Analysis for strategy of closed-loop supply chain with dual recycling channel

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Analysis for strategy of closed-loop supply chain with dual recycling channel Min Huang, Min Song, Loo Hay Lee, Wai Ki Ching

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S0925-5273(13)00141-2 http://dx.doi.org/10.1016/j.ijpe.2013.04.002 PROECO5394

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Int. J. Production Economics

Received date: 28 May 2012 Accepted date: 21 March 2013 Cite this article as: Min Huang, Min Song, Loo Hay Lee, Wai Ki Ching, Analysis for strategy of closed-loop supply chain with dual recycling channel, Int. J. Production Economics, http://dx.doi.org/10.1016/j.ijpe.2013.04.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Analysis for Strategy of Closed-loop Supply Chain with Dual Recycling Channel Min Huanga , Min Song∗,a , Loo Hay Leeb,c , Wai Ki Chingd a

College of Information Science and Engineering, Northeastern University; State Key Laboratory of Synthetical Automation for Process Industries (Northeastern University), Shenyang, Liaoning, 110819, China b Department of Industrial and Systems Engineering, National University of Singapore, Singapore c Department of Logistics Engineering College, Shanghai Maritime University, Shanghai, 201306, China d Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong, China

Abstract This paper investigates optimal strategies of a closed-loop supply chain (CLSC) with dual recycling channel, within which the manufacturer sells products via the retailer in the forward supply chain, while the retailer and the third party competitively collect used products in the reverse supply chain. Based on game theory, we characterize the supply chain performance in terms of the pricing decisions and the recycling strategies for both the decentralized and the centralized channel scenarios. By comparing this work with the existing optimal strategies of the CLSC with single recycling channel (the retailer or the third party), we derive the parameter domain which is defined as the set of competing intensities for which the CLSC with dual recycling channel outperforms the CLSC with single recycling channel from the perspectives of the manufacturer and the consumers, respectively. Moreover, we give some suggestions, which will present paramount social value, to the macro-control policy making by exhaustive numerical analysis. The results in this paper can be used as a reference for choosing recycling model, the single recycling channel or the dual recycling channel, for collecting used products. Key words: ∗

Corresponding author. Tel.:+86-24-83671469; fax:+86-24-83688608. Email address: [email protected] (Min Song)

Preprint submitted to International Journal of Production Economics

April 17, 2013

Closed-loop supply chain, Dual recycling channel, Remanufacturing, Game theory 1. Introduction The economical and environmental benefits of product remanufacturing have been widely recognized during the past fifteen years and closed-loop supply chains (CLSCs) thus have gained considerable attention in industry and academia (Daniel and Guide, 2009; Kannan et al., 2010; Zhang et al., 2012). CLSCs consist of both a forward supply chain and a reverse supply chain. The forward supply chain essentially involves the movement of products from upstream suppliers to downstream customers, while the reverse supply chain involves the movement of used products from customers to upstream suppliers, as shown in Figure 1 (Amaro and Barbosa-Povoa, 2009). Different perspectives of CLSCs have been extensively investigated, such as production planning and inventory management (Kenn´e et al., 2012), design of reverse distribution networks (Kusumastuti et al., 2008), channel management(Cheng et al., 2011) etc. Figure 1: The CLSC system

Among these studies, channel management of CLSCs is one of the most important topics in CLSCs. Savaskan et al. (2004) analyzed the problem of choosing the appropriate reverse channel structure for collecting used products from customers under the bilateral monopoly situation. Three recycling channels were compared in terms of their retail prices, collection rates and profits. Comparison results indicated the optimality of the structure with the retailer recycling under the context they considered. Yao and Chen (2007) further extended the CLSCs models proposed by Savaskan et al. (2004) by allowing the manufacturer to sell products to consumers and conducted exhaustive analysis on different recycling models from the perspective of the manufacturer. Latest research on channel management of CLSCs also considers the situation of uncertain context. Shi et al. (2011) addressed the optimal production and the pricing policies in a CLSC with uncertain demand and return. Though the above works provide methods for choosing recycling channels, they ignore the effect of competition on optimal strategies of CLSCs.

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To address this limitation, a lot of researchers have been devoted to the research on channel management of CLSCs with the presence of competition (Savaskan and Wassenhove, 2006; Han and Dong, 2010; Majumder and Groenevelt, 2001; Ferrer and Swaminathan, 2006; Wei and Zhao, 2010; Kaya, 2010). Savaskan and Wassenhove (2006) studied the interaction between a manufacturer’s recycling channel choice to collect post-consumer goods and strategic product pricing decisions in the forward channel when the competition among retailers is allowed. Han and Dong (2010) considered the optimal recycling channel decision of CLSCs in competitive manufactures-common retailer markets. Majumder and Groenevelt (2001) analyzed a two-period model within which one Original Equipment Manufacturer (OEM) and one remanufacturer competed in the sales market and aimed to find the optimal prices and the optimal quantities in a deterministic setting. Ferrer and Swaminathan (2006) extended the work of Majumder and Groenevelt (2001) to multi-period models and investigated the effects of various parameters on the system. Wei and Zhao (2010) investigated the optimal pricing decision problem of a fuzzy CLSC with retail competition, where they assumed that fuzziness existed in the customer demand, the remanufacturing cost and the collecting cost. By assuming stochastic demand and considering partial substitution of demand in the model operations, Kaya (2010) considered the optimal incentive determination problem in addition to determining the optimal production quantities in the manufacturing/remanufacturing industry. In summary, the models discussed in (Savaskan and Wassenhove, 2006; Han and Dong, 2010; Majumder and Groenevelt, 2001; Ferrer and Swaminathan, 2006; Wei and Zhao, 2010; Kaya, 2010) are more practical due to the consideration of retail competition in the forward supply chain. However, the recycling competition in the reverse supply chain generally exists in practice but, to the best of authors’ knowledge, is still unexplored, which motivates us to work on this problem. In order to study the effects of recycling competition on the pricing and the recycling strategies, we investigate a CLSC consisting of a manufacturer, a retailer and a third party, where in the reverse supply chain the retailer and the third party compete for collecting used products. We aim to obtain optimal pricing and recycling strategies of the CLSC with dual recycling channel for both the decentralized and the centralized channel scenarios, and to answer the question that if the dual recycling channel has a competitive edge over the single recycling channel. By using game theory, we derive closed-form optimal decisions that show how the manufacturer, the retailer 3

and the third party react to the wholesale price, the collection rate and the retail price. Based on the comparison results between the optimal strategies in our work and those in Savaskan et al. (2004), we derive the competing intensity domain for which the dual recycling channel outperforms the single recycling channel. In addition, all theoretical results are followed by elaborate illustrations, which may facilitate the making of macro-control policies. The contribution of this paper is three fold: (i) to obtain optimal strategies of the CLSC with dual recycling channel for both the decentralized and the centralized channel scenarios; (ii) to characterize the competing intensity domain for which the CLSC with dual recycling channel outperforms the CLSC with single recycling channel; (iii) to provide insights for future macrocontrol policy making. The rest of this paper is organized as follows. Problem formulation is presented in Section 2. Optimal strategies of the CLSC with dual recycling channel are obtained in Section 3. Section 4 compares our results with those in Savaskan et al. (2004), where CLSCs with single recycling channel are considered, for both the decentralized and the centralized channel scenarios. Finally section 5 concludes this paper. 2. Problem Formulation 2.1. Definition of symbols w p b cm cr ∆ D φ β A I τ CL α

Unit wholesale price, Unit retail price, Unit transfer price, Unit cost of producing end products from original materials, Unit cost of producing end products from returns, Saving unit cost by recycling, ∆ = cm − cr , Demand for the new product in the market, Market size, Sensitivity of consumers to the retail price, Average recycling price for used products, Investment in used product collection, Collection rate, Scalar parameter, the exchanging coefficient between the collection rate and the investment, Competing coefficient between the third party and the retailer, 4

Π Profit

.

We note that the collection rate denotes the fraction of the current generation product supplied from returns and thus remanufactured units. For ease of presentation, let superscripts Dd, Dc, Sdr, Sdt, and Sc denote the CLSC with the retailer and the third party recycling simultaneously in the decentralized scenario, the CLSC with the retailer and the third party recycling simultaneously in the centralized scenario, the CLSC with the retailer recycling alone in the decentralized scenario, the CLSC with the third party recycling alone in the decentralized scenario, and the CLSC with single recycling channel in the centralized scenario, respectively. Let subscripts M, R, and T denote the manufacturer, the retailer, and the third party, respectively. In this paper we investigate a CLSC with a manufacturer, a retailer and a third party. In the forward supply chain, the manufacturer sells products to the retailer with a wholesale price w and the retailer makes profits by selling with a retail price p to consumers, while in the reverse supply chain, the manufacturer collects used products from the retailer and the third party with different transfer prices bR and bT (see Figure 2). The manufacturer can either produce end products from original materials with the unit cost of production cm , or extract useful parts from returns and produce end products using these parts with the unit cost of remanufacturing cr . This paper only considers the strategies of CLSCs in a single period, namely the research on the delayed effect of the last period on the current period is out of the scope of this paper. Figure 2: The CLSC system with dual recycling channel

2.2. Basic assumptions Assumptions in this paper are similar to those in Savaskan et al. (2004). (1) No difference between remanufactured products and new products. (2) The information is symmetric. (3) The unit cost of producing end products from returns is lower than that from new materials, i.e., cr < cm . The unit cost of recycling is lower than that of remanufacturing, i.e., A < ∆. (4) All agents of the CLSC have interest in cooperating as an integral system. More specifically, p > w > 0, bR > A > 0, bT > A > 0, ∆ ≥ bR , and ∆ ≥ bT . 5

(5) The market demand is a linear function of the retail price, i.e., D(p) = φ − βp. (6) The collection rate is usually modeled as a function of the investment in used product collection (Savaskan et al., 2004). Due to the fact that the retailer and the third party compete for collecting used products in this paper, the collection rate reasonably depends on the investments of these two competing sides. More specifically, the collection rate of one side is reasonably formulated as a monotonic increasing function of its own investment and a monotonic decreasing function of the investment of its competitor. In this paper the collection rates of the retailer and the third party are formulated as follows: s s 0 0 IR IT τR = and τT = CL CL 0

0

0

where IR and IT , whose formulations are given as follows: IR = IR −α1 IT and 0 IT = IT − α2 IR , denote the effective investment of the retailer and the third party, respectively. In practice the investment of one side usually has greater influence on its own effective investment than that of its competitor does, thus we assume 0 ≤ αi < 1, i = 1, 2. Further, we assume that there is no difference between the retailer and the third party in influence, namely αi = α, i = 1, 2, where α completely characterizes the recycling competition in the reverse supply chain. For this reason, α is hereafter called the competing intensity. Then the collection rates of the retailer and the third party can be simplified as follows: r r IR − αIT IT − αIR τR = and τT = . CL CL In addition, τR and τT are also subject to the physical constraint, i.e., 0 ≤ τR + τT ≤ 1. 3. Supply Chain Models with Remanufacturing 3.1. Decentralized channel In the decentralized channel, all of the manufacturer, the retailer, and the third party are independent decision makers. Each of them aims to maximize its own profit. Based on the above problem formulation and assumptions in Section 2, profits of the manufacturer, the retailer and the third party in the decentralized channel are given as follows: Dd Dd Dd Dd Dd ΠDd − cm + (∆ − bDd M = (φ − βp )[w R )τR + (∆ − bT )τT ],

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(1)

Dd Dd Dd ΠDd − w Dd )(φ − βpDd ) + (bDd ) − IRDd , R = (p R − A)τR (φ − βp

(2)

Dd Dd Dd ΠDd ) − ITDd . T = (bT − A)τT (φ − βp

(3)

We assume that the Stackelberg game occurs between the manufacturer and other agents, where the manufacturer behaving as the Stackelberg leader (L) dominates the supply chain system, and other agents (the retailer and the third party) are followers. The retailer (F1 ) and the third party (F2 ) make the best response to the optimal decision by the manufacturer as follows:  Dd Dd Dd Dd Dd Dd Π (L) : maxwDd ,bDd , bR , bT ; p , τRDd , τTDd )  M (w  R ,bT   Dd  s.t.(pDd , τRDd ) = arg maxpDd ,τRDd ΠDd , τRDd ; w Dd , bDd  R (p R ) Dd Dd Dd Dd τT = arg maxτTDd ΠT (τT ; bT ) (4)  Dd Dd Dd Dd Dd   (F1 ) : maxpDd ,τ Dd ΠR (p , τR ; w , bR )    (F ) : max Dd ΠRDd (τ Dd ; bDd ) 2 τT T T T In addition, a non-cooperative game occurs between the retailer and the third party. By the backward induction method, we achieve results below. Proposition 1. In the decentralized channel, if 2CL > β(∆ − A)2 (1 − α2 ) + (φ − βcm )(1 − α2 )(∆ − A), then the optimal retail price pDd∗ and the collection rate of the retailer τRDd∗ are given by pDd∗ =

2CL (6φ + 2βcm ) − 5βφ(1 − α2 )(∆ − A)2 β[16CL − 5β(1 − α2 )(∆ − A)2 ]

(5)

2(φ − βcm )(1 − α2 )(∆ − A) . 16CL − 5β(1 − α2 )(∆ − A)2

(6)

and τRDd∗ =

Proof. See Appendix A. Proposition 2. In the decentralized channel, if 2CL > β(∆ − A)2 (1 − α2 ) + (φ − βcm )(1 − α2 )(∆ − A), then the optimal collection rate of the third party τTDd∗ is given by τTDd∗ =

(φ − βcm )(1 − α2 )(∆ − A) . 16CL − 5β(1 − α2 )(∆ − A)2 7

(7)

Proof. See Appendix A. Proposition 3. In the decentralized channel, if 2CL > β(∆ − A)2 (1 − α2 ) + (φ − βcm )(1 − α2 )(∆ − A), then the optimal wholesale price w Dd∗ and the transfer prices to the retailer Dd∗ and the third party bR and bTDd∗ are given by w Dd∗ =

8CL (φ + βcm ) − β(∆ − A)2 (1 − α2 )(3φ + 2βcm ) , β[16CL − 5β(∆ − A)2 (1 − α2 )]

and

A+∆ . 2 to ΠDd M yields

Dd∗ bR = ∆, bTDd∗ = Dd∗ Substituting w Dd∗ , bR and bTDd∗ Dd∗ ΠM =

2CL (φ − βcm )2 . β[16CL − 5β(∆ − A)2 (1 − α2 )]

(8)

(9)

(10)

Proof. See Appendix A. Obviously, an immediate consequence of above propositions is the following corollary. Corollary 1. In the decentralized channel, the maximal profit of the Dd∗ manufacturer ΠM , the optimal collection rates of the retailer τRDd∗ and the third party τTDd∗ are monotonic decreasing functions of α and in contrast, the optimal wholesale price w Dd∗ and the optimal retail price pDd∗ are monotonic increasing functions of α. To illustrate results above, we present the following Figures 3-61 showing how the competing intensity affects the optimal strategies. Figure 3: The profit of the manufacturer versus the competing intensity in the decentralized channel Figure 4: The collection rate versus the competing intensity in the decentralized channel Figure 5: The wholesale price versus the competing intensity in the decentralized channel 1

CL = 1000, φ = 100, β = 0.3, cm = 20, ∆ = 15, A = 5. These parameters will be adopted throughout this paper.

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Figure 6: The retail price versus the competing intensity in the decentralized channel

The fundamental reason of the results in Corollary 1 lies in that in the CLSC with dual recycling channel, the marginal collection rate to the investment in used product collection decreases as the competing intensity increases. It is obvious that the larger the competing intensity is, in order to get the same collection rate as in the context without competition, the more one has to invest in. This means that the marginal collection rate to the investment in used product collection decreases as the competing intensity increases. This will result in a direct decrease in collection rates of the retailer and the third party as competing intensity increases (see Figure 4). Then, as the result of chain reaction, the decreasing collection rates will lead to an increase in the cost, the wholesale price (see Figure 5) and the retail price (see Figure 6). However, due to the fact that the increase in the cost outweighs the increase in the income, the profit of the manufacturer still decreases (see Figure 3). 3.2. Centralized channel In the centralized channel, the manufacturer, the retailer and the third party cooperate with each other and behave as an integrated firm. The system profit for the CLSC can be expressed as follows: Dc Dc ΠDc − cm + (τRDc + τTDc )(∆ − A)] − (IRDc + ITDc ) C = (φ − βp )[p

(11)

Here we aim to find a retail price pDc , a collection rate of the retailer τRDc and a collection rate of the third party τTDc to maximize ΠDc C , namely max

Dc ,τ Dc pDc ,τR T

ΠDc C

Dc

:= (φ−βp )[p

Dc

[(τRDc )2 Dc Dc −cm +(τR +τT )(∆−A)]−

+ (τTDc )2 ]CL 1−α (12)

Proposition 4. In the centralized channel, if 2CL > β(∆ − A)2 (1 − α) + (φ − βcm )(∆ − A)(1 − α), then the optimal retail price pDc∗ , the optimal collection rate of the retailer τRDc∗ , and the optimal collection rate of the third party τTDc∗ are given by pDc∗ =

CL (φ + βcm ) − βφ(1 − α)(∆ − A)2 , β[2CL − β(1 − α)(∆ − A)2 ] 9

(13)

τRDc∗ =

(φ − βcm )(1 − α)(∆ − A) , 2[2CL − β(1 − α)(∆ − A)2 ]

(14)

τTDc∗ =

(φ − βcm )(1 − α)(∆ − A) . 2[2CL − β(1 − α)(∆ − A)2 ]

(15)

Substituting pDc∗ , τRDc∗ and τTDc∗ to ΠDc C yields ΠDc∗ = C

CL (φ − βcm )2 . 2β[2CL − β(1 − α)(∆ − A)2 ]

(16)

Proof. See Appendix B. Similar to Section 3.1, we have the following corollary for the centralized channel. Corollary 2. In the centralized channel, the maximal profit of the CLSC Dc∗ ΠDc∗ and the third party C , the optimal collection rates of the retailer τR Dc∗ τT are monotonic decreasing functions of α and in contrast, the optimal retail price pDc∗ is a monotonic increasing function of α. In order to show the result intuitively, we also present the following Figures 7-9 showing how the competing intensity affects optimal strategies. Figure 7: The system profit versus the competing intensity in the centralized channel Figure 8: The collection rate versus the competing intensity in the centralized channel Figure 9: The retail price versus the competing intensity in the centralized channel

The implications of Corollary 2 are similar to that of Corollary 1. The analysis is therefore omitted here. 4. Strategies for Choosing Recycling Channel of CLSCs In this section, we compare our results with those in Savaskan et al. (2004) for a better understanding of the recycling channels. We tabulate the manufacturer’s profits and the retail prices for two recycling models (see Table 1). Table 1: Comparison of the results for the decentralized channel

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4.1. Strategies for choosing recycling channel of CLSCs in the decentralized channel 4.1.1. Strategies for choosing recycling channel from the perspective of the manufacturer Based on the results in Table 1, the following proposition is derived. Proposition 5. In the decentralized channel, the relations of the optimal Dd∗ profit of the manufacturer in the CLSC with dual recycling channel ΠM , the optimal profit of the manufacturer in the CLSC with single recycling channel Sdr∗ (the retailer) ΠM , as well as the optimal profit of the manufacturer in the CLSC with single √recycling channel (the third party) ΠSdt∗ are given by M 5 Dd∗ Sdr∗ Sdt∗ (1) If 0 ≤ α < 5 , ΠM > ΠM > ΠM , √ Dd∗ Sdr∗ (2) If √α = 55 , ΠM = ΠM > ΠSdt∗ M , √ 5 2 5 Sdr∗ Dd∗ (3) If 5 < √ α < 5 , ΠM > ΠM > ΠSdt∗ M , 2 5 Sdr∗ Dd∗ Sdt∗ (4) If α√ = 5 , ΠM > ΠM = ΠM , Sdr∗ Dd∗ (5) If 255 < α ≤ 1, ΠM > ΠSdt∗ > ΠM . M Proof. See Appendix C. For a better understanding of Proposition 5, we display the above comparison results in Figure 10. Figure 10: Comparison of the profit of the manufacturer in the decentralized channel

Next, we analyze the strategies of the manufacturer for choosing recycling channel of CLSCs in the decetralized scenario. Proposition 5 tells us that the single recycling channel model with the retailer collecting used products is always better than that with the third party collecting products in terms of Sdr∗ the profit of the manufacturer, i.e., ΠM > ΠSdt∗ M . Therefore, in order to answer the question whether it is beneficial to build the dual recycling channel, we just need to compare the optimal profit of the manufacturer in the CLSC with dual recycling channel with the optimal profit of the manufacturer in the CLSC with single recycling channel (the retailer). (1) The CLSC with dual recycling channel dominates the CLSC with √ 5 single recycling channel (the retailer) when 0 ≤ α ≤ 5 . From Observation 1(Savaskan et al., 2004) and Proposition 3, we conclude that in the CLSC with dual recycling channel, the manufacturer can achieve cost savings from remanufacturing due to the participation of the third party in collecting used products, which means that the average cost of the manufacturer in the CLSC with dual recycling channel is lower than that 11

in the CLSC with single recycling channel (the retailer). The low average cost will result a lower wholesale price in the CLSC with dual recycling channel than that in the CLSC √with single recycling channel (the retailer) (see Figure 11). When 0 ≤ α ≤ 55 , although the collection rate of the retailer in the CLSC with dual recycling channel is smaller than that in the CLSC with single recycling channel (the retailer) (see Figure 12), the cost saving from low wholesale price outweighs the profit loss from low collection rate, which further implies that the retail price in the CLSC with dual recycling channel is lower than that in the CLSC with single recycling channel (the retailer) (see Figure 14). In this case, the demand in the CLSC with dual recycling channel is larger than that in the CLSC with single recycling channel (the retailer). Therefore, the profit of the manufacturer in the CLSC with dual recycling channel is larger than that in the CLSC with single recycling channel (the retailer), which implies the benefit of building the dual recycling channel. (2) The CLSC with single recycling channel (the retailer) dominates the √ 5 CLSC with√ dual recycling channel when 5 < α ≤ 1. When 55 < α ≤ 1, the average cost of the manufacturer in the CLSC with dual recycling channel is still lower than that in the CLSC with single channel (the retailer). However, due to the fact that the collection rate of the retailer in the CLSC with dual recycling channel is much smaller than that in the CLSC with single recycling channel (the retailer) (see Figure 12), the cost saving from low wholesale price cannot offset the profit loss from low collection rate, which implies that the retail price in CLSC with dual recycling channel is higher than that in CLSC with single recycling channel. In this case, the demand in the CLSC with dual recycling channel is smaller than that in the CLSC with single recycling channel. Therefore, the profit of the manufacturer in the CLSC with dual recycling channel is smaller than that in the CLSC with single recycling channel, which implies it is non-beneficial to build the dual recycling channel. Figure 11: Comparison of the wholesale prices in the decentralized channel Figure 12: Comparison of the collection rates in the decentralized channel

4.1.2. Strategies for choosing recycling channel from the perspective of consumers Based on the results in Table 1, we have the following proposition. 12

Proposition 6. In the decentralized channel, the relations of the optimal retail price in the CLSC with dual recycling channel pDd∗ , the optimal retail price in the CLSC with single recycling channel (the retailer) pSdr∗ , as well as the optimal retail price in the CLSC with single recycling channel (the third party) pSdt∗ √ are given by (1) If 0 ≤ α < 55 , pDd∗ < pSdr∗ < pSdt∗ , √ = pSdr∗ < pSdt∗ , (2) If √α = 55 , pDd∗ √ (3) If 55 < √ α < 255 , pSdr∗ < pDd∗ < pSdt∗ , (4) If α√ = 255 , pSdr∗ < pDd∗ = pSdt∗ , (5) If 255 < α ≤ 1, pSdr∗ < pSdt∗ < pDd∗ . Proof. See Appendix D. Again, for a better understanding of Proposition 6, we display the above comparison results in Figure 13. Figure 13: Comparison of retail prices in the decentralized channel

4.1.3. Strategies for choosing recycling channel from the perspective of policymakers Figure 14 shows how the system profits of the CLSCs with single and dual recycling channel vary with the competing intensity. From Figure 14 we observe that the system profit of the CLSC with dual recycling channel is always larger than that of the CLSC with single recycling channel when the competing intensity is relatively small (approximately less than 0.4), which guarantees the feasibility of the dual recycling channel. In this situation, by designing an effective revenue-sharing contract, the dual recycling channel always allows each agent within the CLSC (the manufacturer, the retailer and the third party) to achieve a larger profit than the single recycling channel does. We conclude that, given the simulation parameters, it is possible to target some competing intensity domain for which the CLSC with dual recycling channel is superior to the CLSC with single recycling channel. More specifically, both the profit and the collection rate of the CLSC with dual recycling channel are larger than those of the CLSC with single recycling channel, and what is more, the retail price in the CLSC with dual recycling channel is lower than its counterpart in the CLSC with single recycling channel. In summary, from the perspective of policymakers, the dual recycling channel enables all the agents involved in CLSCs to achieve additional benefits if the competing intensity is appropriately determined by macro-control 13

policies. Figure 14: Comparison of the system profits in the decentralized channel

4.2. Strategies for choosing recycling channel of CLSCs in the centralized channel When 2CL > [(φ − βcm )(∆ − A) + β(∆ − A)2 ](1 − α) and 4CL > (φ − βcm )(∆ − A) + β(∆ − A)2 hold at the same time, the comparison of optimal strategies for the centralized channel for two recycling channel models is listed in Table 2. Table 2: Comparison of the results for the centralized channel

Proposition 7. In the centralized channel, the relations of the optimal system profit of the CLSC with dual recycling channel ΠDc∗ and the optimal C system profit of the CLSC with single recycling channel ΠSc∗ are given by C β(∆−A)2 Dc∗ Sc∗ (1) If 0 ≤ α < 8CL , ΠC > ΠC , β(∆−A)2 , ΠDc∗ = ΠSc∗ C C , 8CL β(∆−A)2 < α ≤ 1, ΠDc∗ < ΠSc∗ C C . 8CL

(2) If α =

(3) If Proof. See Appendix E. Proposition 8. In the centralized channel, the relations of the optimal system collection rate of the CLSC with dual recycling channel τCDc∗ and the optimal system collection rate of the CLSC with single recycling channel τCSc∗ are given by (1) If 0 ≤ α < 12 , τCDc∗ > τCSc∗ , (2) If α = 12 , τCDc∗ = τCSc∗ , (3) If 21 < α ≤ 1, τCDc∗ < τCSc∗ . Proof. See Appendix F . Proposition 9. In the centralized channel, the relations of the optimal retail price in the CLSC with dual recycling channel pDc∗ and the optimal retail price in the CLSC with single recycling channel pSc∗ are given by (1) If 0 ≤ α < 12 , pDc∗ < pSc∗ , (2) If α = 12 , pDc∗ = pSc∗, (3) If 12 < α ≤ 1, pDc∗ > pSc∗ . Proof. See Appendix G. 14

Figures 15, 16 and 17 depict the comparison diagrams of the system profits, the collection rates, and the retail prices in the centralized channel, respectively. The analysis for the centralized channel is similar to that for the decentralized channel and thus is omitted for brevity. Figure 15: Comparison of system profits in the centralized channel Figure 16: Comparison of collection rates in the centralized channel Figure 17: Comparison of retail prices in the centralized channel

5. Conclusions In this paper, we analyzed strategies of a CLSC with dual recycling channel. Based on game theory, we derived the optimal pricing and recycling strategies for both the decentralized and the centralized scenarios. Comparing our results with the optimal strategies of CLSC with single recycling channel, we find that the beneficial choice of recycling channels depends on the competing intensity no matter from the perspective of the manufacturer or the perspective of consumers. The dual recycling channel dominates the single recycling channel under the only context that the competition in recycling in the reverse supply chain is not very strong. The results in this paper can be used as a reference to choose recycling method, the single recycling channel or the dual recycling channel, for collecting used products. We shall extend our result in two ways. One is the contract design of revenue sharing in dual recycling channel. The second one is the performance comparison of different combinations of recycling channels. Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant No.71071028, No.71021061, No.70931001 and No.61070162, the National Science Foundation for Distinguished Young Scholars of China under Grant No.61225012; Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20110042110024 and No.201000421 10025, the Fundamental Research Funds for the Central Universities under Grant No.N110204003; The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.

15

A. The proofs of Proposition 1,2 and 3 Concavity plays a key role in the deduction of optimal strategies. For Dd this reason, we first prove the concavity of ΠDd T and ΠR . The third party’s problem Dd Taking derivative of ΠDd yields T with respect to τT 2CL Dd ∂ΠDd T Dd = (bDd )− τ , T − A)(φ − βp Dd 1 − α2 T ∂τT ∂ 2 ΠDd 2CL T =− < 0, Dd 2 1 − α2 ∂(τT ) which implies the concavity of ΠDd T . The retailer’s problem For constructing the Hessian matrix of ΠDd R , we carry out the following calculations: ∂ΠDd R = φ − 2βpDd + βw − βτRDd (bDd R − A), Dd ∂p ∂ΠDd 2CL Dd R Dd τ . = (bDd )− R − A)(φ − βp Dd 1 − α2 R ∂τR The resulting Hessian matrix of ΠDd R is given by   −2β −β(bDd Dd R − A) HR = . 2CL −β(bDd − 1−α 2 R − A) When 2CL > (1 − α2 )[2β(∆ − A)2 + (φ − βcm )(∆ − A)] and 0 < α < 1 hold, it is obvious that 2CL > β(∆ − A)2 (1 − α). Because bDd R < ∆, we 2 Dd know 4CL > β(bDd − A) (1 − α), which means that H is negative definite, R R Dd Dd Dd namely, ΠR is concave with respect to (p , τR ). By solving the following equations  ∂ΠDd R  =0   ∂τRDd ∂ΠDd R =0 , ∂pDd  Dd  ∂Π  T =0 ∂τTDd

16

we obtain the optimal response functions of the retailer and the third party Dd

pˆ

2 2CL (φ + βw Dd ) − βφ(1 − α2 )(bDd R − A) = , 2 β[4CL − β(1 − α2 )(bDd R − A) ]

τˆRDd =

(φ − βw Dd)(1 − α2 )(bDd R − A) , Dd 4CL − β(1 − α2 )(bR − A)2

τˆTDd =

(φ − βw Dd )(1 − α2 )(bDd T − A) . Dd 4CL − β(1 − α2 )(bR − A)2

2 2 Dd Dd Let Y = 4CL − β(bDd R − A) (1 − α ), S = (∆ − bR )(bR − A) + (∆ − Dd bDd ˆDd , τˆRDd and τˆTDd into the profit function of the T )(bT − A). Substituting p Dd ˆ as follows: manufacturer, we achieve Π M Dd 2 Dd ˆ Dd = 2CL (φ − βw ) [w Dd − cm + S(φ − βw )(1 − α ) ]. Π M Y Y

The manufacturer’s problem ˆ Dd , we carry out the following For constructing the Hessian matrix of Π M calculations: ˆ Dd ∂Π 2CL 2β(1 − α2 )(φ − βw Dd )S M Dd = [(φ + βc − 2βw ) − ], m ∂w Dd Y Y ˆ Dd 2CL (φ − βw Dd)2 (1 − α2 )(∆ + A − 2bDd ∂Π T ) M = . Dd 2 Y ∂bT ˆ Dd For given bDd R , the resulting Hessian matrix of ΠM is given by: " 2 Dd 2 Dd 2

Dd HM =

−4βCL {4CL −β(1−α )[(bR −A) +S]} Y2 −4βCL (1−α2 )(φ−βw Dd )(∆+A−2bDd T ) Y2

−4βCL (1−α )(φ−βw )(∆+A−2bDd T ) Y2 −4CL (φ−βw Dd )2 (1−α2 ) Y2

#

.

When 2CL > (1 − α2 )[2β(∆ − A)2 + (φ − βcm )(∆ − A)] holds, it is obvious that 4CL > β(1 − α2 )4(∆ − A)2 . Since ∆ > bDd > A, ∆ > bDd > A, R T 2 2 2 Dd 2 4CL > β(1 − α )4(∆ − A) > β(1 − α )[(bR − A) + S] trivially holds. Then, 2 Dd 2 it is obvious that 4CL > β(1 − α2 )[(bDd R − A) + S − (∆ + A − 2bT ) ], which Dd ˆ Dd means that for given bR , ΠM is jointly concave with respect to w Dd and bDd T . From the first-order conditions we get wˆ Dd =

2β(1 − α2 )φS − (φ + βcm )Y 2β 2 (1 − α2 )S − 2βY 17

and Substituting wˆ Dd and ˆbDd T

ˆbDd = A + ∆ . T 2 ˆ Dd yields into Π M

ˆ Dd = Π M

CL (φ − βcm )2 . 2β[Y − β(1 − α2 )S]

Because ˆ Dd ∂Π CL (1 − α2 )(φ − βcm )2 (∆ − A) M = > 0, 2[Y − β(1 − α2 )S]2 ∂bDd R ˆ Dd is a monotonic increasing function of bDd . The optimal which means that Π M R Dd∗ transfer price is obviously equal to its upper bound of ∆, i.e., bR = ∆. Dd∗ Dd Substituting bR into wˆ , we get w Dd∗ =

8CL (φ + βcm ) − β(∆ − A)2 (1 − α2 )(3φ + 2βcm ) . β[16CL − 5β(∆ − A)2 (1 − α2 )]

Substituting w Dd∗ into pˆDd , τˆRDd and τˆTDd , we get p

Dd∗

2CL (6φ + 2βcm ) − 5βφ(1 − α2 )(∆ − A)2 = , β[16CL − 5β(1 − α2 )(∆ − A)2 ]

τRDd∗ =

2(φ − βcm )(1 − α2 )(∆ − A) , 16CL − 5β(1 − α2 )(∆ − A)2

τTDd∗ =

(φ − βcm )(1 − α2 )(∆ − A) . 16CL − 5β(1 − α2 )(∆ − A)2

If 2CL > (1 − α2 )[2β(∆ − A)2 + (φ − βcm )(∆ − A)], then it is easy to verify that both ∂ΠDd R | Dd Dd∗ < 0 ∂τRDd τR =1−τT and ∂ΠDd T | Dd Dd∗ < 0 Dd τT =1−τR ∂τT ∂ 2 ΠDd

∂ 2 ΠDd

2CL hold. In addition, due to the fact that ∂(τ DdR)2 = ∂(τ DdT )2 = − 1−α 2 < 0, we R T have ∂ΠDd ∂ΠDd R R | < | Dd Dd∗ = 0, Dd Dd∗ ∂τRDd τR =1−τT ∂τRDd τR =τR

18

∂ΠDd ∂ΠDd T T | Dd Dd∗ < | Dd Dd∗ = 0, Dd τT =1−τR Dd τT =τT ∂τT ∂τT which implies 0 ≤ τRDd∗ + τTDd∗ < 1. This completes the proof. B. The proof of Proposition 4 Similar to the proof in Appendix A, for building the Hessian matrix of we carry out the following calculations:

ΠDc C ,

∂ΠDc C = −2βpDc + φ + βcm − β(τRDc + τTDc )(∆ − A), ∂pDc 2CL Dc ∂ΠDc C = (φ − βpDc )(∆ − A) − τ , Dc ∂τR 1−α R ∂ΠDc 2CL Dc C = (φ − βpDc )(∆ − A) − τ . Dc ∂τT 1−α T The resulting Hessian matrix of ΠDc C is given by   −2β −β(∆ − A) −β(∆ − A) 2CL . − 1−α 0 HCDc =  −β(∆ − A) 2CL −β(∆ − A) 0 − 1−α

When 2CL > β(∆ − A)2 (1 − α) + (φ − βcm )(∆ − A)(1 − α) holds, it is easy to see that 2CL > β(∆ − A)2 (1 − α), which implies that the ΠDc C is a Dc Dc Dc concave function of (p , τR , τT ). From the first-order conditions, we get p

Dc∗

CL (φ + βcm ) − βφ(1 − α)(∆ − A)2 = , β[2CL − β(1 − α)(∆ − A)2 ]

τRDc∗ =

(φ − βcm )(1 − α)(∆ − A) , 2[2CL − β(1 − α)(∆ − A)2 ]

τTDc∗ =

(φ − βcm )(1 − α)(∆ − A) . 2[2CL − β(1 − α)(∆ − A)2 ]

Following the discussion in Appendix A, the physical constraint 0 ≤ < 1 trivially holds in the centralized channel. This completes the proof of Proposition 4. τRDc∗ + τTDc∗

19

C. The proof of Proposition 5 In the decentralized channel, when 2CL > (1 − α2 )[(φ − βcm )(∆ − A) + 2β(∆ − A)2 ] and 4CL > (φ − βcm )(∆ − A) + β(∆ − A)2 hold, Dd∗ Sdr∗ • Comparing ΠM and ΠM , we √achieve 2 Sdr∗ Dd∗ (1) If 5(1 − α ) > 4 (0 < α√ < 55 ), ΠM < ΠM , 5 2 Sdr∗ Dd∗ (2) If 5(1 − α ) = 4 (α = 5 ), ΠM = ΠM , √ 5 2 Sdr∗ Dd∗ (3) If 5(1 − α ) < 4 ( 5 < α < 1), ΠM > ΠM . Dd∗ • Comparing ΠM and ΠSdt∗ , we achieve M √ Dd∗ (1) If 5(1 − α2 ) > 1 (0 < α√< 2 5 5 ), ΠSdt∗ < ΠM , M 2 5 2 Sdt∗ Dd∗ (2) If 5(1 − α ) = 1 (α√= 5 ), ΠM = ΠM , Dd∗ (3) If 5(1 − α2 ) < 1 ( 2 5 5 < α < 1), ΠSdt∗ > ΠM . M Sdr∗ Sdt∗ Since ΠM > ΠM (Savaskan et al., 2004), from the discussion above, we get √ Dd∗ Sdr∗ > ΠM > ΠSdt∗ (1) If 0 < α√ < 55 , ΠM M , 5 Dd∗ Sdr∗ Sdt∗ (2) If α = 5 , ΠM√ = ΠM > ΠM , √ 5 Sdr∗ Dd∗ (3) If 5 < √ α < 2 5 5 , ΠM > ΠM > ΠSdt∗ M , 2 5 Sdr∗ Dd∗ Sdt∗ (4) If α√= 5 , ΠM > ΠM = ΠM , Sdr∗ Dd∗ (5) If 2 5 5 < α < 1, ΠM > ΠSdt∗ > ΠM . M This completes the proof of Proposition 5. D. The proof of Proposition 6 In the decentralized channel, when 2CL > (1 − α2 )[(φ − βcm )(∆ − A) + 2β(∆ − A)2 ] and 4CL > (φ − βcm )(∆ − A) + β(∆ − A)2 hold, pSdr∗ − pDd∗ =

βCL (∆ − A)2 (φ − βcm )(1 − 5α2) , β[4CL − β(∆ − A)2 ][16CL − 5β(1 − α2 )(∆ − A)2 ] 20

It is obvious that: √ (1) If 1 − 5α2 > 0 (0 < α√ < 55 ), pSdr∗ − pDd∗ > 0, (2) If 1 − 5α2 = 0 (α = 55 ), pSdr∗ − pDd∗ = 0, √ (3) If 1 − 5α2 < 0 ( 55 < α < 1), pSdr∗ − pDd∗ < 0; pSdt∗ − pDd∗ =

4CL (∆ − A)2 (φ − βcm )(4 − 5α2) , [16CL − β(∆ − A)2 ][16CL − 5β(∆ − A)2 (1 − α2 )]

It is obvious that: √ (1) If 4 − 5α2 > 0 (0 < α√< 2 5 5 ), pSdt∗ − pDd∗ > 0, (2) If 4 − 5α2 = 0 (α√= 2 5 5 ), pSdt∗ − pDd∗ = 0, (3) If 4 − 5α2 < 0 ( 2 5 5 < α < 1), pSdt∗ − pDd∗ < 0. Since pSdr∗ < pSdt∗ (Savaskan et al., 2004), from the discussion above, we get √ (1) If 0 < α√ < 55 , pDd∗ < pSdr∗ < pSdt∗ , (2) If α = 55 , pDd∗ = pSdr∗ < pSdt∗ , √ √ (3) If 55 < √ α < 2 5 5 , pSdr∗ < pDd∗ < pSdt∗ , (4) If α√= 2 5 5 , pSdr∗ < pDd∗ = pSdt∗ , (5) If 2 5 5 < α < 1, pSdr∗ < pSdt∗ < pDd∗ . This completes the proof of Proposition 6. E. The proof of Proposition 7 In the centralized channel, when 2CL > (1 − α2 )[(φ − βcm )(∆ − A) + 2β(∆ − A)2 ] and 4CL > (φ − βcm )(∆ − A) + β(∆ − A)2 hold, Dc∗ ΠSc∗ = C − ΠC

CL β(∆ − A)2 (φ − βcm )2 [8αCL − β(∆ − A)2 ] . 2β[4CL − β(∆ − A)2 ]2 [2CL − β(∆ − A)2 (1 − α)]

It is obvious that : 2 (1) If 0 ≤ α < β(∆−A) , ΠDc∗ > ΠSc∗ C C , 8CL β(∆−A)2 , ΠDc∗ = ΠSc∗ C C , 8CL 2 β(∆−A) < α ≤ 1, ΠDc∗ < ΠSc∗ C C . 8CL

(2) If α =

(3) If This completes the proof of Proposition 7. 21

F. The proof of Proposition 8 In the centralized channel, the optimal collection rate of the CLSC with dual recycling channel τCDc∗ is given by τCDc∗ = τrDc∗ + τTDc∗ =

(φ − βcm )(∆ − A)(1 − α) . 2CL − β(∆ − A)2 (1 − α)

The quotient of τCDc∗ and τCSc∗ is given by 4CL (1 − α) − β(∆ − A)2 (1 − α) τCDc∗ = . τCSc∗ 2CL − β(∆ − A)2 (1 − α) It is obvious that: (1) If 4CL (1 − α) > 2CL (α < 21 ), (2) If 4CL (1 − α) = 2CL (α = 12 ),

Dc∗ τC Sc∗ τC Dc∗ τC Sc∗ τC Dc∗ τC Sc∗ τC

> 1, = 1,

< 1. (3) If 4CL (1 − α) < 2CL (α > 12 ), This completes the proof of Proposition 8. G. The proof of Proposition 9 In the centralized channel, when 2CL > (1 − α2 )[(φ − βcm )(∆ − A) + 2β(∆ − A)2 ] and 4CL > (φ − βcm )(∆ − A) + β(∆ − A)2 hold, p

Sc∗

−p

Dc∗

βCL (∆ − A)2 (φ − βcm )(1 − 2α) = . β[4CL − β(∆ − A)2 ][2CL − β(∆ − A)2 (1 − α)]

It is obvious that : (1) If 0 ≤ α < 21 , pSc∗ > pDc∗ , (2) If α = 12 , pSc∗ = pDc∗ , (3) If 12 < α ≤ 1, pSc∗ < pDc∗ . This completes the proof of Proposition 9.

22

References Amaro, A.C.S., Barbosa-Povoa, A.P.F.D., 2009. The effect of uncertainty on the optimal closed-loop supply chain planning under different partnerships structure. Computers & Chemical Engineering 33(12), 2144-2158. Cheng, Z.J., Feng, X., Zhao, H.S., 2011. Pricing strategy in a dual-channel and remanufacturing supply chain system. International Journal of Scystems Science 41(7), 909-921. Daniel, V., Guide, R., 2009. The evolution of closed-Loop supply chain research. Operations Research 57(1), 10-18. Ferrer, G., Swaminathan, J.M., 2006. Managing new and remanufactured products. Management Science 52(1), 15-26. Han, X.H., Dong, Z.N., 2010. Reverse channel decision analysis for bilateral competing closed-loop supply chain. Industrial Engineering Journal 13(4), 23-27. Kusumastuti, R.D., Piplani, R., Lim, G.H., 2008. Redesigning closed-loop service network at a computer manufacturer: A case study. International Journal of Production Economics 111(2), 244-260. Kannan, G., Sasikumar, P., Devika, K., 2010. A genetic algorithm approach for solving a closed loop supply chain model: a case of battery recycling. Applied Mathematical Modelling 34(3), 655-670. Kaya, O., 2010. AIncentive and production decisions for remanufacturing operations. European Journal of Operational Research 201(2), 442-453. Kenn´e, J.P., Dejax, P., Gharbi, A., 2012. Production planning of a hybrid manufacturing-remanufacturing system under uncertainty within a closedloop supply chain. International Journal of Production Economics 135(1), 81-93. Majumder, P., Groenevelt, H., 2001. Competition in remanufacturing. Production and Operations Management 10(2), 125-141. Savaskan, R.C., Bhattacharya, S., Wassenhove, L.N.V., 2004. Closed-loop supply chain models with product remanufacturing. Management Science 50(2), 239-252. 23

Savaskan, R.C., Wassenhove, V.L.N., 2006. Reverse channel design: the case of competing retailers. Management Science 52(1), 1-14. Shi, J. M., Zhang, G.Q., Sha, J.C., 2011. Optimal production and pricing policy for a closed-loop system. Resources, Conservation and Recycling 55(6), 639-647. Wei, J., Zhao, J., 2010. Pricing decisions with retail competition in a fuzzy closed-loop supply chain. Expert Systems with Applications 38(9), 1120911216. Yao, W.X., Chen M.M., 2007. Comparison among closed-loop supply chain models. Commercial Research (1), 51-53. Zhang, Z. H., Jiang, H., Pan, X.Z., 2012. A Lagrangian relaxation based approach for the capacitated lot sizing problem in closed-loop supply chain. International Journal of Production Economics 140(1), 249-255.

24



Figure 1: The CLSC system

25

bT

w

bR

WT

p

WR

Figure 2: The CLSC system with dual recycling channel

26

The profit of the manufacturer for the dual recycling channel

3720

Decentralized Channel

3710 3700 3690 3680 0

0.2

0.4 α 0.6

0.8

1

Figure 3: The profit of the manufacturer versus the competing intensity in the decentralized channel

27

The collection rate for the dual recycling channel

0.12

Decentralized Channel Dd

τR

0.1

Dd T

τ

0.08 0.06 0.04 0.02 0 0

0.2

0.4 α 0.6

0.8

1

Figure 4: The collection rate versus the competing intensity in the decentralized channel

28

The wholesale price for the dual recycling channel

176.7

Decentralized Channel

176.65 176.6 176.55 176.5 176.45 176.4 0

0.2

0.4

α

0.6

0.8

1

Figure 5: The wholesale price versus the competing intensity in the decentralized channel

29

The retail price for the dual recycling channel

255.2

Decentralized Channel

255 254.8 254.6 254.4 254.2 0

0.2

0.4

α

0.6

0.8

1

Figure 6: The retail price versus the competing intensity in the decentralized channel

30

The system profit for the dual recycling channel

7480

Centralized Channel

7460 7440 7420 7400 7380 7360 0

0.2

0.4 α 0.6

0.8

1

Figure 7: The system profit versus the completing intensity in the centralized channel

31

The system collection rate for the dual recycling channel

Centralized Channel

0.25 0.2 0.15 0.1 0.05 0 0

0.2

0.4 α 0.6

0.8

1

Figure 8: The collection rate versus the competing intensity in the centralized channel

32

The retail price for the dual recycling channel

177

Centralized Channel

176.5 176 175.5 175 174.5 174 0

0.2

0.4

α

0.6

0.8

1

Figure 9: The retail price versus the competing intensity in the centralized channel

33

Decentralized Channel

The profit of the manufacturer

3720 3715 3710 3705 3700 3695 3690 3685 3680 0

Dual recycling channel

Single recycling channel

ΠDd M Sdr ΠM Sdt ΠM

0.2

0.4

α

0.6

0.8

1

Figure 10: Comparison of the profit of the manufacturer in the decentralized channel

34

Decentralized Channel

The wholesale price

176.7 176.65 176.6 176.55 176.5

wDd Sdr

w

Sdt

w

176.45 176.4 0

0.2

0.4

α

0.6

0.8

1

Figure 11: Comparison of the wholesale prices in the decentralized channel

35

Decentralized Channel Dd

The collection rate

0.15

τR

τDd

0.1

T

Sdr τC

0.05

Sdt τC

0 0

0.2

0.4 α 0.6

0.8

1

Figure 12: Comparison of the collection rates in the decentralized channel

36

Decentralized Channel

The retail price

255.2 255 254.8 254.6

pDd pSdr Sdt

p 254.4 254.2 0

0.2

0.4

α

0.6

0.8

1

Figure 13: Comparison of retail prices in the decentralized channel

37

Decentralized Channel

5590

Dd

The system profit

Sdr

ΠC

5580

Sdt

ΠC

ΠC

5570 5560 5550 5540 5530 5520 0

0.2

0.4

α 0.6

0.8

1

Figure 14: Comparison of the system profits in the decentralized channel

38

Centralized Channel

The system profit

7480

Dc C Sc Π C

7460

Π

7440 7420 7400 7380 7360 0

0.2

0.4

α

0.6

0.8

1

Figure 15: Comparison of system profits in the centralized channel

39

Centralized Channel

The collection rate

0.5

τDc C

0.4

τSc C

0.3 0.2 0.1 0 0

0.2

0.4

α

0.6

0.8

1

Figure 16: Comparison of collection rates in the centralized channel

40

Centralized Channel

177

The retail price

176.5

p

Dc

pSc

176 175.5 175 174.5 174 0

0.2

0.4 α 0.6

0.8

1

Figure 17: Comparison of retail prices in the centralized channel

41

Table 1: Comparison of the results for the decentralized channel Single recycling channel(Savaskanet al., 2004) Dual recycling channel The retailer The third party p∗ Π∗M

[3CL −β(∆−A)2 ]φ+βCL cm β[4CL −β(∆−A)2 ] 2CL (φ−βcm )2 β[16CL −4β(∆−A)2 ]

(φ−βcm )(∆−A)2 4[16CL −β(∆−A)2 ] 2CL (φ−βcm )2 β[16CL −β(∆−A)2 ]

3φ+βcm 4β

−

42

2CL (6φ+2βcm )−5βφ(1−α2 )(∆−A)2 β[16CL −5β(1−α2 )(∆−A)2 ] 2CL (φ−βcm )2 β[16CL −5β(∆−A)2 (1−α2 )]

Table 2: Comparison of the results for the centralized channel Single recycling channel(Savaskanet al., 2004) Dual recycling channel p∗

φ+βcm 2β

−

(∆−A)2 (φ−βcm ) 2[4CL −β(∆−A)2 ]

τ∗

(φ−βcm )(∆−A) 4CL −β(∆−A)2

∗ πC

2 (φ−βc )2 4CL m β[4CL −β(∆−A)2 ]2

CL (φ+βcm )−βφ(∆−A)2 (1−α) β[2CL −β(∆−A)2 (1−α)] (φ−βcm )(∆−A)(1−α) τRDc∗ 2[2CL −β(∆−A)2 (1−α)] (φ−βcm )(∆−A)(1−α) τTDc∗ 2[2CL −β(∆−A)2 (1−α)] CL (φ−βcm )2 2β[2CL −β(∆−A)2 (1−α)]

43