Price and service competition between new and remanufactured products in a two-echelon supply chain

Int. J. Production Economics 140 (2012) 496–507

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Price and service competition between new and remanufactured products in a two-echelon supply chain Cheng-Han Wu * Department of Industrial Engineering and Management, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan, ROC

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 July 2011 Accepted 27 June 2012 Available online 5 July 2012

In this paper, we consider a supply chain consisting of two manufacturers and a retailer. The first manufacturer is a traditional manufacturer that produces the new product, while the second manufacturer operates a reverse channel producing remanufactured products from used cores. Both manufacturers bundle their products with services, including warranty and advertisement, and they sell through the same retailer, which independently determines the sales prices. We assume that the second manufacturer invests extra effort in facilitating the remanufacturing process. In this study, we identify the equilibrium characteristics with respect to the remanufacturer’s effort and price and service decisions for all members of the supply chain. We also investigate the profits of chain members by considering different interactions between prices and service. Based on the theoretical and numerical analyses, we derive economic and managerial insights for chain members. & 2012 Elsevier B.V. All rights reserved.

Keywords: Game theory Pricing Remanufacturing Service Supply chain management

1. Introduction Remanufacturing is the process whereby some components of used products are disassembled, cleaned, reprocessed, inspected, and then reassembled to be used again. Consumer awareness, oversight from non-governmental organizations, and legislative pressures have encouraged manufacturers to produce green and eco-friendly products, and thus, more and more manufacturers now build reverse channels to recycle used products for remanufacturing. However, rather than environmental concerns, the economic benefits that accompany remanufacturing is the main consideration for manufacturers. Because of decrements in the costs associated with raw material production, a remanufacturing system provides an opportunity to reduce not only the environmental burden but also production costs. According to a recent report by Global Industry Analysts (2010), global automotive manufacturing is growing, and by 2015, it is forecasted to reach US$104.8 billion. Such a strong growth in remanufacturing is also present in other industries, such as toner and inkjet cartridges, electrical equipments, consumer electronics, and furniture (Hauser and Lund, 2008). In practice, it is important for manufacturers to adjust their sales strategies in response to the introduction of remanufacturing. For instance, the large personal computer manufacturer, HP Inc., has adopted a remanufacturing program called ‘‘HP Renew Program’’ for recycling and selling the remanufactured or refurbished products. Its remanufacturing program certifies that the remanufactured products

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performing well and can substitute new products at lower prices, and it also provides warranty and service for the remanufactured products. Thus far, not all HP’s competitors have followed suit, and therefore, the consumers must choose between remanufactured and new products based on product information of prices and service offerings. In the context of this business model, a comparative investigation of price and service for new versus remanufactured products is useful for manufacturers as well as retailers, as it should provide valuable insights into the interactions between price and service, sales decisions, and performance. To the best of our knowledge, all the previous studies examining remanufacturing are models that only consider the remanufactured product while ignoring the competition, or they are competitive models with a single price attribute. In this study, we incorporate the competitive nature of products and service into our supply chain model of a common retailer and two manufacturers. One manufacturer produces the new product, and the other produces the remanufactured product under the framework of product competition. In our model, we analyze the effects of competition, and thus, we consider four competitive interactions: the presence of price and service competition, price competition only, and service competition only as well as the absence of both price and service competition. Note that the interaction with both price and service competition can be viewed as a general model such that the demand functions, chain members’ profit functions, and equilibrium decisions for other interactions can be obtained from this competitive model. Our analyses reveal that when remanufacturing leads to more cost-savings, the remanufacturer will provide a higher service level to the customers than the traditional manufacturer. Moreover,

C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

production cost always has negative effects on the chain members’ decisions; however, the effects of the costs of recycling and service investment on equilibrium decisions depend on the intensities of price and service competition, especially in determining the newproduct manufacturer’s equilibrium decisions. Comparing the different interactions, we find that in the presence of competition, the manufacturer has an incentive to remanufacture, yet the opposite is the case in the absence of competition. Moreover, for the retailer, price competition generally enhances profits from the remanufactured product, and thus benefits the remanufacturer. Meanwhile, service competition is profitable for the retailer yet detrimental to both manufacturers. However, the remanufacturer is likely to engage in service competition when cost-savings from remanufacturing are significant or recycling costs are low. Moreover, remanufacturing is an effective strategy in a highly price-sensitive market, even in the absence of price competition. The remainder of this paper is organized as follows. Section 2 surveys the related literature while emphasizing the contribution of our work. In Section 3, we formalize price- and servicesensitive market demands and chain members’ profit functions, and we derive their best response functions. Then, we solve for the member equilibrium decisions under the four competitive interactions. Section 4 presents analytical analyses of chain members’ equilibrium decisions with respect to the cost parameters, and then it describes numerical studies that examine the effects of cost- and demand-related parameters on chain members’ equilibrium profits. The final section concludes with a brief summary, including suggestions for future research.

2. Literature review This study relates to two streams of literature: one examines the dynamics of chain members’ service decisions in the face of service incentive demand, and the other explores supply chains with remanufacturing. Previous studies in the first stream of literature have concentrated on service interactions in supply chains under game theoretical models (Bernstein and Federgruen, 2007; Cohen and Whang, 1997; De Borger and Van Dender, 2006; Desiraju and Moorthy, 1997; Ray, 2005; Tsay and Agrawal, 2000; Winter, 1993; Xia and Gilbert, 2007; Xiao and Yang, 2008; Zhen, 2012). Cohen and Whang (1997) used a product life cycle model to study the sales prices and service qualities in a supply chain with a manufacturer and a service shop. In this model, the manufacturer sells the product as well as provides the after-sales service, while the service shop only provides service. They adopted a timedependent utility function to construct market demand, and then sequentially analyzed the price and service decisions of members. Tsay and Agrawal (2000) studied a supply chain with two competing retailers with a common manufacturer in which both the retailers provide products as well as service to customers. Thus, retailers sell the same products and compete along price and service. Xia and Gilbert (2007) focused on a supply chain model with two substitutable products in which the product service levels are provided by a manufacturer, and the prices are determined by a dealer (i.e., a retailer). This setting is applicable to consumer electronics, computer devices and automobiles, which widely adopt remanufacturing. Many models have also addressed service competition, such as Ba et al. (2008), Bernstein and Federgruen (2007), De Borger and Van Dender (2006), Darian et al. (2005), Dan et al. (2012), and Kurata and Nam (2010); however, only forward channels are examined in these studies. More recently, studies have begun to take into account remanufacturing as a part of production. However, these studies have focused on price interaction within a closed-loop supply

497

chain in the absence of service interactions, and often analyzed the recycling effort or profitability associated with remanufacturing (Atasu et al., 2008; Debo et al., 2005; Ferrer and Swaminathan, 2006; Guide and Van Wassenhove, 2009; Hsueh, 2011; Liang et al., 2009; Mitra and Webster, 2008; Pokharel and Liang, 2012; Robotis et al., 2005; Robotis et al., in press; Savaskan et al., 2004; Savaskan and Van Wassenhove, 2006; Vadde et al., 2011). Savaskan et al. (2004), for instance, studied price decisions and collection effort in a supply chain with three types of reverse channels for remanufacturing in comparison with a centralized system; the study examined the incentives provided to the customer in order to improve the collection of used items in the absence of the competition. Savaskan and Van Wassenhove (2006) further extended the previous closed-loop supply chain models by considering a duopolistic competition with a retailer; it also discussed the interactions surrounding price decisions with respect to forward and reverse channels. Atasu et al. (2008) examined remanufacturing as a marketing strategy by considering several factors, including cost savings, green-market segment, competition with new products, and product life cycle. The study found that a remanufactured product is more profitable under monopolistic competition, even in the absence of a green market segment. Moreover, the study also pointed out that smart price decision-making is important as the manufacturer cannibalizes used products. A few recent studies including Atasu et al. (2008), Debo et al. (2005), Ferrer and Swaminathan (2006), Majumder and Groenevelt (2001), Mitra and Webster (2008), and Savaskan and Van Wassenhove (2006) have shed light on competition in closed-loop supply chains with remanufacturing, but they all focus exclusively on price competition. This paper differs from the aforementioned studies in two aspects. First, we assume that manufacturers compete in a duopoly, and second, we incorporate service competition between manufacturers. As discussed in Atasu et al. (2008), remanufactured products usually have lower customer valuations, and thus, manufacturers with remanufacturing (e.g. HP Inc.) may provide service (e.g., product information, after-sales maintenance, etc.) for customers to stimulate sales. Moreover, the remanufactured versus new products possess different characteristics in terms of cost structures and interactive effects such that remanufactured products directly relate to the amount a manufacturer invests in remanufacturing. Thus, a study of the interactions between price and service decisions and an analysis of profitability of these two products should provide useful insights to supply chain members.

3. The model Consider a supply chain comprised a manufacturer that produces new products from raw materials, a remanufacturer1 that produces remanufactured products from used items directly collected from customers (Savaskan et al., 2004; Savaskan and Van Wassenhove, 2006), and a common retailer that sells the products of these two manufacturers on the market. The two products are functionally the same. However, the customers treat them differently as new and remanufactured products. The retailer independently determines the sales prices of the two products. Each manufacturer produces only one product, and each has the opportunity to enhance demand by bundling the product with service, such as sales effort (e.g., advertisement, product information) (Tsay and Agrawal, 2000) or 1 In many instances, we use ‘‘remanufacturer’’ specifically to refer to the manufacturer producing the remanufactured products. In other instances, we refer to both the manufacturer and the remanufacturer as the ‘‘two manufacturers.’’ Moreover, we refer to the retailer as female and the manufacturers as males.

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after-sales service (e.g., maintenance and warranty repair agreements) (Cohen and Whang, 1997; Xia and Gilbert, 2007). However, to facilitate the remanufacturing process, the remanufacturer has to decide its recovering effort, which may lead to additional costs and savings. We discuss costs and savings later in the model. All chain members’ decisions must be made prior to the selling season. The sequence of events is as follows. Let the subscript j take the values of r and n, denoting the remanufactured and new products, respectively. Because the production plan should be made before the sale decisions, the remanufacturer first determines its remanufacturing effort t, 0 r t r1, where t can be considered the portion of recovered materials used to produce a unit product by the remanufacturer. Then, both manufacturers choose their service package levels sj. Finally, given the wholesale prices wj of manufacturer j, the retailer chooses the sales prices pj for these two products, and orders the required quantities from the two manufacturers. Before the start of the selling season, the price and service information for the products is announced to the market. This scenario is common in real life, for instance, the remanufactured and new toner cartridges are usually sold through a common retailer (e.g., Amazon.com) in a market. In this example, the cartridge manufacturers provide the service for their products, and Amazon.com decides the sales prices. All product and service information is communicated to the customers. We assume that the manufacturers have ample capacity to meet the retailer’s order quantity (Ferrer and Ketzenberg, 2004; Savaskan et al., 2004; Savaskan and Van Wassenhove, 2006). This assumption is plausible to the reality when the recovered materials are common modules, which can be collected from different types of products (e.g., aftermarket automotive parts) or from different brands (e.g., computer parts), and when the source of recovered materials is from popular or long-life products, for example, XPS Inc., a remanufacturer of toner cartridges, has sufficient capacity of remanufactured cartridges for some popular printers (www.xps-lasercartridge.com). The manufacturers are able to deliver the order quantities to the retailer before the selling season. The retailer is the monopolist in the considered market area, and other products in this market area or different market areas under consideration, while the other products in this market area or different market have no effect on the demand of these two products. This assumption allows us to investigate competition between the new and remanufactured products. We also consider the chain members to be independent, risk-neutral, and profit-maximizing. Moreover, the chain members choose their decisions sequentially in a manufacturer-Stackelberg game (Lau and Lau, 2002; Savaskan et al., 2004; Savaskan and Van Wassenhove, 2006; Trivedi, 1998) under a level playing field, and they have complete information about the other members (Cachon and Lariviere, 1999; Choi, 1991, 1996; Goyal and Netessine, 2007; Lee and Staelin, 1997; Trivedi, 1998). The notation used throughout this paper is summarized in Appendix A.

3.1. Demand and profit functions

where aj , bp , bs 4 0, and gp , gs Z0, j ¼ fr,ng, and k ¼ fr,ng\j. aj describes the market base parameterizing the potential market size for product j when both prices at zero and no service is offered. bp and bs represent the price and service elasticities, respectively, on market demand. dj is downward sloping along its price pj yet upward sloping along its service sj; i.e., the higher the price (service) is, the lower (higher) is the demand. gp and gs measure the price and service competition, respectively, between the new and remanufactured products, and a higher gp (gs ) magnifies the influence of price (service) competitively. As discussed in Tsay and Agrawal (2000), lowering pj by one unit will attract bp þ gp ð@dj =@pj ¼ bp þ gp Þ more customers with demand (dj), if all else parameters are held equal; that is, bp is the additional customers induced from a decrease in product j’s price, and gp ð@dj =@pk ¼ gp Þ is additional customers diverted due to an increase in the other product’s price. Therefore, the higher gp leads to severe price competition between the products, and gs has similar implications in service competition. (Refer to Tsay and Agrawal, 2000, for more details about the demand functions with competition.) Given market demand dj, we can formulate the members’ profit functions. The new product manufacturer carries the production cost and service cost. The total production cost is cdj , and the service cost is ðms2j Þ=2, where c is the unit production cost and m is the ultimate cost of service as in Tsay and Agrawal (2000). Therefore, the manufacturer’s profit function is

PM n ¼ ðwn cÞdn 

ms2n : 2

ð2Þ

The remanufacturer must account for collection cost, production cost, and service cost. Following Debo et al. (2005) and Savaskan and Van Wassenhove (2006), we assume that the recycling process incurs a total collection cost of ðbt2 Þ=2 for the remanufacturer, where b is a recycling cost. Note that the quadratic cost functions of service and collection imply the increasing marginal costs, that is, the subsequent improvements of unit service and collection are progressively more difficult (for example, the manufacturer bears the higher cost to enhance the service level from 0.8 to 0.9 than does from 0.1 to 0.2.). This means that the chain members will target the ‘‘lowest-hanging fruit’’ (Tsay and Agrawal, 2000) in determining service and remanufacturing levels. Let cr be the complete remanufacturing cost, and let d  ccr 4 0 be the cost-savings from the remanufacturing (Atasu et al., 2008; Ferrer and Swaminathan, 2006; Savaskan et al., 2004; Savaskan and Van Wassenhove, 2006). In practice, cost reduction is a main reason for many manufacturers (e.g., Caterpillar, HP, Xerox) to engage in remanufacturing, which provides 30–70% cost savings in comparison to the usage of new cores (Gray and Charter, 2007). Then, the unit remanufacturing cost with returned items can be expressed as cð1tÞ þcr t ¼ cdt. Similar cost structure is also adopted in the literature (Gupta and Loulou, 1998; Savaskan et al., 2004; Savaskan and Van Wassenhove, 2006) in the formulation of fixed costs and cost differences. The remanufacturer’s profit function can be described by the following:

PM r ¼ ðwr c þ dtÞdr 

ms2r bt2  : 2 2

Let dj denote the market demand for product j A fr,ng. We consider demand dj to be a linear function of prices and service levels, which is commonly adopted in the literature on price competition (Choi, 1991, 1996; Lee and Staelin, 1997; McGuire and Staelin, 1983; Trivedi, 1998) and price and service competition (Tsay and Agrawal, 2000; Xia and Gilbert, 2007). We define the general demand function that captures both product and service competition as follows:

Assume both manufacturers’ prices to the retailer wr and wn are predetermined (Hsieh and Wu, 2009; Lee et al., 2000; Tsay and Agrawal, 2000). Then, the retailer’s profit function PR is the sum of the profits PRr and PRn from the remanufactured and new products, respectively:

dj ¼ aj bp pj þ gp ðpk pj Þ þ bs sj gs ðsk sj Þ,

PR ¼ PRr þ PRn ¼ ðpr wr Þdr þ ðpn wn Þdn :

ð1Þ

ð3Þ

ð4Þ

C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

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3.2. Strategic interactions

results in the following:

We now examine chain members’ decisions by maximizing their profits in (2)–(4). Because the remanufacturing effort is related to the remanufacturer’s production plan, we consider that the remanufacturer chooses the remanufacturing effort prior to sale decisions. Afterwards, both the manufacturers determine their service levels, and the retailer determines the prices for the remanufactured and new products, depending on their service levels. Note that the proofs of the propositions presented throughout this paper are included in Appendix A. The following proposition characterizes chain members’ profit functions with respect to their decisions.

s[r ¼

ðwr c þ dtÞLs 2m

and

s[n ¼

ðwn cÞLs : 2m

ð6Þ

In (6), we find that s[n is independent of t; i.e., s*n ¼ s[n . Yet, s[r is increasing in t, meaning that the higher the remanufacturing effort is, the higher is the service level provided by the remanufacturer. Moreover, if dt 4 wn wr , then s[r 4 s[n . This indicates that the remanufacturer would provide a better service level for his product than that offered with the new product when remanufacturing strategy leads to higher cost-savings. Finally, with the best response functions for prices in (5) and service levels in (6), the remanufacturer’s best response in terms of his remanufacturing effort can be obtained by solving t* A

Proposition 1. (i) The remanufacturer’s profit function PM is r concave with respect to its remanufacturing effort t and service level sr. (ii) The manufacturer’s profit function PM n is concave with respect to its service level sn. (iii) The retailer’s profit function PR is concave with respect to sales prices pr and pn.

[ [ [ [ arg maxt PM r ðt9sr ,sn ,pr ,pn Þ, which yields the equilibrium remanu-

Proposition 1 shows that each chain member has an optimal choice to maximize heir profits, and this also implies that each member has the optimal strategies in response to the decisions of other chain members in competition. In pursuit of equilibrium decisions, we use backward induction; that is, each chain member formulates its best reactions, denoted by the superscript ‘‘[’’, as a function of the decisions of the prior movers. Let the superscript ‘‘*’’ denote the equilibrium value. Up to the remanufacturer’s equilibrium decision of t* is obtained, and all chain members’ equilibrium decisions are revealed by taking the remanufacturer’s equilibrium decision into the best response functions for the service levels and prices. First let us derive the last mover’s strategy, i.e., the best

With t* in (7), we can derive all equilibrium service levels and prices s*r , p*r , and p*n in different scenarios (i.e., gs , gp a0 presents the presence of price and service competition, gs ¼ 0 no service competition, gp ¼ 0 no price competition, and gs , gp ¼ 0 no price and service competition), as shown in Table 1.

response functions of the retailer’s prices p[r ,p[n A arg maxpr ,pn PR ðpr ,pn 9t,sr ,sn Þ. By solving the first conditions @PR =@pr ¼ 0 and @PR =@pn ¼ 0, we obtain p[r

facturing effort t* :

t* ¼

dð2mðar þwn gp Þ þ Ls ðcbs þwn gs Þ þwr ð2mLp L2s ÞÞ 2

4bm þ d L2s

:

ð7Þ

4. Analysis We now perform further analyses of the equilibrium results. We focus on the effects of cost- and demand-related parameters on equilibrium decisions, and then we explore their influences on both manufacturers’ equilibrium profits. However, due to analytical intractability of chain members’ equilibrium profits, we use a numerical approach for providing more insights. 4.1. Effects of costs on equilibrium decisions

ar Lp þ an gp þðsr sn Þbp gs þ wr bp ðgp þ Lp Þ þ bs ðsn gp þ sr Lp Þ , ¼ 2bp ðgp þ Lp Þ ð5Þ

Based on Table 1, we analyze the equilibrium decisions of the remanufacturing effort, service levels, and sales prices with respect to costs.

an Lp þ ar gp þ ðsn sr Þbp gs þwn bp ðgp þ Lp Þ þ bs ðsr gp þ sn Lp Þ : 2bp ðgp þ Lp Þ

Proposition 2. All the equilibrium decisions t* ,s*r ,s*n ,p*r , and p*n are decreasing in c.

Note that Lp  bp þ gp is the overall effect of the product’s price on its demand. Focusing on the impact of service levels on the retailers’ best response functions, we derive the first differential of p[j with respect to the service levels, i.e., @p[j =@sj ¼ ðbs gp þ

Proposition 2 shows that greater unit production cost leads to lower equilibrium decisions. From the proof shown in Appendix, we observe that product cost has a direct effect on the equilibrium service levels, yet it has an indirect effect on equilibrium prices and remanufacturing effort. When production cost is higher, the manufacturers will provide less service to lower costs, and decreasing service levels decreases equilibrium prices and the remanufacturing effort.

p[n ¼

bp Ls Þ=ð2bp ðbp þ Lp ÞÞ 4 0 and @p[j =@sk ¼ ðbs gp bp gs Þ= ð2b2p þ 4bp gp Þ, where Ls  bs þ gs represents the overall effect of service on

demand, k aj, and j ¼ fr,ng. The higher the service level of product j is, the higher is the sales price of product j charged by the retailer. However, the influence of the competitor’s service level on the sales price depends on the competition intensities gj (j ¼ r,n). If bs gp 4 bp gs , then @p[j =@sk 40. This indicates that if the price competition is severer (i.e., gp is greater) or service competition is milder (i.e., gs is smaller), then the higherservice level of product k increases the sales price of product j; otherwise, the result is reversed. Each manufacturer’s best response functions for service level s[j includes the retailer’s best reaction in (5); that is, [ [ [ s[j A arg maxsj PM j ðsj ,sk 9t,pr ,pn Þ,k a j and j ¼ fr,ng. Thus, solving M [ [ [ [ the first conditions, @ðPM r 9pr ,pn Þ=@sr ¼ 0 and @ðPn 9pr ,pn Þ=@sn ¼ 0, of both manufacturers’ profits with respect to their service levels

Proposition 3. (i) t* is decreasing in b but increasing in d. (ii) s*r is decreasing in b but increasing in d, but s*n is independent of b and d. (iii) p*r is decreasing in b but increasing in d. Whenever bs gp Z bp gs , p*n is decreasing in b but increasing in d; otherwise, p*n is increasing in b but decreasing in d. Proposition 3 shows that the equilibrium decision trends behave more complexly with respect to remanufacturing-related costs b and d than unit cost c. We observe that the remanufacturer invests a lower remanufacturing effort in production while bearing the higher cost in remanufacturing, leading to the lower service levels provided by the remanufacturer (recall that sr is positively related to t, as discussed before). Intuitively, the retailer charges a lower price, as the remanufactured product

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C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

Table 1 Equilibrium decisions under the different competition scenarios. Scenarios

gs , gp a 0

Equilibrium decisions

tn ¼

snr ¼

snn ¼

dð2mðar þ wn gp Þ þ Ls ðcbs þ wn gs Þþ wr ð2mLp L2s ÞÞ 2

4bm þ d L2s

Ls ð4bcm þ wr ð4bm2md2 Lp Þþ d2 ð2mðar þ wn gp Þ þ ðcwn Þgs Ls ÞÞ 2

2mð4bmd L2s Þ ðwn cÞLs 2m 2

2

pnr ¼ ½8bm ðan gp þ ar Lp þ wr ðg2p þ L2p ÞÞ þ 2mgs ðgp ð2bc þ 2bwr þ d ðar þ wn gp ÞÞ 2

2

ð2bc2bwn þ d wr gp ÞLp ÞLs þ ð2md

2 wr 2p þ 4mLp ðbc þ wr ðb þ d Lp ÞÞ

2

g

2

þ gp ð4bcm þ d ð2man þ cg2s Þwn ð4bm þ d ðg2s þ 2mLp ÞÞÞÞL2s 2

2

þ d ðc þ wn Þgp L4s =½4mðg2p L2p Þð4bmd L2s Þ

2

2

2

pnn ¼ ½8bm ðar gp þ an Lp þ wn ðg2p þ L2p ÞÞþ 2mgs ðLp ð2bc þ d ar þ wr ð2bd Lp ÞÞ 2

2

þ gp ð2bc þ wn ð2b þ d Lp ÞÞÞLs þ ð4mgp ðbc þ bwr þ d wn gp Þþ ð2mð2bc 2

2

2

2bwn þ d ðan þ wr gp ÞÞþ d ðcwn Þg2s ÞLp þ 2md wn L2p ÞL2s 2

2

þ d ðc þ wn ÞLp L4s =½4mðg2p L2p Þð4bmd L2s Þ

gs ¼ 0

tn ¼ 

snr ¼

snn ¼

dð2mar þ ðc þ wr Þb2s þ 2mðwn gp wr Lp ÞÞ 2

2

4bm þ d bs

bs ð2bcd2 ar d2 wn gp þ wr ð2b þ d2 Lp ÞÞ 2

2

4bm þ d bs ðwn cÞbs 2m 2

2

2

2

pnr ¼ ½ð4bmd bs Þgp ð2man þ ðcwn Þbs þ 2mwr gp Þ2mð4bmar þ bs ð2bc þ 2bwr 2

2

2

2

2

þ d wn gp ÞÞLp þ 4mwr ð2bm þ d bs ÞL2p =½4mð4bmd bs Þðg2p L2p Þ 2

2

2

2

2

2

2

2

2

pnn ¼ ½4mgp ð2bmar þ bðwr cÞbs þ wn ðd bs 2bmÞgp Þ þ ðð4bmd bs Þð2man þ ðwn 2

2

2

cÞbs Þþ 2md wr bs gp ÞLp þ 2mwn ðd bs 4bmÞL2p =½4mð4bmd bs Þðg2p L2p Þ

gp ¼ 0

tn ¼ 

snr ¼

snn ¼

dð2mar Ls ððc þ wn Þgs þ cLs Þ þ wr ð2mbp þ L2s ÞÞ 2

4bm þ d L2s

Ls ð4bcm þ wr ð4bm2md2 bp Þ þ d2 ð2mar þ ðcwn Þgs Ls ÞÞ 2

2mð4bmd L2s Þ ðwn cÞLs 2m 2

pnr ¼

pnn ¼

2bmðar þ wr bp Þþ bðcwn Þgs Ls ðbc þ wr ðb þ d bp ÞÞL2s 2

bp ð4bmd 1

L2s Þ

2

2

4mbp ð4bmd L2s Þ

2

2

ð8bm ðan þ wn bp Þ þ 2mð2bcd ar þ wr ð2b þ d bp ÞÞgs Ls

2

2

2

2

ð2mð2bc þ d an þ wn ð2b þ d bp ÞÞ þ d ðcwn Þg2s ÞL2s þ d ðcwn ÞL4s Þ

gp , gs ¼ 0

tn ¼

dð2mar þ cb2s þ wr ð2mbp b2s ÞÞ 2

2

4bm þ d bs 2

2

ð2bcd ar þ wr ð2b þ d bp ÞÞbs snr ¼  2 2 4bmd bs snn ¼

ðwn cÞbs 2m

C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

501

Table 1 (continued ) Scenarios

Equilibrium decisions 2

pnr ¼

Proposition 4. (i) , are decreasing in m. (ii) Let Fr  @s*r =@m * and Fn  @sn =@m. Whenever bs p 4 ½bp ðFn s Fr Ls Þ=ðFn þ Fr Þ, p*r decreases in m; otherwise, p*r increases in m. Whenever bs p 4 ½bp ðFn Ls Fr s Þ=ðFn þ Fr Þ, p*n decreases in m; otherwise, p*n

t

s*r ,

s*n

g

g

2

2

2man cbs þ wn ð2mbp þ bs Þ 4mbp

bundled with the lower service level. In contrast, when remanufacturing results in higher cost-savings, more effort is invested in production such that the higher service level and price are chosen by the remanufacturer and the retailer, respectively. Note that when d ¼ 0, the remanufacturer will abandon the remanufacturing (i.e., t* ¼ 0) and behaves as a traditional manufacturer. The equilibrium service level s*n of the new product is independent of the remanufacturing costs, but the value of the remanufacturer’s service level is influenced by b and d. Moreover, the effects of d and b on p*n depend on price and service competition. When the intensity of price competition gp is sufficiently large (i.e., gp 4 bp gs =bs ), decreasing p*r stimulates p*r to decrease. Alternatively, when the intensity of service competition gs is sufficiently large (i.e., gs 4 bs gp =bp ), the retailer chooses a higher sales price of the new product. Hence, in the absence of price competition, p*n always increases in b yet decreases in d. In the absence of service competition, p*n decreases in b yet increases in d. Moreover, in the absence of the price and service competition, p*n is independent of b and d. Next, we characterize chain members’ equilibrium decisions with respect to the investment of service m as follows: *

2

bp ð4bmd2 b2s Þ 2

pnn ¼

2

2bmar bcbs þ wr ðbbs þ bp ð2bmd bs ÞÞ

g

g

increases in m. Proposition 4 first shows that both manufacturers’ service levels are decreasing in m in equilibrium, and in turn, the decreasing service levels lead to a decrease in the remanufacturing effort. This indicates that increasing service cost indirectly crowds out the remanufacturer’s investment in remanufacturing. The effects of m on equilibrium prices depend on the intensities of competition as well as the effects Fr and Fn of m on equilibrium service levels. It is intuitive that severe price competition stimulates the retailer to choose lower equilibrium prices of the products. Moreover, p*r and p*n behave differently with respect to the effect of m on its equilibrium service level. If Fr is greater, increasing m likely leads the retailer to select the smaller p*r ; however, if Fn is greater, the retailer is likely to select the greater p*n in response to the increasing m. Considering different interactions, we find that in the absence of price interaction, both equilibrium prices will increase in m; and in the absence of service interaction, p*r always decreases in m, but p*n increases in m when gp o½bp Fn =ðFn þ Fr Þ. Thus, in the scenario with the moderate price interaction, greater m causes p*r to decrease and p*n to increase, leading p*n to be less competitive than p*r . 4.2. Competitive effects on the equilibrium decisions of the manufacturers Regarding the effects of the competition on the two manufacturers’ equilibrium decisions (i.e., the remanufacturing effort and service levels), we first consider t* trends with respect to gp and gs .

Proposition 5. (i) t* is increasing in gp ; and (ii) t* is increasing in gs whenever pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bmððwn cÞgs þðc þ wn 2wr ÞLs Þ d Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Ls ð4mðar þ wn gp wr Lp Þðwn cÞgs Ls þ ðwn cÞL2s Þ

t* is decreasing in gs , otherwise. Proposition 5 shows that price competition induces the remanufacturer to improve his remanufacturing effort, because remanufacturing leads to price advantages due to cost-savings. However, service competition does not inevitably increase the remanufacturing effort. If the remanufacturer has sufficient costsavings, service competition also improves t* ; otherwise, it reduces t* . Hence, the remanufacturer possessing the superiority in terms of cost-savings would choose the higher remanufacturing effort when either price competition or service competition is severer, that is, the remanufacturing is an effective competitive advantage (as in the base example shown in Table 2). Proposition 6. (i) s*r and s*n are increasing in gs . (ii) s*r is increasing in gp , and s*n is independent of gp . We intuitively observe in Proposition 6 that if the two manufacturers engage in more intensive service competition, they choose higher service levels. Moreover, price competition only has an influence on the remanufacturer’s service level, which increases in the intensity of price competition. 4.3. Comparisons between the equilibrium decisions for the two products One incentive of the remanufacturing is cost-savings, and thus, we consider compare equilibrium decisions between the new and remanufactured products with regard to cost-savings. First, in Proposition 7, we compare the relationship between the equilibrium prices of the two products, and we obtain a threshold for d, p denoted by d , below which the equilibrium price of the remanufactured product is smaller than that of the new product. Because of the lower wholesale price of the remanufactured product, the retailer consequently chooses the lower prices, leading the remanufactured product to be more competitive in terms of price. However, when d exceeds the threshold, p*r is greater than p*n . Because when the remanufacturer has superiority in terms of cost-savings, he chooses a better service level, as stated in Proposition 8. Then, higher service level of the remanufactured product mitigates by its pressure from price competition, inducing the retailer to shift the price of remanufactured product. Proposition 7. Let C1 ¼ an 2mar þ ðwn wr Þ½2mðgp þ Lp Þ þ Ls ðgs þ Ls Þ and C2 ¼ ðLs ð2mgs ðar þ wn gp wr Lp Þ þ ð2man þ cg2s 2 3 2 mwr ðgp þ 2Lp þ wp p Þffiffiffiffiffiffiffi n ð4m ffiffiffiffiffiffiffi gp gs þ2mLp ÞÞLs þ ðc þ wn ÞLs ÞÞ. p * * * * If d Z d ¼ 2 C1 = C2 , pr Z pn ; otherwise, pr opn . s

In Proposition 8, the threshold of d for s*r Z s*n is designated by d .

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C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

Table 2 Parameter values and the equilibrium results. Interactions

Decisions

t* gp ¼ gs ¼ 0 gp ¼ 0, gs ¼ 3 gp ¼ 2, gs ¼ 0 gp ¼ 2, gs ¼ 3

Demand s*r

s*n

p*r

p*n

*

Profits *

dr

dn

PM r *

PM n *

PRr *

PRn *

PR

*

0.7155

0.4527

0.5000

3.9310

4.2000

7.1553

6.000

7.8206

8.2500

10.2393

7.2000

17.4393

0.7773

0.8100

0.8750

4.0545

4.3695

7.7726

6.8475

7.3145

7.9743

12.0827

9.3775

21.4602

0.7672

0.4612

0.5000

3.9379

4.1966

7.6724

5.5000

8.5059

7.5000

11.0324

6.5810

17.6134

0.8329

0.8263

0.8750

4.0767

4.3538

8.3295

6.3231

8.0158

7.1878

13.1332

8.5601

21.6934

s

Proposition 8. If d Z d , s*r Zs*n ; otherwise, s*r o s*n ; where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bmðwn wr Þ s d ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2mðar þwn gp wr Lp Þ þ ðcwn Þgs Ls þðc þ wn ÞL2s

Remanufacturing is more effective in saving production cost, the remanufacturer would provide a higher service level than the s manufacturer. This indicates that if d o d , the remanufactured product is inferior to the new product in terms of service s competition. We further analyze d by taking the first derivatives p of d with respect to the service investment, recycling cost, and the intensities of price and service competition, and we obtain s s s s @d =@m 4 0, @d =@b 40, @d =@gs 40, and @d =@gp o 0. As a result, the remanufacturer bears higher costs in service or recycling investments, and thus, he provides a lower service level than the manufacturer. Moreover, when service competition is severe, the more cost-savings are required for the remanufacturer to provide a higher service level. However, price competition behaves s inversely on d ; namely, the remanufacturer may wish to provide a higher service level than the manufacturer under more intense price competition. With the aid of Proposition 7, we know that the remanufactured product is competitive in terms of price competition, and thus, the fierce price competition is more profitable to the remanufacturer, leading to a higher service level. 4.4. Comparisons between the manufacturers’ equilibrium profits The two manufacturers’ equilibrium profits are presented in (A.1). We investigate the relationship between the two manufacturers’ profit with respect to unit cost-savings. If the equilibrium profit of the remanufacturer is greater than that of the manufacturer, the remanufacturing strategy is more attractive to the manufacturers. Proposition 9. (i) The equilibrium profit PM r * is increasing in d, yet y M M PM n * is decreasing in d. (ii) If d Z d , Pr * Z Pn *; otherwise, M M Pr * o Pn *. Note that

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 bm½4mððwn cÞan ðwr cÞar Þðwn wr ÞO1  ffi, dy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4m2 ðar þwn gp wr Lp Þ2 þ ðwn cÞO2 L2s ðwn cÞ2 L4s

O1 ¼ 4mðwn þ wr ÞLp ðwn þwr 2cÞL2s 4cmgs , O2 ¼ 4mðan þ wr gp wn Lp Þðwn cÞg2s : M As can be seen in Proposition 9, PM r * increases in d, yet Pn * decreases in d. The value of d is directly profitable to the remanufacturer, yet it is indirectly detrimental to the manufacturer due to price competition. The remanufacturer obtains a higher profit than the manufacturer when the unit cost-savings y from the remanufacturing is above a threshold d , which

y

Fig. 1. Influences of gp and gs on d with the different values of m(‘‘—’’: effect of gs when gp ¼ 2; and ‘‘- - -’’: effect of gp when gs ¼ 3, under the setting as ar ¼ an ¼ 25, b¼ 5, m¼ 6, c ¼1.5, d ¼ 0:5, bp ¼ 5, and bs ¼ 4.)

intuitively increases in the remanufacturer’s recycling cost. That is, the remanufacturer will keep producing the remanufactured product when the cost-savings from the remanufacturing comy y pensate for recycling cost. Setting wr ¼wn in d , we obtain d ¼ 0, meaning that when the two manufacturers obtain equal unit revenues from the retailer, the remanufacturer obtains greater profits than the manufacturer, because the remanufacturer grabs all the benefits from remanufacturing. Such a scenario with identical wholesale prices is common in the real life when the remanufacturer has the capability to produce the remanufactured products as well as new products (i.e., they are indistinguishable to the consumers), such as the case of one-time use cameras where remanufacturing is profitable to remanufacturers (Majumder and Groenevelt, 2001; Mukhopadhyay and Ma, 2009; Savaskan et al., 2004; Savaskan and Van Wassenhove, 2006). To provide more insight regarding price and service, Fig. 1 y depicts the influences of price and service competition on d . As service competition becomes severer, the remanufacturer requires greater cost-savings from remanufacturing to gain superior profit as compared to the manufacturer. This indicates that under the intense service competition, the remanufacturer should improve cost-savings by remanufacturing. Conversely, severe y price competition leads d to decrease because of price advantage of the remanufactured products. Thus, remanufacturing becomes more beneficial under more intense price competition. Moreover, improving the service investment cost is more favorable to the y manufacturer than the remanufacturer, and thus, d increases with lower m. Proposition 10. The equilibrium profit PM r * is increasing in gp , but PM n * is decreasing in gp . Proposition 10 states that price competition is beneficial to the remanufacturer yet harmful to the manufacturer. This echoes with Proposition 7; i.e., the remanufacturer is competitive in terms of price competition.

C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

5. Numerical experiments Thus far, we have investigated the closed forms of chain members’ equilibrium decisions with respect to the parameters, and we have compared the equilibrium decisions and profits between the two manufacturers. Now, we turn to numerical analyses of the parameters and competition influences on chain members’ equilibrium profits. A selected set of parameters is as follows: wn ¼3, wr ¼2.5, ar ¼ an ¼ 252, b¼5, m¼ 6, c¼1.5, d ¼ 0:5, gp ¼ 2, gs ¼ 3, bp ¼ 5, and bs ¼ 4. The results of the base example are summarized in Table 2. Comparing the four interactions, we observe that the sales price of the remanufactured product is lower than that of the new product, because of the cost-savings of the remanufactured product from remanufacturing. (From Proposition 7, we see that p the setting is below the threshold d ¼ 0:5 o d ¼ 0:7943.) In the case with no price and service competition (i.e., gp ¼ gs ¼ 0), the remanufactured product’s price competitiveness for stimulating demand vanishes, and the remanufacturer chooses the lowest effort in remanufacturing. Under service competition, both manufacturers enhance their efforts in terms of service and thus provide higher service levels. Moreover, we observe that s*n 4 s*r under the four interactions, which corresponds to the results following from Proposition 8 that d ¼ 0:5 is smaller than ds ¼ 0:5872 with neither price nor service competition, ds ¼ 0:5590 without price competition, ds ¼ 0:5680 without sers vice competition, and d ¼ 0:5423 with both price and service competition. Service competition increases the retailer’s profit, but it is harmful to the two manufacturers’ profits, because the service costs are born by the manufacturers. However, under price competition, the remanufactured product’s price advantage compels the new product’s price to decrease, but the remanufactured product’s price increases to gain additional revenue. Hence, price competition increases the remanufacturer’s profit and the retailer’s profits from the remanufactured product, but it is detrimental to the manufacturer’s profit (as stated in Proposition 10) as well as the retailer’s profit from the new products. In addition, the remanufacturer’s profit is greater than the manufacturer’s profit in the presence of price competition, but the result is reversed in the absence of price competition. This M finding that PM r * 4 Pn * can be derived from Proposition 9 y because d ¼ 0:54 d ¼ 0:3974 in the presence of price competiy M tion; however, PM r * o Pn * because d ¼ 0:5o d ¼ 0:5715 in the absence of price competition. The example reveals that chain members’ equilibrium decisions and profits behave differently with respect to price and service competition. Now, we examine the effects of cost-savings, recycling cost, service investment cost, and demand elasticities on chain members’ profits under the four competitive interactions. Effects of unit cost-savings d. Fig. 2 shows that as d increases, the remanufacturer’s and retailer’s profits from the remanufactured product increase, but the manufacturer’s and retailer’s profits from the new products decrease. With the aid of y M Proposition 9, the threshold d for PM r * Z Pn * can be obtained, as shown in the figure. In addition, the presence of service competition amplifies the effects of d. This finding indicates that remanufacturing is more effective in improving profit; as the cost-savings increase, the remanufactured product becomes more competitive than the new product, and moreover, service competition magnifies the remanufacturer’s superiority in terms of cost-savings. For the retailer, increasing d is overall beneficial,

2 To avert the effects of the difference between the primary demands for the two products, we set ar ¼ an . Note that this would not affect the trends in the equilibrium decisions and profits with respect to the parameters.

503

because the profit increase from the remanufactured product offsets the profit decrease from the new product. In addition, the retailer prefers to sell the remanufactured products in the presence of service competition. Effects of recycling cost b. It is intuitive that increasing the recycling cost b is detrimental to the profits of the remanufacR tured product (i.e., PM r * and Pr *), and conversely, it is beneficial R to the profits from the new product (i.e., PM n * and Pn *), as shown in Fig. 3. However, the positive effect of b on the manufacturer’s profit and the retailer’s profit from the new products is insignificant in the absence of service competition. Furthermore, we note that the two manufacturers’ and the retailer’s profits for the two products converge in b, implying that if remanufacturing is uneconomic, the remanufacturer will abandon remanufacturing process and acts as a traditional manufacturer. Effects of service investment cost m. As shown in Fig. 4, in the absence of price competition, all chain members’ profits decrease in service investment cost m. In the presence of price competition, the remanufacturer’s and retailer’s profits still decrease in m, but while the manufacturer’s profit initially increases and then decreases in m. In summary, increasing m is unfavorable not only for the two manufacturers but also for the retailer. This is because that the two manufacturers provide lower services to customers under the higher m, which in turn lowers retail sales. Effects of elasticities on demand bp and bs . From Figs. 5 and 6, we observe that all chain members’ equilibrium profits decrease in bp , yet they increase in bs . The retailer’s profit is more sensitive to price and service elasticities than that of the manufacturers, because she is closer to the market. With respect to the effects of bp , the manufacturer’s decreasing profit rate is greater than that of the remanufacturer. Thus, in the presence of price competition, the remanufacturer’s profit is always higher than the manufacturer’s profit, and in the absence of price competition, the remanufacturer’s profit is higher than the manufacturer’s profit as long as bp is sufficiently large. This result indicates that the remanufacturing process enables the remanufacturer to mitigate the negative influence of increasing price elasticity, and moreover, even when price competition is absent, remanufacturing is profitable in the face of a highly price-sensitive market. In contrast, while all chain members’ profits increase in bs , bs does not change the relations of the profits between the new and remanufactured products. Namely, the manufacturer’s profit is larger than that of the remanufacturer in the absence of price competition, while the contrary is true in the presence of price competition. This finding indicates that the service sensitivity of market demand does not change the manufacturer’s attitude regarding whether to adopt remanufacturing.

6. Summary and future research In this study, we proposed a supply chain model with a common retailer and two manufacturers—one manufacturer produces a new product and the other produces a remanufactured product. Both manufacturers provide service for their products, and the retailer determines the prices of the products for the market, which is incentive in terms of price and service. Four competitive interactions are considered: no price and service competition, price competition only, service competition only, and both price and service competition. Furthermore, our model enables us to examine the influences of price and service competition on chain members’ interactions with respect to decisions and profitabilities. Our contribution is substantive as no prior research has considered service competition in a closedloop supply chain with remanufacturing. More importantly, our

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C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

Fig. 2. Influence of cost-savings on chain members’ equilibrium profits.

Fig. 3. Influence of recycling cost on chain members’ equilibrium profits.

Fig. 4. Influence of service investment cost on chain members’ equilibrium profits.

Fig. 5. Influence of price elasticity on chain members’ equilibrium profits.

Fig. 6. Influence of service elasticity on chain members’ equilibrium profits.

results also provide guidelines for choosing marketing strategies for price and service decisions under different interactions. Our analysis yields the following insights. The intensities of price and service competition affect the trends in equilibrium decisions with respect to recycling cost and service investment, especially for the new-product manufacturer’s equilibrium decisions. The

remanufacturer delegates more effort to remanufacturing under competition, while the reverse is hold without competition. Price and service have different impacts on chain members’ profits. Price competition generally increases the remanufacturer’s and the retailer’s profit from the remanufactured product because of lower cost of that product. Service competition is profitable for the retailer yet

C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

detrimental to both manufacturers; this is because service costs are born by the manufacturers. However, if cost-savings are significant and recycling cost is low, remanufacturing is an effective strategy for the remanufacturer under service competition with the traditional manufacturer. Moreover, remanufacturing is also beneficial to the remanufacturer under a highly price-sensitive market, even in the absence of price competition. Nonetheless, this paper has made several assumptions; relaxing these assumptions may allow us to better comprehend the interactive dynamics in a closed-loop supply chain. For instance, wholesale prices are considered to be predetermined, as in the previous studies (Hsieh and Wu, 2009; Lee et al., 2000; Tsay and Agrawal, 2000). Thus, an extension of this paper would be to allow the manufacturers to choose their wholesale prices, yet this may lead the model to be very complex and, thus, analytically intractable. Second, we assumed that the demand is deterministic, and if the model is treated as stochastic, more findings might be obtained, for example, regarding the influence of random factor of demand on the remanufacturing effort, chain members’ interactions, and performances. Third, it is insightful to consider the availability of collection, which may constrain the remanufacturing and change the interaction between the products (Atasu et al., 2008; Ferrer and Swaminathan, 2006). Finally, the power structure between the manufacturers and the retailer should be taken into account, as it may shed lights on whether the current results will hold if the retailer has a greater power than the manufacturers or if all chain members have equal power.

p, s

price and service, respectively

Superscripts R, M [ *

retailer and manufacturers, respectively best response functions Nash equilibrium

A.2. Equilibrium profits of the two manufacturers The remanufacturer’s and manufacturer’s equilibrium profits are as follows: 2 2 2 2 2 2 PM r * ¼ 1=ð8mð4bmd Ls ÞÞ½4m d ar þ 16bcm wr Lp 2

2

16bm w2r Lp þ 4m2 d w2r L2p 2

2

8bc mgs Ls þ 8bcmwr gs Ls 4cmd wr gs Lp Ls 2

þ 4bc mL2s 8bcmwr L2s 2

2

2

A.1. Summary of notation

2

þ 2wn ð2mgp gs Ls Þð4bcm þwr ð4bm2md Lp Þ þ cd gs Ls Þ 2

4mar ðwr ð4bm þ 2md Lp Þ 2

2

þ d wn ð2mgp þ gs Ls Þ þ cð4bmd gs Ls ÞÞ, 2 2 2 PM n * ¼ 1=ð8mð4bmd Ls ÞÞ½ðcwn Þ ð16bm wn Lp 8bcmgs Ls 2

2

þ 4md ar gs Ls þ4md wn gp gs Ls þ4bcmL2s 4bmwn L2s 2

Appendix A

2

þ 4bmwr L2s þc2 d g2s L2s þ d w2n ð2mgp þ gs Ls Þ2

2

2

2

þ 2cd g2s L2s 2d wn g2s L2s 4md wn Lp L2s cd L4s

Acknowledgment The author thanks the anonymous referee for the constructive comments and suggestions that significantly enhanced the paper. This research was supported by National Science Council, Taiwan, ROC, under Grant #NSC-99-2410-H-224-028-MY2.

505

2

2

þ d wn L4s þ4man ð4bm þ d L2s Þ 2

2

4mwr ðgs ð2b þ d Lp ÞLs þ gp ð4bmd L2s ÞÞÞ:

ðA:1Þ

Proof of Proposition 1. (i) Assuming PM r in (3) is not a hyper2 bolic function; then, the inequality bm 4 d ðbs þ gs Þ2 must hold. The Hessian matrix of PM with respect to t and sr is r 0 2 M 2 M1 ! @ Pr @ Pr b dðbs þ gs Þ @t@sr C B @t2 , HM ¼ ¼ @ A M M 2 2 r @ Pr @ Pr dðbs þ gs Þ m 2 @sr @t

@sr

Symbols j b c dj m pj sj wj

d

t P

aj bp , bs

gp , gs Lk

indicator of product j A fr,ng remanufacturer’s recycling cost each manufacturer’s product cost market demand of product j A fr,ng each manufacturer’s service investment cost retailer’s sales price of product j A fr,ng sales level of product j A fr,ng wholesale price of product j A fr,ng cost-savings from remanufacturing remanufacturer’s effort in remanufacturing (0 r t r1) Profit function market base of product j A fr,ng price and service elasticities on market demand, respectively intensities of price and service competition, respectively overall effects of k A fp,sg on demand (Lk ¼ bk þ gk )

Subscripts r, n

remanufactured and new products, respectively

2

2 and the determinate of the Hessian det HM r ¼ bmd ðbs þ gs Þ 4 0, M i.e., negative definite Hessian. Hence, Pr is concave in t and sr. (ii) The second derivative of PM in (2) with respect to sn is n M 2 @2 PM n =@sn ¼ m o0, and thus Pn is concave in sn. (iii) The R determinate of the Hessian of P in (4) is 0 2 R 1 ! @ P @2 PR 2gp 2ðbp gp Þ @pr @pn C B @p2r R det H ¼ det@ @2 PR ¼ det A @2 PR 2gp 2ðbp gp Þ 2 @pn @pr

@pn

¼ 4bp ðbp þ 2gp Þ 4 0, and hence, PR is concave in pr and pn.

&

Proof of Proposition 2.. The parameters b are assumed to be 2 2 large enough so that 4bmd gs Ls d L2s Z 0. The first derivatives of the equilibrium decisions with respect to c are as follows: @t* dbs Ls ¼ r0, 2 @c 4bm þ d L2s

@s*r ¼ @c

ðbs þ gs Þ 1þ

d2 bs Ls 2 4bm þ d L2s

2m

! r0,

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C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

@s*n b þ gs ¼ s r 0, @c 2m

@p*r ¼ @c

bs ðbs þ gs Þ 1 þ

d2 ðbs gp þ bp ðbs þ gs ÞÞLs

!

2

ðbp þ2gp Þð4bm þ d L2s Þ 4mbp 2

2

r0, 2

2

bs ðbs þ gs Þðgp ð8bm þ d bs Ls 2d L2s Þ þ bp ð4bmd gs Ls d L2s ÞÞ @p*n ¼ r0: 2 @c 4mbp ðbp þ 2gp Þð4bmd L2 Þ s

Hence, all equilibrium decisions are decreasing in c.

&

Proof of Proposition 3. Let O  2mðar þwn gp Þ þ Ls ðcbs þwn gs Þ þwr ð2mLp L2s Þ. Then, t* in (7) can be simplified as 2 ðdOÞ=ð4bm þ d L2s Þ. Because t* Z 0, we obtain O r0. We first consider the derivatives with respect to d. The first derivative of t* with respect to d is 2

@t* Oð4bm þ d L2s Þ ¼ Z0: 2 @d ð4bm þ d L2 Þ2 s

Taking the derivatives of s*r , s*n , p*r , and p*n with respect to d then yields   @t* ð b þ g Þ t þ d s s @s*r @s[ @t* @s[r @d Z 0, ¼ r þ ¼ @d @t @d @d 2m @s*n ¼ 0, @d ½bs gp þ bp ðbs þ gs Þ @s*r @p*r @p[ @s* @p[ @s* @p[ ¼ r rþ r nþ r ¼ Z0, @d @sr @d @sn @d @d 2bp ðbp þ 2gp Þ @d ðbs gp bp gs Þ @s*r @p*n @p[ @s* @p[ @s* @p[ ¼ n rþ n nþ n ¼ : 2bp ðbp þ 2gp Þ @d @d @sr @d @sn @d @d s*r and p*r are increasing in d. If bs gp Z bp gs , p*n is increasing in d; otherwise, p*n is decreasing in d. We now turn our attention to the derivatives of the equilibrium decisions with respect to b. The first derivative of t* with respect to b is @t* 4mdO ¼ r 0: 2 @b ð4bm þ d L2s Þ2 The derivatives of @s*r @b

¼

@s[r

*

@s[r

s*r , s*n ,

p*r ,

Note that @t* =@m is less than or equal to zero, except when wn is extremely large. However, this extreme case is not examined in this paper. The derivatives of s*r and s*n with respect to m are given by     @t*  L d m t cÞ ðw s r @s*r @s[ @t* @s[r @m ¼ r þ ¼ r 0, @m @t @m @m 2m2 @s*n @s[ @t* @s[n Ls ðcwn Þ ¼ n þ ¼ r 0: @m @t @m @m 2m2 The proof of (i) is completed. Taking the derivatives of p*r and p*n with respect to m yields @p*r @p[ @s* @p[ @s* @p[ ¼ r r þ r nþ r @m @sr @m @sn @m @m bs gp ðFn þ Fr Þ þ bp ½Fr bs þ ðFr Fn Þgs  ¼ , 2bp ðLp þ gp Þ @p*n @p[ @s* @p[ @s* @p[ ¼ n r þ n nþ n @m @sr @m @sn @m @m bs gp ðFn þ Fr Þ þ bp ½Fn bs þ ðFn Fr Þgs  : ¼ 2bp ðLp þ gp Þ Hence, if bs gp 4½bp ðFn gs Fr Ls Þ=ðFn þ Fr Þ, p*r decreases in m; otherwise, p*r increases in m. If bs gp 4 ½bp ðFn Ls Fr gs Þ=ðFn þ Fr Þ, p*n decreases in m; otherwise, p*n increases in m. The proof of (ii) is completed. & Proof of Proposition 5. Taking the first derivative of t* with respect to gp , we obtain @t* =@gs ¼ ½ð2mdðwn wr Þ= ð4bm d2 L2s Þ Z 0, and thus, t* is increasing in gp , as stated in (i). Next, we take the first derivative of t* with respect to gs : @t* 1 2 2 ¼ ð4mdðbc þ d ar þ wn ðb þ d gp Þ 2 @gs ð4bm þ d L2s Þ2 2

3

2

þwr ð2bd Lp ÞÞLs þ d ðc þ wn ÞL3s þ dðcwn Þgs ð4bm þ d L2s ÞÞ, and then we derive the conditions of d by solving @t* =@gs ¼ 0. We note that if pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bmððwn cÞgs þðc þ wn 2wr ÞLs Þ d Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Ls ð4mðar þ wn gp wr Lp Þðwn cÞgs Ls þ ðwn cÞL2s Þ

t* is increasing in gs ; otherwise, t* is decreasing in gs . The proof of and

p*n

with respect to b are as follows:

*

@t dLs @t þ ¼ r 0, @t @b @b 2m @b

(ii) is completed.

&

Proof of Proposition 6. Consider the following derivatives: @s*r @s[ @s[ @t* ¼ r þ r , @gs @gs @t @gs

@s*n ¼ 0, @d

dLs ðbs gp þ bp Ls Þ @t* @p*r @p[ @s* @p[ @s* @p[ ¼ r rþ r nþ r ¼ r 0, @b @sr @b @sn @b @b 4mbp ðbp þ2gp Þ @b dLs ðbs gp bp gs Þ @t* @p*n @p[ @s* @p[ @s* @p[ ¼ n rþ n nþ n ¼ : 4mbp ðbp þ 2gp Þ @b @b @sr @b @sn @b @b Contrary to the influences of d, s*r and p*r are decreasing in b. If bs gp Z bp gs , p*n is decreasing in b; otherwise, p*n is increasing in b. & Proof of Proposition 4. The first derivative of t* with respect to m is as follows: 2

2dLs ðd Ls ðar þwn gp wr Lp Þ2bðwr Ls wn gs cbs ÞÞ @t* ¼ r 0: 2 @m ð4bm þ d L2 Þ2 s

@s*n wn c 4 0, ¼ 2m @gs 2

@s*r d ðwn wr ÞLs ¼ 40, 2 @gp 4bmd L2s @s*n ¼ 0: @gp

ðA:2Þ

It is straightforward that s*n is increasing in gs , s*r is increasing in gp , and s*n is independent to gp . Observing @s*r =@gs , we note that the first term of (A.2) is always positive, but the second term depends on @t* =@gs . With the aid of Proposition 5, we find that @t* =@gs decreases in d, we then input d ¼ 0 into (A.2), and obtain @s*r =@gs ¼ ðwr cÞ=2m 40. Hence, s*r is increasing in gs , and the proof is completed. &

C.-H. Wu / Int. J. Production Economics 140 (2012) 496–507

Proof of Proposition 7. Taking the first derivatives of p*r p*n with respect to d yields @ðp*r p*n Þ 2bLs ðgs þ Ls Þt* 4 0: ¼ 2 @d ðgp þ Lp Þð4bmd L2s Þ p

Thus, solving p*r p*n ¼ 0 for d gives the threshold, denoted by d , as pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi dp ¼ 2 C1 = C2 , where C1 ¼ an 2mar þðwn wr Þ½2mðgp þ Lp Þ þ Ls ðgs þ Ls Þ and C2 ¼ ðLs ð2mgs ðar þ wn gp wr Lp Þ þð2man þ cg2s  2mwr ðgp þ2Lp Þ þ wn ð4mgp g2s þ2mLp ÞÞLs þ ðc þ wn ÞL3s ÞÞ. Hence, pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p if d Z d ¼ 2 C1 = C2 , p*r r p*n ; otherwise, p*r 4p*n . & Proof of Proposition 8. With the aid of Proposition 3, we obtain the threshold of d for s*r Z s*n by solving s*r s*n ¼ 0 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bmðwn wr Þ d Z ds ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : & 2mðar þ wn gp wr Lp Þ þ ðcwn Þgs Ls þ ðc þ wn ÞL2s Proof of Proposition 9. For (i), the first derivatives of PM r * and 2

* PM with respect to d are @PM and n * r *=@d ¼ bt =d 4 0 2 2 M * respectively. @Pn *=@d ¼ ½2bðwn cÞgs Ls t =ð4bmd Ls Þ o 0,

Note that proof of (ii) is analogous to the proof of Proposition 8. M Thus, solving PM r * Z Pn * for d yields pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bm½4mððwn cÞan ðwr cÞar Þðwn wr ÞO1  ffi, d Z dy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4m2 ðar þ wn gp wr Lp Þ2 þ ðwn cÞO2 L2s ðwn cÞ2 L4s

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