Analyzing the performance of a two-period remanufacturing supply chain with dual collecting channels

Computers & Industrial Engineering xxx (xxxx) xxx–xxx

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Analyzing the performance of a two-period remanufacturing supply chain with dual collecting channels Jie Weia, Yue Wanga, Jing Zhaob, Ernesto D.R. Santibanez Gonzalezc,

⁎

a

School of Economics and Management, Hebei University of Technology, Tianjin 300401, PR China School of Science, Tianjin Polytechnic University, Tianjin 300387, PR China c Department of Industrial Engineering, University of Talca, Curicó, Chile b

ARTICLE INFO

ABSTRACT

Keywords: Remanufacturing Dynamic Dual collecting channels Supply chain

Collecting channel of used products, dynamic nature of product’s life-cycle and firm’s profit discount over the time are remanufacturer’s key concerns to make decisions. This paper studies a remanufacturing supply chain with dual collecting channels under dynamic setting and establishes three two-period game models by considering both the profit discount and competition between two collecting channels. We analyze the effects of profit discount and collection competition on firms’ pricing decisions, collection rates and profits, and reveal the remanufacturer’s optimal strategies of maximizing its profit or of maximizing the collection rate. Analysis and comparison of the model solutions are performed through analytical approach and numerical studies. Some new insights are as follows: (1) The profit discount has no effect on the optimal retail/wholesale prices for the second period and only affects the optimal decisions for the first period. (2) As the profit discount increases, the remanufacturer’s profit increases very rapidly regardless of the remanufacturer’s options for collecting used products. (3) The retailer and the third party will make the same optimal collection rate when they jointly collect used products, which means the collection competition does not generate differences on their individual optimal collection rates.

1. Introduction Remanufacturing is the industrial process of collecting used products, extracting useful parts from used products and reusing these useful parts in the production of new products. Possible cost reductions, consumer’s environmental consciousness and stringent environmental legislations have made more and more manufacturers pay increasing attention to the remanufacturing of used products in recent years. In practice, many manufacturers (for example, Apple, Hewlett-Packard, Xerox) have incorporated a remanufacturing process for used products into their original production systems. It is very common to encounter the inherent uncertainties in the collecting process (for example, time, location, quantity and quality of used products). The remanufacturers will inevitably meet with difficulties in collecting sufficient used products for remanufacturing demand and difficulties in acquiring a scale effect of remanufacturing. How to design an appropriate collecting channel for used products is a very important decision for remanufacturers. With the intent to ensure remanufacturing run smoothly, many remanufacturers have adopted dual collecting options to collect used

⁎

products. For example, Xerox Corporation not only contracts the collection of used products to its retailer but also prepays mailboxes so that customers can directly return their used products to Xerox without incurring any costs (Hong, Wang, Wang, & Zhang, 2013). Apple Corporation recycles its iPhone and iPad through its retailers and its official website (www.applepoweredbysims.com) simultaneously. ReCellular Inc., the largest cell phone remanufacturer in the USA, collects used phones not only from cellular airtime providers but also from third party collectors. In view of the above practices, this article formulates, from the remanufacturer’s viewpoint, three two-period dynamic game models where a manufacturer produces new products only with raw materials in the first period, and has the option of producing new products in the second period with raw materials or with used products which are collected through two independent parties. Different to the singleperiod model, the two-period models keep the analysis tractable that gives deeper theoretic and managerial insights. In addition, it characterizes the product’s life-cycle and the discount of profit in subsequent period. Profit discount in subsequent period may affect remanufacturer’s pricing strategy for preceding period, and the price in

Corresponding author. E-mail address: [email protected] (E.D.R. Santibanez Gonzalez).

https://doi.org/10.1016/j.cie.2018.12.063

0360-8352/ © 2018 Published by Elsevier Ltd.

Please cite this article as: Wei, J., Computers & Industrial Engineering, https://doi.org/10.1016/j.cie.2018.12.063

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the preceding period may also influence available used-products in subsequent period and firms’ profits. Our models analyze the effects of profit discount and collection competition on firms’ pricing decisions, collection rates and profits, and reveal the remanufacturer’s optimal strategies of maximizing its profit or maximizing the collection rate. To the best of our knowledge, no researcher has studied these problems of remanufacturing supply chain with dual collecting channels under twoperiod setting. This article tries to fill the research gap. The rest of this paper is organized as follows. Section 2 discusses the related literature. Section 3 formulates three models and gives their main solutions. In Section 4, we analyze the model solutions and make further analysis about discount factor and collection competition. The final section summarizes our study briefly and addresses the potential future research directions.

products can be collected through three dual formats. However, in Hong et al. (2013), the closed-loop supply chain is assumed to run over one period, and the transfer price paid by the manufacturer to the collector is an exogenous constant. Huang et al. (2013) also studied the optimal decisions on pricing and collection rate of a closed-loop supply chain where the retailer and the third party are simultaneously in charge of collecting used products from end customers. However, they also assumed the closed-loop supply chain run over one period, and they did not make comparison between different dual-collecting scenarios. Savaskan and Van Wassenhove (2006) studied the reverse channel design and optimal pricing decisions of a closed-loop supply chain where used products can be collected by two competing recyclers, however, there is no collection competition between the two recyclers. Zhao et al. (2017) studied the decisions on collecting channel design, optimal pricing and optimal collection rate for a remanufacturing supply chain with dual collecting models under one-period. To our knowledge, all existing researches on remanufacturing with dual collecting channels assume that the remanufacturing supply chain is operated in one period, and do not consider the dynamic nature of product-life period and firm’s profit discount over the time. Motivated by the researches (e.g., Savaskan et al., 2004; Savaskan & Van Wassenhove, 2006; Huang et al., 2013), we study three dual-collecting channels for used products in two-period scenario. This article is the first one to study remanufacturing supply chain with dual collecting channels under dynamic setting. The main contributions are listed as follows: First, through considering the manufacturer’s three possible dual-collecting options, we formulate three dynamic game models which characterize both the competition between two collecting channels and the profit discount of subsequent period, which extends Savaskan et al. (2004) by considering both the collection competition and the dynamic nature of product’s life-cycle, and generalizes De Giovanni and Zaccour (2014) by considering dual-collecting channels with collection competition. Second, we derive the analytical solutions of these models and provide analytical insights of our results. Third, we not only examine how the collection competition and profit discount affect the dynamics of wholesale/retail prices and firms’ profits across periods but also study how the collection competition and profit discount affect manufacturer’s dual-collecting option and firms’ collection efforts under dynamic scenario. Some new results which different from those of Savaskan et al. (2004) and De Giovanni and Zaccour (2014) are obtained. For example, Savaskan et al. (2004) showed that the optimal collection effort is obtained when only the retailer collects the used products, and De Giovanni and Zaccour (2014) showed the reverse logistics gets the equivalent performance when only the retailer or only the third party collects used products. However, our results show that (1) the total collection rate gets the smallest value when both the retailer and the third party are in charge of collecting used products simultaneously, and the collection competition does not bring differences between their own optimal collection rates; (2) although the optimal wholesale/retail prices for the second period are independent of both the manufacturer’s collection options and the collection competition, the optimal wholesale/retail prices for the first period depends on both the manufacturer’s collection options and the collection competition; (3) as the profit discount increases, the total collection rate increases slightly in three models, and the manufacturer’s profit increases very rapidly regardless of the manufacturer’s collection options for used products. Table 1 shows the main difference between this study and the most related literature on pricing and collecting of used products.

2. Literature review The existing literature on remanufacturing supply chain mainly focuses on the pricing, inventory, product design and competition between new and remanufacturing products under one-period, twoperiod, multi-period and infinite-period scenarios (e.g., Savaskan, Bhattacharya, & Van Wassenhove, 2004; Zhao, Wei, & Li, 2017; Gaur, Amini, & Rao, 2017; Webster & Mitra, 2007; Wei, Govindan, Li, & Zhao, 2015; Wu, 2013; Chen & Chang, 2013). For a comprehensive review on this topic, we refer the reader to Govindan, Soleimani, and Kannan (2015), and Guide and Van Vassenhove (2006). Webster and Mitra (2007) formulated a general two-period model to study the effect of take-back laws which require firms take responsibility for the costs of collecting/disposing of their products. Choi (2017) considered the optimal pricing and brand investment decisions of a fashion company who collects, recycles and sells remanufactured fashion products. Wu (2013) considered the decisions of product design for an original equipment manufacturer and the pricing strategies of the original equipment manufacturer and its competitor through formulating a two-period supply chain model. Chen and Chang (2013) explored the pricing behavior over time with respect to the parameters (substitutability, market property, and return rate) by using Lagrangean relaxation and dynamic programming approaches. Wei and Zhao (2013) considered the optimal decisions on reverse channel choice for a closed-loop supply chain through considering the fuzzy uncertainty. Agrawal, Atasu, and Ittersum (2015) explored the effects of the remanufactured products and the remanufacturer’s identity on the perceived value of new products by using behavioral experiments. Their results show that the remanufactured products sold by the original equipment manufacturer can reduce the new products’ perceived values. Cai, Lai, Li, Li, and Wu (2014) formulated a stochastic dynamic programming model to characterize a production planning and acquisition problem for a finiteperiod manufacturing/remanufacturing system which has two core inventories with two quality conditions and a serviceable inventory. They derived the optimal dynamic pricing and production policy of this model, and analyzed the effects of system parameters on the pricing policy and production quantities. Savaskan et al. (2004) is one earlier research that considered the collecting channel design for used products in one-period scenario. Recently, De Giovanni and Zaccour (2014) followed the remanufacturing supply chain structure of Savaskan et al. (2004), and also considered the collecting channel design for used products in two-period scenario. However, all the above researches either assume that only a single firm is in charge of collecting used products or do not consider the problem of collecting channel design for used products. In recent years, few researchers established some results on remanufacturing under the scenario where two or more firms are simultaneously in charge of collecting used products (e.g., Huang, Song, Lee, & Ching, 2013; Hong et al., 2013; Savaskan & Van Wassenhove, 2006). Specifically, Hong et al. (2013) studied the optimal decisions on pricing and collection rates of a closed-loop supply chain where used

3. Model assumptions and notations We consider a remanufacturing supply chain where a manufacturer sells its product to consumers through one retailer. To capture the dynamic properties of product life-cycle, we establish three two-period game models in which firms choose sales and collecting decisions to 2

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which is the retailer’s decision variable; unit wholesale price of the new product in period k, which is the manufacturer’s decision variable. We assume the manufacturer has sufficient channel power over the retailer and the third party and acts as channel leader. Let subscripts m , r , and t denote the manufacturer, the retailer, and the third party, respectively. The reverse channel performance of the manufacturer, the retailer, and the third party is characterized by parameter i (i = m , r , t ) , respectively. Parameter i can be regarded as the fraction of new products made through remanufacturing used products which are sold in the first period and collected in the second period by channel 1, i also can be regarded as the channel member member i, so, 0 i i ’s collection rate on used product, and the collectors decide their collection rates in the second period. Moreover, the i (i = m , r , t ) are 1, i , j {m , r , t }, i j. subject to the physical constraint 0 i + j Similar to the Assumption 9 of Savaskan et al. (2004), we can see that 1 hold if the scalar parameter cL is suffi0 1 and 0 i + j i ciently large. In order to establish our models and obtain some interesting insights, we make the following assumptions in our models.

Table 1 Articles on pricing and collecting of used products. Articles

wk

Collecting channel

Period

Wei and Zhao (2013)

single

single

×

Choi (2017)

single

single

×

Wu (2013)

single

single

×

Savaskan et al. (2004)

single

single

×

Savaskan and Van Wassenhove (2006) Huang et al. (2013)

dual

single

×

dual

single

Hong et al. (2013)

dual

single

Zhao et al. (2017)

dual

single

single

two

dual

two

De Giovanni and Zaccour (2014) Present study

Pricing

Collection competition

×

maximize their own profits. In the first period, the manufacturer produces new products directly from raw materials at unit manufacturing cost cm and sells them to the retailer at unit wholesale price w1, the retailer then sells them to consumers at unit retail price p1. In the second period, some consumers who have purchased new products in the first period return their used products. We assume that the manufacturer has incorporated a remanufacturing process for used products into its original production system, so the manufacturer can produce new product directly from raw materials at unit manufacturing cost cm , or from used products at unit remanufacturing cost cr . Following Ferrer and Swaminathan (2006) and Savaskan et al. (2004), we assume there is no differences between the remanufactured and manufactured products, and the manufacturer sells new products to the retailer at unit wholesale price w2 , and the retailer sells them to consumers at unit retail price p2 . In the second period, the manufacturer has three options for collecting used products: (1) the manufacturer and the retailer collect the used products from the customers, simultaneously; (2) the manufacturer and the third party collect the used products from the customers, simultaneously; (3) the retailer and the third party collect the used products from the customers, simultaneously. To take the used products back from the retailer or the third party, the manufacturer will pay a transfer price br to retailer or bt to third party per product, respectively. We summarize some main parameters and decision variables used in our models as follows. Main parameters

cm cr

bc Ii

(i) Manufacturing a new product with used products is less costly than manufacturing a new one directly from raw materials, i.e., cr < cm and cr is the same for all remanufactured products. Let = cm cr which denotes the unit cost savings from manufacturing a new br > bc , bt > bc , bc is product with used products, and the unit average collecting price for used products paid to the consumer by the collector. (ii) Similar to Savaskan et al. (2004) and Huang et al. (2013), the market demand Dk (pk ) in period k for new products is linear in price pk , and

Dk (pk ) = a

br (bt )

pk

k = 1, 2,

(1)

where the parameter a represents the potential demand for new products, and the parameter is the self-price elastic coefficient. (iii) Similar to Hong et al. (2013), Zhao et al. (2017) and Huang et al. (2013), firm i ’s collection investment Ii for used products is modeled as a function of its own and its competitor’s collection rates, and the channel member has the same influence of its investment on its competitor’s investment. For example, if firms i and j jointly collect used products, their collection investments (Ii and I j ) and collection rates ( i and j ) have the following relationships (Huang et al., 2013; Hong et al., 2013):

unit manufacturing cost of new product; unit remanufacturing cost of new product; unit cost savings from manufacturing a new product from br > bc , bt > bc ; used products, = cm cr , and unit average collecting price for used products paid to the consumer by the collector; firm i ’s collection investment for used products, i {m , r , t } ; the common discount factor of the second-period profit.

i

=

j

=

Ii

Ij cL

Ij

Ii cL

,

,

(2)

(3)

j , the parameter cL is scalar parameter where i , j {m , r , t } , and i which is an exchanging coefficient between the collection rate and the investment (we can refer to Savaskan et al. (2004) and Huang et al. (2013) for more details about the relationship between investment and < 1) is competing coefficollection rate), and the parameter (0 cient of two firms’ collection activities which shows the intensity of collection competition between the players who are in charge of collecting used products (Huang et al., 2013; Hong et al., 2013). Specifically, the larger the , the more intense the competition. To ensure that the various profit expressions possess the unique optimum, similar to Savaskan et al. (2004), we assume that the scalar parameter cL is sufficiently large and impose the following condition on the parameter 2 )( cL: cL > (1 bc ) 2 , this assumption ensures that remanufacturing is sufficiently costly so that it is not economically viable to manufacture all units from used products (Savaskan et al., 2004).

Decision variables i

pk ,

collection rate for used product of firm i (i {m , r , t }) , which is a decision variable of the firm i; unit transfer price paid to the retailer (the third party) by the manufacturer, which are the manufacturer’s decision variables; unit retail price of the new product in period k (k = 1, 2) , 3

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We will establish three two-period game models by considering the above three collection scenarios in the following. Specifically, (1) the MR model where the manufacturer and the retailer collect used products in the second period, simultaneously; (2) the RT model where the retailer and the third party collect used products in the second period, simultaneously; (3) the MT model where the manufacturer and the third party collect used products in the second period, simultaneously.

r

m,

br ) =

p2 , r ) =

m1 (w1)

+

m2 (w 2, m,

br ),

r 1 (p1 )

+

r 2 (p2 , r ),

max

max

p1, p2 , r

m (w1,

r (p1 ,

p2 ,

w 2,

m,

w2 =

m

=

br = p1 =

aA1

(4)

(5)

m (w1,

,

r (p1 ,

2 )(

(1

bc )(a A1

cm)

+ ( + 1) bc , 2+ A1 a

2cL (a A1

,

t ( t)

,

7 2

2 3)

4

4 2

3)

(2 + )2

bc )2 (1

, A2 .

(2 + )2

w2,

m,

bt ) =

m1 (w1)

+

m2 (w 2, m,

bt ),

(13)

p2 ) =

r1 (p1 )

+

(14)

r 2 (p2 ),

= (a

p1 )(bt

bc )

It .

t

(15)

Proposition 2. In the MT model, the manufacturer’s optimal wholesale price w1 for the first period, optimal wholesale price w2 , optimal collection rate m , and optimal transfer price bt for the second period are given in Eqs. (16)–(19), respectively. The retailer’s optimal retail prices p1 and p2 for the first and second periods are given in Eqs. (20) and (21), respectively, and the third party’s optimal collection rate t for the second period is given in Eq. (22).

(7)

w1 =

(8)

w2 =

(9)

cm)

3

p1 )(p1 w1) is the retailer’s profit obtained in where r1 (p1 ) = (a p2 )(p2 w2) is the retailer’s profit the first period, and r2 (p2 ) = (a obtained in the second period. The third party’s total profit t ( t ) is

(6)

a + cm , 2

(1

2 )(

(12)

bc )2 (8

p1 )(w1 cm) is the manufacturer’s profit obtained where m1 (w1) = (a p2 )(w2 cm) in the first period, and m2 (w2, m, bt ) = (a + ( bt )(a p1 ) t + ( bc )(a p1 ) m Im is the manufacturer’s profit obtained in the second period. The retailer’s total profit r (p1 , p2 ) is

br ),

cm )

2 )(

The MT model characterizes the markets where the manufacturer produces new products directly from raw materials and sells the new products at unit price w1 to the retailer who then sells them to consumers at unit price p1 in the first period; In the second period, the manufacturer produces new products not only directly from raw materials but also from used products, and both the manufacturer and the third party collect used products from the customers simultaneously. In this scenario, the manufacturer needs to decide the wholesale price w1 for the first period, the wholesale price w2 , transfer price bt , and collection rate m for the second period. The retailer needs to make the retail prices p1 and p2 for the first and second period respectively, the third party decides collection rate t for the second period. The firms’ profit functions are given as follows, respectively. The manufacturer’s total profit m (w1, w2, m, bt ) is

r ).

A2 (a A1

(1

,

3.2. The MT model

Proposition 1. In the MR model, the manufacturer’s optimal wholesale price w1 for the first period, optimal wholesale price w2 , optimal collection rate m , and optimal transfer price br for the second period are given in Eqs. (6)–(9), respectively. The retailer’s optimal retail price p1 for the first period, optimal retail price p2 , and the optimal collection rate r for the second period are given in Eqs. (10)–(12), respectively.

w1 =

cm)

The proof of Proposition 1, as well as the proofs of the other propositions, are given in Appendix.

p1 )(p1 w1) where is the retailer’s profit r1 (p1 ) = (a p2 ) obtained in the first period, and r 2 (p2 , r ) = (a (p2 w2) + (br bc ) r (a p1 ) Ir is the retailer’s profit obtained in the second period. The optimization problems of the manufacturer and the retailer are as follows: w1, w 2, m, br

bc )(a (2 + ) A1

Remark 1. Eqs. (8) and (12) tell us that when the manufacturer and the retailer collect used products simultaneously, the manufacturer’s optimal collection rate is (2 + ) times higher than the retailer’s optimal collection rate.

p1 )(w1 cm) is the manufacturer’s profit obwhere m1 (w1) = (a p2 )(w2 cm) tained in the first period, and m2 (w2, m, br ) = (a + (( br ) r + ( bc ) m)(a p1 ) Im is the manufacturer’s profit obtained in the second period, and the parameter is the common discount factor of the second-period profit. The retailer’s total profit ( r (p1 , p2 , r ) ) is r (p1 ,

2 )(

(1

= 4cL

The MR model characterizes the following scenario: the manufacturer produces new products directly from raw materials and sells the new products at price w1 to the retailer who then sells them to consumers at price p1 in the first period; in the second period, the manufacturer produces new products not only directly from raw materials but also from used products, and both the manufacturer and the retailer collect used products from the customers, simultaneously. In this scenario, the manufacturer needs to decide the wholesale price w1 for the first period, the wholesale price w2 , transfer price br , and collection rate m for the second period. The retailer needs to make the retail price p1 for the first period, the retail price p2 , and collection rate r for the second period. According to the above description, we can formulate the channel members’ profit functions as follows. The manufacturer’s total profit ( m (w1, w2, m, br ) ) is

w2,

=

(11)

where A1 = 8cL

3.1. The MR model

m (w1,

3a + cm , 4

p2 =

m

=

bt =

(10) 4

A3 a

4cL (a A3

cm)

,

(16)

a + cm , 2 (1

2 )(

(17)

bc )(a A3

+ (1 + ) bc , 2+

cm)

,

(18) (19)

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A3 a

p1 =

=

cm)

,

(1

respectively. The retailer’s optimal retail price p1 for the first period, and optimal retail price p2 , optimal collection rate r for the second period are given in Eqs. (33)–(35), respectively, and the third party’s optimal collection rate t for the second period is given in Eq. (36).

(20)

3a + cm , 4

p2 =

t

2cL (a A3

(21)

2 )(

bc )(a (2 + ) A3 (1

where A3 = 8cL

cm ) 2 )(

w1 =

,

(22)

bc )2 (3 + )

2+

.

Remark 2. Eqs. (18) and (22) tell us that when the manufacturer and the third party collect used products from the customers simultaneously, the optimal collection rate of the manufacturer is (2 + ) times higher than that of the third party. Which means that the new products made through remanufacturing used products collected by the manufacturer is (2 + ) times higher than that of the third party. Using Eqs. (13)–(22), we can obtain the maximum profits of the manufacturer, the retailer, and the third party as follows, respectively m

r

t

=

= =

cL (a

cm

)2

A3 4cL2 (a

cm) 2 A32

cL (1

+

(a 8

,

cm) 2

(a

+

2 )(

cm

)2

16

,

bc ) 2 (a cm) 2 (1 (2 + ) 2A32

4

4

2

3)

.

m1 (w1)

+

m 2 (w 2 ,

br , bt ),

p2 , r ) =

r 1 (p1 )

+

r 2 (p2 , r ),

= (a

p1 )(bt

bc )

t

It .

br =

+ bc , 2

(32)

=

=

aA 4

2cL (a A4

cm)

,

(33)

3a + cm , 4 (1

2 )(

(1

2 )(

(34)

bc )(a 2A 4

cm)

bc )(a 2A 4

cm ) 2 )(

(1

,

(35)

,

(36)

bc )2 (3

)

2

, A5 = 4cL

(1

2 )(

bc )2 (1 4

)

.

Remark 4. The optimal wholesale price w2 has the same value in the three models, which is also valid for the optimal retail price p2 , this is because the used-product’s collection is independent of the sales quantity in the second period. So, both the manufacturer and the retailer need to make their decisions to maximize their own profits facing the same deterministic demand in three models. Using Eqs. (26)–(36), we can obtain the maximum profits of the manufacturer, the retailer, and the third party as follows, respectively

(26)

m

=

r

=

t

(27)

=

cm) 2

cL (a A4

cm )2

cL A5 (a A42

cL (1

2 )(

cm )2

(a

+

8 +

,

cm) 2

(a 16

) 2 (1

bc 4A42

)(a

(37)

,

(38)

cm

)2

.

(39)

4. Analysis of solutions and managerial insights

p1 )(p1 w1) where is the retailer’s profit r1 (p1 ) = (a p2 ) obtained in the first period, and r 2 (p2 , r ) = (a (p2 w2) + (br bc )(a p1 ) r Ir is the retailer’s profit obtained in the second period. The third party’s total profit t ( t ) is t ( t)

(31)

Remark 3. When the retailer and the third party collect used products from the customers simultaneously, Eqs. (31) and (32) tell us that, the manufacturer will make the same optimal transfer price for the retailer and the third party, this is because the retailer and the third party have the same unit average collecting price for used products. Interestingly, Eqs. (35) and (36) show that the retailer and the third party will make the same optimal collection rate in the RT model. The collection competition does not generate differences between their optimal collection rates.

p1 )(w1 cm) is the manufacturer’s profit obwhere m1 (w1) = (a p2 )(w2 cm ) tained in the first period, and m2 (w2, br , bt ) = (a + ( br )(a p1 ) r + ( bt )(a p1 ) t is the manufacturer’s profit obtained in the second period. The retailer’s total profit r (p1 , p2 , r ) is r (p1 ,

+ bc , 2

where A 4 = 8cL

The RT model characterizes the markets where the manufacturer produces new products directly from raw materials and sells the new products at unit price w1 to the retailer who then sells them to consumers at unit price p1 in the first period. In the second period, the manufacturer produces new products not only directly from raw materials but also from used products, and both the retailer and the third party collect used products from the customers, simultaneously. In this scenario, the manufacturer needs to decide the wholesale price w1 for the first period, the wholesale price w2 , transfer prices br and bt for the second period. The retailer needs to decide the retail prices p1 and p2 for the first and second period respectively, and collection rate r for the second period. The third party decides collection rate t for the second period. The firms’ profit functions are given as follows respectively. The manufacturer’s total profit m (w1, w2, br , bt ) is

w2, br , bt ) =

(29)

bt =

t

3.3. The RT model

m (w1,

,

(30)

r

(25)

cm )

a + cm , 2

p2 =

(24)

A5 (a A4

w2 =

p1 =

(23)

aA 4

In this section, we make some analysis and comparison of the results obtained in this paper. For convenience and clarity, the optimal decisions and channel members’ maximal profits obtained in our models are listed in Table 2. With Table 2, one can see that the optimal retail/wholesale prices for the second period are independent of the manufacturer’s options for collecting used products and are independent of the competition intensity between two collecting channels, although the optimal wholesale/retail prices for the first period are dependent of both the

(28)

Proposition 3. In the RT model, the manufacturer’s optimal wholesale price w1 for the first period, optimal wholesale price w2 and optimal transfer prices br and bt for the second period are given in Eqs. (29)–(32), 5

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Table 2 Optimal decisions and maximum profits in three models. MR model

p1

A1 a

cm)

A3 a

A2 (a cm) A1 a + cm 2

A3 a

2cL (a A1 3a + cm 4

p2 aA1

w1 w2 m

(1

2 )(

r

(1

2 )(

bc )(a (2 + ) A1

cm)

bt

N/A

cL (a

m

cm ) A1

2

+

cL A2 (a cm)2 A12

r

Note: (1

2 )(

2 )(

(1

A1 = 8cL bc )2 (1 4

)

bc )2 (8 3 (2 + )2

cm)2

(a

+

cL (a

8 cm)2

(a

4cL2 (a

16

7 2

cL (1

2 3)

, A2 = 4cL

(1

2 )(

bc )2 (1 4 (2 + )2

2 )(

4 2

=

cm)2 A32

(1

2 )(

bc )(a 2A 4

cm)

(1

2 )(

bc )(a 2A 4

cm)

+ bc 2 + bc 2

+

(a

+

(a

cm)2

4 2

4

(1

, A3 = 8cL

cm ) A4

2

cL A5 (a cm)2 A 42

cm)2 16

cm )2 (1 (2 + )2A32

3)

cL (a

8

bc )2 (a

p1mt

3)

2 )(

bc )2 (3 + )

2+

,

cL (1

2)(

(1

2 )(

A 4 = 8cL

+

cm)2

(a 8

cm )2

(a

+

16

bc )2 (1 4A 42

)(a

bc )2 (3

)

2

cm )2

,

A5 =

A4 )

=

cL (a

cm)(1

2 )(

2cL (a

< 0.2327, p1mr = p1rt if

cm )(A3 A3 A 4

A4 )

=

cL (a

= 0.2327 , and

2 )( cm)(1 bc )2 ( (2 + ) A3 A4

2

)

< 0,

Theorem 1 shows that the optimal retail price for the first period depends on both the manufacturer’s options for collecting used products and the competition intensity between two collecting channels. Specifically, the optimal retail price for the first period reaches the largest value when the retailer and the third party collect used products simultaneously. However, if the competition between two collecting channels is not higher, the optimal retail price for the first period gets its smallest value in the MR model, and if the competition between two collecting channels is not too low, the optimal retail price for the first period gets its smallest value in the MT model. This means that the consumers benefit from the manufacturer’s action on collecting used products. Theorem 2. w1mr is the optimal wholesale price for the first period in MR model, w1mt is the optimal wholesale price for the first period in MT model, and w1rt is the optimal wholesale price for the first period in RT model. We have the following results.

< 0.2056 , then w1mr < w1mt < w1rt ; (1) If 0 (2) If = 0.2056, then w1mr = w1mt < w1rt ; (3) If 0.2056 < < 1, then w1mt < w1mr < w1rt . Proof. With Table 2, we have w1mr

2 )( cm )(A1 A3 ) 4c ( a cm )(1 bc )2 ( 3 + 4 2 + 4 1) = L . A1 A3 (2 + )2A1 A3 mr mt mr < p1 < 0.2056, p1 = p1mt that p1 when 0 and p1mr > p1mt when 0.2056 < < 1.

cm )(A1 A1 A 4

p1rt =

so p1mt < p1rt . # □

2cL (a

2cL (a

cm)

So, we have p1mr < p1rt if 0 p1mr > p1rt if 0.2327 < < 1.

Proof. With Table 2, we have

p1rt

N/A

cm)

.

< 0.2056 , then p1mr < p1mt < p1rt ; If 0 If = 0.2056, then p1mr = p1mt < p1rt ; If 0.2056 < < 0.2327 , then p1mt < p1mr < p1rt ; If = 0.2327 , then p1mt < p1mr = p1rt ; If 0.2327 < < 1, then p1mt < p1rt = p1mr .

p1mr

bc )(a A3

2

cm)

a + cm 2

bc )(a (2 + ) A3

cm ) A3

cm)

A5 (a A4

N/A

Theorem 1. p1mr is the optimal retail price for the first period when the manufacturer and the retailer collect the used products simultaneously, p1mt denotes the optimal retail price for the first period when the manufacturer and the third party collect the used products simultaneously, p1rt is the optimal retail price for the first period when the retailer and the third party collect the used products simultaneously.

It is easy to see when = 0.2056,

2cL (a A4

+ (1 + ) bc 2+

Using Table 2 and some algebraic calculations, the following results about the optimal retail price for the first period are obtained.

p1mt =

aA 4

2 )(

(1

5. Analysis of parameter

p1mr

cm)

N/A

manufacturer’s collecting channel options and the collection competition. Moreover, the manufacturer will pay the same optimal transfer price to the retailer and the third party in the MT and MR models. However, the manufacturer will pay higher transfer price to the third party in the MT than in the RT models, and the manufacturer will pay higher transfer price to the retailer in the MR model than in the RT model.

(1) (2) (3) (4) (5)

2 )(

(1

N/A

t

aA 4

3a + cm 4

4cL (a A3

+ (1 + ) bc 2+

br

cm)

a + cm 2

cm)

N/A

2cL (a A3

RT model

3a + cm 4

bc )(a A1

t

4cL

MT model

bc )2 (3 3 + 13 2 + 14

(2 + )2A1 A 4

4)

w1mt =

(a

cm)(4cL A1 A1 A3

It is easy to see that = 0.2056, and w1mr > We can have w1mr

.

6

w1rt =

(a

cm )(A1 A5 A1 A 4

A2 A3 )

w1mr w1mt A2 A 4 )

=

2 (a

cm)(1

2)(

bc )4 (3 + )( 3 + 4 2 + 4 (2 + )3A1 A3

< < 0.2056, if 0 if 0.2056 < < 1. w1mt

=

(a

cm) B [32cL

w1mr

B (2 + 12 2 + 3 3) + (16cL

4 (2 + )2A1 A 4

=

1)

.

w1mt

15B ]

.

if

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Fig. 1. Changes of profit of retailer with ρ.

w1rt < 0 when cL > So, it is easy to see w1mr mr rt namely w1 < w1 . w1rt w1mt = We can have B (a

cm)(16cL B (3 + )(1 ) > 0, 4 (2 + ) A3 A 4 2 )( (1 bc ) 2 . # □

B=

so

Fig. 2. Changes of profit of third party with ρ.

(a

w1rt >

bc ) 2 ,

2 )(

(1

cm)(4cL A 4 A3 A 4 w1mt ,

A3 A5)

Theorem 4. mmr is the manufacturer’s maximum profit in the MR model, mt is the manufacturer’s maximum profit in the MT model, and mrt is the m manufacturer’s maximum profit in the RT model. We can get the following results.

=

where

(1) (2) (3) (4) (5)

Theorem 2 shows that the optimal wholesale price for the first period also depends on both the manufacturer’s options for collecting used products and the competition intensity between two collecting channels. Specifically, the optimal wholesale price for the first period obtains its largest value in the RT model. However, if the competition between two collecting channels is not higher, the optimal wholesale price for the first period gets its biggest value in the MR model, and if the competition between two collecting channels is not too low, the optimal wholesale price for the first period gets its biggest value in the MT model. This means that the retailer benefits from the low wholesale price when the manufacturer participates in collecting used products.

Proof. With Table 2, we have mr m

Proof. With Table 2, we have

mr

easy to prove that mr < mt if 0 and mr > mt if 0.2056 < < 1. mr

=

rt

we have 8cL mr

> mt

rt .

rt

have 8cL where B =

=

B1 (8cL

B (2 + 15 12 2 + 3 3) ) 2(2 + )

(2 + ) A1 A 4 B (2 + 15 12 2 + 3 3) 2(2 + ) B1 (8cL

(3 + )(1 2

(2 + ) A3 A4 (3 + )(1 )B > 0, 2 2 )( (1

)B

)

mr ;

=

2BB1 (3 + )(1

< 0.2056,

mr

=

, since cL >

> 0 , which means , since cL >

which means bc ) 2 , B1 = (1

mt 2 )(

4 2

4

(2 + )3A1 A3

(1 rt

if

mt

rt

2 )(

. It is

= 0.2056, 2 )(

(1 mr

3)

=

cL (a

cm )(A3 A1 A3

A1)

=

2cL B (a

cm)2 (1

;

4

4 2

3)

(2 + )2A1 A3 mr = mmt m

. It is easy

Theorem 4 means that the manufacturer’s maximum profit in two periods not only depends on the manufacturer’s options for collecting used products but also depends on the competition intensity between two collecting channels. Because of the complexities, we cannot analyze the effects of the competition intensity’s changes on the retailer’s and the third party’s profits and collection rates through analytical approach. Figs. 1–4 are obtained through numerical studies where the parameters take the cL = 1000, a = 100, = 0.3, cm = 20, cr = following values 5, bc = 5, = 0.4, [0.10, 0.90]. The following management applications are obtained with Figs. 1–4.

mt . mt

mt m

mt m

< 0.2056, to see mmr > mmt when 0 when = 0.2056, and mmr < mmt when 0.2056 < < 1. c (a cm )2 (A1 A 4 ) c B (a cm )2 (3 3 + 13 2 + 14 4) rt mr = L = L . So, m m A1 A4 2 (2 + )2A1 A4 mr rt mr rt < 0.2327, m = m when > m when 0 = 0.2327 , and m mr < mrt when 0.2327 < < 1. m cL (a cm )2 (A4 A3 ) c B (a c )2 (3 + 2) mt mt rt so = L 2 (2 +m ) A A > 0, m m m = A3 A4 3 4 > mrt .# □

Theorem 3. mr denotes the total collection rate in MR model, mt is the total collection rate in MT model, and rt is the total collection rate in RT model. The following results can be obtained from Table 2.

< 0.2056 , then rt < mt < (1) If 0 (2) If = 0.2056, then rt < mt = mr ; (3) If 0.2056 < < 1, then rt < mr <

< 0.2056 , then mrt < mmt < mmr ; If 0 If = 0.2056, then mrt < mmt = mmr ; If 0.2056 < < 0.2327 , then mrt < mmr < If = 0.2327 , then mrt = mmr < mmt ; If 0.2327 < < 1, then mmr < mrt < mmt .

bc ) 2 ,

> 0 , namely bc ) 2 , we

> 0 , namely mt > rt , bc )(a cm) . # □

From Theorem 3, one can derive the results: the total collection rate also depends on both the manufacturer’s options for collecting used products and the competition intensity between two collecting channels; the total collection rate gets its smallest value in RT model regardless of the competition between two collecting channels. However, if the competition between two collecting channels is not higher, the total collection rate gets its biggest value in the MR model, and if the competition between two collecting channels is not too low, the total collection rate gets its biggest value in the MT model. So, it is good for the manufacturer to improve remanufacturing rate when he is in charge of collecting used products.

Fig. 3. Changes of collection effort of retailer with ρ.

7

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for period one decreases, however the changes of the optimal retail price for period one still depend on the competition intensity between two collecting channels. Specifically, as the profit discount increases, the optimal retail price for period one decreases when the competition intensity between two collecting channels is relatively low, and it will increase when the competition intensity between two collecting channels is relatively high. This means that, as the profit discount increases, the manufacturer will decrease the optimal wholesale price for period one to increase product sales in period one. Moreover, the increase of profit discount does not affect the optimal retail/wholesale price for period two, and has no effect on the manufacturer’s optimal transfer price, which are consistent with our intuitions. (2) In the MT and RT models, as the profit discount increases, the channel members’ collection rates increase; However, in the MR model, as the profit discount increases, the changes of each channel member’s collection rate still depend on the two collectors’ competition intensity. Specifically, as the profit discount increases, the manufacturer’s and the retailer’s collection rates increase when the competition intensity is relatively low, and collection rates will decrease when the competition intensity is relatively high. This tells us that the increase of profit discount always benefits the collection of used products as long as one of the two collectors is the third party. (3) Table 3 shows that the changes of profit discount do not affect the unit transfer price paid to the retailer (or the third party) by the manufacturer, which is because the transfer price occurs in second period. (4) The increase of profit discount always benefits the retailer, regardless of the collection scenarios. However, the increase of profit discount will always benefit the manufacturer as long as one of the two collectors is the third party, and the increase of profit discount will always benefit the third party as long as the manufacturer does not collect used products directly.

Fig. 4. Changes of collection effort of third party with ρ.

(1) Whatever the manufacturer’s options for collecting used products are, the retailer’s profit decreases as the competition intensity increases. As long as the competition intensity between two collecting channels is not too low, the retailer’s best choice is not in charge of collecting used products. (2) The third party gets higher profit when the retailer and the third party are in charge of collecting used products than when the manufacturer and the third party are in charge of collecting used products, however, the retailer gets higher profit when the manufacturer and the third party are in charge of collecting used products than when the retailer and the third party are in charge of collecting used products. The third party cannot get positive revenue when the competition intensity between two collecting channels is not too low in the MT model. (3) The retailer’s collection rate is higher in the RT model than in the MR model. Similarly, the third party will also make more effort to collect used products in the RT model than in the MT model. Moreover, all firms’ collection rates will decrease as the competition intensity between two collecting channels increases.

In order to get more managerial applications about discount factor , we also give Figs. 5–9 through numerical studies where the parameters take the following values cL = 1000, a = 100, = 0.3, cm = 20, cr = 5, bc = 5, = 0.2, [0.10, 0.60]. The following management applications are obtained with Figs. 5–9 and Table 2. The optimal retail/wholesale prices for the first period is affected by the profit discount , and the differences of the effects of profit discount on the optimal retail prices of three models are minimal. As the profit discount increases, the total collection rate increases in the MT and RT models very slowly, however, the changes of total

5.1. Analysis of parameter With some algebraic manipulations, it is possible to provide some insights on how the supply chain members would behave with different profit discounts. The corresponding results are listed in Table 3 as follows. (1) In the MT and RT models, as the profit discount increases, the optimal wholesale price for period one decreases, which directly leads to the decreases of optimal retail price for period one. In the MR model, as the profit discount increases, the optimal wholesale price Table 3 Effects of the parameter

on the optimal solutions. MR model

MT model

0<

< 0.805,

;0.805 <

< 1,

m

0<

< 0.805,

;0.805 <

< 1,

r

0<

< 0.805,

;0.805 <

< 1,

p1

RT model

p2

w1 w2

N/A

t

N/A

bt

N/A

br 0<

m

r

;0.805 <

< 1,

N/A

t

Note:

< 0.805,

: increasing;

: decreasing;

N/A

N/A

0<

: constant; N/A: not applicable. 8

< 0.206,

;0.206 <

< 1,

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Fig. 5. Changes of retail price in first period with δ.

Fig. 9. Changes of profit of third party with δ.

collection rate depend on the competition intensity between two collecting channels in the MR model. increases, the manufacturer’s profit inAs the profit discount creases very rapidly in the MT and RT models. The differences of the effects of profit discount on manufacturer’s profit of three models are very minimal. However, the third party’s profit increases very rapidly in the RT model, and increases very slowly in the MT model when the competition intensity between two collecting channels is relatively low. 6. Conclusions and suggestions for further research This paper considers a two-period remanufacturing problem where a manufacturer produces new product in the first period only with raw material and produces the new product either with raw material or with used products in the second period, the manufacturer has three options of collecting used products in the second period, and two of the three firms (e.g., the manufacturer, the retailer, and the third party) are in charge of collecting the used products. We formulate three dynamic game models by considering the dynamic nature of product life-cycle and firm’s profit discount over the time with the intent to illustrate the manufacturer’s potential benefit of adopting remanufacturing process and dual collecting channels. We get the analytical solutions of these models and provide the analysis and comparison of the results obtained in this paper. As we have shown in our analysis, the optimal retail price for the first period depends on both the manufacturer’s options for collecting used products and the competition intensity between two collecting channels, and the profit discount has no effect on the optimal retail/wholesale prices for the second period, and has no effect on the manufacturer’s optimal transfer price. This paper is formulated under some assumptions which allow us to consider several areas in future to better understand the remanufacturing. First, a natural extension to our models is the uncertainty issues (e.g., in demand, in collecting of used products). This is a relevant aspect in many practical remanufacturing problems. Second, we assume that there is no differences between the remanufactured and manufactured products. To capture the impact of competition between the manufactured and remanufactured product markets, the model can be extended by considering an industry in which the manufactured and remanufactured products are different. Finally, the inclusion of uncertainty in this problem also causes the inclusions of symmetric information and risk pooling as this is a feature that under an uncertainty setting may strongly influence the firms’ decisions.

Fig. 6. Changes of wholesale price in first period with δ.

Fig. 7. Changes of total collection effort with δ.

Fig. 8. Changes of profit of manufacturer with δ.

9

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Acknowledgments

anonymous referees for their constructive comments and suggestions on the paper. We gratefully acknowledge the support of National Natural Science Foundation of China, Nos. 71371186, 71840005.

The authors wish to express their sincerest thanks to the editors and Appendix A

Proof of Proposition 1. The MR model is a two-period Stackelberg game, there is a Stackelberg game in each period. To obtain a subgame-perfect Stackelberg equilibrium, we have to first solve the Stackelberg game in the second period, and then solve the Stackelberg game in the first period as follows. The second period. In the second period, the manufacturer is leader and the retailer is follower. So, we first derive the retailer’s best response functions given manufacturer’s earlier decisions. The retailer needs choose the optimal retail price p2 and the optimal collection rate r to optimize its second-period profit, i.e.,

max

r 2 (p2 , r ).

p2 , r

Using the expression of profit function r2

p2 r2 r

= (a

2 p2 + w2),

= (br

bc )(a

r 2 (p2 , r ) ,

the first order partial derivatives of

r 2 (p2 , r )

to p2 and

r

can be shown as: (a1)

2cL 1

p1 )

r 2

,

(a2)

The second order partial derivatives to check for the optimality are as follows: 2

r2

=

p22 2

r2

p2

2 2

=

2

< 0,

r

r2

2cL 1

=

< 0,

2

(a3)

= 0.

p2

r

r2 2 r

(a4)

Eqs. (a3) and (a4) indicate that the Hessian matrix of r2 to p2 and r is negative definite. Namely, r (p2 , r ) is jointly concave in p2 and r . Hence, the following response functions can be obtained by setting Eqs. (a1) and (a2) to zero and solving them simultaneously.

a + w2 , 2

p2 =

(br

=

r

(a5) 2 )(a

bc )(1

p1 )

2cL

.

(a6)

Having the information about the retailer’s response functions, the manufacturer would then use them to maximize its profit of the second period, namely, the manufacturer optimizes its profit of the second period with respect to w2, m , and br , taking into account the retailer’s response functions, i.e.,

max

m 2 ( w 2, m ,

w 2, m, br

br ).

(a7)

Using Eqs. (a5), (a6) and (4), the first and second order partial derivatives of m2

w2 m2

(a + cm 2

=

= (

bc )(a

m m2

br 2

m2

w22 2

m2

br2 2

=

m2

2cL 1

p1 ) 2 (

=

2

< 0,

m2 2 m

2 )(a

(1

m , 2

2

m2 m

=

2br

=

p1 2 m

m2

br

The Hessian matrix Hm of

m

and br can be shown as:

(a9)

br + bc + bc )

2cL 1

) 2 (2

2

+ )

=

2 m

m2

w2

m2 (w 2, m,

,

(a10)

< 0,

(a11)

< 0,

2cL w2

br ) to w2,

(a8)

2cL

=

w2 br

,

p1 )

2 )(a

(1

=

2 w2)

m2 (w 2, m,

=

(a12) 2

br

m2

=

m

br ) to w2,

m

2

m2

br w 2

= 0.

(a13)

and br is given as:

10

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0 Hm =

0

2cL 2 1

0 0

0 (1

0

.

2 )(a

p1)2 (2 + )

(a14)

2cL

One can see that Hm is a negative definite Matrix, which indicates that m2 (w2, m, br ) is jointly concave in w2, zero and solving them simultaneously, we can obtain the following best response functions:

and br . By setting Eqs. (a8)–(a10) to

a + cm , 2

w2 =

m

m

2 )(

(1

=

(a15)

bc )(a

p1 )

2cL

,

(a16)

+ ( + 1) bc . 2+

br =

(a17)

Using Eqs. (a15) and (a5), we get the retailer’s optimal retail price for the second period as follows:

p2 =

3a + cm . 4

(a18)

The first period. Moving to the first period, the manufacturer also is leader and the retailer is follower. So, we still first derive the retailer’s best response functions given manufacturer’s earlier decisions. The retailer chooses the optimal retail price p1 to maximize its total profit of the two periods. Using Eqs. (5), (a15)–(a18), we obtain the first and second order derivatives of r to p1 as: r

2

(A2

=

p1 r

+ w1,

2cL

(a19)

A2 < 0, 2cL

=

p12

A2 p1

2cL) a 2cL

(a20) (1

where A2 = 4cL first period as follows:

p1 (w1) =

A2 a

2 )(

bc )2 (1

4

4 2

3)

(2 + )2

. By setting Eq. (a19) to zero and solving it, we can obtain the retailer’s best response function for the

2cL a + 2 cL w1 . A2

(a21)

Having the retailer’s response functions, the manufacturer would then use them to maximize its profit, the manufacturer chooses the optimal wholesale price w1 to maximize its total profit of the two periods. Using Eqs. (5), (a15)–(a18), and (a21), we obtain the first and second order derivatives of m to w1 as: m

w1

=

2

m w12

2cL A2 (2 + )(a + cm

=

2 cL [2(2 + ) A2

Since 2cL >

(1

2 )(

2 w1)

2cL (1 (2 + ) A22

2 )( (1 2 (2 + ) A2

bc

)2,

2 )(

bc )2 (3 + )]

then 2(2 + ) A2

(1

bc )2 (3 + )(a

w1)

,

(a22)

.

(a23) 2 )(

bc

) 2 (3

+ )] > 0 , which means that

2

m w12

< 0 , namely, the manufacturer’s total

profit m is concave in w1. By setting Eq. (a22) to zero and solving it, we can obtain the manufacturer’s optimal wholesale price for the first period as follows:

w1 =

A1 a

A2 (a A1

where A1 = 8cL

cm) (1

,

(a24)

2 )(

bc )2 (8

3

7 2

2 3)

b )2 (1

4

4 2

3)

(2 + )2

(1

2 )(

, A2 .

c = 4cL (2 + )2 With Eqs. (a21) and (a24), we obtain the retailer’s optimal retail price for the first period as follows:

p1 =

A1 a

2cL (a A1

cm)

.

(a25)

Using Eqs. (a5), (a15)–(a18), (a24) and (a25), one can get the results of Proposition 1. Thus, Proposition 1 is proved. □ Proof of Proposition 2. The MT model also is a two-period Stackelberg game, similar to the Proof of Proposition 1, we can prove Proposition 2 as follows. The second period. In the second period, the manufacturer first makes its decisions, then the retailer gives its decisions, and at last the third party decides its collection rate. So, we need first derive the third party’s response function given manufacturer’s and retailer’s earlier decisions, then derive the retailer’s best response functions given manufacturer’s earlier decisions, at last give the manufacturer’s decisions as follows. The third party decides optimal collection rate t to optimize its second-period profit, i.e., 11

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max

t ( t ).

t

Using the expression of profit function t ( t)

= (a

p1 )(bt

t 2

t ( t) 2 t

2cL 1

=

bc )

(a

=

p1 )(bt

t 2

t ( t)

in Eq. (15), the first and second order derivatives of

t ( t)

to

t

can be shown as:

,

(a26)

< 0.

2

(a27)

Eq. (a27) indicates that t

2cL 1

t ( t ) is concave in t . Hence, the following response function can be obtained by setting Eq. (a26) to zero and solving it. 2)

bc )(1

2cL

.

(a28)

Having the third party’s response function, the retailer chooses the optimal retail price p2 to optimize its second-period profit, i.e.,

max

r 2 (p2 )

p2

= max (a

p2 )(p2

p2

w2).

The first and second order derivatives of r2 (p2 )

= (a

p2 2

r 2 (p2 ) p22

=

2 p2 + w2),

2

(a29)

< 0.

(a30)

Eq. (a30) indicates that

p2 =

to p2 can be shown as:

r2 (p2 )

r (p2 ) is concave in p2 . Hence, the following response function can be obtained by setting Eq. (a29) to zero and solving it.

a + w2 . 2

(a31)

At last, the manufacturer optimizes its profit of the second period with respect to w2, party’s reaction functions, i.e.,

max

m 2 (w 2 , m ,

w 2, m, bt

m,

and bt , taking into account both the retailer’s and the third

bt ).

(a32)

Using Eqs. (a28), (a31) and (13), we have the first and second order partial derivatives of m2

w2 m2

(a

=

2 w2 + cm) , 2

= (

bc )(a

m m2

bt 2

m2

m2

2

m2

w2

p1 2

< 0,

)2 (

m2 2 m

2 )(a

(1

=

bt2

2cL 1

m , 2

2bt + bc 2cL 1

=

2

p1 ) 2 (2 + ) 2cL

=

m

2

m2

w2 bt

2

= 0,

m2

m

The Hessian matrix Hm of

0 0

Hm =

0

m

and bt as follows:

(a34)

bt + bc )

2cL

=

w22 2

(1

=

bt ) to w2,

(a33)

p1 )

2 )(a

m2 (w 2, m,

w2

2

=

m

m2 (w 2,

,

(a35)

< 0,

(a36)

< 0, m2

bt

(a37)

= 0,

m, bt ) to w2,

2

m2

bt w2 m

=

2

bt

m2

= 0.

(a38)

m

and bt is given as:

0

2cL 2 1

0 (1

0

2 )(a

. p1)2 (2 + )

(a39)

2cL

One can see that Hm is a negative definite Hessian Matrix, which indicates that m2 (w2, m, bt ) is jointly concave in w2, m and bt . By setting Eqs. (a33)–(a35) to zero and solving them simultaneously, we can obtain the manufacturer’s best response functions for the second period as follows:

w2 = m (p1 )

bt =

a + cm , 2 =

(1

(a40) 2 )(

bc )(a 2cL

p1 )

,

(a41)

+ (1 + ) bc . 2+

(a42) 12

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Using Eqs. (a31) and (a40), we get the retailer’s optimal retail price for the second period as follows:

3a + cm . 4

p2 =

(a43)

Using Eqs. (a40) and (a43), we get the retailer’s optimal profit in the second period as follows:

cm)2

(a

=

r2

.

16

(a44)

Using Eqs. (a28), and (a40)–(a43), we get the manufacturer’s optimal profit in the second period as follows: 2 )(

(1

=

m2

bc ) 2 (a p1 ) 2 (3 + ) + 4cL (2 + )

cm) 2

(a 8

.

(a45)

The first period. Moving to the first period, the manufacturer also is leader and the retailer is follower. So, we still first derive the retailer’s best response functions given manufacturer’s earlier decisions. The retailer chooses the optimal retail price p1 to maximize its total profit of the two periods. Using Eqs. (14) and (a24), we obtain the first and second order derivatives of r to p1 as: r1 (p1 )

p1 2

r 1 (p1 ) p12

=a

=

2 p1 + w1,

(a46)

2 < 0.

(a47)

By setting Eq. (a46) to zero and solving it, we can obtain the retailer’s best response function for the first period as follows:

p1 (w1) =

a + w1 . 2

(a48)

Having the retailer’s response function, the manufacturer chooses the optimal wholesale price w1 to maximize its total profit of the two periods. Using Eqs. (13), (a40)–(a45), and (a28), we obtain the first and second order derivatives of m to w1 as: m

=

w1 2

m

w12

A3 a

=

4cL (a 8cL

cm)

A3 w1 , 8cL

(a49)

A3 < 0, 8cL

where A3 = 8cL Since 2cL >

(a50) 2 )(

(1

bc )2 (3 + )

2+

bc ) 2 , then 8cL >

2 )(

(1

.

(1

2 )(

bc )2 (3 + ) , which means that

2

m

w12

< 0 , namely, the manufacturer’s total profit

m

is

concave in w1. By setting Eq. (a49) to zero and solving it, we can obtain the manufacturer’s optimal wholesale price for the first period as follows:

w1 =

A3 a

4cL (a A3

cm)

.

(a51)

With Eqs. (a48) and (a51), we obtain the retailer’s optimal retail price for the first period as follows:

p1 =

A3 a

2cL (a A3

cm)

.

(a52)

Using Eqs. (a41), (a52), (a28) and (a42), one can get m

=

t

=

(1

(1

2 )(

bc )(a A3

cm)

bc )(a (2 + ) A3

cm )

2 )(

,

(a53)

.

(a54)

Thus, Proposition 2 is proved. □ Proof of Proposition 3. The proof of Proposition 3 is more similar to the proof of Proposition 2, the RT model is also a two-period Stackelberg game. We can prove Proposition 3 as follows. The second period. In the second period, the manufacturer first makes its decisions, then the retailer gives its decisions, and at last the third party decides its collection rate. So, we need first derive the third party’s response function given manufacturer’s and retailer’s earlier decisions, then derive the retailer’s best response functions given manufacturer’s earlier decisions, at last give the manufacturer’s decisions. The third party decides optimal collection rate t to optimize its profit, i.e.,

max t

t ( t ).

Using the expression of profit function t t

= (bt

bc )(a

p1 )

2cL 1

t 2

t ( t)

in Eq. (28), the first and second order derivatives of

,

t ( t)

to

t

can be shown as: (a55)

13

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J. Wei et al. 2

t 2 t

2cL 1

=

< 0.

2

(a56)

Eq. (a56) indicates that

(bt

=

t

bc )(a

t ( t ) is concave in t . Hence, the following response function can be obtained by setting Eq. (a55) to zero and solving it. 2)

p1 )(1 2cL

.

(a57)

Having the third party’s response function, the retailer chooses the optimal retail price p2 and optimal collection rate period profit, i.e.,

max

r 2 (p2 )

p2

= max( (a

p2 )(p2

p2

w2) + (br

bc )(a

The first and second order partial derivatives of r2

p2 r2 r 2

= a+

w2

= (br

bc )(a

r2

=

p22 2

r2

p2

2

=

r

r2

Ir ).

r

to p2 and

r

can be shown as: (a58)

2cL 1

p1 ) 2

r2 2 r

r 2

2cL 1

=

,

(a59)

2

< 0,

(a60)

= 0.

p2

r

r2 (p2 )

to optimize its second-

2 p2 ,

< 0, 2

p1 )

r

(a61)

Eq. (a60) and (a61) indicate that r (p2 , r ) is jointly concave in p2 and r . Hence, the following response functions can be obtained by setting Eqs. (a58) and (a59) to zero and solving them simultaneously.

a + w2 , 2

p2 =

2 )(b r

(1

=

r

(a62)

bc )(a

p1 )

2cL

.

(a63)

The manufacturer then optimizes its profit of the second period with respect to w2, br , and bt , taking into account both the retailer’s and the third party’s reaction functions, i.e.,

max

m2 (w 2,

w 2, br , bt

br , bt ).

(a64)

Using Eqs. (a57), (a62)–(a64), we have the first and second order partial derivatives of m2

w2 m2

br m2

bt 2

=

m2

p1 )2 (1

m2

2 )(

p1 )2 (1

(a

2br + bc )

2 )(

2bt + bc )

2cL

=

w2 br

=

br , bt ) to w2, br and bt as follows: (a65)

2cL

=

m2 (w 2,

2 w2 + cm) , 2

(a

=

w22 2

(a

< 0, 2

m2

w2 bt

2

m2

br2

= 0,

= 2

2

m2

bt2 m2

br w2

=

(a66)

,

(a67)

(a

= 2

,

p1

)2 (1

2)

cL m2

br bt

= 0,

2

m2

bt w2

=

< 0, 2

m2

bt br

(a68)

= 0.

(a69)

With Eqs. (a68) and (a69), one can see that the Hessian matrix of m2 (w2, br , bt ) to w2, br and bt is a negative definite, which indicates that br , bt ) is jointly concave in w2, br and bt . By setting Eqs. (a65)–(a67) to zero and solving them simultaneously, we can obtain the manufacturer’s best response functions for the second period as follows: m2 (w 2,

w2 =

a + cm , 2

(a70)

br =

+ bc , 2

(a71)

bt =

+ bc . 2

(a72)

Using Eq. (a62) and (a70), we get the retailer’s optimal retail price for the second period as follows:

p2 =

3a + cm . 4

(a73)

14

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The first period. Moving to the first period, the retailer chooses the optimal retail price p1 to maximize its total profit of the two periods. Using Eq. (27), and the results obtained in period 2, we obtain the first and second order derivatives of r to p1 as: r

2

A5 a

=

p1 r

w1)

A5 p1

2cL

,

(a74)

A5 < 0. 2cL

=

p12

2cL (a

(a75)

By setting Eq. (a74) to zero and solving it, we can obtain the retailer’s best response function in the first period as follows:

p1 (w1) =

A5 a

2cL (a A5

w1) 2 )(

(1

,

(a76)

bc )2 (1

)

where A5 = 4cL . 4 Having the retailer’s response function, the manufacturer chooses the optimal wholesale price w1 to maximize its total profit of the two periods. Using Eq. (a76), and the results obtained in period 2, we obtain the first and second order derivatives of m to w1 as: m

w1 2

=

m

w12

2cL (aA 4

=

A5 (a cm) A52

A 4 w1)

,

(a77)

2 cL A 4 < 0, A52

(a78) 2 )(

(1

bc )2 (3

)

(1

2 )(

bc )2 (1

)

, A5 = 4cL where A 4 = 8cL . 2 4 By setting Eq. (a77) to zero and solving it, we can obtain the manufacturer’s optimal wholesale price for the first period as follows:

w1 =

A4 a

A5 (a A4

cm )

.

(a79)

With Eqs. (a76) and (a79), we obtain the retailer’s optimal retail price for the first period as follows:

p1 =

A4 a

2cL (a A4

cm)

.

(a80)

Using Eqs. (a57), (a63), (a71), (a72) and (a80), one can get r

=

t

=

(1

(1

2 )(

2 )(

bc )(a 2A 4

cm)

bc )(a 2A 4

cm )

,

(a81)

.

(a82)

Thus, Proposition 3 is proved. □

471–472. Hong, X., Wang, Z., Wang, D., & Zhang, H. (2013). Decision models of closed-loop supply chain with remanufacturing under hybrid dual-channel collection. The International Journal of Advanced Manufacturing Technology, 68, 1851–1865. Huang, M., Song, M., Lee, L., & Ching, W. (2013). Analysis for strategy of closed-loop supply chain with dual recycling channel. International Journal of Production Economics, 144, 510–520. Savaskan, R., Bhattacharya, S., & Van Wassenhove, L. (2004). Closed-loop supply chain models with product remanufacturing. Management Science, 50, 239–252. Savaskan, R., & Van Wassenhove, L. (2006). Reverse channel design: The case of competing retailers. Management Science, 52, 1–14. Webster, S., & Mitra, S. (2007). Competitive strategy in remanufacturing and the impact of take-back laws. Journal of Operations Management, 25(6), 1123–1140. Wei, J., Govindan, K., Li, Y., & Zhao, J. (2015). Pricing and collecting decisions in a closed-loop supply chain with symmetric and asymmetric information. Computers & Operations Research, 54, 257–265. Wei, J., & Zhao, J. (2013). Reverse channel decisions for a fuzzy closed-loop supply chain. Applied Mathematical Modelling, 37(3), 1502–1513. Wu, C. (2013). OEM product design in a price competition with remanufactured product. Omega, 41(2), 287–298. Zhao, J., Wei, J., & Li, M. (2017). Collecting channel choice and optimal decisions on pricing and collecting in a remanufacturing supply chain. Journal of Cleaner Production, 167(20), 530–544.

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