Joint replenishment and carbon trading in fresh food supply chains

European Journal of Operational Research 277 (2019) 561–573

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing, Transportation and Logistics

Joint replenishment and carbon trading in fresh food supply chains Min Wang a, Lindu Zhao a,∗, Michael Herty b a b

School of Economics and Management, Southeast University, No.2 Sipailou, Nanjing 210096, PR China Institute for Geometry and Applied Mathematics, RWTH Aachen University, Templergraben 55, 52074 Aachen, Germany

a r t i c l e

i n f o

Article history: Received 1 November 2017 Accepted 1 March 2019 Available online 8 March 2019 Keywords: Supply chain management Joint replenishment Carbon trading Fresh food Bargaining game

a b s t r a c t We investigate a fresh food supply chain comprising a large-scale supplier and multiple small-scale retailers under a carbon cap-and-trade policy. Retailers’ joint replenishment and the carbon trading behavior of supply chain members are studied. We assume that three replenishment strategies are available for the supply chain: (1) separate replenishment; (2) joint replenishment: a leader-follower relationship among retailers; and (3) joint replenishment: the coalition of retailers. Under each strategy, a bargaining framework for supply chain members is set up to maximize their profits, where the price of the refrigerated transportation services provided by the supplier and the retail price of fresh food are optimized. The optimal decisions are analyzed to provide insights into logistics pricing and retail pricing strategies. Through comparing three replenishment strategies, we also identify the optimal replenishment strategies from the perspectives of the supplier, retailers and a carbon emission optimizer. Moreover, we investigate the role of the carbon cap-and-trade policy by comparing the cases with and without a carbon cap-and-trade policy. It is noteworthy that the goals of profit growth and emission reduction are simultaneously achieved under the carbon cap-and-trade policy. © 2019 Elsevier B.V. All rights reserved.

1. Introduction It is well recognized that collaboration in supply chains brings competitiveness and enhances profitability (Horvath, 2001). Cleophas, Cottrill, Ehmke, and Tierney (2019) differentiate the direction of collaboration as vertical, i.e., between upstream and downstream firms (Derrouiche, Neubert, & Bouras, 2008) and horizontal, i.e., between companies in the same level of the logistics network (Soosay & Hyland, 2015). Our research focuses on the horizontal collaboration of retailers in a single-supplier multi-retailer fresh food supply chain. The horizontal collaboration of retailers takes different forms. Retailers may make joint procurement to reduce purchasing cost through quantity discount or enhanced bargaining power (Hsu, Lai, Niu, & Xiao, 2016; Krichen, Laabidi, & Abdelaziz, 2011), or adopt the strategy of centralized inventory to reduce lost sales and ensure their smooth operation (Chen & Zhang, 2009). In centralized inventory systems, the practice of transshipment is extensively studied (Paterson, Kiesmüller, Teunter, & Glazebrook, 2011; van Wijk, Adan, & van Houtum, 2019). Moreover, retailers may collaborate in transportation by sharing transportation facilities or bundling their shipment requests to

∗

Corresponding author. E-mail addresses: [email protected] (M. Wang), [email protected] (L. Zhao), [email protected] (M. Herty). https://doi.org/10.1016/j.ejor.2019.03.004 0377-2217/© 2019 Elsevier B.V. All rights reserved.

make joint replenishment (Guajardo, Rönnqvist, Flisberg, & Frisk, 2018; Özener & Ergun, 2008). We are interested in retailers’ joint replenishment in the form of bundling the long-haul shipments from the supplier to their warehouses. This paper’s fresh food supply chain comprises a large-scale supplier and multiple small-scale retailers. Because the supplier possesses many refrigerated trucks, it takes the role as a third-party logistics (3PL) company by providing refrigerated transportation services for retailers. When retailers make orders from the supplier, their shipment requests are sent to the logistics department of the supplier. Since fresh food easily becomes rotten, retailers make frequent and small-quantity orders from the supplier, which leads to less-than-truckload shipments and the increased fixed costs of trucks. To cope with this dilemma, retailers consolidate their shipments aiming for exploiting the economies of scale of joint replenishment. To analyze retailers’ behavior of joint replenishment, we assume that three replenishment strategies are available, including the strategy of separate replenishment (as a benchmark case) and two joint replenishment strategies. Meanwhile, climate change has highlighted the importance of carbon emission reduction, which encourages governments to implement carbon policies, including the carbon cap-and-trade policy. Under this policy, firms are initially allocated with carbon emission permits denoted as their carbon caps, and then, they trade (buy or sell) emission permits in the carbon market if necessary. In addition to joint replenishment, we are curious about supply

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chain members’ behavior of carbon trading. Through investigating replenishment strategies under a carbon cap-and-trade policy, we will address issues including the optimal replenishment strategies from different perspectives and the influences of a carbon cap-andtrade policy on supply chain members and carbon emissions. In this paper, a bargaining framework is set up for supply chain members to maximize their profits, where the price of the refrigerated transportation services provided by the supplier and the retail price of fresh food are jointly optimized. On this basis, recommendations about logistics pricing and retail pricing are provided for supply chain members. Through comparing three replenishment strategies, we further identify the optimal replenishment strategies from perspectives of the supplier, retailers and a carbon emission optimizer, respectively. Furthermore, the role of the carbon cap-and-trade policy in decision-making, profit growth and emission reduction is investigated. The remainder of this paper is organized as follows. Relevant literature is reviewed in Section 2. In Section 3, we describe the problem to be solved and introduce the necessary assumptions and notations. The optimal price of refrigerated transportation services and the retail price under different replenishment strategies are derived and analyzed in Section 4. Section 5 discusses the optimal replenishment strategies from different perspectives. The role of the carbon cap-and-trade policy is revealed in Section 6. In Section 7, concluding remarks as well as future work are stated. 2. Literature review 2.1. Joint replenishment The joint replenishment problem (JRP) describes that a subset of agents simultaneously place an order aiming for total cost reduction, mainly in the form of a single retailer ordering multiple items or multiple locations (retailers) ordering the same type of items (Fiestras-Janeiro, García-Jurado, Meca, & Mosquera, 2011; Nguyen, Dessouky, & Toriello, 2014). There has been much research on multi-item JRP and comprehensive reviews have been conducted (Aksoy & Selcuk Erenguc, 1988; Khouja & Goyal, 2008). Many factors have been incorporated in the basic model, such as dynamic demand (Kang, Lee, Wu, & Lee, 2017), product substitution (Salameh, Yassine, Maddah, & Ghaddar, 2014), resource and shipment constraints and defective items (Ongkunaruk, Wahab, & Chen, 2016), and expedited and regular shipment orders (Satır, Erenay, & Bookbinder, 2018). Our research is limited to multi-retailer JRP. Relevant research is also conducted in the fields of the onewarehouse multi-retailer (OWMR) problem (Levi, Roundy, Shmoys, & Sviridenko, 2008), the joint replenishment and delivery (JRD) problem (Qu, Wang, & Zeng, 2013), and collaborative transportation (Guajardo et al., 2018). Joint replenishment provides retailers with a cost saving opportunity. On the one hand, a quantity discount is offered to retailers if their orders are bundled; on the other hand, the consolidated shipment decreases the total fixed transportation cost (Kang et al., 2017). The joint cost structure known as first-order interaction is common in JRP. In this structure, the major setup cost refers to the fixed ordering or transportation cost each time an order is placed, and the minor setup cost is the variable ordering or transportation cost associated with the order quantity (He, Sethuraman, Wang, & Zhang, 2017; Zhang, 2009). Our research focuses on the saving of major setup cost in the joint replenishment of retailers. The majority of previous studies are conducted using cooperative approaches. For example, Elomri, Ghaffari, Jemai, and Dallery (2012) research on the coalition formation and cost allocation problem in a multi-retailer joint replenishment system and discuss both superadditive and non-superadditive games. Li, Cai, and Zeng (2016) solve the transportation facility selection and cost allocation

problem in the less-than-truckload collaboration among perishable product retailers. Non-cooperative games are also used to solve the problem of joint replenishment. For instance, Körpeog˘ lu, S¸ en, and Güler (2013) consider asymmetric information in retailers’ joint replenishment and obtain the Bayesian Nash equilibrium in a non-cooperative game. He et al. (2017) adopt the allocation rule in which the major setup cost is split equally among retailers and determine each retailer’s optimal replenishment policy. However, to the best of the authors’ knowledge, the joint replenishment simultaneously considering leader-follower relationship and coalition formation has not been discussed yet.

2.2. Carbon cap-and-trade policy Carbon emission reduction has become a critical issue worldwide, which encourages the implementation of carbon policies such as a carbon cap-and-trade policy (Tang, Wang, Cho, & Yan, 2018). Under the carbon cap-and-trade policy, firms are initially allocated with carbon emission permits as their carbon caps, and then, firms are free to trade emission permits in the carbon market. Various methods of emission permit allocation have been proposed, and their performances are evaluated (Ji, Zhang, & Yang, 2017; Morrell, 2007). For example, Qiu, Xu, and Zeng (2017) develop a mixed mechanism for governments to allocate carbon emission allowances to airlines operating on the routes. Its practicality and efficiency in mitigating carbon emissions are verified through a case study. In addition, a series of research is conducted to analyze the influences of carbon prices (Benjaafar, Li, & Daskin, 2013). For example, Luo, Chen, and Wang (2016) note that a higher carbon price causes higher retail prices and the manufacturers’ green technology investments first increase and then decrease with the increase in the carbon price. Rezaee, Dehghanian, Fahimnia, and Beamon (2017) study a green supply chain design problem with an uncertain carbon price and observe that it highly depends on the probability distribution of the carbon price. Other literature relevant to our research is divided into two streams. One stream integrates environmental considerations into the operation of supply chains with temperature-sensitive products or deteriorating items. For instance, Bozorgi, Pazour, and Nazzal (2014) derive the optimal order quantity based on the cost and emission functions associated with creating a temperaturecontrolled environment. Saif and Elhedhli (2016) study the cold supply chain design problem with the purpose of minimizing the total operational cost and the global warming impact. Another stream incorporates environmental concerns in joint replenishment or consolidated shipment. For instance, Dem and Singh (2015) study multi-item joint replenishment in a productioninventory system adopting green remanufacturing and recycling approaches. In addition, Zhou and Zhang (2017) investigate the effects of different carbon policies on operational decisions in airfreight shipment with integration and consolidation, including carbon tax, carbon cap and carbon cap-and-trade policies. However, there is little research considering multi-retailer JRP, carbon capand-trade and fresh food simultaneously. On this basis, the contributions of our paper are specified. Theoretically, we complement the extant research by integrating multi-retailer JRP, carbon cap-and-trade and fresh food. The integration is necessary due to the emission-intensive characteristic of refrigerated transportation, the emission reduction opportunity in joint replenishment along with joint replenishment’s advantages in the frequent and small-quantity deliveries of fresh food. We also make contributions by considering two joint replenishment strategies with a leader-follower relationship among retailers or the coalition formation of retailers. Moreover, our paper has practical interests because findings about the carbon cap-and-trade

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563

Fig. 1. Three replenishment strategies.

policy and the optimal replenishment strategy are useful for governments and supply chains. 3. Problem description, assumptions and notations 3.1. Problem description The fresh food supply chain under consideration comprises a large-scale supplier and N small-scale retailers. Retailers are dispersedly located and independently serve consumers. Since fresh food easily becomes rotten, retailers need to make frequent and small-quantity replenishment orders. There are three replenishment strategies to choose from. (1) Separate replenishment (strategy 1), which is the benchmark for all replenishment strategies. (2) Joint replenishment: a leader-follower relationship among retailers (strategy 2), in which a retailer is selected by the supplier as the leader and makes joint replenishment for all retailers. (3) Joint replenishment: the coalition of retailers (strategy 3), in which retailers form a coalition and make joint replenishment. These replenishment strategies are depicted in Fig. 1, where the subscript i stands for retailer i (i = 1, 2, . . . , N). The subscripts x and c stand for the leader and the coalition of retailers, respectively. 3.2. Assumptions and notations 3.2.1. Market demand Following Li, Sethi, and Zhang (2013), we assume market demand is negatively correlated with retail price. More precisely, the demand of retailer i is expressed as Di = v − a pi , where v is the potential market size, a is the price sensitivity coefficient of consumers, and pi is the retail price determined by retailer i. We also assume that 0 < pi < av , which ensures the non-negativity of retail price and market demand. 3.2.2. Order quantity The inventory of retailer i at time t is denoted as Ii (t ). It dedI (t )

i teriorates during the sales cycle according to dt = −Di − uIi (t ), where u is the deterioration rate of fresh food and u > 0 (Dye & Hsieh, 2012). Assuming that food is sold out at the end of the

sales cycle (Ii (T ) = 0, T is the length of sales cycle), we derive eu(T −t ) −1 ( v − a pi ). u euT −1 Ii (0 ) = u (v − a pi ).

that Ii (t ) =

Thus, the order quantity of retailer i uT

is Qi = By letting λ = e u−1 , we obtain that Qi = λ(v − a pi ). The non-negativity of order quantity is guaranteed by the condition that 0 < pi < av . 3.2.3. Refrigerated transportation services As we introduced in Section 1, the supplier provides refrigerated transportation services for retailers. It is reasonable that the supplier bears relevant cost and carbon emissions. Based on Bernstein and Federgruen (2003) and Zhang, Tang, and Zhou (2017), we assume both the cost and emissions of refrigerated transportation comprise fixed and variable parts. Following their assumption that the variable part is proportional to the amount of the shipment, we further consider the influence of the distance between the supplier and retailers. More specifically, we use Gsi (Qi ) = Fs + ct di Qi to represent the cost of the transportation from the supplier to retailer i, where Fs is the fixed cost of transportation, ct is the unit transportation cost, and di is the distance from the supplier to retailer i. Likewise, Usi (Qi ) = Es +χ di Qi denotes the emissions of the transportation from the supplier to retailer i, where Es is the fixed emissions of transportation, and χ is the emission coefficient of transportation. Meanwhile, retailers need to pay for the refrigerated transportation services provided by the supplier, including refrigeration costs and shipping costs. We use Gri (Qi ) = ki di Qi to represent retailer i’s expense of refrigerated transportation services. ki is the price of the refrigerated transportation services provided for retailer i. It is determined by the supplier and referred to as the logistics price hereinafter. 3.2.4. Multiple retailers In joint replenishment, the transshipment among retailers incurs fixed cost and carbon emissions (Noham & Tzur, 2015). Fr and Er stand for the cost and the emissions of transshipment, respectively. Fr < Fs and Er < Es because transshipment usually takes the form of short-haul shipment, which incurs less cost and emissions than long-haul transportation. In addition, we assume the cost and

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Table 1 Carbon revenue (or cost) of supply chain members. The supplier N 

Strategy 2

   1 i=1 χ di λ v − api + Es    N  sCs − s χ dx λ i=1 v − ap2i + Es

Strategy 3

sCs − s Nχ dλ(v − ap3c ) + Es

Strategy 1

sCs − s





Retailer i (i = 1, 2, . . . , N) sCr sCr − s(N − 1 )Er (leader) sCr (follower) sNCr − s(N − 1 )Er N

Note: d = min{d1 , d2 , . . . , dN } .

emissions of transshipment are borne by the leader of retailers under strategy 2 or are shared by all retailers under strategy 3. Two scenarios where retailers are heterogeneous or homogeneous are discussed separately. In the scenario of heterogeneous retailers, we assume the distance from the supplier and the bargaining power vary from retailer to retailer. However, those parameters are the same for different retailers in the scenario of homogeneous retailers. 3.2.5. Carbon cap-and-trade policy Under the carbon cap-and-trade policy, firms are initially allocated with carbon emission permits referred to as their carbon caps. Each retailer’s carbon cap is denoted as Cr , and Cs is the supplier’s carbon cap. Following Yenipazarli (2016) and Luo et al. (2016), we assume supply chain members are free to trade in the carbon market at a certain price. The trading (buying or selling) price of carbon emission permits is denoted as s and referred to as the carbon price hereinafter. Whether carbon trading brings revenue or cost depends on the actual emissions of supply chain members. The carbon revenue (or cost) of supply chain members is summarized in Table 1, where a positive value indicates that carbon revenue has been generated, and a negative value stands for carbon cost. 3.2.6. Other notations Other notations used in the model include the unit production cost c, the wholesale price w, and retailer i’s bargaining power over the supplier αi (0 < αi < 0.5 due to a dominant position of the supplier). In addition, s and ri stand for the profits of the supplier and retailer i, respectively. Note that all the notations used in this model are non-negative.

where the three terms are the sales revenue, the expenses of refrigerated transportation services and the carbon revenue, respectively. On the basis of Hsu et al. (2016), we develop a two-stage optimization model. In the first stage, retailer i optimizes its retail price based on the subsequent bargaining outcome. By backward induction, retailer i negotiates the logistics price with the supplier in the second stage. To model the negotiation process, we use the generalized Nash bargaining (GNB) framework (Nagarajan & Bassok, 2008). If the negotiation between the supplier and retailer i does not succeed, the supplier cannot gain profits from trading with retailer i, and retailer i’s profit is zero. They are denoted as disagreement points in the bargaining framework. In other words, the GNB model is formulated as max (1ri − 0 )αi (1s i∈L − 1s i∈/ L )1−αi (L is the set of retailers). Thus, the optimization problem under strategy 1 is formulated as follows:



max 1ri k1i , p1i

 

s.t. k1i p1i = arg max

p1i =

v

+

2a

cλ (ct + sχ )λdi + . 2T 2T

p1i > 0 is assumed according to the non-negativity of all parameters, and p1i < av holds true if parameters satisfy the condition that cλ T

k1i

( c + sχ ) λ d

i + t T < av . The optimal logistics price set by the supplier for retailer i is

=

aT 2

v

a

  + λTc − 2λTw + (ct +sTχ )λdi av − λTc − (ct +sTχ )λdi + 2sCr

+sCs − s

   χ di λ v − ap1i + Es ,

v

a

  adi λ av − λTc − (ct +sTχ )λdi

2 − λTc − (ct +sTχ )λdi + 2sCr − 2(sEs + Fs )   adi λ av − λTc − (ct +sTχ )λdi

αi ,

(5)

where parameters satisfy the condition that k1i > 0.

 = 1 s

N  i=1



aT (1 − αi ) 4



v a

(1)

i=1

which is the sum of the revenue from supplying fresh food, the revenue from providing refrigerated transportation services and the carbon revenue (or cost). Retailer i’s profit is given as follows:

       1ri = p1i T v − ap1i − wλ v − ap1i − k1i di λ v − ap1i + sCr , (2)

 = αi 1 ri

−

λc T

−

(ct + sχ )λdi

2

T

+ sCr − (sEs + Fs )

i=1 N  

aT 2

−

N         k1i − ct di λ v − ap1i − Fs (w − c )λ v − ap1i +

i=1

(4)

Then, the supplier’s profit and retailer i’s profit are derived as

We first study a benchmark case in which retailers separately make replenishment orders. In the replenishment of retailer i, only the supplier and retailer i are involved. The supplier’s profit is expressed by Eq. (1): N 

 α   1−αi 1ri k1i , p1i i 1si k1i , p1i , (3)

Proposition 1. The optimal retail price of retailer i is

4.1. Separate replenishment of retailers

1s =



where 1si (k1i , p1i ) = 1s i∈L −1s i∈/ L = (w −c )λ(v −ap1i ) + (k1i − ct )di λ (v − ap1i ) − Fs − sχ di λ(v − ap1i ) − sEs . To solve this problem, we first derive the optimal response function of logistics price to retail price by solving the GNB model, and then retailer i determines its retail price by solving its profit maximization problem. The optimal solutions are presented in Proposition 1, and the detailed solution procedure is in the Appendix.

4. Replenishment strategies In this section, we will derive the optimal logistics price and retail price under different replenishment strategies. The superscripts 1, 2 and 3 stand for strategies 1-3, respectively.



aT 4



v a

−

λc T

−

+ sCs ,

(ct + sχ )λdi T

(6)

2

+ sCr − (sEs + Fs ) . (7)

If retailers are homogenous, they have equal bargaining power, and their distances from the supplier are the same. With αi = α and di = d for i = 1, 2, . . . , N, the optimal decisions and profits in the scenario of homogeneous retailers will be derived based on the above results, which are omitted here for brevity.

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565

4.2. Joint replenishment: a leader-follower relationship among retailers

Proposition 2. The optimal retail prices of the leader and the followers are:

We assume the supplier is allowed to select a leader among retailers as its distributor. The leader of the retailers monopolizes the replenishment channel and makes joint replenishment for all retailers (Zhang et al., 2017). In other words, all food is transported to the leader, after which the followers are replenished through transshipment at a wholesale price equal to the leader’s retail price. Subscripts x and i (i = x) stand for the leader and the followers of the retailers, respectively.

p2x =

4.2.1. Heterogeneous retailers Similar to Section 4.1, we express the profits of supply chain

v

(N+1 )aT 2

k2x =

+ cTλ −

a

v

(N+1 )aT 2

−

(N + 1)aλdx   (ct +sχ )λdx 2

v

 v − ap2i +

 



k2x − ct dx λ

i=1



χ dx λ

+sCs − s

3v cλ (ct + sχ )λdx + + , 4a 4T 4T

(12)

for

i = 1, 2, ..., N and i = x, (13)

where p2x > 0 and p2i, i=x > 0 are assumed according to the nonnegativity of all parameters. In addition, p2x < av and p2i, i=x < av hold true if parameters satisfy the condition that cTλ + (ct +sTχ )λdx < av . The optimal logistics price is:

+ 4 sCr + sCs − (N − 1 )(sEr + Fr ) − (sEs + Fs ) − 1s − 1rx

T

members in Eqs. (8)–(10): N  

cλ (ct + sχ )λdx + , 2T 2T

N 



 − cTλ − (ct +sTχ )λdx

a

(N + 1 )aλdx

2s = (w − c )λ

+

    + (ct +sTχ )λdx av − cTλ − (ct +sTχ )λdx + 4 sCr − (N − 1 )(sEr + Fr ) − 1rx

2 wλ T

− cTλ −

a

p2i =

v 2a

 2

v

a

 − cTλ − (ct +sTχ )λdx

N  

 v − ap2i − Fs



v − api + Es ,

(8)

,

(14)

αx

where parameters satisfy the condition that ensures the nonnegativity of logistics price. Then, the profits of supply chain members are obtained as



i=1





2s = (1 − αx )

 (N + 1)aT v 8

a

−

cλ (ct + sχ )λdx − T T

i=1

 = 2 rx

N 

p2x T



v−

ap2i



− wλ

N 

i=1



v−

i=1



− k2x dx λ

N  

ap2i

+ sCr − (N − 1 )(sEr + Fr ) − (sEs + Fs )



 =

p2i

−

p2x

 

(15)

v − api + (N − 1 )Fr + sCr − s(N − 1 )Er ,

T

v−

ap2i



+ sCr



for

 = αx 2 rx

i = 1, 2, ..., N and i = x.

 (N + 1 )aT v 8

a

−

cλ (ct + sχ )λdx − T T

In the first stage, the leader of the retailers optimizes its retail price based on the subsequent bargaining outcome and the reactions of the other retailers. In the second stage, the leader negotiates the logistics price with the supplier, and meanwhile, the followers optimize their retail prices. If the negotiation between the supplier and the leader of the retailers does not succeed, all retailers have to make separate replenishment, meaning that the profits of supply chain members under strategy 1 are regarded as their disagreement points. In other words, the GNB model is formulated as max (2rx (k2x , p2x , p2i, i=x ) − 1rx )αx (2s (k2x , p2x , p2i, i=x ) − 1s )1−αx . Thus, the optimization problem under strategy 2 is formulated as follows:











  αx 2rx k2x , p2x , p2i, i=x − 1rx    1−αx × 2s k2x , p2x , p2i, i=x − 1s .      p2i, i=x k2x , p2x = arg max 2ri, i=x k2x , p2x , p2i, i=x (11)

s.t. k2x p2x , p2i, i=x = arg max

Using backward induction, we solve the optimization problem and obtain the optimal decisions, which are shown in Proposition 2. The proof of Proposition 2 is in the Appendix.

2

+sCr − (N − 1 )(sEr + Fr ) − (sEs + Fs ) −

N 

1si ,

(16)

i=x

(10)

max 2rx k2x , p2x , p2i, i=x

1si + sCs ,



 2

(9)



N  i=x

i=1

2 ri

+ αx

2



2ri =

cλ aT v (ct + sχ )λdx − − 16 a T T

2 + sCr ,

for

i = 1, 2, . . . , N and i = x.

(17)

Lemma 1. In the scenario of heterogeneous retailers, the leader makes more profits than the followers if the condi1 vT −aλc 4 tions that αx > αx = 2(N+1 ) and dx < dx = (ct +sχ )aλ − (ct +sχ )λ  T [αx (N−1 )(sEr +Fr )+αx (sEs +Fs )+αx

N

[2(N+1 )αx −1]a

1 i=x si + (1−αx )sCr ]

hold true.

The proof of Lemma 1 is in the Appendix. It is manifest that if dx ≤ 0, none of retailers is willing to become the leader, meaning that strategy 2 cannot be implemented. As a result, we assume parameters satisfy the condition that ensures dx > 0 to make the bound of dx feasible. Then, we infer from Lemma 1 that the leader of the retailers should have strong bargaining power and should not be too distant from the supplier to gain profit advantage over followers. Based on the conditions in Lemma 1, we derive Corollary 1, which illustrates the influences of the distance between the leader and the supplier. Its proof is given in the Appendix.

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Corollary 1. The supplier’s profit and the leader’s profit advantage over the followers increase with the decrease in the distance between the leader and the supplier.



cλ aT v (ct + sχ )λd  = − − 16 a T T

Corollary 1 indicates that the supplier will choose the nearest retailer as the leader of the retailers. Meanwhile, a shorter distance from the supplier can yield the leader more profits than the

(N−3 )A p 2(N−1 )

α>ξ =

(N−2 )(B p −D p )

−

+ Cp +

N−1



(3−N )A p 2(N−1 )

N−2 )(B p −D p ) + ( N−1 − Cp

4.2.2. Homogeneous retailers If retailers are homogenous, they have equal bargaining power and their distances from the supplier are the same. With αi = α and di = d for i = 1, 2, . . . , N, the optimal decisions and profits in the scenario of homogeneous retailers are derived based on Proposition 2. Proposition 3. The optimal retail prices of the leader and the followers are,

p2x =

p2i =

v 2a

+

cλ (ct + sχ )λd + , 2T 2T

(18)

3v cλ (ct + sχ )λd + + , 4a 4T 4T

for i = 1, 2, . . . , N and i = x, (19)

where > 0 and > 0 are assumed. In addition, < av and v 2 pi, i=x < a hold true if parameters satisfy the condition that cTλ + (ct +sχ )λd < v. p2x

p2i, i=x

p2x

2

A +D −B p )(4D p +A p ) + ( p p N−1

in which A p = and D p = sCr .

(N+1 )aT 2

v

a

+ cTλ −

2 wλ T

 (2α −1)(N+1 )aT v 2

−

a

−

 (N + 1)adλ av −  (ct +sχ )λd 2 cλ T

−

cλ T



(N + 1)aT v 8

 (N + 1 )adλ av −

cλ T

 = 2 s

a

−

cλ (ct + sχ )λd − T T

2

+ sCr + (N − 1 ) − (N − 1 )(sEr + Fr ) − (sEs + Fs )

+ sCs , (21)

 =α

 (N + 1)aT v 8

a

−

 (2N + 1)aT v 16

cλ (ct + sχ )λd − − a T T

2

+sCs − (N − 1 )(sEr + Fr ) − (sEs + Fs ),

 − (ct +sTχ )λd



[ av − cTλ − (ct +sTχ )λd ]2 , B p = sEs + Fs , C p = sEr + Fr

+ 4[1 − (N + 1 )(1 − α )][sCr − (sEs + Fs )] − 4(N − 1 )(sEr + Fr )

T

1 ri

2 rx

(24)

 − (ct +sTχ )λd

Based on Proposition 3, we derive the profits of supply chain members as follows:

2s = (1 − α )

aT 4

,

(25)

  + (ct +sTχ )λd av − cTλ − (ct +sTχ )λd + 4sCr − 4(N − 1 )(sEr + Fr )

where parameters satisfy the condition that ensures the nonnegativity of logistics price.



(23)

The proof of Lemma 2 is in the Appendix. We infer from Lemma 2 that: (1) if ξ ≤ 0, it is advantageous for retailers to be the leader, no matter how dominant the supplier is; (2) if 0 < ξ < 0.5, the leader makes more profits than the followers in the case that each retailer’s bargaining power exceeds ξ ; and (3) if ξ ≥ 0.5, strategy 2 is impossible to be adopted because none of retailers wants to become the leader. Assuming parameters in Eq. (24) satisfy the condition that ξ < 0.5, we will discuss the strategic actions of these homogeneous retailers. In this case, the advantage of being the leader motivates retailers to compete for the leader position. Since the leader is chosen by the supplier, paying the supplier a negotiation fee is an alternative method for retailers to obtain the leadership. As a result, the equilibrium state is achieved when all retailers gain equal profits, that is to say, the negotiation fee is equal to the leader’s profit advantage over each follower (Nagarajan & Bassok, 2008). On this basis, we derive the profits of supply chain members as follows:

The optimal logistics price is,

k2x =

for

Lemma 2. The leader gains more profits than the followers if the bargaining power of each retailer exceeds a threshold:

a

T

+ sCr ,

i = 1, 2, . . . , N and i = x.

2 (A p + D p − B p )

followers, which makes the nearest retailer volunteer to be chosen as the leader.

2

2 ri

cλ (ct + sχ )λd − T T

2

− (1 − α )(N − 1 )1ri , (22)

α

2 + sCr .

(26)

4.3. Joint replenishment: the coalition of retailers Forming a coalition is another approach for retailers to make joint replenishment. Based on the assumption that the bargaining power of a coalition increases with its size (Nagarajan & Bassok, 2008), we define the bargaining power of the coalition of retailers b−1 1  b , which is αc (α ) = N b α in the scenario of as αc (αi ) = N i=1 αi /N homogeneous retailers. b is a parameter greater than zero, which guarantees that αc (α ) > α > 0 and ∂ α∂cN(α ) > 0. This means that retailers can gain stronger bargaining power by forming a coalition and the bargaining power of the coalition increases with its size. In addition, knowing that α < 0.5 (as mentioned in Section 3.2.6), we 1

+ sCr − (N − 1 )(sEr + Fr ) − (sEs + Fs )



cλ aT v (ct + sχ )λd  = − − 16 a T T 2 ri

(20)

derive the condition that N b < 2 which links N and b to guarantee that αc (α ) < 1. This condition shows that if b is deterministic, the coalition of retailers cannot form in a supply chain with the number of retailers N exceeding 2b , or if Nmax denoting the maximal number of retailers is given, the coalition cannot form with N the coefficient b lower than log2 max .

M. Wang, L. Zhao and M. Herty / European Journal of Operational Research 277 (2019) 561–573

4.3.1. Heterogeneous retailers The profits of the supplier and the coalition are as follows:

         = N (w − c )λ v − ap3c + Nd k3c − ct λ v − ap3c − Fs     +sCs − s N dχ λ v − ap3c + Es ,

 = 3 rc

(27)

 3

(28) If the negotiation between the supplier and the coalition does not succeed, all retailers have to make separate replenishment, meaning that their profits under strategy 1 are regarded as the disagreement points. In other words, the GNB model is forb−1 N  b 1 i=1 αi /N mulated as max (3rc (k3c , p3c ) − N (3s (k3c , p3c ) − i=1 ri ) N

i=1 αi /N

b−1 b

. Therefore, the optimization problem is formu-

lated as follows:

max 

3 rc



k3c ,

p3c



s.t. kc pc = arg max



× 

3 s

⎧  ⎪ ⎨ ⎪ ⎩



k3c ,

N    3rc k3c , p3c − 1ri

Ni=1 αi /N b−1b

i=1

p3c





1 1− s

−

N

i=1

αi /N

b−1 b

N

 αi NaT v

b−1 b



a

cλ (ct + sχ )λd − T T

2

2

+(N − 1 )(sEs + Fs − sEr − Fr )}.

(29)

To solve this problem, we first derive the optimal response function of logistics price to retail price by solving the GNB model, and then the coalition determines the retail price by solving its profit maximization problem. The optimal solutions are given by Proposition 4, and its proof is in the Appendix.

(33)

Based on the above results, we will analyze whether retailers are willing to form a coalition. As defined in Section 4.1, L is the set of retailers. For each subset H of L, v(H ) represents the worth of H. If a coalition H forms, it can divide its worth v(H ) in any possible way among its members. In other words, H can achieve every  payoff vector xi (i ∈ H), which satisfies i∈H xi ≤ v(H ). In addition, Definition 1 is needed. Definition 1. A game is weakly superadditive if v(H ∪ {g}) > v(H ) + v({g} ) for all H ⊆ L, g ∈ L and g ∈/ H. Lemma 3. The coalitional game of heterogeneous retailers is not superadditive. Therefore, they will not form a coalition.

3rc,H∪{g} − 3rc,H − 1rg > 0, ∀H ⊆ L, g ∈

condition

L, g ∈ / H is essential to guarantee that the coalitional game is weakly superadditive. Let |H | = h. We obtain that

3rc,H∪{g} − 3rc,H − 1rg

 2 h+1 1 (h + 1)aT v cλ (ct + sχ )λ min {di |i ∈ H ∪ g } i=1 αi b = (h + 1 ) − − h+1

,

4

−

N aT  v λc (ct + sχ )λdi − − 4 a T T

Proof. The



 3 3

4

a



h+1 aT  v cλ (ct + sχ )λdi − − − 4 a T T i=1

1

h

−h b

i=1

h

αi

haT 4



v a

−

T

T

2

+ h(sEs + Fs − sEr − Fr )

cλ (ct + sχ )λ min {di |i ∈ H } − T T



h aT  v cλ (ct + sχ )λdi − − − 4 a T T

2

2

+ (h − 1 )(sEs + Fs − sEr − Fr ) .

i=1

Proposition 4. The optimal retail price is

p3c

 +

i=1

i=1

  3rc = N p3c T v − apc − Nwλ v − ap3c     − N dk3c λ v − ap3c + (N − 1 )Fr + s[NCr − (N − 1 )Er ].

1s )1−

−

N

1 ri

i=1

3 s



N 

567

(34)

cλ (ct + sχ )λd = + + , 2a 2T 2T

v

(30)

We infer from Eq. (34) that 3rc,H∪{g} − 3rc,H − 1rg > 0 may not h+1

p3c

where > 0 is assumed according to the non-negativity of all parameters, and p3c < av holds true if parameters satisfy the condition

α

h

α

i i hold if hi=1 < i=1 . In the scenario of heterogeneous retailers, +1 h the condition that retailer g has stronger bargaining power than all

that cTλ + (ct +sTχ )λd < av . The optimal logistics price is

k3c = −

NaT 2

NaT 2

v

a

v

a

     + cTλ − 2λTw + (ct +sTχ )λd av − cTλ − (ct +sTχ )λd + 2 NsCr − (N − 1 )(sEr + Fr ) − Ni=1 1ri   Naλd av − cTλ − (ct +sTχ )λd

, 2    − cTλ − (ct +sTχ )λd + 2 NsCr + sCs − (N − 1 )(sEr + Fr ) − (sEs + Fs ) − Ni=1 1ri − 1s N α i=1 i   b−1 N b Naλd av − cTλ − (ct +sTχ )λd

where parameters satisfy the condition that ensures the nonnegativity of logistics price. Then, the profits of the supplier and the coalition are obtained

(31)

retailers in the subset H never holds in all possible cases. Hence, we conclude that the coalitional game is not superadditive. 

as

 3s =

N 1−

α

i=1 i b−1 N b



NaT 4



v a

−

cλ (ct + sχ )λd − T T

+sCs − (N − 1 )(sEr + Fr ) − (sEs + Fs ) −

N  i=1

2 + NsCr



1 ri

N +

α

i=1 i b−1 N b

1s ,

(32)

4.3.2. Homogeneous retailers With αi = α and di = d for i = 1, 2, ...N and following the same solution procedure, we obtain the optimal solutions based on Proposition 4.

568

M. Wang, L. Zhao and M. Herty / European Journal of Operational Research 277 (2019) 561–573

Proposition 5. The optimal retail price is

p3c =

v 2a

+

so that each retailer gains the same profit. Then, the profit of each retailer is obtained as

cλ (ct + sχ )λd + , 2T 2T

(35)

where p3c > 0 is assumed, and p3c < av holds true if parameters satisfy the condition that cTλ + (ct +sTχ )λd < av . The optimal logistics price is

k3c =

aT 2

v

−

a

+ cTλ −

aT 2

v

−

a

2 wλ T

cλ T

−

   + (ct +sTχ )λd av − cTλ − (ct +sTχ )λd + 2 sCr −

 aλd av −   (ct +sχ )λd 2

cλ T

−



 (ct +sχ )λd T

1

1



T





cλ (ct +sχ )λd − − a T T

v

1

+sCr

+ sCs + (Nα

aT 4

 = Nα



v a

−

cλ (ct + sχ )λd − T T

2 + sCr − sEs − Fs

+N

1 −1 b

(N − 1 )(sEs + Fs − sEr − Fr ) .

(38)

Based on the above results, we will analyze whether retailers are willing to form a coalition. The findings are presented in Lemma 4. Lemma 4. The coalitional game of homogenous retailers is weakly superadditive. Therefore, it is essential for retailers to form a coalition. Proof. It is derived that

  1 1 3rc,H∪{g} − 3rc,H − 1rg = (h + 1) b h − h b (h − 1 ) (sEs + Fs − sEr − Fr )α . (39) Since Es > Er and Fs > Fr (as mentioned in Section 3.2.4), we infer that 3rc,H∪{g} − 3rc,H − 1rg > 0 is guaranteed. In addition, we derive that

aT 4

 = Nα 3 rc



cλ (ct + sχ )λd − − a T T

v

2 + sCr − sEs − Fs

+N

1 −1 b

> Nα

aT 4

(N − 1 )(sEs + Fs − sEr − Fr )



v cλ (ct + sχ )λd a

−



3ri = 3rc N

=

α

aT 4



v a

−

(36)

α,

cλ (ct + sχ )λd − T T

2 + sCr − sEs − Fs

+N

1 −1 b

(N − 1 )(sEs + Fs − sEr − Fr ) .

(41)

1

(37)

3 rc



1

2

−1 + N b α − N 1+ b α )(sEs + Fs ) − (1 − N b α )(N − 1 )(sEr + Fr ), 1

1

  aλd av − cTλ − (ct +sTχ )λd

Based on Proposition 5, we obtain the profits of the supplier and the coalition as follows:

aT 3s = N (1 − α ) 4

 (sEr + Fr )

+ 2 sCr − N b − N b −1 (sEr + Fr ) − 1 − N b + N b −1 (sEs + Fs )

where parameters satisfy the condition that ensures the nonnegativity of logistics price.



N−1 N

T

−

T

2 + sCr −sEs −Fs

= N 1ri , (40)

which means that the coalition formation of retailers generates cooperation surplus.  According to Lemma 4, retailers will form a coalition if they are homogeneous. Due to their homogeneity, the cooperation surplus generated from coalition formation will be equally allocated

We conclude that retailers will not form a coalition if they are heterogeneous, while if they are homogeneous, a coalition will form and all retailers earn the same profit. Consequently, it is inferred that heterogeneous retailers have strategies 1 and 2 to choose from, and all replenishment strategies are available for homogeneous retailers. 4.4. The analysis of optimal decisions under different replenishment strategies In this subsection, we will analyze the influences of selected parameters on decision variables. In addition, decision variables will be compared among different replenishment strategies. 4.4.1. Retail price The influences of parameters on retail price are analytically analyzed, as shown in Proposition 6. Note that parameters about the carbon cap-and-trade policy are omitted here because they will be separately analyzed in Section 6 to reveal the role of the carbon cap-and-trade policy. ∂ p1i ∂ p1i ∂ p1i ∂ p2x ∂ p2x ∂ di > 0, ∂ αi = 0 and ∂ u > 0; ∂ dx > 0, ∂ αx = 0 ∂ p2 ∂ p2 ∂ p2 ∂ p3 ∂ p3 ∂ p2 and ∂ ux > 0; ∂i,di=x > 0, ∂i,αi=x = 0 and ∂i,ui=x > 0; ∂ dc > 0, ∂ αc = x x i ∂ p3 0 and ∂ uc > 0.

Proposition 6.

The proof of Proposition 6 is in the Appendix. We see from Proposition 6 that retailers should raise retail prices if their distances from the supplier become longer or food deteriorates at a faster rate. This is because either longer distance or a faster deterioration rate incurs higher cost, which drives retailers to increase retail prices. Proposition 7 is obtained from the comparison of retail prices under different replenishment strategies. Proposition 7. For heterogeneous retailers, p2i, i=x > p2x and p1i ≥ p2x .

In addition, p2i, i=x > p2x = p1i = p3c in the scenario of homogeneous retailers. The proof of Proposition 7 is in the Appendix. Retail pricing strategies are concluded from Proposition 7. The retail price of the leader is lower than that of the followers under strategy 2. In other words, the leader of the retailers has an obvious advantage over

M. Wang, L. Zhao and M. Herty / European Journal of Operational Research 277 (2019) 561–573

569

1.8

Logistics price

Logistics price

2 Nearest retailer (strategy 1) Other retailers (strategy 1) Strategy 2

1.6 1.4 16

17

18

19

Strategy 1 Strategy 2 Strategy 3

2.5

2

1.5

20

10

12

Followers' distance from the supplier (a)

16

18

20

2

1.8

Strategy 1 Strategy 2 Strategy 3

1.6

1.4 0

0.01

0.02

0.03

0.04

Logistics price

2

Logistics price

14

Retailers' distance from the supplier (b)

1.8

Strategy 1 Strategy 2 Strategy 3

1.6

1.4 0.45

0.46

Deterioration rate (c)

0.47

0.48

0.49

0.5

Bargaining power of retailers (d)

Fig. 2. Influences of parameters on the logistics price.

the followers in enhancing market demand. Moreover, if retailers are homogeneous, the retail price under strategy 2 is the highest and the pricing decisions under strategies 1 and 3 make no difference. 4.4.2. Logistics price It is challenging to analytically analyze logistics prices due to their complex expressions. Hence, we conduct numerical experiments to make sensitivity analysis and comparative analysis, as shown in Fig. 2. The basic dataset is that v = 10, a = 0.2, u = 0.02, T = 1, c = 3, w = 5, s = 0.04, ct = 1, χ = 0.1, Fs = 5, Es = 10, Fr = 1, Er = 2, b = 10 0 0, N = 5, Cs = 50, Cr = 10 and α = 0.48. Fig. 2(a) depicts the influence of the distance from the supplier in the scenario of heterogeneous retailers, in which retailer 1 is assumed to be the leader and d1 = 15, and the followers’ distance from the supplier d j ( j = 2, 3, ..., N) is varied within the interval of [16, 20]. Figs. 2(b)−2(d) are about the scenario of homogeneous retailers. We vary the distance from the supplier d within the interval of [10, 20], set d = 15 and vary deterioration rate u within the interval of (0, 0.04], and set u = 0.02 and vary the bargaining power α within the interval of [0.45, 0.5) in Figs. 2(b)−2(d), respectively. All these numerical results are checked to ensure that conditions in Propositions 1–5 and Lemmas 1 and 2 hold true so that retail prices are within the interval of (0, av ), logistics prices and profits of supply chain members are non-negative, the distance of the leader of the retailers from the supplier is within its bound, the bargaining power of retailers is within its bound, and the leader of the retailers gains more profits than the followers. It is clear in Fig. 2 that the logistics price decreases with the increase in the distance from the supplier, which matches the reality that the logistics price of long-haul transportation is usually lower than that of short-haul transportation. We also see that the logistics price decreases with the increase in the deterioration rate. A faster deterioration rate results in more food losses and a larger order quantity. Due to the economies of scale of transportation, the logistics price is lowered. In addition, it is shown that the supplier reduces the logistics price if retailers have stronger bargaining power, which reveals the significant role of bargaining power in decreasing the logistics price. Furthermore, recommendations about how to obtain a lower logistics price are provided for retailers. The logistics price under strategy 3 is the lowest, indicating that retailers can obtain a low

logistics price by forming a coalition. In addition, the logistics price under strategy 2 is the highest. Thus, retailers can cooperate to prevent the formation of a leader-follower relationship so that a low logistics price is maintained. 5. Optimal replenishment strategy In this section, we will compare three replenishment strategies to identify the optimal replenishment strategies from different perspectives. 5.1. The perspective of the supplier In the scenario of heterogeneous retailers, strategy 2 is the supplier’s optimal replenishment strategy because its profits under strategy 1 are disagreement points in the bargaining framework. Thus, we will focus on the scenario of homogeneous retailers in the following discussion. Since profits under strategy 1 are disagreement points in the bargaining game in Section 4.2, 2s > 1s holds before the reallocation of profits among the supplier and retailers. As we mentioned in Section 4.2.2, the leader of the retailers pays a negotiation fee to the supplier to obtain the leadership, which makes the supplier’s profit under strategy 2 become higher. As a result, 2s > 1s still holds true. In addition, it is obvious that:

  1 3s − 1s = 1 − N b α (N − 1 )(sEs + Fs − sEr − Fr )

= (1 − αc (α ) )(N − 1 )(sEs + Fs − sEr − Fr ) > 0.

(42)

Assuming 2s = 3s , we derive the following critical threshold based on Eqs. (25) and (37):

ϕ

s 23

 2 (1 − 2N + 4Nα )aT v cλ (ct + sχ )λd = − − 16N (1 − α )s a T T

α (sEs + Fs ) N b (N − 1 )α + (sEs + Fs − sEr − Fr ). N (1 − α )s (1 − α )s 1

−

(43)

s , 3 ≥ 2 , or if C < ϕ s , 3 < 2 . If Cr ≥ ϕ23 r s s s s 23

5.2. The perspective of retailers Similar to the supplier, the leader of the retailers prefers strategy 2 to strategy 1 in the scenario of heterogeneous retailers. Due

570

M. Wang, L. Zhao and M. Herty / European Journal of Operational Research 277 (2019) 561–573 700 Pr1=Pr2 Pr2=Pr3 Ps2=Ps3

600

Each retailer's carbon cap

500

C r=0

Region I

Pr2>Pr1

Region II

Pr1>Pr2

Pr2>Pr3 Pr3>Pr2

400

300

Region III 200

100 Ps3>Ps2

Ps3>Ps1 Ps2>Ps1

0

Pr3>Pr1

Region IV

Ps2>Ps3

Region V 0.03

0.032

0.034

0.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

Carbon price

Fig. 3. The critical thresholds of each retailer’s carbon cap. Table 2 The optimal replenishment strategies of supply chain members and retailers’ optimal reaction. Regions

The profit of the supplier 3 s 3 s 3 s 2 s

Region I Region II Region III Region IV

> > > >

2 s 2 s 2 s 3 s

> > > >

1 s 1 s 1 s 1 s

The supplier’s optimal strategy

The profit of each retailer 2 ri 3 ri 3 ri 3 ri

Strategy 3 Strategy 3 Strategy 3 Strategy 2

Region V

> > >

> > > >

1 ri 1 ri 2 ri 2 ri

Retailers’ optimal strategy

Retailers’ optimal reaction

Strategy 2

May accept

Strategy 3

Accept

Strategy 3

Accept

Strategy 3

Not accept

Infeasible

to the follower position, other retailers have to choose strategy 2. Thus, retailers’ optimal replenishment strategy is strategy 2 in the scenario of heterogeneous retailers. If retailers are homogeneous, their optimal replenishment strategy is sensitive to their carbon cap as analyzed below. Based on Eqs. (7), (26) and (41), we will investigate the relationship among each retailer’s profit under strategies 1-3. First, it is obvious that:

3ri − 1ri = N b −1 (N − 1 )α [s(Es − Er ) + (Fs − Fr )] > 0. 1

(44)

We then derive a critical threshold of each retailer’s carbon cap by assuming 1ri = 2ri :

 2 α (sEs + Fs ) (4α − 1 )aT v cλ (ct + sχ )λd ϕ = − − − . (45) 16(1 − α )s a T T (1 − α )s r 12

r , 2 ≥ 1 , or if C < ϕ r , 2 < 1 . If Cr ≥ ϕ12 r 12 ri ri ri ri Moreover, 2ri ≥ 3ri is interpreted as follows: r Cr ≥ ϕ23

>

3 ri 2 ri 1 ri 1 ri

5.3. The perspective of a carbon emission optimizer

 2 (4α − 1 )aT v cλ (ct + sχ )λd = − − 16(1 − α )s a T T α (sEs + Fs ) N b (N − 1 )α + (sEs + Fs − sEr − Fr ), N (1 − α )s ( 1 − α )s 1

−

supply chain members. Moreover, we provide the optimal reaction strategy for retailers facing an aggressive supplier. Table 2 shows that the supplier chooses strategy 3 and retailers prefer strategy 2 in region I, where retailers have a relatively high carbon cap. In this region, the supplier’s optimal strategy is not the best but not the worst strategy for retailers, meaning that retailers may accept the supplier’s decision. In regions II and III, the supplier’s optimal strategy perfectly matches that of retailers. This implies that retailers are willing to accept the supplier’s decision when they have a medium-high carbon cap. In region IV, we see that the supplier’s optimal replenishment strategy is the worst strategy for retailers, which is not acceptable for retailers. Region V is infeasible because each retailer’s carbon cap is non-negative. In summary, disagreements over the decision on replenishment strategy may arise in a single-supplier multi-retailer supply chain, particularly when each retailer’s carbon cap is very low.

Based on Section 4, we derive the amount of carbon emissions as follows:



Et1

(46) r , 2 < 3 . and if Cr < ϕ23 ri ri In addition to the above analytical analysis, we conduct numerical experiments as supplements. Using the basic dataset in Section 4.4.2 and a carbon price s varying within the interval of [0.03, 0.05], we plot the critical thresholds of each retailer’s carbon cap derived above in Fig. 3. Note that Ps1 , Ps2 and Ps3 are the supplier’s profit under strategies 1-3, and Pr1 , Pr2 and Pr3 stand for each retailer’s profit under strategies 1-3 in Fig. 3, respectively. As we mentioned above, Ps2 > Ps1 , Ps3 > Ps1 and Pr3 > Pr1 always hold. The characteristics of regions divided in Fig. 3 are summarized in Table 2, which reveal the optimal replenishment strategies of



N cλ aλχ  v (ct + sχ )λdi = − − di + N Es , 2 a T T

(47)

i=1

Et2 =

 (N + 1)aλχ v 4

a



−

cλ (ct + sχ )λdx − dx + Es + (N − 1 )Er , T T (48)

Et3

Naλχ = 2





cλ (ct + sχ )λd − − d + Es + (N − 1 )Er . a T T

v

(49)

The comparative results about carbon emissions are shown in Proposition 8.

M. Wang, L. Zhao and M. Herty / European Journal of Operational Research 277 (2019) 561–573

1.43

1.48 1.475 1.47 1.465

Logistics price

1.9

Logistics price

Logistics price

1.485

1.895

1.89

1.885

1.46 0.06

Carbon price

0

0

Each retailer's carbon cap

1.42

0.06 20

0.04

10

0.02

1.425

1.415

0.06 20

0.04

571

10

0.02

Carbon price

(a) Strategy 1

0

0

Each retailer's carbon cap

20

0.04 10

0.02

Carbon price

(b) Strategy 2

0

0

Each retailer's carbon cap

(c) Strategy 3

Fig. 4. The influences of the carbon price and carbon caps on the logistics price.

Proposition 8. In the scenario of heterogeneous retailers, Et1 > Et2 if the condition that s < T ( av − cTλ ) − cχt (d = max{d1 , d2 , ..., dN }) 2λχ d

holds. In the scenario of homogeneous retailers, Et1 > Et3 > Et2 always holds. The proof of Proposition 8 is in the Appendix. Proposition 8 shows that the largest amount of emissions is produced if strategy 1 is adopted. This implies that joint replenishment is an effective approach to control carbon emissions. Referring to supply chain members’ optimal replenishment strategies in Table 2, it is noteworthy that they all prefer joint replenishment strategies to separate replenishment strategy. In other words, the goal of emission reduction can be achieved in such a bargaining framework. Furthermore, we observe that strategy 2 causes the least emissions in both scenarios. Since supply chain members do not agree on which joint replenishment strategy to choose, we suggest that carbon emission optimizers such as governments should take measures to coordinate the supply chain so that strategy 2 is acceptable for all supply chain members. For instance, governments can set a low carbon cap for retailers and meanwhile provide subsidies for retailers choosing strategy 2.

plier’s carbon cap is set to be Cs = 5Cr . The results are plotted in Fig. 4. According to Proposition 9 and Fig. 4, the retail price and logistics price increase if a carbon cap-and-trade policy is implemented. Moreover, both the retail price and logistics price are sensitive to the carbon price, while carbon caps only affect the logistics price. It implies that adjusting the carbon price is more effective than adjusting carbon caps in terms of controlling the retail price and logistics price. 6.2. Profits and carbon emissions Using the dataset in Section 6.1, we plot the profits of supply chain members in Fig. 5. As shown in Fig. 5, the implementation of a carbon cap-andtrade policy gives rise to profit growth in the supply chain, especially when carbon caps are relatively high. This is because supply chain members not only earn profits from fresh food business but also from carbon trading. In addition, the influences of the carbon price and carbon caps on carbon emissions are analytically analyzed in Proposition 10. The proof of Proposition 10 is in the Appendix. ∂ Et1 ∂ Et1 ∂ Et1 ∂ Et2 ∂ Et2 ∂ s < 0, ∂ Cr = 0 and ∂ Cs = 0; ∂ s < 0, ∂ Cr = 0 ∂ E3 ∂ E3 ∂ E3 ∂ E2 and ∂ Ct = 0; ∂ st < 0, ∂ Ct = 0 and ∂ Ct = 0. s r s

Proposition 10. 6. The role of the carbon cap-and-trade policy In this section, the case without a carbon cap-and-trade policy is introduced through equating the carbon price and carbon caps at zero. Then, we will identify the role of the carbon cap-and-trade policy by comparing the cases with and without a carbon cap-andtrade policy. For simplicity, we will focus on the scenario of homogeneous retailers. 6.1. Decision variables Based on Eqs. (4), (18), (19) and (35), the influences of the carbon price and carbon caps on the retail price are analyzed in Proposition 9. The proof of Proposition 9 is in the Appendix. ∂ p1i ∂ p1i ∂ p1i ∂ p2x ∂ s > 0, ∂ Cr = 0 and ∂ Cs = 0; ∂ s > 0, ∂ p2 ∂ p2 ∂ p2 ∂ p3 ∂ p2 and ∂ Cx = 0; ∂i,si=x > 0, ∂i,Ci=x = 0 and ∂i,Ci=x = 0; ∂ sc > 0, s r s 3 ∂p and ∂ Cc = 0. s

Proposition 9.

∂ p2x ∂ Cr = 0 ∂ p3c ∂ Cr = 0

With regard to the logistics price, we perform numerical experiments using the basic dataset in Section 4.4.2, except that the carbon price s is varied within the interval of [0, 0.05], each retailer’s carbon cap Cr is varied within the interval of [0, 20], and the sup-

Proposition 10 implies that the implementation of a carbon capand-trade policy leads to the reduction in carbon emissions, which verifies the effectiveness of this policy. In particular, it is the carbon price that takes effect in emission reduction, which highlights the importance of carbon pricing. We conclude that the goals of profit growth and carbon emission reduction are simultaneously achieved under a carbon capand-trade policy. With such advantages, a carbon cap-and-trade policy is strongly recommended to be adopted by governments. 6.3. The decision on the replenishment strategy Furthermore, we are curious about whether the implementation of a carbon cap-and-trade policy affects the decision on the replenishment strategy. We start from the perspectives of supply chain members. With the increase in the carbon price or each retailer’s carbon cap, strategy 3 is more possible to be the optimal replenishment strategy for the supplier, while retailers are more likely to choose strategy 2 (see Fig. 3). Moreover, it is observed from Fig. 3 and Table 2 that the possibility of "not accept" decreases with the increase in the carbon price or each retailer’s carbon cap. It indicates that a higher

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Fig. 5. The influences of the carbon price and carbon caps on profits.

carbon price or a higher carbon cap of each retailer contributes to the coordination of the supply chain. According to Proposition 8, the implementation of a carbon cap-and-trade policy does not structurally change the optimal replenishment strategy from the perspective of a carbon emission optimizer. That is, strategy 2 is the most environment-friendly replenishment strategy regardless of whether a carbon cap-and-trade policy is adopted.

7. Conclusions In this study, we attempt to investigate retailers’ joint replenishment as well as the carbon trading behavior of supply chain members. We assume that three replenishment strategies are available for the supply chain: (1) separate replenishment; (2) joint replenishment: a leader-follower relationship among retailers; and (3) joint replenishment: the coalition of retailers. Under each strategy, the price of the refrigerated transportation services provided by the supplier and the retail price of fresh food are optimized and analytically analyzed, which provide insights into the logistics pricing and retail pricing strategies. On this basis, the optimal replenishment strategies from the perspectives of the supplier, retailers and a carbon emission optimizer are identified. We observe that disagreements over the decision on the replenishment strategy exist in the supply chain, particularly when each retailer’s carbon cap is extremely low. Moreover, it is found that the joint replenishment strategy with a leader-follower relationship among retailers causes the least carbon emissions. Thus, we suggest that governments should set a low carbon cap for retailers and adopt a subsidy policy to coordinate the supply chain so that the joint replenishment strategy with a leader-follower relationship among retailers is acceptable for all supply chain members. We further investigate the role of the carbon cap-and-trade policy through comparing the cases with and without a carbon capand-trade policy. We conclude that the goals of profit growth and carbon emission reduction are simultaneously achieved under a carbon cap-and-trade policy. With such advantages, a carbon capand-trade policy is strongly recommended to be adopted by governments. Moreover, it is proven that the implementation of a carbon cap-and-trade policy does not structurally change the optimal replenishment strategy.

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