible for remanufacturing. In Section 6, we discuss one firm

BSH Bosch und Siemens Hausgeräte GmbH), which is currently in-

estigating the possibility of developing in-house remanufacturing

apability for their commercial returns. Other examples of firms that

1 Available at http://ec.europa.eu/environment/waste/weee/index_en.htm.

EURO) within the International Federation of Operational Research Societies (IFORS).

W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819 805

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o not use third-party remanufacturing services include Caterpillar,

ewlett-Packard and Xerox.

Understanding the difference in the customers’ valuation of the

ew and the remanufactured products is crucial for assessing the

rofit potential of remanufacturing. Though customers differ in how

uch they are willing to pay for the remanufactured goods, the ini-

ial (and often uninformed) perceptions of products containing used

omponents are generally negative (Ferrer & Whybark, 2000).2 In

ddition, the OEMs constantly introduce new products in an effort

o sustain/increase their market share, which pronounces the cus-

omers’ depreciation of the value of the remanufactured products.

or example, the introduction of energy-efficient appliances under-

ines the market value of remanufactured merchandise. Thus, OEMs

sually have poor knowledge of their potential secondary market

emand.

A firm’s product recovery strategy specifies its level of commit-

ent to each recovery option; particularly, remanufacturing and re-

ycling. This decision is intertwined with the configuration of the

rm’s closed-loop supply chain (CLSC) under the extended producer

esponsibility laws. Atasu and Wassenhove (2012) point out that the

-waste network design problem is strongly restricted by environ-

ental legislations, such as recycling technology standards and land-

ll bans. In this paper, we focus on return streams with both a healthy

econdary market for remanufactured products and a profit potential

n the recycling market. Such return streams typically include end-of-

se and commercial returns. In this context, CLSC design is a complex

roblem that comprises determining the optimal number and loca-

ion of the distribution centers (DCs) and the return centers (RCs), as

ell as the extent of the OEM’s involvement in remanufacturing and

ecycling activities.

The primary objective of this paper is to develop an integrative,

et practical, decision support tool for the formation of a product re-

overy strategy under the variety of uncertainties faced by the OEM’s

op management. We utilize a profit-maximization framework so as

o incentivize the OEM’s voluntary engagement in remanufacturing

nder uncertainty. This requires the estimation of the potential rev-

nues from the primary and secondary markets as well as the costs

f the development and operation of the firm’s CLSC. Traditionally,

he marketing aspects mentioned above have been studied through

tylized models, whereas the network design problems have been

epresented by mathematical programming formulations. In this pa-

er, we integrate these two modeling approaches by incorporating a

tylized representation of the firm’s primary and secondary markets

n the mathematical programming formulation developed for CLSC

esign under uncertainty. Without this integration one would fail to

onsider the potential profitability of remanufacturing in making the

trategic product recovery and CLSC design decisions.

We seek answers to the following primary research questions:

ow do uncertainties influence the profitability of closed-loop supply

hains? and How does variable cost structure as well as consumer valua-

ion of remanufactured products influence the product recovery strategy

n uncertain business environments? In addition, we explore the ro-

ustness of the CLSC configuration under uncertainties. We propose

n integrated stochastic CLSC design model (IS-CLSC) that explicitly

onsiders different sources of uncertainty as well as the difference in

ustomer valuation of new and remanufactured products. The pro-

osed model enables us to optimize the extent of an OEM’s involve-

ent in each product recovery option and to determine the most ap-

ropriate facility network configuration for supporting this strategy.

One of the main contributions of our work is the incorporation of

oth primary and secondary markets as well as the product recov-

ry choices in a detailed network design model under uncertainty.

2 Some exceptions are retreaded tires for commercial fleet companies and Kodak’s

ingle-use camera (Atasu, Guide, & Wassenhove, 2010; Esenduran, Kemahlioglu-Ziya,

Swaminathan, 2012; Souza, 2008).

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s we summarize in the next section, to the best of our knowledge all

he prevailing work on CLSC design under uncertainty focuses on cost

inimization, and hence ignores the impact of the product markets.

he literature that explicitly represents the market segments for the

ew and remanufactured products and the recovery options, how-

ver, mostly comprises stylized models based on broad assumptions

nd lacking a detailed representation of the dynamics of the network

esign decisions. The arising stochastic mixed integer quadratic for-

ulation is not amenable to solution by commercial software. As a

ethodological contribution, we propose an integration of the inte-

er L-shaped decomposition with sample average approximation that

nables us to use a large number of scenarios in our analyses.

We also developed a new case based on BSH Bosch und Siemens

ausgeräte GmbH’s operation in Germany so as to illustrate the pro-

osed methodology and develop managerial insights. BSH is cur-

ently the largest manufacturer of home appliances in Europe and

ne of the leading companies in the sector worldwide. The case study

s inspired by a real-life problem encountered by BSH concerning

he decision whether or not to offer remanufactured products within

ermany’s unique WEEE take-back scheme. The results provide a

olid understanding of the impact of different sources of uncertainty

n the CLSC configuration. Also, we shed light on the conditions un-

er which BSH needs to develop in-house remanufacturing capabili-

ies in response to the trend of tightening environmental regulations.

The remainder of this paper is organized as follows. In the next

ection, we position our research in the context of the relevant liter-

ture. In Section 3 the IS-CLSC model is presented. We describe the

roposed solution method in Section 4 and report on its performance

n Section 5. In Section 6, we present the BSH case study in order

o highlight the features of the proposed model and show the im-

act of uncertainties on the performance of a real supply chain. Un-

ike a significant majority of the earlier papers, this new case study

s presented at a level of detail that would enable the readers to re-

onstruct the problem instances. In Section 7, we analyze the impact

f uncertainties on network structure, overall profitability and recov-

ry strategy. The computation experiments reported in this section

nable us to provide substantial answers to the research questions

tated above. The paper ends with managerial insights and our con-

luding remarks in Section 8.

. Overview of the literature

Our research draws on two separate streams of literature: reman-

facturing and CLSC network design. In this section, we provide a

eview of the prominent research in each stream and position our

esearch at the point of their intersection. We begin with an overview

f the relevant remanufacturing literature.

Remanufacturing has received attention in the academic litera-

ure for more than a decade. The existing literature has addressed a

ariety of problems spanning from strategic to tactical level issues. In

rder to satisfy the immediate need for firms, the operational aspects

f remanufacturing such as inventory control (Ahiska & King, 2010;

eCroix & Zipkin, 2005; Toktay, Wein, & Zenios, 2000), production

lanning (Ferguson, Guide, Koca, & Souza, 2009; Guide, Jayaraman,

Linton, 2003a) and logistic network design (Fleischmann, Beullens,

loemhof-Ruwaard, & Wassenhove, 2001; Salema, Barbosapovoa,

Novais, 2007) have received the most attention. Other research

fforts have considered remanufacturing from a more strategic

erspective. Significant portion of these papers adapted a game-

heoretic approach. For instance, Majumder and Groenevelt (2001)

odel the competition between an OEM and a local remanufacturer

n a two-period game setting. Ferrer and Swaminathan (2006)

xpand the above model and characterize the optimal strategies

n monopoly and duopoly environments for two-period, multi-

eriod and infinite-period settings. They prove the existence of a

emanufacturing threshold policy in the second period based on

806 W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819

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remanufacturing cost saving. Ferrer and Swaminathan (2010) further

expand the monopoly model by relaxing the perfect substitution

assumption of new and remanufactured products. For the two-period

case, a two-level remanufacturing threshold policy in the second

period based on remanufacturing cost saving is identified.

Remanufactured products are offered as an alternative to the new

products with lower price. Therefore, they may cannibalize the sales

of new products but also may be able to extend the customer base

of the firm attracting price-conscious customers, who previously did

not consider purchasing a product of that brand (Thiel, 1994). A few

recent papers consider the distinguished nature of remanufactured

products. Debo, Toktay, and Wassenhove (2005) investigate joint

pricing and production technology selection decisions faced by

an OEM that considers introducing a remanufacturable product in

a market, which consists of heterogeneous consumers. Ferguson

and Toktay (2006) explore the strategic role of OEM remanufac-

turing as an entry deterrent to local remanufacturing, but with

concerns that the remanufactured product will cannibalize sales of

the higher-margin new product. Atasu, Sarvary, and Wassenhove

(2008) investigate the profitability of remanufacturing system in the

presence of environmentally conscious demands, peer competition

and product life-cycle effects. Örsdemir, Kemahlıoglu-Ziya, and

Parlaktürk (2014) extend Ferguson and Toktay (2006) by capturing

the endogenous quality decision of the OEM. Galbreth, Boyacı, and

Verter (2013) study the required technology innovation affects

reuse decisions. To this end, these recent papers allow the price of

new and remanufactured products to be endogenously determined

via a market-clearing mechanism. In particular, the price-demand

functions for new and remanufactured products are interdependent

and derived based on utility theory; and therefore the potential

cannibalization effect is captured. In the same vein, motivated by

the lack of information for academics studying such problems, Guide

and Li (2010) provide the first attempt to empirically examine the

cannibalization of new product sales by remanufactured goods. To

this end, the authors use Internet auctions to investigate consumer

valuation differences for new and remanufactured products.

Most of these studies mentioned about are characterized by a rel-

atively compartmentalized approach, putting a strong emphasis on

one of the different parts of the system such as production plan-

ning, inventory control, logistic network, distribution channel design,

product acquisition or market segmentation(Geyer, Wassenhove, &

Atasu, 2007). By decoupling these strongly dependent issues, these

studies have two major shortcomings: (i) they fail to consider the in-

tegrated nature of remanufacturing systems and (ii) they are inca-

pable of answering the fundamental question of whether offering a

remanufactured product is profitable, because they presuppose that

the profitability of remanufacturing is obvious (Guide et al., 2003b).

We contribute to this emerging stream of literature by investigat-

ing the profitability of remanufacturing through integration of mar-

ket segmentation and closed-loop network design in the presence of

uncertainty.

A number of researchers address problems with respect to CLSC

network design. Despite the significance of uncertainty in CLSCs, it

has not been adequately analyzed in the past, possibly because of

the difficulty in handling a lot of interdependent involved factors

simultaneously. To the best of our knowledge, Salema et al., 2007

is the first paper that studied a general CLSC network design model

taking uncertainty into account. Assuming that customer’s demand

and return are scenario-dependent, a stochastic programming-based

MILP is proposed. Listes (2007), Lee, Dong, and Bian (2010), Pishvaee,

Jolai, and Razmi (2009) and Zeballos, Gomes, Barbosa-Povoa, and

Novais (2012) also presented two-stage stochastic programs to ex-

plicitly include uncertainty. Though sources of uncertainty vary, the

principle of the stochastic models is the same. The primary objective

of these models is to manage the CLSC to meet customer demands as

cost effectively as possible. With this intent, all the prevailing work

n CLSC design under uncertainty focuses on cost minimization, and

ence ignores the impact of the product markets. This is presumably

ecause e-waste has been traditionally viewed by the industry as

waste rather than a resource. As remanufacturing shows a vast

conomic potential in today’s business, however, it becomes nec-

ssary for us to view CLSCs from a value-creation perspective. Only

hen can the hidden value be released from the system. Although

t is out of the scope of this paper we would like to point out that

obust optimization is the other main approach of dealing with

ptimization under uncertainty. For such an approach the reader can

efer to Realff, Ammons, and Newton (2004).

In our analysis, we assume that new and remanufactured prod-

cts are imperfect substitutes. We further assume that the associated

rices of them are endogenously determined via a market clearing

echanism. Our findings differ from (or complement) the two

treams of literature on remanufacturing in a CLSC network design

etting. To this end, we investigate the profitability of remanufactur-

ng as one of the product recovery strategies, and under what condi-

ions it is preferable. From a modeling perspective, the incorporation

f a stylized representation of the primary and secondary markets

inks our CLSC network design model under uncertainty with the

arket segmentation models such as Ferguson and Toktay (2006),

tasu et al. (2008), Galbreth et al. (2013) and Örsdemir et al. (2014).

. Model development

In this section, we formulate a generic model for a single prod-

ct category by considering a monopolist who makes both new and

emanufactured products, which are clearly differentiable to cus-

omers, and sells them via different retailing channels under price

iscrimination. We focus on the market, where consumers are inter-

sted in a specific product but are still deciding whether to buy a new

r remanufactured product. Such an internal and imperfect compe-

ition between new and remanufactured products is also considered

y Vorasayan and Ryan (2006).

In terms of the design of the network, we consider a CLSC network

hat is capable of dealing with product returns via several recovery

ptions: remanufacturing/reuse, recycling and proper disposal. New

nd remanufactured products are produced in plants with separate

anufacturing and remanufacturing capacities, and then shipped to

arkets through distribution centers in order to satisfy the different

emand segments. To comply with e-waste laws, all the returns have

o be collected from collection centers located in each market point

nd sent to return centers for sorting and inspection purposes. After

he testing performed in the return centers, the OEM identifies the

eturns that are profitable to remanufacture and ships them to one

f the plants. The remaining returns are either shipped to recycling

acilities or properly disposed.

We model this environment by describing a monopolistic OEM

hat is responsible for the design and operation of its CLSC under un-

ertainties in (i) market size; (ii) return quantity and (iii) return qual-

ty. In general, durable goods are categorized by their long useful life.

or such products, the total return volume during a year is not neces-

arily determined or constrained by the sales from the previous year.

or instance, economic incentives through product acquisition cam-

aigns might substantially influence the consumers’ decisions con-

erning the timing of returns. We observed this phenomenon in the

SH case, with regards to the sales and return volumes of refriger-

tors and washing machines from 2005 to 2011. In some particular

ears, the return volume is even higher than the sales volume from

he previous year. Thus, we opted to represent the key problem dy-

amics in a single-period setting.

We aim to determine the best way for the OEM to be involved

n product recovery by optimizing (i) the number and location of

Cs and RCs (i) the manufacturing and remanufacturing quantities

nd (iii) the flow of products through the CLSC. The OEM gains

W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819 807

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evenues from primary and secondary markets, and occasionally

rom recycling when the recycled raw material prices are higher than

he recycling costs. We do not explicitly differentiate between recy-

ling and disposal in the model because usually both are cost factors.

nder this circumstance, the OEM will certainly choose the cheaper

ay between recycling and proper disposal to the extent permitted

y law. In the event that there is potential revenue from selling re-

ycled materials, this is represented by a negative recycling/disposal

ost. Such a representation is also used by Alumur, Nickel, da Gama,

nd Verter (2012).

.1. Notation

ets

= {1, . . . , i} set of plants

J = {1, . . . , j} set of potential DCs as well as potential RCs

K = {1, . . . , k} set of customer zones

L = {n, r} set of products, where n represents new product and

r represents remanufactured product

s = {1, . . . , s} set of all possible scenarios of the uncertain

parameters

ncertain parameters

M(s) = total market size of the company under scenario s

r(s) = total volume of returns under scenario s

α(s) = recovery rate under scenario s

eneral parameters

βk = fraction of the total population living in customer zone k

sli= capacity of producing type l product at plant i, for l = n, r

arketing-related parameters

ϕ = customer’s reservation price

a = lower limit of reservation price

b = upper limit of reservation price

δ = customers’ relative willingness-to-pay for remanufactured

roduct

osts

cl = unit production cost of product type l for l = n, r

ch = unit handling cost in DCs

cs = unit sorting cost in RCs

cd = unit recycling/disposal cost (cd < 0 indicates recycling

evenue)

fj = (annualized) fixed cost of opening a DC at site j

gj = (annualized) fixed cost of opening a RC at site j

cij = cost of shipping one unit of product from plant i to DC j

ejk = cost of shipping one unit of product from DC j to customer

one k

c′ji

= cost of shipping one unit of return from RC j to plant i

e′k j

= cost of shipping one unit of return from customer zone k to

C j

ecision variables

qli= quantity of type l product produced in plant i, for l = n, r

j ={

1 if a DC is located at site j;0 otherwise

Tj ={

1 if a RC is located at site j;0 otherwise

Uli j

= quantity of type l product shipped from plant i to DC j ∀i, j, l

Xljk

= quantity of type l product shipped from DC j to customer

one k ∀j, k, l

Wkj = quantity of return shipped from customer zone k to RC

∀j, k

Vji = quantity of return shipped from RC j to plant i ∀i, j

.2. Inverse demand function

Following the market segmentation literature, we assume that

ustomer’s willingness-to-pay is heterogeneous and represented by

is/her reservation price for a product. To this end, a parameter that is

niformly distributed within [0,1] is commonly used to represent the

ustomers’ heterogeneity in the market. It is also common to normal-

ze the market size to 1. These assumptions, of course, offer analyti-

al simplicity without affecting the nature of the analysis and results.

ormalization, however, is not desirable in our work because one of

ur goals is to provide a solid understanding of the influence of the

ifferent sources of uncertainty. Thus, we work with the actual mar-

et size and use uniformly distributed reservation prices between a

nd b (a < b) to represent the customer’s willingness-to-pay. In most

ases, the customers distinguish the remanufactured products from

he new products, and thus the firm chooses differentiated prices to

erve the market. Each consumer’s evaluation for a remanufactured

roduct is a fraction δ (0 < δ < 1) of their evaluation for the new

roduct. Therefore, a consumer who has a reservation price of ϕ for a

ew product is willing to pay δϕ for the remanufactured product. We

erive the inverse demand functions from consumers’ utility func-

ions (See Appendix A for all proofs), which leads to the following

roposition:

roposition 1. Let qn and qr denote the production quantity for new

nd remanufactured products, respectively. The inverse demand func-

ions of the new and remanufactured products are linear, and

pn = b − b − a

Mqn − δ

b − a

Mqr (1)

pr = δ

(b − b − a

Mqn − b − a

Mqr

). (2)

Although the market area comprises a number of customer zones,

e assume that the firm will charge the same price across the mar-

et. The inverse demand functions (1) and (2) map the quantities of

utput demanded to the market prices for the output. Here, we im-

licitly assume that the market clearing mechanism is applied. That

s, at the market clearing prices of the new and remanufactured prod-

cts (pn and pr), the quantities produced (qn and qr, respectively) will

e equal to the quantities demanded for the new and remanufactured

roducts, respectively. As a result, the market clearing prices pnopt and

propt are functions of decision variables qn and qr and thus will be op-

imized from the IS-CLSC model proposed in next section. Note that a

umber of authors, including Ferguson and Toktay (2006), Atasu et al.

2008) and Galbreth et al. (2013) have used demand functions of sim-

lar forms. The difference in Proposition 1 is that we incorporate the

ctual market size and use reservation prices rather than resorting to

ormalization as described above. The fact that we do not normalize

he parameters in (1) and (2) enables us to incorporate the inverse

emand function in an applied setting.

.3. The IS-CLSC model

The IS-CLSC model can be formulated as follows. Note that

etwork configuration is decided under uncertainty and can-

ot be changed after the realization of the uncertain param-

ters concerning the market size, return volume and recovery

ate.

ax E[Qs(Y, T)] −(∑

j

f jYj +∑

j

g jTj

)(3)

.t.

j, Tj ∈ {0, 1}∀ j (4)

808 W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819

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where the profit function Q(Y, T) for scenario s is defined as

Qs(Y, T) = max (pn − cn)qn + (pr − cr)qr

−(∑

i

∑j

∑l

ci jUli j +

∑j

∑k

∑l

e jkXljk

+∑

k

∑j

e′k jWk j +

∑j

∑i

c′jiVji

)

− ch(qn + qr) − csr(s) − cd(r(s) − qr) (5)

s.t.

qli ≤ sl

i ∀i, l (6)

ql =∑

i

qli ∀l (7)

qr ≤ α(s)r(s) (8)

qn + qr ≤ M(s) (9)

qli =

∑j

Uli j ∀i, l (10)

∑i

Uli j =

∑k

Xljk ∀ j, l (11)

∑j

X ljk = qlβk ∀l, k (12)

∑j

Wk j = r(s)βk ∀k (13)

α(s)∑

k

Wk j ≥∑

i

Vji ∀ j (14)

qri =

∑j

Vi j ∀i (15)

Xljk ≤

(∑i

sli

)Yj ∀ j, k, l (16)

k j ≤ r(s)βkTj ∀ j, k (17)

qli,Ul

i j, Xljk,Wk j,Vji ≥ 0 ∀i, j, k, l. (18)

The first-stage problem is modeled by (3)–(4), while the second-

stage problem is defined by (5)–(18). The first stage corresponds to

the investments that must be made for establishing facilities namely,

DCs and RCs prior to the actual realization of the random parameters;

whereas, the second stage involves the allocation of product flows

through the established network and the decisions pertaining the

quantity of the new and remanufactured products to be produced,

based on the realized uncertain scenario. The location variables

are assigned as the first-stage decision variables and the allocation

variables and production quantities are assigned as the second-stage

decision variables. Note that the optimal value of the second stage

problem Q∗s (Y, T) is a function of the first-stage decision variables.

The first-stage objective function (3) maximizes the expected net

profit, which is given by the difference of expected sales profit and

the fixed costs of establishing facilities. The second-stage objective

function (5) maximizes the operating profit for any realized uncertain

scenario s. The first two terms represent the sales revenues of the

new and remanufactured products, respectively. The four terms in the

second row denote the transportation costs from plants to DCs, from

DCs to customer zones, from customer zones to RCs and from RCs to

plants, respectively. The last three terms relate to the handling cost in

DCs, the sorting cost in RCs and the disposal cost, respectively.

Constraints (6) are capacity constraints with respect to manufac-

turing and remanufacturing. Constraints (7) calculate total produc-

tion quantity of the new and remanufactured products, respectively.

onstraint (8) ensures that the amount of remanufactured products

roduced does not exceed the total amount of remanufacturable

eturns. By Constraint (9) the total production quantity of the two

roducts does not exceed the number of potential buyers in the mar-

et since we assume that each customer will purchase at most one

nit. Constraints (10)–(12) are flow balance constraints for plants,

Cs and the entire market, respectively. Constraints (13) ensure that

ll returns must be collected in every customer zone. Constraints

14) are the maximum throughput constraints guaranteeing that the

otal outflow from RCs cannot exceed the inbound return handling.

onstraints (15) ensure that all the returns sent to remanufacturing

lants will be remanufactured. Constraints (16) and (17) guaran-

ee that flows can only be assigned to open DCs and RCs. Finally,

onstraint (4) and (18) are domain constraints.

Plugging the price functions (1) and (2) into (5), we can get the

rofit function for any possible scenario s as follows:

s(Y, T) = max −b − a

M(s)(qn)2 − 2δ

b − a

M(s)qnqr − δ

b − a

M(s)(qr)2

+ (b − cn − ch)qn + (δb − cr − ch + cd)qr

−(∑

i

∑j

∑l

ci jUli j +

∑j

∑k

∑l

e jkXljk +

∑j

∑i

c′jiVji

+∑

k

∑j

e′k jWk j

)− (cs + cd)r(s). (19)

hus, we have a quadratic objective function.

roposition 2. The objective function in (19) is concave in the profit

unction Qs(Y, T) for any given s is concave in (qn, qr) throughout δ ∈ (0,

).

Proposition 2 ensures the existence of an optimal solution, which

erves as a starting point of the development of our solution method

n next section.

The proposed IS-CLSC model is integrated in the sense that it in-

orporates the major product recovery options i.e., remanufacturing,

ecycling and proper disposal, for OEMs to fulfill their environmental

esponsibilities. Thus, the model is applicable for various industries.

n Section 6, we present an application of the model on a case study

ocusing on refrigerators (that are in the large household and refrig-

ration appliances category of WEEE) with the purpose of illustrating

he implications of these options in a realistic setting.

. Solution method

In this section, we propose a solution method that integrates the

nteger L-shaped method and the sample average approximation

SAA) technique for solving the stochastic mixed-integer quadratic

rogram presented in Section 3. The motivation for developing

uch an algorithm is two-fold: First, our IS-CLSC model contains

quadratic resource function and there is no existing commercial

oftware that can be used for solving stochastic quadratic programs

irectly. Second, we prefer to use SAA so that we can avoid using

limited number of scenarios to represent the real life setting. In

rder to focus on the decomposition itself and organize ideas in a

ransparent manner, we restrict our discussion to a single plant.

For the purpose of illustration, the resource function Qs(Y, T) can

e written concisely as

inx

1

2x′Hx + g′x

.t.

A1x ≥ b1 − Dy : z1

A2x = b2 : z2

x ≥ 0 : π

W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819 809

w

r

Ll

m

s

θ

Y

o

a

a

p

a

c

c

o

L

t

f

W

s

w

b

4

s

t

F

m

s

a

m

e

c

c

p

c

b

t

a

o

s

S

S

S

S

S

S

S

4

t

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t

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2

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4

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I

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S

S

S

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S

5

c

here y represents the first-stage decision variable vector and x

epresents the second-stage decision variable vector. By defining

(Y, T) = E[Qs(Y, T)], the first-stage problem is equivalent to the fol-

owing reformulation in which a new variable θ is introduced:

ax −θ −(∑

j

f jYj +∑

j

g jTj

)(20)

.t.

≥ −L(Y, T) (21)

j, Tj ∈ {0, 1}∀ j. (22)

As illustrated by Listes (2007), IS-CLSC has two potential sources

f difficulty. First, constraint (21) cannot be used computationally as

constraint since L(Y, T) is not available in a closed analytical form

nd is only implicitly defined. The L-shaped decomposition method

rovides the means to create an outer-approximation of it by using

series of tangent hyperplanes (Kulkarni & Shanbhag, 2012), also

alled optimality cuts (Birge & Louveaux, 1997). This method is appli-

able to problems with continuous variables. Consequently, the sec-

nd issue is the integrality of the first-stage variables. Combining the

-shaped method for quadratic programs (see Appendix B for the de-

ailed derivation) with the well-known branch-and-bound method

or mixed integer programs results in the integer L-shaped method.

e start by brief descriptions of the integer L-shaped method for

tochastic MIQPs (SMIQPs) and the SAA technique; and in Section 4.3

e present the solution method we used to solve the IS-CLSC model

y integrating the SAA scheme with the integer L-shaped algorithm.

.1. Integer L-shaped method for SMIQPs

First, we ignore the integrality constraints (22) and allow the first-

tage variables to be continuous in the interval [0,1]. Let it be the ini-

ial feasible set, and denoted by

0 = {(Y, T, θ)|θ ∈ �,Yj, Tj ∈ [0, 1] ∀ j}.Let f = ( f1, f2, . . . , f j) and g = (g1, g2, . . . , g j). Such L-shaped

ethod proceeds iteratively by solving the first-stage problem with a

eries of additional constraints i.e., optimality cuts, which can define

monotonically decreasing feasible set F1 such that the problem

ax {−θ − fY − gT |(Y, T, θ) ∈ F0 ∩ F1} (23)

ventually yields a solution that satisfies constraint (21). With a

oncave second-stage problem, the optimality cuts can be efficiently

onstructed from the corresponding optimal dual solutions. For any

articular F1 during the iterative process, problem (23) is called the

urrent problem (Listes, 2007).

As already mentioned, the integer L-shaped method combines the

ranch-and-bound scheme for the first-stage problem with the itera-

ive cutting planes procedure of the L-shaped method. Thus, we oper-

te with a list of active nodes, each of which corresponding to a form

f the current problem. The procedure consists of the following main

teps.

Integer L-shaped method for SMIQPs

tep 1.0: Set ν := 0, ω := 0, L := 0. The list consists of one node cor-

responding to the initial current problem.

tep 1.1: Choose a node from the list. If the list is empty, vm = L, m :=m + 1, and return to Step 1.0.

tep 1.2: Let ν := ν + 1. Solve the current problem and denote an op-

timal solution by (Yν , Tν , θν ).

tep 1.3: If −θν − (fY + gT) < L, fathom the current node and return

to Step 1.1.

tep 1.4: If there are unsatisfied integrality constraints, pick a vari-

able with fractional value and create two new nodes corre-

sponding to setting its value at 0 or 1. Replace the current

node by the two new nodes in the list and return to Step 1.1.

tep 1.5: L-shaped SQP Algorithm (Kulkarni & Shanbhag, 2012).

tep 1.6: If θν ≥ −L(Yν , Tν), fathom the current node and return

to Step 1.1. Otherwise, impose the optimality cut E1mω+1 +

E2mω+1 + θ ≥ zm

ω+1, set ω := ω + 1 and return to Step 1.2.

.2. The SAA scheme

The basic idea of SAA is that a random sample is generated and

he expected value function is approximated by the corresponding

ample average function. The obtained SAA problem is solved and

he procedure is repeated until a stopping criterion is satisfied. In the

AA scheme, a random sample of N realizations of the random vec-

or ξ = (M, r, α) is generated, and the expected future profit is then

pproximated by the sample average function:

[Q(Y, T, ξ )] = 1

N

N∑s=1

Q(Y, T, ξs). (24)

Z independent sample sets of random parameters each size N are

enerated in Step 1 to select potential optimal solutions of the first-

tage problem. The true objective function value is estimated in Step

with much bigger sample size N′. For more details of SAA the reader

an refer to Kleywegt, Shapiro, and Homem-de Mello (2002).

.3. Solution scheme for IS-CLSC model

In this subsection, the integer L-shaped method for SMIQP and the

AA scheme are integrated to develop the solution algorithm for the

S-CLSC model. The overall procedure is as follows:

tep 0: Generate Z independent sample sets of random parameters

each of size N, i.e., (ξ 1,z, … , ξN,z) for z = 1, . . . , Z and s =1, . . . , N.

tep 1: For z = 1 : Z, obtain the location, production and flow deci-

sion by the Integer L-shaped method for SMIQPs. Let vz and (Y,

T)z, be the corresponding objective value and the location so-

lution for a given sample z, respectively.

tep 2: Calculate vZ,N = 1Z

∑Zz=1 vz. It provides a upper statistical

bound for v∗.

tep 3: For each feasible solution (Y, T)z of the first-stage problem

in Step 1, generate a sample (ξ1, . . . , ξN′) of size N′ indepen-

dently of the samples used to obtain the feasible solution.

Typically, N′ is set to be much bigger than the sample size

N used in Step 1. Using the dual formulation of the second-

stage problem to determine Q(Yz, Tz, s) for every s. Estimate

the true objective function value F(Yz, Tz, qn, qr, U, X, W, V) as

follows:

F(Y z, T z, qn, qr,U, X,W,V) = 1

N′N′∑

s=1

Q(Y z, T z, ξs) − (fY z + gT z).

(25)

tep 4: Choose the feasible solution (Y , T) with the biggest esti-

mated objective value FN′(Y , T) is an unbiased estimator of

F(Y , T). Since (Y , T) is a feasible solution of the true problem,

F(Y , T) ≤ v∗. Therefore, FN′(Y , T) is an estimate of an lower

bound on v∗.

tep 5: Calculate an estimate of the optimality gap of the solution

(Y , T) using the lower bound estimate and the objective func-

tion value estimate as follows:

εZ,N,N′(Y , T) = vZ,N − FN′(Y , T). (26)

. Performance of the algorithm

In this section, our goal is to show that our solution method

an solve the model to optimality in reasonable time. We illustrate

810 W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819

Table 1

Computational results for J = 5, 10, 15.

J Time (second) Nodes Optimality cuts

5 2.8 259 102

10 29.2 3,360 702

15 2,065.4 60,772 9,579

B

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a

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1

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3 We divide the total amount of demand in tons from BSH data, by the sales in units

given by Anonymous (2012) to estimate the average weight of a refrigerator.

the computational performance of the proposed methodology in a

smaller-scale numerical setting generated from the real case pre-

sented in the next session. The specific setting we use is as follows.

We consider problem instances with J = 5, 10 and 15 candidates of

facilities (i.e., DC/RCs) where K is fixed at 15 customer zones. The

15 most populated cities in Germany are identified as the customer

zones. To generate the problem instances, the alternative facility loca-

tions are randomly chosen among the 15 customer zones with equal

probability. We generate 3 problem instances for each problem size.

For each problem instance, we produce 3 independent sample sets of

uncertain parameters each of sample size 9. In all problem instances

we consider one fixed plant (that is a randomly chosen location). The

remaining parameters, such as demand, return and cost structure, are

fixed as outlined in Section 6.

We take the runs on a server with 2.80 gigahertz Intel Xeon pro-

cessor and 4 gigabyte of RAM and we used the optimization software

CPLEX version 12.4. All the runs are solved to optimality. Pilot tests

show that the CPU time of solving the optimal location from step one

of the algorithm predominate over the CPU time of estimating the

optimal total profit from step two. An increased number of poten-

tial facilities will further increase this predominant effect. Given that

the basic procedures of our solution method are facility-based, the

number of potential facilities has stronger effect on solution time.

Table 1 presents computational results with varying number of po-

tential facilities given Z = 1, N = 9 and N′ = 0 in order to focus on

performance of the integer L-shaped algorithm. The table reports the

CPU running time (in seconds) as well as the number of nodes and

optimality cuts generated during the solution process. The CPU times

include both the branch-and-cut part of the procedure and the build-

ing of the problem objects for the current problem and for the dual

second-stage problem in step one. The figures reported in Table 1 are

the overall averages for each problem instance.

For J = 15, the model contains 30 binary first-stage variables and

722 continuous second-stage variables for each scenario (resulting in

a total of 6498 continuous variables for all scenarios). Although a rel-

atively large number of nodes are investigated and a significant num-

ber of optimality cuts are generated, the computational time is less

than 35 minutes. The results show that the number of optimality cuts

has a stronger effect on solution time. It is also worthy to mention

that the significant monetary magnitude difference between produc-

tion related parameters (qn, qr, pn, pr) and logistics related parameters

induces numerical difficulties during computation. Through further

analysis, we observe that the CPU time of solving large-scale prob-

lems (e.g. 40 × 40) can reduce from hours to seconds when the values

of these parameters are closer an order of magnitude. This demon-

strates the efficiency of our solution method itself.

6. An illustrative case: BSH Bosch und Siemens Hausgeräte GmbH

We apply the IS-CLSC model on a new case based on BSH’s op-

erations in Germany, focusing on the remanufacturing of refrigera-

tors under EPR. The case study is inspired by the need for BSH to

decide whether to offer remanufactured products under Germany’s

unique WEEE take-back scheme. Our goal is to highlight the features

of the proposed model and also to show the impact of uncertain-

ties on the performance of real supply chains. The information in this

case study is developed through telephone interviews and follow-up

email exchanges with a number of managers in BSH Germany and

SH Spain. In addition, we interviewed two of the main reverse lo-

istics partners of BSH in Germany. Secondary data was collected

rom BSH concerning sales and recycled volume of refrigerators in

011 as well as the associated recycler matrix. We also benefited from

he market research reports concerning the household appliances in

ermany.

The CLSC network structure in this case study represents BSH’s

xisting supply chain. We assume that refrigerators are sold and

ollected from 40 retailer points located in the 40 most populated

erman cities (i.e., customer zones). The locations of the cities on

he map of Germany are depicted in Fig. 1. To obtain a list of facility

andidates, we start with the capital city of each federal state in Ger-

any (16 in total). Potsdam, Schwerin and Saarbrücken are excluded

rom this list since they are scarcely populated. Furthermore, Berlin,

amburg and Munich are also excluded due to the high real estate

ost concerns as well as the environmental and safety concerns as-

ociated with the hazardous materials contained in the refrigerators.

onsequently, we have 10 alternative facility sites which are (1)

tuttgart, (2) Düsseldorf, (3) Bremen, (4) Dresden, (5) Hanover, (6)

iesbaden, (7) Kiel, (8) Magdeburg, (9) Erfurt and (10) Mainz.

BSH has a single plant of producing refrigerators in Germany.

ince it is used to supply the whole Western Europe, we assumed that

ll demands from the German market will be satisfied. The returns

n this case are referred to those commercial returns and end-of-use

eturns collected in major retailer points, which account for about

0 percent of the total returns for BSH. The quality of these returned

roducts are much better than those collected from municipality col-

ection points in general and therefore it might be profitable for BSH

o consider remanufacturing instead of recycling. We take 10 percent

f the total return to estimate the average amount of returns in tons

enerated in retailer points. In order to convert weight to the total

umber of returned refrigerators, the average weight per refrigerator

s estimated by 50 kilogram.3 We assume that each returned refrig-

rator contains only one core. It is verified by BSH that demands for

ew and remanufactured products and return at each customer zone

re allocated proportionally according to population density.

We estimate market price of the new product by dividing BSH’s to-

al monetary sales by the associated total sales volume (Anonymous,

012). The manufacturing cost is estimated by 25 percent of its profit

argin i.e., 80 percent of the market price of the new product. The

elative willingness-to-pay for BSH remanufactured refrigerators is

stimated by customer’s relative willingness-to-pay for LG remanu-

actured refrigerators from eBay (www.ebay.com), which is 62 per-

ent on average. Taking into account the larger market share and

igher reputation in the German market, we use 70 percent as the

elative willingness-to-pay for BSH. Then we estimate the remanu-

acturing cost by 70 percent of its corresponding market price.

Refrigerators are appliances that contain refrigerants and there-

ore special certification requirements are needed for their reverse

ogistics. Alumur et al. (2012) provide estimates for the costs in-

urred at RCs for washing machines. We extrapolate the sorting cost

n RCs as well as recycling/disposal cost for the refrigerators from this

eference.

The basis of annualized fixed costs of opening a DC is estimated

y the rental of the only warehouse, which is located in Nauren.

his property is not owned by BSH. The company informed us that

auren arguably has the lowest fixed cost comparing to the other

ities in Germany. Since land prices are higher in the populated cities,

parameter γ is generated in order to reflect the differences on fixed

osts depending on the land prices at the candidate locations. As

sed in Alumur et al. (2012), for each candidate location, this pa-

ameter is equal to the ratio of the population of the city to the total

W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819 811

Fig. 1. The stochastic solution based on the 40 most populated cities.

p

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Table 2

Value of cost parameters.

Parameter Value

Unit manufacturing cost cn €370

Unit remanufacturing cost cr €225

Set-up costs

For DCs fj €270,000γ

For RCs gj €400,000γ

Operational costs per unit

Handling cost for DCs ch €10

Sorting/inspecting cost for RCs cs €20

Disposal/recycling cost cd €50

Transportation costs per refrigerator per kilometer

Plant-DC €0.01125

DC-customer €0.025

Customer-RC €0.0625

RC-plant €0.0281

t

e

a

a

s

opulation of these 40 cities multiplied by 100. We adapted the base

f annualized fixed cost of opening a RC used in Alumur et al. (2012)

nd adjusted this number by land price parameter to get the fixed

ost for each potential RC.

A distance matrix is generated between the 40 cities by using the

hortest suggested route on the road network from maps.google.com.

iesel consumption of Volvo heavy duty 2010 model truck is used to

stimate the net transportation cost. We multiply this number by 1.25

o estimate the unit transportation cost per truck per kilometer. In or-

er to convert transportation cost per truck to transportation cost per

efrigerator, we assume 80 units of refrigerators per truck to estimate

he truck load. These give us the unit transportation cost between

otential plants and DCs. Next, we apply the same scale among dif-

erent unit transportation costs used in the copier remanufacturing

ase study by Fleischmann et al. (2001) to estimate the unit trans-

ortation cost between DCs and customer zones. We assume that the

nit transportation costs on the reverse network are 2.5 times more

xpensive than their counterparts on the forward network due to the

azardous materials content of the returned refrigerators, their trans-

ortation is regulated and hence more expensive. Due to the regu-

atory difference, BSH also uses separate trucks for backhauling the

eturned refrigerators.

Finally, the lower limit and upper limit of the reservation price are

aken as 370 and 600 euros. And the customer’s relative willingness-

o-pay for remanufactured goods is estimated as 0.7. The cost param-

ters used in the case study are summarized in Table 2.

As indicated in Section 3, the market size, total return volume

nd recovery rate are uncertain, each assumed to be independent

nd identically distributed. In the context of this case, we further as-

ume that they follow triangular distributions with modes taken from

812 W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819

Table 3

Base-case uncertain parameters.

Description Parameter Mode Uncertainty level

(percent)

Market size M 2,648,785 10

Return volume r 114,840 75

Recovery rate α 0.7 25

Table 4

Summary of 40 optimal scenario solutions and

stochastic solution.

Solution Number of scenarios

Open DCs Open RCs

1, 6, 8 8, 10 19

1, 6, 8, 9 8, 10 11

1, 6, 8 9 6

1, 6, 8 6, 8, 10 2

1, 6, 8, 9 9 1

1, 6, 8, 9 6, 8, 10 1

1, 6, 8 8, 10 Stochastic

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BSH’s business, and both lower and upper limits as percentage of the

associated mode. In particular, the actual total return volume of BSH

is taken as mode of r. Since BSH has not yet included remanufactur-

ing in their business i.e., qr = 0 in Eq. (1), M can be expressed as a

function of pn and qn:

M = qn

1 − pn

b−a

(27)

where qn is taken from the real sales data.

7. Results and discussion: refrigerator recovery

In this section, we investigate the impact of the uncertainties

in market size M, return quantity r and return quality α on the

performance of the BSH supply chain, where a full range of product

recovery options are available. We also explore the characteristics of

cost structures and customer perceptions that would encourage (or

discourage) remanufacturing under uncertainty in Section 7.2. We

introduce effective recovery rate, as a measure to facilitate the com-

parison of different recovery strategies. In Section 7.3, we investigate

the impact of the uncertainties associated with disposal/recycling

cost. In the last subsection, we study the benefit of the integrated

network design approach compared to the sequential approach

under the market, return and recovery rate uncertainties.

7.1. Impact of uncertainty on configuration and profitability

In this subsection, we seek answers to the following questions:

How robust is the network configuration under uncertainty? How do un-

certainties influence the profitability of CLSC systems?

We start with the formal definition of the term, uncertainty level,

used in this paper. The uncertainty level of a stochastic parameter

represents the half-width of its range as percentage of the mode of

this parameter. Table 3 provides a summary of the uncertain parame-

ters used in the base-case, where all parameters are (symmetric) tri-

angular distributed.

We generate 40 scenarios based on random realizations of the

uncertain parameters M, r and α. The stochastic solution and the

optimal solutions for each of the 40 scenarios, each solved as a

deterministic problem, are depicted in Table 4, which also shows the

number of times each network configuration constitutes the optimal

solution for one of the scenarios. Although there are six alternative

network structures, each being optimal to at least one of the 40

cenarios; two network structures are optimal 75 percent of the sce-

arios. Interestingly, the optimal stochastic network configuration is

he same as the optimal deterministic solution that is repeated most

ften i.e., DCs at Stuttgart, Wiesbaden and Magdeburg and RCs at

agdeburg and Mainz. This stochastic solution is shown in Fig. 1.

In general, forward supply chains tend to be spatially decentral-

zed compared with their reverse logistics counterparts. This is pre-

umably due to the smaller volume of commodities that flow through

he reverse network. In the case of refrigerators, however, this dispar-

ty is much smaller mainly because of the high transport costs associ-

ted with returned refrigerators that include hazardous materials. It

s evident from Table 2 that shipping returns to RCs is 2.5 times more

xpensive than shipping products from DCs to the customers, due

o the safety guidelines across the reverse network for this product

ategory.

Another interesting observation from Table 4 pertains to the sets

f optimal DCs under different realizations of the uncertain param-

ters. Note that the set of optimal DCs with three facilities (i.e., {1,

, 8}) remains in the solution when it becomes necessary for BSH to

pen four DCs (i.e., {1, 6, 8, 9}). Although theoretically this is not true

n general, it is certainly helpful in practice to know that BSH has a

iable strategy option to start with DCs at sites 1, 6, and 8, and open

n additional DC at site 9 if it becomes optimal depending on the

cenario they end up facing. It is worthwhile to note that the same

roperty does not hold for the reverse network. That is, BSH cannot

tart with a single RC at site 9 and expect that the set of RCs can be

imply expanded and will remain optimal as the scenario in effect is

evealed.

We now turn to investigating the sensitivity of the solution to the

hanges in the distribution of each uncertain parameter. To this end,

e keep the base-case settings (see Table 3) for two of the param-

ters (e.g., r and α) while we alter the distribution of the third pa-

ameter (M for this example). In ascertaining the overall impact of

ach uncertain parameter, we study the effect of varying (i) the un-

ertainty level, (ii) the mode of the distribution, and (iii) the upper

imit of the triangular distribution. For each parameter, such effects

re studied via Test 1, Test 2 and Test 3, respectively. In Tables 5–7, the

ower limit, mode and upper limit of the triangular distribution used

n each test are represented as a percent deviation from the base-case

ode of the associated parameter. For example, the distribution of M

n the fifth row of Table 5 is [0,+10 percent,+20 percent] indicating

hat the market size is triangular distributed with [2, 648, 785, 2, 913,

64, 3, 178, 542] for this test. The profit growth column in Tables 5–7

rovides a comparison with the base case. For each problem instance,

e also report the mean and standard deviation of total profit as well

s the stochastic solution.

Table 5 reports on the experiments with varying the distribution

f market size M, while r and α were kept at their base case distri-

utions. In Test 1, we increased the uncertainty level of M without

ltering the mode of the distribution. Interestingly, the profit and the

etwork configuration seems to be rather robust to changes in the

ncertainty level of up to 30 percent. In Test 2, we keep the uncer-

ainty level at 10 percent and study the impact of varying the average

arket size. The problem instances with 5 percent and 10 percent

ncrease in the average M, requires that a new DC is open at Erfurt

site 9) to address the anticipated market growth. More importantly,

hese two instances show that the expected percent increase in profit

s larger than the percent increase in the average market size. Test 3

n Table 5 depicts that increasing the upside uncertainty level – while

he downside uncertainty level remains the same – is likely to lead to

otable growth in profits.

Tables 6 and 7 show that total profit is quite robust with respect

o changes in the distribution of return quantity r and return rate α,

espectively. For example, in Tables 6, when we increase the upper

ound of r by roughly 30 percent, total profit increases less than only

percent (see Test 3). This robustness can be due to two reasons:

W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819 813

Table 5

Solutions of the model with increasing M.

Test Distribution of M Profit × 103 Profit Relative standard Open DC Open RC

growth (percent) deviation (percent)

Base case [−10%, 0,+10%] 125,086 – 0.421 1, 6, 8 8, 10

1 [−30%, 0,+30%] 124,971 −0.09 1.658 1, 6, 8 8, 10

[−40%, 0,+40%] 123,540 −1.24 4.486 1, 6, 8 1, 6, 7, 8

2 [−5%,+5%,+15%] 131,654 5.25 0.306 1, 6, 8, 9 8, 10

[0,+10%,+20%] 137,983 10.31 0.547 1, 6, 8, 9 8, 10

[−15%,−5%,+5%] 118,358 −5.38 0.548 1, 6, 8 8, 10

[−20%,−10%, 0] 112,003 −10.46 0.692 1, 6, 8 8, 10

3 [−10%, 0,+20%] 129,777 3.75 0.378 1, 6, 8 8, 10

[−10%, 0,+30%] 133,041 6.36 1.280 1, 6, 8, 9 8, 10

Table 6

Solutions of the model with varying r.

Test Distribution of r Profit × 103 Profit Relative standard Open DC Open RC

growth (percent) deviation (percent)

Base case [−75%, 0,+75%] 125,086 – 0.421 1, 6, 8 8, 10

1 [−50%, 0,+50%] 125,012 −0.06 0.277 1, 6, 8 8, 10

[−100%, 0,+100%] 124,868 −0.17 0.400 1, 6, 8 8, 10

2 [−50%,+25%,+100%] 124,158 −0.74 0.473 1, 6, 8 8, 10

[−25%,+50%,+125%] 123,478 −1.29 0.269 1, 6, 8 6, 8, 10

3 [−75%, 0,+100%] 125,661 0.46 0.582 1, 6, 8 8, 10

[−75%, 0,+125%] 126,224 0.91 0.269 1, 6, 8 8, 10

Table 7

Solutions of the model with varying α.

Test Distribution of α Profit × 103 Profit Relative standard Open DC Open RC

growth (percent) deviation (percent)

Base case [−25%, 0,+25%] 125,086 – 0.421 1, 6, 8 8, 10

1 [−10%, 0,+10%] 124,869 −0.17 0.817 1, 6, 8 8, 10

[−40%, 0,+40%] 124,836 −0.20 0.232 1, 6, 8 8, 10

2 [−15%,+10%,+35%] 125,700 0.49 0.758 1, 6, 8 8, 10

[0,+20%,+40%] 126,020 0.75 0.660 1, 6, 8 8, 10

3 [−25%, 0,+33%] 125,449 0.29 0.418 1, 6, 8 8, 10

[−25%, 0,+41%] 125,699 0.47 0.369 1, 6, 8 8, 10

F

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irst, the return volumes are rather small, since only 10 percent of

SH’s total returns are collected through the retail stores. Second, the

ase study is confined to the activities of BSH within Germany, where

he transportation distances are relatively short, and hence the lo-

istics costs are dominated by the production costs; weakening the

mpact of r and α.

It might seem counter-intuitive that in Table 6, the total profit de-

reases while the mode of r increases (see Test 2). In general, higher

eturn volumes result in larger amounts of unrecoverable returns

hat have to be either disposed off or recycled. In Germany, however,

isposal is discouraged due to the high priority of environmental

rotection. For instance, even the disposal costs of shredder residue

urpassed 180 € a ton in 2001 (Lambert & Stoop, 2001). Consequently,

he end-of-life treatment costs can adapt to 50 € per refrigerator in

ermany. Although, there is a profit growth from remanufacturing

nder increased r, this seems to be offset by the high costs of the

andated environmentally-friendly end-of-life treatment.

In closing this subsection, we remark that the primary obser-

ations reported above validate the proposed model in light of a

ealistic case. Perhaps more importantly, some of our less intuitive

ndings – in the context of BSH – dispel the “one-size-fits-all” type

pproaches to the design of CLSCs under uncertainty.

.2. Recovery strategies in the presence of uncertainty

In this subsection, we explore the impact of key parameters that

ould encourage (or discourage) remanufacturing under uncertainty.

n particular, we focus on the following three parameters:

• cd: the disposal/recycling cost;• cr: the remanufacturing cost;• δ: customer’s valuation of the remanufactured product.

Evidently, the impact of cd and cr depends on the magnitude

f these costs relative to the cost of manufacturing a new product,n. Also it is important that these three parameters are not directly

nd fully controlled by the OEM. We ask the following questions:

hat is the impact of the variable cost structure on the extent of the

EM’s involvement in remanufacturing? How does consumer’s relative

aluation of remanufactured products influence the preferred product

ecovery alternative?

In order to facilitate the comparison of the firm’s involvement in

emanufacturing under different scenarios, we define the effective re-

overy rate for each scenario (α∗e f f

(s)). Let qr∗(s) denote the optimal

emanufacturing quantity under scenario s. Omitting the scenario in-

ex for ease of exposition, the effective recovery rate is:

∗e f f = qr∗

α r. (28)

We make the following observation by analyzing the sensitivity

f the effective recovery rate to the change of cd in a deterministic

etting.

bservation 1. Given a realization of the uncertain parameters, there

xists a two-level threshold policy for the dominant recovery alter-

ative that varies with the recycling/disposal cost and the customer’s

aluation of the remanufactured product.

814 W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819

Table 8

Solutions of the deterministic model with cn = 370€

and cr = 225€.

δ cd α∗eff

δ cd α∗eff

0.3 ≤ 130 0 0.7 ≤ −25 0

131 0.25 −24 0.28

132 0.59 −23 0.62

133 0.93 −22 0.96

≥ 134 1 ≥ −21 1

0.5 ≤ 52 0 0.9 ≤ −103 0

53 0.08 −102 0.29

54 0.37 ≥ −101 1

55 0.65

56 0.94

≥ 57 1

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That is, there exists a lower-threshold value for cd, below which

the OEM should not engage in remanufacturing and have all the re-

turns recycled or properly disposed. There is also a higher-threshold

and if cd is above this level the firm should remanufacture all the re-

turns. In the event that the recycling/disposal cost is between these

two thresholds then remanufacturing a certain portion of the returns

is the best option for the OEM. Naturally, the values of these thresh-

olds also depend on the values of the other problem parameters, par-

ticularly cn and cr.

Table 8 presents the results for BSH when the uncertain param-

eters are at the mode of their distributions (see Table 3), cn = 370€,

cr = 225€ and δ takes the values 0.3, 0.5, 0.7 and 0.9 (i.e., in increas-

ing order of customer valuation for remanufactured products). For

δ = 0.5, for example, BSH should remanufacture all the returns if cd

≥ 57€ and do not remanufacture at all when cd ≤ 52€. Interestingly,

the two thresholds are within 5€ of each other in all quadrants of

Fig. 2. The positi

he table. This suggests that the preferred recovery option is very

ensitive to the recycling/disposal cost for BSH. Here, we remind the

eader that negative values of cd represent revenue from recycling.

vidently, even for higher values of δ, the firm could still have the

ncentive to recycle all the returns (i.e. α∗eff

= 0) if recycling is prof-

table enough. Table 8 also depicts that the customers’ valuation of

emanufactured products, δ, has a profound effect on the threshold

evels. As long as the clients are willing to pay somewhat more than

alf of the price of a new refrigerator to purchase a remanufactured

ne, it seems that BSH should consider recycling as an option only

hen it generates revenues. That is, in a deterministic setting, when

≥ 0.7 BSH should remanufacture all the returns as long as cd ≥ 0.

Expanding the sensitivity analysis above, Fig. 2a depicts the two

hresholds for the entire range of (cd, δ) values. In Fig. 2, region III

epresents the parameter range where BSH should remanufacture all

he returns and region I represents the range where all returns should

e recycled or properly disposed. Region II in the figure corresponds

o the partial remanufacturing option. Indeed, we made a similar ob-

ervation through a sensitivity analysis on the remanufacturing costr that is depicted in Fig. 2b.

bservation 2. Given a realization of the uncertain parameters, there

xists a two-level threshold policy for the dominant recovery alter-

ative that varies with the remanufacturing cost and the customer’s

aluation of the remanufactured product.

The above observations under a deterministic setting i.e., for a

iven realization of (M, r, α), provide the basis for studying the recov-

ry options in the presence of uncertainty. To this end, we perform a

onte Carlo simulation by generating 100 realizations of (M, r, α), us-

ng their base-case triangular distributions (see Table 3). This enables

s to estimate α∗eff

as an average of the α∗eff

values calculated for each

f the 100 realizations. Figs. 2a and b show that not only the two-level

on of α∗eff .

W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819 815

Table 9

Summary of 100 optimal scenario solutions of cd .

Solution Number of scenarios Regret (percent)

Open DCs Open RCs

cd > €15 1, 6, 8, 9 1, 6, 8 54 –

1, 6, 8, 9 1, 6, 8, 10 28 0.5

€−13 < cd ≤ €15 1, 6, 8, 9 1, 6, 7, 8 13 0.2

cd ≤ €−13 1, 6, 8 1, 6, 7, 8 5 3.8

Table 10

Integrated network design verses sequential network design.

Profit (×108) Profit decrease Open DC Open RC pn

(percent)

Integrated 1.539 – 1, 6, 8, 9 1, 6, 8, 10 497.9

Sequential with fixed pn 1.019 34 1, 6, 8 1, 6, 8 493.5

Sequential without fixed pn 1.510 2 1, 6, 8 1, 6, 8, 10 497.6

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hreshold policy generalizes to the problem under uncertainty, but

lso both the threshold levels are quite robust to uncertainty for the

ermany business of BSH (i.e., the three regions almost overlap). We

lso drew Fig. 2a for cr = €175 and €370 as well as Fig. 2b for cd = 0

nd €−50, and found similar results. More importantly, we make the

ollowing observation based on the fact that region II is quite narrow

n Figs. 2a and b.

bservation 3. Under the uncertain parameters (M, r, α), the extent

f the firm’s involvement in remanufacturing is sensitive to changes

n disposal/recycling cost cd, remanufacturing cost cr and customer’s

aluation on remanufactured products δ.

Here, we remark that the above observations are valid for BSH’s

ermany business, where transportation distances are not too long.

e carried out similar commutations with quadrupling the trans-

ortation costs to develop a rough understanding of whether our

esults hold true for larger markets, such as the entire Europe. Our

ndings are depicted in Figs. 2c and d.

bservation 4. Under the uncertain parameters (M, r, α), as the mar-

et territory is expanded, remanufacturing only a portion of the re-

urns is more likely to be the preferred recovery option.

To conclude, we remark here that it is quite important for the

anagers to accurately estimate the costs associated with product re-

overy options under uncertainty. We also find that under highly un-

ertain environments, the managers are less likely to have the easier

ptions of “recycle all the returns” or “remanufacture all the returns”

s the most appropriate strategy. For the BSH case, we excluded a

umber of cost items such as utilities, indirect labor, etc., and focused

n real estate costs, primarily due to the unavailability of data. Pre-

umably, including these costs would not alter the main insights, but

ay results in few sites being opened.

.3. Uncertainty in the disposal/recycling cost

As noted in Observation 3, the net cost (or profit) of recycling is

ighly dependent on the market prices of recycled material, which

re subject to frequent frustrations. Therefore, it will also be of inter-

st to a firm to consider not only the value of the disposal/recycling

ost cd but also the uncertainty associated with it. Mean reverting

rocess is widely used in finance to model commodity prices stochas-

ically and commodity prices are commonly assumed to be lognor-

ally distributed, as the commodity price should be retained as

ositive (Chan, Karolyi, Longstaff, & Sanders, 1992; Schwartz & Smith,

000; Uhlenbeck & Ornstein, 1930). In our case, however, the normal

istribution assumption enables us to capture the possibility of dis-

osal/recycling being a cost (cd > 0) and that being a revenue (cd < 0).

teel, copper and aluminum are the three major materials recovered

rom used refrigerators. Based on the bill of materials (BOM) of a

++ Siemens refrigerator, which is very popular in Germany (Chen,

ucukyazici, & Sàenz, 2014), we use the weighted average of the vari-

nces of the prices of steel, copper and aluminum in the secondary

aterial markets so as to estimate the overall variance of cd.

We perform a Monte Carlo simulation by generating 100 real-

zations of cd, using the normal distribution with mean € 50 per

efrigerator and variance 35. Meanwhile, we keep the base-case set-

ing for three uncertain parameters. When we consider the current

eturns collected from retail stores (i.e., 10 percent of BSH’s total

eturn), we find that the impact of uncertainty in cd is negligible. It

s important to note that Observation 3 still holds and the value of cd

s still significant. Further computation experiments reveal that the

ncertainty plays an increasingly important role as the proportion of

ommercial returns increases. Table 9 summarizes the results of the

cenario where BSH is able to collect 75 percent of its total returns

rom retail stores.

When cd > €15 per unit i.e., remanufacturing is quite profitable

iven a high disposal/recycling cost, reverse network tends to be

loser to Giengen, where the production facility is located. When

−13 < cd ≤ €15 i.e., there is no huge difference in terms of profitabil-

ty between remanufacturing and recycling, reverse network tends

o be more evenly spread. When cd ≤ €−13 i.e., recycling is more

rofitable, forward network shrinks as less DC capacity is needed to

andle remanufactured products. Assuming that BSH will implement

ost common supply chain configuration associated with the 100

cenarios tested, the regret of opening DC at {1, 6, 8, 9} and RC at {1,

, 8} is negligible except for the case where recycling is considerably

rofitable (i.e., cd ≤ €−13).

.4. Sequential decision making

Given the established forward network, a company may realize

he economic value associated with remanufacturing or may be

orced to deal with product returns after the enforcement of the

ake-back legislation. In this subsection, we study the benefit of

he integrated network design approach compared to the sequential

pproach under the market, return and recovery rate uncertainties. To

his end, two variations of the sequential approach were considered,

he main difference being the pricing strategy concerning the new

roduct: In strategy 1, the price of the new product is determined

ased on the forward network and cannot be changed afterward;

hile in strategy 2, the price of the new product can be altered when

he reverse network is launched. Table 10 summarizes the results of

he case where we assume 75 percent of BSH’s returns are collected

n retail stores. Sequential decision making (with either strategy)

816 W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819

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results in a reduction in the number of DCs i.e., the DC in Erfurt

(i.e., site 9) is closed. This is because the sequential decision making

does not take into account of the increase of flow volume due to the

existence of remanufactured products that will also be redistributed

through the DCs. Moreover, in the integrated design as well as the

sequential Strategy 2, where pn and pr are determined simultane-

ously, a new RC is open in Mainz (i.e., site 10). Such market prices

maximally absorb the demands for the remanufactured products and

therefore opening RCs closer to remanufacturing facility in Giengen

can effectively reduce the increasing outbound transportation costs

of RCs. Perhaps more interestingly, the sequential strategy 1 achieves

an expected profit of only 66 percent of the integrated solution;

while that of Strategy 2 is 98 percent. The intuition is the following:

With a higher pn in Strategy 2, the firm obtains more profit from the

primary market demand and reduces the potential cannibalization

effect from the sales of remanufactured products. On the other

hand, the fixed pn in Strategy 1 sets an “anchor” for the price of the

remanufactured product via the inverse demand functions. Such

a suboptimal (pn, pr) pair fails to extend the firm’s customer base

furthest nor to contain the cannibalization effect to the least extent.

8. Conclusions

Managers are increasingly finding it necessary to ponder the ex-

tent of their company’s involvement in remanufacturing. This of-

ten involves deliberations concerning the establishment of in-house

remanufacturing capability. Considering the uncertainties they face

pertaining to the market size, return volume and recovery rate, we

study the profitability of the firm’s product recovery options as well

as the associated network configuration. An overwhelming majority

of the prevailing network design literature aims at minimizing the

total cost of establishing and operating the CLSC, which amounts to

viewing uncertainty as a nuisance. In contrast, we perceive the un-

certainties mentioned above as an opportunity and present an in-

tegrated network design model to assist the firm in unlocking this

potential. Hence, the proposed model incorporates a market clearing

mechanism and aims at maximizing the firm’s profit.

Inspired by a real-life problem pertaining to refrigerator reman-

ufacturing, we study the choice between remanufacturing and recy-

cling (or proper disposal) in highly uncertain settings. Our model cap-

tures the market characteristics and the overall cost structure as the

two key factors driving the choice of recovery strategy. Motivated by

practice, we focus on return streams with both a healthy secondary

market for remanufactured products and a profit potential in recy-

cling market for the OEM. This enables us to represent the benefits

(or costs) of remanufacturing relative to those of recycling that have

not been explicitly explored in previous research.

The proposed IS-CSLSC model and the solution method that inte-

grates the SAA and integer L-shaped decomposition techniques are

generic. The managerial insights developed in Section 7, however, are

specific to the BSH case. Nevertheless, it is important to note that

there are a large number of OEMs that operate in the WEEE domain,

whose only source of remanufacturable returns is retail stores. For

such firms, the return streams through the municipal collection cen-

ters are often directed to recycling; a process that the firm usually

perceives as a cost factor and delegates to a take-back scheme. It is

safe to argue that our findings can be generalized to firms in this cat-

egory. Our key results are summarized below.

First of all, integrating uncertainty and differences in customer

valuation in the same model enables us to identify market size un-

certainty as a primary lever for overall profitability and supply chain

design. This is a rather unique finding: On the one hand, the closed-

loop supply chain design models under uncertainty (e.g., Lee et al.,

2010; Listes, 2007; Salema et al., 2007) focus on cost minimization,

and hence do not incorporate market clearance for the new and the

remanufactured products. On the other hand, the models that do

ncorporate market clearance (e.g., Ferguson and Toktay, 2006 and

albreth et al., 2013) do not focus on supply chain design, and hence

eport no findings concerning its impact in this domain.

We find that the configuration of the CLSC is rather robust to un-

ertainties. We also find that the integrated network design approach

erforms better than the sequential approach under the market, re-

urn and recovery rate uncertainties. This has been also observed by

ee et al. (2010). In our work, however, this is due to a better pric-

ng strategy, rather than the cost sharing by utilizing hybrid facili-

ies as in Lee et al. (2010). Investment on network infrastructure is

medium-to-long-term decision. For those firms that add reverse

ogistics initiatives onto their existing forward network, an almost-

ptimal profit can be achieved by coordinating the prices of new and

emanufactured products during the development of their reverse

etwork. Thus, flexible pricing can be an effective way for them to

eact to the uncertain business environment. In addition, due to the

egulatory difference, the trucks used for return shipments are differ-

nt than those used for delivering the new and remanufactured re-

rigerators, which weakens the tendency of co-locating DCs and RCs.

We observe the existence of cost thresholds that make remanufac-

uring a profitable alternative, which is consistent with Ferguson and

oktay (2006) and Atasu et al. (2008). We find that the consumers’

aluation of remanufactured products increases the economic via-

ility of remanufacturing, which is also consistent with the above

wo papers. Perhaps more importantly, we also find that the extent

f the firm’s involvement in remanufacturing is very sensitive to the

elative values of the remanufacturing and recycling costs as well

s the customers’ willingness-to-pay for remanufactured products.

he likelihood for remanufacturing a certain portion of the returns

o be the most appropriate choice for the firm increases under un-

ertainty. The prevailing studies on closed-loop supply chain design

nder uncertainty mostly focus on model and solution methodology

evelopment and offer limited managerial insights. Therefore, our

ndings concerning the sensitivity associated with the variable cost

tructure and the customer valuation of remanufactured products are

nique.

Our work also reveals the significance of secondary markets for

n OEM in the context of product recovery. To the extent that the

rm can influence the customers’ perception and valuation of re-

anufacturing products, it will be able to position itself in a situ-

tion where simply remanufacturing all viable returns is the best

ption. Considering that remanufacturing only a portion of the re-

urns is more likely the best option for larger market regions, we

iew the efforts of an OEM to improve the marketing of its reman-

factured products as an important lever under highly uncertain

nvironments.

An important extension of the setting considered in this paper is

he inclusion of a regulator’s perspective in the analytical framework.

his could lead to some results that are helpful to policy makers. Nev-

rtheless, this extension is far from straightforward and hence out

f the scope of this paper – since it requires a multi-stage decision

aking formulation of the dynamics of the interaction between the

egulator and the OEM.

cknowledgments

The authors are particularly grateful to Carrie Maher, Head of the

etrologistics Appliances and Packaging Department of BSH Bosch

nd Siemens Hausgeräte GmbH, for the insights she provided as well

s her help with the data collection process. Also, the input of Javier

hocarro, Christian Dworak, Anja Stumpf, Marcus Greving, Gerhard

okic, Jörg Westerfeld improved the case study presented in this pa-

er. Feng Zhao’s professional and timely support with the computer

mplementation is greatly appreciated. Last but not least, the authors

ppreciate the insightful comments from the two anonymous review-

rs and the editor.

W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819 817

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ppendix A. Proofs

1. Proof of Proposition 1

We assume that each customer buy at most one unit. We also as-

ume that reservation price ϕ ∈ [a, b](a < b) is uniformly distributed

n the interval. Furthermore, a consumer who has a reservation price

f ϕ for a new product has δϕ for a remanufactured product. The

tility that each consumer gains from purchasing a product is given

y the difference of their reservation price and market price. The net

tility, NU, from using a unit is

Ul = δmϕ − pl (29)

here m is an indicator variable such that m = 0 if the product is new,

= 1 if the unit is remanufactured.

We also assume that reservation price ϕ ∈ [a, b] is uniformly dis-

ributed in the interval.

Consider the problem facing consumers. Each consumer has to

hoose from one of the three strategies: (i) buy a new product (N);

ii) buy a remanufactured product (R) and (iii) buy nothing (X). From

he net utility perspective, the reservation price of customers who

ollow a N strategy is higher than that of consumers who follow a R

trategy, that of who is higher than that of consumers who follow a X

trategy.

Now consider the consumer who adopts a R strategy with the

owest reservation price. This consumer’s reservation price is ϕ =+ (M−qn−qr)(b−a)

M and is indifferent between following a R and an X

trategy. From (A1), this customer’s net utility from a R strategy is

Ur = δϕ − pr = δ

[a + (M − qn − qr)(b − a)

M

]− pr (30)

nd the utility from following an X strategy is 0. Equating these two

ives a price for the remanufactured product of

pr = δ

(b − b − a

Mqn − b − a

Mqr

). (31)

ext, consider the consumer who follows a N strategy with the low-

st reservation price. This consumer has a reservation price of ϕ =− b−a

M qn and is indifferent between the N and R strategies. The net

tility from a N strategy is

Un = ϕ − pn = b − b − a

Mqn − pn. (32)

The net utility from a R strategy is

Ur = δϕ − pr = δb − δb − a

Mqn − pr. (33)

Equating these two net utilities gives a price of the new product

f

pn = b − b − a

Mqn − δ

b − a

Mqr. (34)

2. Proof of Proposition 2

θ (Y, T) = max − b − a

M(θ)(qn)2 − 2δ

b − a

M(θ)qnqr − δ

b − a

M(θ)(qr)2

+ (b − cn − ch)qn + (δb − cr − ch + cd)qr

−(∑

i

∑j

∑l

ci jUli j +

∑j

∑k

∑l

e jkXljk +

∑j

∑i

c′jiVji

+∑

k

∑j

e′k jWk j

)− (cs + cd)

∑k

rk(θ).

The Hessian is 2(b−a)M (1 δ

δ δ) whose determinant − 2(b−a)M [(δ − 1

2 )2 +14 ] is negative for any δ. Thus, the Hessian is negative definite and the

rofit function is concave in (qn, qr) for every given δ.

ppendix B. L-shaped method for SQPs

Step 1: Initialization.

1 = {(Y, T, θ)|θ ≥ θ0,Yj, Tj ∈ [0, 1] ∀ j}here θ0 is a upper bound for L. We will show its existence later.

e solve the current problem corresponding to this F1. Denote the

urrent optimal solution by (Y , T , θ ).

Step 2: Consider the current optimal first-stage solution (Y , T , θ )esulted from the last current problem solved. Clearly in our case

(Y , T , s) is finite for every scenario s, because for any feasible first-

tage solution, qn = 0, qr, U = 0, X = 0, V = 0, We = r (where e is an

ll ones vector) is always a feasible second-stage solution. Thus, L is

lso finite.

Observe that the Hessian of second-stage objective function is

ositive definite and the constraints are linear, so for any scenario

, the second-stage problem is concave. Therefore, the resource func-

ion Q s(Y, T) can also be expressed as the optimal value of the dual

roblem associated with the second-stage problem in (5)–(19). More

recisely, if γ l(s), ε(s), η(s), π l(s), ρ lj(s), φl

k(s), χ k(s), κ j(s), ψ(s),

ljk(s), ξ kj(s), ζ l(s), ιl

j(s), τ l

jk(s), ς kj(s), oj(s) are dual variables corre-

ponding, respectively, to the constraints (6), (8)–(9), (11)–(19), then

Qs(Y, T) = max −b − a

M(s)(ζ n(s))2 − 2δ

b − a

M(s)ζ n(s)ζ r(s)

− δb − a

M(s)(ζ r(s))2 −

∑l

γ l(s)sl − α(s)r(s)ε(s)

− M(s)η(s) −∑

j

∑k

∑l

slYjμljk(s)

−∑

j

∑k

rk(s)Tjξk j(s) +∑

k

rk(s)χk(s) (35)

.t.

γ n(s) − η(s) + πn(s) +∑

k

βk(s)φnk (s) − b − a

M(s)ζ n(s)

− δb − a

M(s)ζ r(s) ≤ −(b − cn − ch) (36)

γ r(s) − ε(s) − η(s) + π r(s) +∑

k

βk(s)φrk(s) + ψ(s)

− δb − a

M(s)ζ n(s) − δζ r(s) ≤ −(δb − cr − ch + cd) (37)

π l(s) + ρ lj(s) ≤ c j ∀ j, l (38)

μljk(s) − ρ l

j(s) − φ lk(s) ≤ e jk ∀ j, k, l (39)

(s)κ j(s) − ξ jk(s) + χk(s) ≤ e′k j ∀ j, k (40)

κ j(s) − ψ(s) ≤ c j ∀ j (41)

l(s), ε(s), η(s), κ j(s),μljk(s), ξ jk(s) ≥ 0 ∀i, j, k, l (42)

r concisely as

max1,z2,π

−1

2π ′Hπ + (b1 − Dy)′z1 + b′

2z2

.t.

′1z1 + A′

2z2 − Hπ ≤ f

1 ≥ 0.

Now, if θ ≥ −L(Y , T), we are done: we stop with (Y , T) being an

ptimal solution. If θ < −L(Y , T), then (Y , T) is not optimal. In this

ase, for every scenario s, let (γ l(s), ε(s), η(s), κ j(s), μljk(s), ˆξk j(s),

818 W. Chen et al. / European Journal of Operational Research 247 (2015) 804–819

)

w

L

θ

R

A

A

A

A

A

A

A

B

C

C

D

D

E

F

F

F

F

F

F

G

G

G

G

G

K

K

L

L

L

M

Ö

π l(s), ρ lj(s), φl

k(s), χk(s), ψ(s), ζ l(s), ιl

j(s), ˆτ l

jk(s), ςk j(s), o j(s)) be

an optimal dual solution of the second-stage problem corresponding

to (Y , T). Observe that the feasible set of the dual second-stage prob-

lem does not depend on (Y, T), that is, for any first-stage decision the

resource function is optimized over the same feasible region for any

scenario s. Using this argument, we can construct the following opti-

mality cut:

θ ≥ −∑

s

psg

[b−a

M(s)(ζ n(s))2+2δ

b−a

M(s)ζ n(s)ζ r(s)+δ

b−a

M(s)(ζ r(s))2

+∑

l

γ l(s)sl + α(s)r(s)ε(s) + M(s)η +∑

j

∑k

∑l

slYjμljk

+∑

j

∑k

rk(s)Tjξk j −∑

k

rk(s)χkg

](43

which must be satisfied by any optimal solution (Y, T, θ ), but is not

satisfied by the non-optimal solution (Y , T , θ ). This inequality can be

re-written as follows:

θ ≥ −(b − a)Esg

[(ζ n(s))2

M(s)g

]− 2δ(b − a)Esg

[ζ n(s)ζ r(s)

M(s)g

]

− δ(b − a)Esg

[(ζ r(s))2

M(s)g

]−

∑l

Es[γl(s)]sl − Es[α(s)r(s)ε(s)] − Es[M(s)η(s)]

−∑

j

∑k

∑l

slEs[μ

ljk(s)]Yj −

∑j

∑k

Es[rk(s)ξk j(s)]Tj

+∑

k

Es[rk(s)χk(s)] (44)

or concisely as

E1Y + E2T + θ ≥ z (45)

where E1, E2 are vectors of corresponding sizes and z is a real value.

Redefine F1 := F1 ∩ {(Y, T, θ)|(34)} and continue with Step 3.

Step 3: Solve the current problem with the updated F1 and return

to Step 2.

We now provide details on the computation of the upper bound

θ0. Define c = min j{c j}, ek = min j{e jk} and e = mink{ek}. Given the

definition of Q(Y, T, s) and the constraints (10), (13)–(16) we get

Q(Y, T, s) ≤ (pn − cn − ch)qn + (pr − cr − ch + cd)qr

− c(qn + qr) −∑

k

ekβk(qn + qr) − cqr −∑

k

ekrk(s)

= −I(s)(qn)2 − 2δI(s)qnqr

+(

b − cn − ch − c −∑

k

ekβk

)qn − δI(s)(qr)2

+(

δb − cr − ch + cd − 2c −∑

k

ekβk

)qr −

∑k

ekrk(s)

≤ −I(s)(qn)2 − 2δI(s)qnqr − δI(s)(qr)2

+ (b − cn − ch − c − e)qn

+ (δb − cr − ch + cd − 2c − e)qr − er(s)

≤ −I(s)(qn + δqr)2 + (b − cn − ch − c − e)qn

+ (δb − cr − ch + cd − 2c − e)qr − er(s)

< (b − cn − ch − c − e)sn

+ (δb − cr − ch + cd − 2c − e)sr − er(s)

here I(s) = (b − a)/M(s). Taking expectations at both sides yields

(Y, T) < (b − cn − ch − c − e)sn

+ (δb − cr − ch + cd − 2c − e)sr − eEs[r]

We define the lower bound θ0 as:

0 = eEs[r]−(b−cn−ch−c−e)sn − (δb − cr − ch + cd − 2c − e)sr.

(46)

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