Effects of carbon emission reduction on supply chain coordination with vendor-managed deteriorating product inventory

Accepted Manuscript Effects of carbon emission reduction on supply chain coordination with vendormanaged deteriorating product inventory Qingguo Bai, Yeming (Yale) Gong, Mingzhou Jin, Xianhao Xu PII:

S0925-5273(18)30447-X

DOI:

https://doi.org/10.1016/j.ijpe.2018.11.008

Reference:

PROECO 7209

To appear in:

International Journal of Production Economics

Received Date: 20 September 2017 Revised Date:

2 November 2018

Accepted Date: 10 November 2018

Please cite this article as: Bai, Q., Gong, Y.(Y.), Jin, M., Xu, X., Effects of carbon emission reduction on supply chain coordination with vendor-managed deteriorating product inventory, International Journal of Production Economics (2018), doi: https://doi.org/10.1016/j.ijpe.2018.11.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Title: Effects of carbon emission reduction on vendor-managed deteriorating product inventory Qingguo Baia,b, Yeming (Yale) Gongb, Mingzhou Jinc,d *, Xianhao Xue a School of Management, Qufu Normal University, Rizhao, Shandong, 276826, China; b EMLYON Business School, 23 Avenue Guy de Collongue, Ecully Cedex 69134, France c Logistics and Transportation College, Central-South University of Forestry and Technology, China d Department of Industrial and Systems Engineering, University of Tennessee, Knoxville, TN, USA e School of Management, Huazhong University of Science and Technology, Wuhan, 430074, China;

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Authors: Author affiliations:

Corresponding Author:

Prof. Mingzhou Jin

Address:

Logistics and Transportation College, Central-South University of Forestry and Technology, China Department of Industrial and Systems Engineering, University of Tennessee, Knoxville, TN, USA [email protected]

E-mail:

+865-974-9992

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Effects of carbon emission reduction on supply chain coordination with vendor-managed deteriorating product inventory

Abstract

While many supply chains use vendor-managed inventory (VMI) to handle deteriorating products, they

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employ special facilities (e.g., cold warehouses), which produce significant carbon emissions. This new

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operational practice motivates us to investigate the effects of carbon emission reduction on a supply chain with one manufacturer and two competing retailers for deteriorating products under VMI, which has not been fully studied in the existing literature. Carbon cap-and-trade regulation and investment in green technologies are used to curb carbon emissions generated by production and inventory holding in this system. To bench mark the performance of the decentralized system, an optimization model for the centralized sys-

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tem is formulated to show that there exists an upper bound on the profit penalty for decentralization and the carbon emissions of the centralized system may be lower than those of the decentralized system.

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revenue-sharing contract is then proposed to improve the profit and emissions of the decentralized system.

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The results show that the supply chain can be coordinated perfectly when the demand depends on the man-

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ufacturer’s green technology level and two competing retailers’ selling prices. Numerical examples with sensitivity analysis are further provided to test the robustness of the supply chain’s operational decisions. Keywords: Supply chain management, carbon emissions, green technology, deteriorating product, VMI

1. Introduction

Managing the supply chain for deteriorating products is challenging and risky because the usefulness of the deteriorating products will decrease during storage and transportation through spoilage, damage, or decay. About 10% of fresh produce and other food products spoil before these deteriorating products are Preprint submitted to Elsevier

November 19, 2018

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sold to consumers (Ketzenberg et al., 2015). The National Supermarket Research Group (2006) also shows that although 39% of total store sales is accounted for deteriorating products, an average of 64% of total

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store shrink is contributed by them. Vendor-managed inventory (VMI) as an effective inventory management strategy is adopted to improve the performance of the supply chain for deteriorating products. Under the VMI system, the vendor (or manufacturer) manages the inventory of its own and its downstream retailer. For example, Barilla from Italy and Kraft Foods from the United States, two leading food manufacturers,

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implement VMI to provide their retailers and customers with foods as fresh as possible (Yu et al., 2012). All

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members of the supply chain benefit from the implementation of VMI since the retailer’s inventory cost is reduced and the manufacturer can adopt the flexible delivery strategy. However, some new challenges have also been brought forth in the supply chain management with deteriorating products. Lekkerland, a German convenience store chain selling vendor-managed deteriorating products such as confectionery and soft drinks, ice cream, frozen foods, fresh bakery products, and hot drinks, is using a distribution strategy called

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multitemperature logistics with special multitemperature and multichamber facilities producing more carbon emissions than standard warehousing and logistics systems with no need for handling vendor-managed deteriorating products, which raises an industrial interest to study the effects of carbon emission reduction

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on vendor-managed deteriorating product inventory.

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To reduce and mitigate carbon emissions, the carbon cap-and-trade regulation as a market-based approach has been adopted by some national and international authorities. Under this regulation, a government agency allocates a predetermined amount of carbon emissions (carbon cap) to a firm, and the firm can sell or buy its surplus or extra carbon emission permits in a carbon trading market such as the European Union Emissions Trading System (EU ETS). On the other hand, a growing number of consumers have strong motivation to purchase eco-friendly products with the rise of environmental awareness (Agatz et al., 2012; Thφgersen et al., 2012; Stiglic et al., 2015). Under pressures from the government and customers, 2

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many responsible firms invest in cleaner technologies to reduce carbon emissions under the cap-and-trade regulation (Drake et al., 2016). In this scenario, when the carbon cap-and-trade regulation is incorporated

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into a two-echelon supply chain with deteriorating products under the VMI system, the decision maker should consider the following key questions: (i) A revenue-sharing (RS) contract is often used to coordinate the supply chain systems with deteriorating products (e.g., Xiao and Xu, 2013). Could it coordinate the supply chain under the constraint of carbon emissions? (ii) When the RS contract is carried out, how much

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does the profit of the decentralized system increase ? Can it lead to reduced carbon emissions ? (iii) What

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are the effects of investing in green technologies on supply chain coordination?

This paper considers a supply chain with one manufacturer and two competing retailers for deteriorating products over an infinite time horizon. VMI is considered in this system, and the manufacturer makes his or her operational decision under the constraint of carbon cap-and-trade regulation because production and inventory holding are two main sources of carbon emissions. The manufacturer also invests in green tech-

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nologies to curb carbon emissions, and the market demand of each retailer depends on the green technology level and retailers’ selling prices. We first formulate centralized and decentralized supply chain models and compare these two models for their operational decisions. We then propose a RS contract to coordinate

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the supply chain. Finally, we use a numerical example with a robust sensitivity analysis to illustrate the

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theoretical results and obtain management insights. The contributions of this paper are as follows: First, we incorporate the cap-and-trade regulation and carbon emission reduction technologies into a two-echelon supply chain for deteriorating products under VMI. Second, we obtain that there exists an upper bound on the profit penalty for decentralization and the carbon emissions of the centralized system may be lower than those of the decentralized system. We also analyze the impacts of the RS contract on the profits of the decentralized system’s members and carbon emissions. Third, we prove that the RS contract can lead to perfect coordination when the market demand 3

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of each retailer depends on the green technologies of the manufacturer and both retailers’ selling prices. We further analyze the conditions under which these three members of the supply chain will accept the

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revenue-sharing contract. The remainder of this paper is organized as follows: In Section 2, we review the related literature. In Section 3, we describe the problem in details and introduce notations used in modeling the problem. In Section 4, we formulate and compare two mathematical models for the decentralized and centralized

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systems, and further propose the revenue-sharing contract to coordinate a decentralized supply chain. In

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Section 5, we use a numerical example with sensitivity analysis to illustrate the models and the theoretical results. In Section 6, we provide our conclusions.

2. Literature review

Our paper is related to the following three streams of research: Supply chain models under VMI, supply

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chain coordination for deteriorating products, and supply chain models under carbon emission regulations.

2.1. Supply chain models under vendor-managed inventory

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VMI is an effective strategy to improve the inventory management performance of a supply chain (Bernstein et al., 2006; Govindan, 2015). Various supply chain models under VMI have been extensively ana-

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lyzed. For example, Yu and Huang (2010) proposed a one-manufacturer and multi-retailer supply chain with multiple substitutable products in the VMI system. They developed a dual Nash game model to analyze the impacts of noncooperative among retailers on a supply chain’s decision. Almehdawe and Mantin (2010) considered a two-echelon supply chain consisting of one capacitated manufacturer and multiple noncompetitive retailers under VMI. They formulated a mixed-integer nonlinear model to maximize each member’s profit and analyzed the impacts of channel power on the operational decisions of the supply chain. Yu et al.

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(2013) used a Stackelberg game to analyze the retailer selection decision for a supply chain under VMI consisting of one vendor and a number of retailers. They developed a hybrid algorithm to solve the optimization

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model. Rahim et al. (2016) developed a one-warehouse and multi-retailer supply chain model under VMI. They designed a heuristic algorithm to minimize the total cost of the supply chain. Several researchers have also considered a two-echelon supply chain system with one supplier and one retailer under VMI (Zanoni et al., 2012; Hariga and Al-Ahmari, 2013; Braglia et al., 2014; Zanoni et al., 2014; Lee et al., 2016).

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The VMI models considered above do not consider deterioration. Several researchers have developed

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supply chain models for deteriorating products under VMI. Yu et al. (2012) developed a mathematical model for an integrated one-vendor and multi-retailer supply chain with deteriorating products under VMI. By proving the convexity of the system’s cost function, they proposed an exact algorithm to find the optimal solution of the model. Chen and Wei (2012) considered a vertically decentralized supply chain for deteriorating products with a single manufacturer and a single retailer under VMI. When the system operated in a

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multiperiod planning horizon, they used the dynamic programming techniques to solve the channel coordination decision problem. Tat et al. (2015) considered a two-echelon supply chain composed of one supplier and one retailer with a type of noninstantaneous deteriorating item. They investigated the performance

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of the VMI system in this supply chain and obtained some management insights with numerical analysis.

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Taleizadeh et al. (2015) developed a VMI model for a two-echelon supply chain for deteriorating products composed of one vendor and several noncompeting retailers. They analyzed the optimal operational decision of maximizing the total profit of the supply chain. Different from the models discussed above, we focus on analyzing the impacts of carbon emission reduction on supply chain coordination with deteriorating products under VMI. We also propose the RS contract to coordinate this low-carbon supply chain and analyze the conditions under which the members of the supply chain will accept this contract. 5

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2.2. Supply chain coordination for deteriorating products Deterioration brings more challenges to supply chain management (Teunter and Flapper, 2003; Bakker

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et al., 2012; Shin and Eksioglu, 2015). Several researchers have proposed contract mechanisms to coordinate the supply chain for deteriorating products. Huang et al. (2011) considered a one-supplier and onebuyer supply chain for deteriorating products with inventory-dependent demand. Assuming that shortages

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were allowed for the supplier and the retailer, they proposed the lead-time discount coordination strategy to determine the optimal order quantity for maximizing the supply chain’s profit and showed that both parties

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of the supply chain could benefit from coordination. Xiao and Xu (2013) considered a two-echelon supply chain for deteriorating products with a single supplier and a single retailer under VMI. They formulated the demand as a linear function of the retailer’s service level and selling price with the assumption that the production cycle and the replenishment cycle would be fixed and have the same length. They also proposed a generalized RS contract to coordinate the supply chain and showed several management insights with a

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numerical example. Zhang et al. (2015) considered a one-manufacturer and one-retailer supplier chain for deteriorating products with price-dependent demand. In their system, both members cooperatively invested

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in preservation technology to control deterioration. They designed two algorithms to solve the pricing and preservation technology investment decisions under the centralized and decentralized scenarios. They also

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proposed a RS and cooperative investment contract to coordinate the decentralized system and showed the conditions for attaining the prefect coordination. Bai et al. (2016) proposed a revenue-sharing contract and a revised revenue-sharing contract to coordinate a supply chain composed of one manufacturer and one retailer for deteriorating products over a finite planning horizon. When the time-varying demand depended on the selling price and the advertising effort, they showed that the revised revenue-sharing contract could perfectly coordinate the supply chain.The literature on inventory models for deteriorating items that published from 2012 to 2015 was reviewed by Janssen et al. (2016) and more recent publications include Chung and 6

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Kwon (2016), Chan et al. (2017), and Zhang et al. (2017). In the literature above, supply chain coordination for deteriorating products does not consider carbon

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emissions. This paper considers a two-echelon supply chain for deteriorating products consisting of one manufacturer and two competing retailers under VMI and the carbon cap-and-trade regulation. We also analyze the conditions under which these three members of the supply chain accept the RS contract.

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2.3. Supply chain models under carbon emission regulations

Incorporating carbon footprint into supply chain management has become one of the most important and

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challenging issues in operations management (Dekker et al., 2012). A growing number of researchers have studied the impact of carbon emission regulations on the supply chain system. Jin et al. (2014) developed mathematical models for the supply network design problem under different carbon regulations such as capand-trade, carbon cap, and carbon tax. They analyzed the impact of carbon regulations on the operational

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decisions of the supply chain. Hammami et al. (2015) developed two multi-echelon supply chain models with lead time constraints under carbon emission tax and carbon emission cap regulations. They solved the optimal production and inventory decisions of the supply chain and provided several management insights.

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Zakeri et al. (2015) considered the impacts of carbon tax and carbon emission trading regulations on a supply chain planning model. They used actual data from an Australian company to compare these two

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regulations. Li et al. (2017) incorporated carbon emissions into a production and transportation operation problem and developed two optimization models for a two-echelon supply chain system under the cap-andtrade regulation and the joint carbon cap-and-trade and tax regulation. They obtained managerial insights by comparing these two carbon regulations. Xu et al. (2017) considered a one-manufacturer and one-retailer supply chain under carbon cap-and-trade regulation in which the manufacturer produced two products. For the make-to-order setting, they derived the optimal production and pricing decisions and analyzed the impacts of cap-and-trade regulation on the profit of either party. Recent related literature includes Jaber et 7

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al. (2013), Baud-Lavigne et al. (2014), Bazan et al. (2015), Toptal and C ¸ etinkaya (2017), Ji et al. (2017), and Sabzevar et al. (2017).

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In the abovementioned literature, various carbon regulations are studied for controling carbon emissions. Many responsible firms who face carbon emission regulations invest in green technologies to reduce carbon emissions (Drake et al., 2016). Several researchers have studied investment decisions for the supply chain system under various carbon emission regulations. For example, Xu et al. (2016) studied a one-supplier and

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one-manufacturer supply chain under the cap-and-trade regulation, in which a certain sustainable technol-

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ogy could be used to control carbon emissions. Under a make-to-order setting, they proposed two contracts to coordinate the supply chain and obtained some management insights using numerical analysis.Yang et al. (2017) developed two competitive two-echelon supply chains with one single manufacturer and one single retailer under the cap-and-trade regulation. In these two supply chain systems, the two manufacturers invest in green technologies to curb carbon emissions and compete on the product greening level. The authors

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considered and compared the vertical and horizontal relationship, and showed that the vertical cooperation leads to higher carbon emission reduction rate and lower retailer prices. In addition, several researchers

Raza et al., 2018).

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studied the related problems from the random demand perspective (Krass et al., 2013; Dong et al., 2016;

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The abovementioned literature does not consider the impact of deterioration on the supply chain coordination model. In this paper, we focus on analyzing supply chain coordination under the constraint of the cap-and-trade regulation and investment in green technologies.

3. Problem description and notations The supply chain considered in this paper consists of a single manufacturer (she) and two homogenous and competing retailers over an infinite time horizon. This supply chain structure is shown in Figure 1. 8

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The manufacturer produces a type of deteriorating item with the machine setup time tm for production and the production rate P. Let cm be the unit production cost of the manufacturer and wi the unit wholesale

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price sold from the manufacturer to the retailer i (he), i = 1, 2. The manufacturer adopts the “lot-forlot” policy to deliver the finished product produced during each production cycle to retailers. Retailer i sells product to satisfy his customers’ demand in each replenishment cycle with the unit selling price pi , i = 1, 2. Let T be the length of the production cycle. With the homogeneity of two retailers, we

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assume that the length of the replenishment cycle for each retailer is equal to that of the production cycle

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for the manufacturer (Xiao and Xu, 2013; Verma et al., 2014; Zhang et al., 2015). A vendor-managed inventory (VMI) is incorporated into this system; that is, the manufacturer manages the inventory at both the manufacturer’s and the retailer’s side. Let θ, (0 < θ < 1), be the deterioration rate of the finished product. Deterioration of product occurs in production and inventory holding, which also leads to more carbon emissions in the two stages. The manufacturer may invest in green technologies to curb carbon

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emissions, and the technology cost per production cycle is 21 ηs2 when the manufacturer provides the green technology level s following the existing literature (Dong et al., 2016; Xu et al., 2016). The carbon capand-trade regulation is assumed to be imposed on the manufacturer, and the objective of the manufacturer

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unit.

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is to determine the optimal wholesale price and green technology level to maximize her total profit per time

When the upstream manufacturer attempts to invest in the green technology to increase the market share of her product, the downstream retailers compete with each other to maximize their own profits by providing an effective pricing strategy to increase their market demands. Let Di (s, p1 , p2 ) be the market demand rate of retailer i, i = 1, 2. Following the existing literature, e.g., Gurnani and Erkoc (2008), Wang et al. (2016a) and Wang et al. (2016b), we model the demand rate functions of both homogenous retailers as the following

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p1

Manufacturer

Retailer 1

D1(s, p1, p2 )

#

w2

p2 D2 (s, p1, p2 ) Retailer 2

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w1

Consumer

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Figure 1: Supply chain structure.

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linear structure:

D1 (s, p1 , p2 ) = a + αs − βp1 + γp2

(1)

D2 (s, p1 , p2 ) = a + αs − βp2 + γp1 ,

(2)

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where a(> 0) is the market base, α(> 0) and β(> 0) measure the elasticity of demand regarding green technology level and selling price, and γ(> 0) measures the effect of the competitor’s selling price on the demand. In the real world, a retailer’s demand is more sensitive to his own selling price than to his

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competitor’s, so we assume β > γ. Moreover, demand information is assumed to be public to all parties, and the objective of each retailer is to determine the optimal selling price to maximize his total profit per

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time unit. The notations used in developing our mathematical models are summarized in Table 1. Carbon emissions per unit of product in the production and inventory holding stages are e1 − b1 s and e2 − b2 s when the green technology level s is invested (Dong et al., 2016). In the following, we assume 0 ≤ s < min{ be11 , be22 } to avoid negative emissions. In addition, all proofs for theorems are provided in the Appendices.

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Table 1. The main parameters and notations in this paper

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Unit selling price of the retailer i, i = 1, 2 Unit wholesale price sold from the manufacturer to the retailer i, i = 1, 2 Fraction of the revenue shared by retailer i in the RS contract, 0 < ρi < 1, i = 1, 2 Green technology level of the manufacturer Total profit per time unit for the retailer i in the decentralized supply chain, i = 1, 2 Total profit per time unit for the retailer i in the RS contract, i = 1, 2 Total profit per time unit for the manufacturer in the decentralized supply chain, i = 1, 2 Total profit per time unit for the manufacturer in the RS contract, i = 1, 2 Total profit per time unit for the system in the centralized supply chain

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C Em (s) η Di (s, p1 , p2 ) a α β γ Decision variables pi wi ρi s Objective functions Πdri (pi ) Πrs ri (pi ) Πdm (s, w1 , w2 ) Πrs m (s, w1 , w2 ) Πc (s, p1 , p2 )

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b2

Deterioration rate of the product, 0 < θ < 1 Unit holding cost at each retailer’s side Unit holding cost at the manufacturer’s side Unit deterioration cost of the product Length of the production cycle for the manufacturer Machine setup time for production, i.e., the starting production time of the manufacturer, 0 ≤ tm < T Production rate of the manufacturer Inventory level at time t of the manufacturer Inventory level at time t of the retailer i Unit trading price of carbon emission permit Carbon emissions per unit in the production stage when the green technology level is zero Carbon emissions per unit in the inventory holding stage when the green technology level is zero Parameter of the green technology effect on reducing carbon emissions generated in the production stage Parameter of the green technology effect on reducing carbon emissions generated in the inventory holding stage Carbon emission cap Total emission amount of the supply chain Manufacturer’s green technology investment cost coefficient, η > 0 Market demand rate of the retailer i, i = 1, 2 Market scale parameter Green technology elasticity parameter of demand Price elasticity parameter of demand Cross-price elasticity parameter of demand

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Parameters θ hr hm hd T tm P Im (t) Iri (t) cp e1 e2 b1

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4. Mathematical models 4.1. Basic model

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According to the aforementioned problem description and notations, the inventory levels follow the pattern depicted in Figure 2. In the first production cycle, the manufacturer starts to produce with zero

"

tm

T T tm

2T

"

t

0

T

2T

t

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0

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I ri (t )

I m (t )

(a) Inventory level of the manufacturer

(b) Inventory level of each retailer

Figure 2: Inventory level of the manufacturer and both retailers.

stock at time tm and stops the production at time T . During [tm , T ], the inventory level of the manufacturer

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gradually increases because of the combined effects of production and deterioration. Hence, the differential equation describing the inventory level at time t in the production cycle is given by

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dIm (t) = P − θIm (t), tm ≤ t ≤ T, dt

(3)

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with the boundary condition Im (tm ) = 0. Solving Eq. (3), we have

Im (t) =

P [1 − eθ(tm −t) ], tm ≤ t ≤ T. θ

(4)

During each production cycle, the sum of inventory holding cost and deterioration cost at the manufac-

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turer will be

Z

T tm

P P Im (t)dt = (hm + θhd ){ (T − tm ) − 2 [1 − eθ(tm −T ) ]}. θ θ

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(hm + θhd )

(5)

In addition, the total quantity produced by the manufacturer in each production cycle is

(6)

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Q = P(T − tm ).

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At time T , the retailer i receives the Im (T ) units from the manufacturer under the zero delivery lead time assumption. The inventory level of retailer i gradually decreases and falls to zero at time 2T when he receives another batch because of the combined effects of demand and deterioration. Hence, the inventory level of the retailer i is governed by the following differential equation:

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dIri (t) = −Di (s, p1 , p2 ) − θIri (t), T ≤ t ≤ 2T, i = 1, 2 dt

(7)

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with the boundary condition Iri (2T ) = 0. Solving Eq. (7), we have

Di (s, p1 , p2 ) θ(2T −t) [e − 1], T ≤ t ≤ 2T, i = 1, 2. θ

(8)

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Iri (t) =

During each replenishment cycle, the sum of inventory holding cost and deterioration cost at both retailers will be

Z 2 X (hr + θhd ) i=1

2

2T

Iri (t)dt = T

X (hr + θhd ) θT (e − θT − 1) Di (s, p1 , p2 ). θ2 i=1

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(9)

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In addition, from Eqs. (4) and (8), using Im (T ) = Ir1 (T ) + Ir2 (T ), we have

(10)

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P[1 − eθ(tm −T ) ] = [D1 (s, p1 , p2 ) + D2 (s, p1 , p2 )](eθT − 1).

Since the manufacturer will invest in green technologies to curb carbon emissions generated by production and inventory holding, the total emission amount in the system during each production cycle is given

T

Im (t)dt + tm

2 Z X

2T

Iri (t)dt].

i=1

(11)

T

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Z Em (s) = (e1 − b1 s)Q + (e2 − b2 s)[

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by

For convenience, we normalize the cycle length T to 1 and define the following equations as

φ2 = φ3 =

(eθ − 1)(1 − tm ) , 1 − eθ(tm −1) eθ − θ − 1 , θ2 1 − tm (eθ − 1) 1 [ − ], θ(t −1) m θ θ 1−e

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φ1 =

(13) (14) (15)

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A = (cm + c p e1 )φ1 + (hr + θhd + c p e2 )φ2 + (hm + θhd + c p e2 )φ3 .

(12)

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From Eqs. (12)–(14), we easily have φ1 > 0, φ2 > 0, and φ3 > 0. In addition, we let φ = φ1 + φ2 + φ3 . In reality, a higher investment leads to improved green technology level, so we assume that the parameter η is high enough such that η >

[α+(β−γ)bc p φ]2 , β−γ

3) where b = b1 φφ1 + b2 (φ2 +φ . φ

4.2. Decentralized and centralized models We consider the decentralized case where the manufacturer is a Stackelberg leader and two competing retailers are followers. The sequence of events is as follows: (1) The manufacturer first sets the wholesale

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price wi (i = 1, 2), to retailer i and determines the green technology level s to maximize her total profit per time unit. (2) Observing the manufacturer’s decisions, retailers 1 and 2 simultaneously choose the selling

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price p1 and p2 to maximize their own total profit per time unit. Obviously, in this case, the retailers compete for the selling price. Under VMI, the manufacturer holds and manages the inventories at both her own and the retailer’s side. Hence, the total profit per time unit of retailer i is

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Πdri(pi ) = (pi − wi )Di (s, p1 , p2 ), i = 1, 2.

(16)

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Since the manufacturer is the main carbon emitter, the profit per production cycle of the manufacturer under the cap-and-trade regulation includes the sales revenue, production cost, holding inventory cost, deterioration cost for deteriorated items, revenue(cost) from selling (buying) the extra carbon emission permits, and investment in green technology. Hence, from Eqs. (5), (6), (9) and (10), we express the total profit per

Πdm (s, w1 , w2 )

=

2 X i=1

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time unit of the manufacturer as

Z 2 X wi Di (s, p1 .p2 ) − cm Q − (hr + θhd ) i=1

2T

Iri (t)dt − (hm + θhd ) T

Z

T

Im (t)dt (17)

tm

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1 − ηs2 − c p [Em (s) − C]. 2

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Note that c p C in Eq.(17) is a constant term and doesn’t play a role in optimization. By using the backward induction technique, we derive the equilibrium strategies of the manufacturer and retailer. From Eqs. (16) and (17), we have the following conclusion: Theorem 4.1. For the decentralized supply chain under VMI, the following holds: d∗ (i)When two retailers compete for the selling price, there exists a unique Nash equilibrium (pd∗ 1 , p2 ), where d∗ pd∗ 1 = p2 =

a(3β − 2γ) + β(β − γ)A + [α(3β − 2γ) − bβc p φ(β − γ)]s∗d . 2(β − γ)(2β − γ)

(18)

(ii)In the Stackelberg game with the manufacturer as the leader, the manufacturer’s best response to two 15

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d∗ ∗ competing retailers’ selling prices is (wd∗ 1 , w2 , sd ), where d∗ wd∗ 1 = w2 =

a + (β − γ)A + [α − bc p φ(β − γ)]s∗d

(19)

2(β − γ)

s∗d =

β[a − (β − γ)A][α + bc p φ(β − γ)] . (2β − γ)(β − γ)η − β[α + bc p φ(β − γ)]2

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and (20)

Theorem 4.1 proves the existence and uniqueness of the Nash equilibrium between two competing retailers

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and the Stackelberg equilibrium between the manufacturer and two retailers, respectively. From Theorem 4.1, we observe that two competing retailers will set the same pricing decision when the manufacturer

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provides the same wholesale price to them under the decentralized scenario. Based on Theorem 4.1, we have the following:

Theorem 4.2. For the decentralized supply chain under VMI, the optimal profits per time unit of the manufacturer and the retailers are, respectively, d∗ d Πdr1 (pd∗ 1 ) = Πr2 (p2 ) =

(β − γ)2 η2 (s∗d )2

4β[α + bc p φ(β − γ)]2

,

η{(2β − γ)(β − γ)η − β[α + bc p φ(β − γ)]2 }(s∗d )2

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d∗ Πdm (s∗d , wd∗ 1 , w2 ) =

2β[α + bc p φ(β − γ)]2

(21) + c pC.

(22)

Moreover, the corresponding total emissions amount is (β − γ)η(e1 φ1 + e2 φ2 − bφs∗d )s∗d . α + bc p φ(β − γ)

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d ∗ Em (sd ) =

(23)

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From Eqs. (21), (22), and (23), we can obtain the following corollary: Corollary 4.3. For the decentralized supply chain under VMI, the following holds: d d∗ d ∗ d∗ d∗ (i) Πdr1 (pd∗ 1 ) and Πr2 (p2 ) do not depend on the parameter C while Πm (sd , w1 , w2 ) is linearly increasing in C. d (s∗ ) is linearly increasing in e or e . (ii) Em 1 2 d From Theorem 4.2 and Corollary 4.3, we observe that the total profit per time unit of each retailer is not affected by the carbon cap C, while the total profit per time unit of the manufacturer increases as C increases. Moreover, the emission amount increases as e1 or e2 increases, while it is not affected by the 16

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carbon cap C. This means that increasing the carbon cap leads the system to obtain more profits while it plays no effects on controlling carbon emissions.

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In a centralized case, the manufacturer and two retailers operate as a whole and jointly find the optimal selling price and green technology level to maximize the total profit per time unit of the supply chain. Under the cap-and-trade regulation, the total profit per time unit of the supply chain is

2 X

pi Di (s, p1 .p2 ) − cm Q −

i=1

Z 2 X (hr + θhd ) i=1

Iri (t)dt − (hm + θhd )

T

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1 − ηs2 − c p [Em (s) − C]. 2

2T

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Πc (s, p1 , p2 ) =

Z

T

Im (t)dt (24)

tm

From Eq. (24), we have the following theorem:

Theorem 4.4. For the centralized supply chain under VMI, the total profit pert time unit of the supply chain is jointly concave in s, p1 , and p2 . Moreover, the corresponding optimal solutions are, respectively,

and

[a − (β − γ)A][α + (β − γ)bc p φ] (β − γ)η − [α + (β − γ)bc p φ]2

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s∗c =

c∗ pc∗ 1 = p2 =

a + (β − γ)A + [α − (β − γ)bc p φ]s∗c . 2(β − γ)

(25)

(26)

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Using Eqs. (25) and (26), we get the following theorem:

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Theorem 4.5. For the centralized supply chain under VMI, the optimal total profit per time unit and the corresponding emission amount of the supply chain are c∗ Πc (s∗c , pc∗ 1 , p2 ) =

and

c ∗ Em (sc ) =

η{(β − γ)η − [α + (β − γ)bc p φ]2 }(s∗c )2 + c p C, 2[α + (β − γ)bc p φ]2

(β − γ)η[e1 φ1 + e2 (φ2 + φ3 ) − bφs∗c ]s∗c . α + bc p φ(β − γ)

(27)

(28)

From Eqs. (27) and (28), we have the following corollary: c∗ Corollary 4.6. For the centralized supply chain under VMI, Πc (s∗c , pc∗ 1 , p2 ) is linearly increasing in C and c ∗ Em (sc ) is linearly increasing in e1 or e2 .

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Theorem 4.5 and Corollary 4.6 show that the effects of carbon parameters C, e1 , and e2 on the profit and emissions amount of the manufacturer in the decentralized system are similar to those on the profit and

From Theorems 4.1 and 4.4, we have the following:

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emissions amount of the centralized supply chain.

Corollary 4.7. For the decentralized and centralized supply chains, the following holds: (2β−γ)(β−γ)η−β[α+(β−γ)bc φ]2 s∗ (i) s∗c = β(β−γ)η−β[α+(β−γ)bc φ]p2 . p

d

=

pd∗ i

−

(β−γ)η{(β−γ)η−2α[α+(β−γ)bc p φ]}s∗d . 2β[α+(β−γ)bc p φ]{(β−γ)η−[α+(β−γ)bc p φ]2 }

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(ii) For i =

1, 2,pc∗ i

Corollary 4.7 implies that s∗c > s∗d because of β > γ. This shows that the manufacturer will provide a

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higher green technology level in the centralized system than in the decentralized system. Corollary 4.7(ii) implies that when (β − γ)η > 2α[α + (β − γ)bc p φ], p∗c < p∗d holds; when (β − γ)η < 2α[α + (β − γ)bc p φ], p∗c > p∗d holds; and when (β − γ)η = 2α[α + (β − γ)bc p φ], p∗c = p∗d holds. This also shows that the optimal selling prices of the decentralized and centralized supply chains can be compared using the values of (β−γ)η

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and 2α[α + (β − γ)bc p φ].

To obtain a fair comparison between the centralized and the decentralized supply chains, we obtain the following:

(ii)

i=1 c (s∗ ) Em c d (s∗ ) Em d

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Theorem 4.8. For the decentralized and the centralized supply chain, the following holds: Πc (s∗c ,pc∗ ,pc∗ ) {(2β−γ)(β−γ)η−β[α+bc p φ(β−γ)]2 }2 1 2 < ψ, where ψ = β{(β−γ)η−[α+bc φ(β−γ)]2 }{(β−γ)(3β−2γ)η−β[α+bc (i) 1 < P2 ; φ(β−γ)]2 } p

d ∗ d∗ d∗ Πdri (pd∗ i )+Πm (sd ,w1 ,w2 )

=

[e1 φ1 +e2 (φ2 +φ3 )−bφs∗c ]s∗c [e1 φ1 +e2 (φ2 +φ3 )−bφs∗d ]s∗d .

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c (s∗ ) ≥ E d (s∗ ). Em m d c

Moreover, when s∗c + s∗d >

p

e1 φ1 +e2 (φ2 +φ3 ) , bφ

c (s∗ ) < E d (s∗ ); otherwise, Em m d c

2 P

d∗ d ∗ d∗ Πdri (pd∗ i ) + Πm (sd , w1 , w2 ).

c∗ Theorem 4.8(i) provides an upper bound of the ratio Πc (s∗c , pc∗ 1 , p2 ) to

i=1

This implies that the total profit per time unit of the supply chain increases by at most (ψ − 1) × 100% when the manufacturer cooperates with the retailer. However, the cooperation decision may lead the retailer to a decrease in profit. For example, when (β−γ)η > 2α[α+(β−γ)bc p φ], we have p∗c < p∗d , which implies that the retailer should provide a lower selling price when he or she accepts the cooperation decision. If the decrease 18

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in selling price is higher than the increase in green technology level charged by the manufacturer, the total profit per time unit of the retailer will decrease. Under this scenario, as the follower, the retailer will reject

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to cooperate with the manufacturer. For the manufacturer, it is important to choose an effective contract in which certain incentives can be provided to the retailer. Theorem 4.8(ii) shows that carbon emissions generated by the centralized and decentralized systems can be compared. Hence, one of the challenge questions is whether the supply chain coordinated by the contracts can increase the profit and reduce carbon

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emissions. In the next section, we will use a revenue-sharing contract to coordinate the supply chain.

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4.3. Coordinating the decentralized system with a revenue-sharing contract

An RS contract is investigated to coordinate the supply chain system studied in this paper, and this contract is described as follows: The manufacturer provides some incentives, such as charging a lower wholesale price and improving the green technology level, to each retailer such that he makes decisions

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consistent with the supply chain total optimization. As the follower, each retailer will provide a fraction of his revenue to the manufacturer. After several negotiations between the manufacturer and the retailers, the corresponding decision variables are established, implying that the supply chain is coordinated. In the RS

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contract, the profits per time unit of these three members are, respectively,

and

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Πrs ri (pi ) = (ρi pi − wi )Di (s, p1 , p2 ), i = 1, 2,

Πrs m (s, w1 , w2 ) =

2 X

wi Di (s, p1 .p2 ) − cm Q −

i=1

−c p [Em (s) − C] +

2 X

2 X 1 (hr + θhd )Iri − (hm + θhd )Im − ηs2 2 i=1

(1 − ρi )pi Di (s, p1 , p2 ).

i=1

19

(29)

(30)

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The following results provide the condition of using an RS contract to coordinate the supply chain developed in this paper.

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Theorem 4.9. For the supply chain considered in this paper, the coordinated result can be achieved by an c∗ rs c∗ ∗ RS contract when prs 1 = p1 ,p2 = p2 , and srs = sc . The corresponding wholesale prices are, respectively, rs wrs 1 ρ2 = w2 ρ1

and

ρ1 {aγ + (β − γ)(2β − γ)A + [αγ − (2β − γ)(β − γ)bcP φ]s∗c }. 2β(β − γ)

(32)

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wrs 1 =

(31)

Theorem 4.9 shows that when investment in green technologies, holding inventory and deteriorating

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costs, and cap-and-trade regulation are incorporated into the supply chain for deteriorating products under VMI, an RS contract can lead the system to a coordination. Eqs. (31) and (32) show that when the supply chain achieves a coordination, the wholesale price charged by the manufacturer to each retailer is increasing in the corresponding revenue fraction for the retailer. This implies that the manufacturer may share more of

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each retailer’s revenue when she reduces the wholesale price.

We further investigate the conditions under which these three members of the supply chain accept the RS contract.

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Theorem 4.10. For these three members of the supply chain, the RS contract is accepted if and only if ρ1 β(β−γ)η−β[α+(β−γ)bc p φ]2 2β{(β−γ)η−[α+(β−γ)bc p φ]2 } and ρ2 satisfy ρi ≥ { (2β−γ)(β−γ)η−β[α+(β−γ)bc }2 , i = 1, 2, and ρ1 + ρ2 ≤ (2β−γ)(β−γ)η−β[α+(β−γ)bc . φ]2 φ]2 p

p

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From Theorems 4.9 and 4.10, we have the following conclusion: Corollary 4.11. For the supply chain considered in this paper, the following holds: (i)The profit per time unit of the supply chain in the RS contract is equal to that in the centralized system, 2 P rs ∗ c∗ c∗ rs rs Πrs that is, Πrs m (srs , w1 , w2 ) + ri (pi ) = Πc (sc , p1 , p2 ). i=1

rs (s ) = E c (s∗ ). (ii) The emission amount in the RS contract is equal to that in the centralized system, Em rs m c (iii) If ρ1 = ρ2 = ρ, then the RS contract is acceptable for each member of the supply chain if and only β(β−γ)η−β[α+(β−γ)bc p φ]2 β{(β−γ)η−[α+(β−γ)bc p φ]2 } if { (2β−γ)(β−γ)η−β[α+(β−γ)bc }2 ≤ ρ ≤ (2β−γ)(β−γ)η−β[α+(β−γ)bc φ]2 φ]2 p

p

Corollary 4.11(i) shows that the RS contract can lead the supply chain to perfect coordination. Corollary 4.11(ii) implies that the decentralized supply chain may attain the same profit and carbon emissions as the 20

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centralized system using the RS contract. From Corollary 4.11(iii) and Eq.(31), we observe that, in the RS contract, if these two competing retailers provide the same fraction of their revenues to the manufacturer,

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the manufacturer will charge the same wholesale price to them. Under this scenario, there exists a value of ρ such that the profits of these three members in the RS contract are higher or equal to those in the decentralized system, respectively.

It should be noted that Cachon and Lariviere (2005) comprehensively analyzed several merits and lim-

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itations of the RS contract, and they verified that the RS contract cannot coordinate the supply chain with

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demand being dependent on the retailer’s effort level. On the other hand, for the supply chain model developed in this paper, we verify that the RS contract can lead to perfect coordination when the demand rate of each retailer is formulated as a linear function of the manufacturer’s green technology level and both retailers’ selling prices.

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5. Numerical experiment

In this section, we conduct a numerical experiment to illustrate the above theoretical results and gain

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managerial insights.

5.1. Numerical example

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The parameters for the benchmark case are as follows: a = 500, α = 0.45, β = 0.85, γ = 0.4, cm = 8, θ = 0.08, hr = 3, hm = 2, hd = 1.5, tm = 0.15, e1 = 50, e2 = 23, c p = 2.5, b1 = 0.45, b2 = 0.34, C = 10000, η = 25. The computational results are summarized in Table 2. Table 2 shows that the profit per unit time and corresponding carbon emissions of the supply chain in the centralized system are 248, 480 and 13, 407, respectively. In the decentralized system, the profit per time unit and corresponding carbon emissions of the supply chain are 214, 772 and 13, 876, respectively. Therefore, we can conclude that cooperation between the manufacturer and these two retailers leads to an 21

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Table 2. The optimal solution for the numerical example

Wholesale price (w1 ;w2 )

Green technology level s

Selling price (p1 ;p2 )

Profit (Retailer 1; Retailer 2; Manufacturer)

Profit of the supply chain

Carbon emission

Centralized system Decentralized system Revenue-sharing contract (ρ1 ; ρ2 ) (0.4;0.4) (0.45;0.4) (0.5;0.45) (0.5;0.5) (0.6;0.5) (0.6;0.6)

(-;-) (637.13,637.13)

59.68 36.50

(625.10;625.10) (813.84;813.84)

(-;-;-) (26,541;26,541;161,690)

248,480 214,772

13,407 13,876

(134.48;134.48) (151.29;134.48) (168.10;151.29) (1668.10;168.10) (201.73;168.10) (201.73;201.73)

59.68 59.68 59.68 59.68 59.68 59.68

(625.10;625.10) (625.10;625.10) (625.10;625.10) (625.10;625.10) (625.10;625.10) (625.10;625.10)

(28,376;28,376;191,728) (31,923;28,376;188,181) (35,470;31,923;181,087) (35,470;35,470;177,540) (42,565;35,470;170,445) (42,565;42,565;163,350)

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Model

13,407 13,407 13,407 13,407 13,407 13,407

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248,480 248,480 248,480 248,480 248,480 248,480

increase of 15.70% in profit and a decrease of 3.38% in carbon emissions. This implies that cooperation

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between these three members of the supply chain can increase the profit and reduce the carbon emissions. ∗ d∗ A comparison of s∗c = 59.68 and pc∗ i = 625.10, i = 1, 2, with sd = 36.50 and pi = 813.84, respectively,

shows that the manufacturer should increase the green technology level, and both retailers should reduce the selling prices when the three members of the supply chain agree to cooperate. In addition, in the centralized

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system, both competing retailers set the same selling price and obtain the same profit. On the other hand, in the decentralized system, when the manufacturer sets the same wholesale price for these competing retailers, they set the same selling price and obtain the same profit.

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From Table 2, we also observe the following:

(i) In the RS contract, when both competing retailers provide the same fraction of their revenues to the

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manufacturer, that is, ρ1 = ρ2 , the manufacturer sets the same wholesale price for these two retailers, which leads to no difference in the selling price and profit between these two competing retailers. Moreover, as ρ1 or ρ2 increases from 0.4 to 0.6, the wholesale price in the RS contract increases, and the selling prices and the green technology level remain unchanged. The wholesale price in the RS contract is less than that in the decentralized system, and the selling prices and the green technology level are equal to those in the centralized system. This means that the RS contract incentivizes the manufacturer and the two competing

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retailers to make decisions coherent with the supply chain total optimization. As ρ1 or ρ2 increases, the profit of the manufacturer decreases, while the profits of both retailers increase. In particular, when ρ1 or

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ρ2 increases from 0.4 to 0.6, the profits of the manufacturer and the two competing retailers are higher than those in the decentralized system. This shows that the RS contract is acceptable for these three members of the supply chain, and the exact range of ρ1 or ρ2 is [0.37, 0.61]. d

r2

10

8

8 6 6 4 Profit

0

r2

∆

0

−2

1

−4 −6 1

2

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Profit

4 2

−2

d

Πrs −Πd

0

0

rs

Πr1−Πr1

4

x 10

∆

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rs

Πm−Πm

4

x 10

0.5 0.8

0.6

0.4

0.2

ρ2

0

0

−4

−6 1

ρ

1

0.8

0.6

0.4 ρ

0.2

1 0.5 0

0

ρ

1

2

d (a) Values of ρi satisfying Πrs m ≥ Πm , i = 1, 2.

d (b) Values of ρi satisfying Πrs ri ≥ Πri , i = 1, 2.

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Figure 3: Profits and values of ρi , i = 1, 2 when the supply chain accepts the RS contract.

(ii) In the RS contract, when the two competing retailers provide different fractions of their revenues to the manufacturer, that is, ρ1 , ρ2 , the manufacturer sets the different wholesale prices for these two

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retailers, and then they set different selling prices to maximize their profits. For example, when ρ1 > ρ2 , the

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manufacturer sets a higher wholesale price for retailer 1 than retailer 2, which implies that the manufacturer provides more incentives to the retailer 2. Under this scenario, the profit of the retailer is higher than that of retailer 1. The opposite occurs for ρ1 < ρ2 . In addition, when these three members of the supply chain accept the RS contract, the values of ρ1 and ρ2 satisfy ρ1 + ρ2 ≤ 1.22 and ρi ≥ 0.37, i = 1, 2. Let ∆0 = 0. The graphical description for determining the values of ρ1 and ρ2 is shown in Figure 3. (iii) In the RS contract, the profit of the supply chain is equal to that in the centralized system and is higher than that in the decentralized system. But the amount of carbon emissions is equal to that in 23

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the centralized system and is less than that in the decentralized system. This means that supply chain coordination with the RS contract leads to higher profits and less carbon emissions.

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5.2. Sensitivity analysis In this section, we first conduct a static sensitive analysis that only varies a single parameter with others being constant to study the effects on the supply chain coordination strategies. Considering the ranges of

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ρi , i = 1, 2 for supply chain coordination, we select ρ1 = 0.5 and ρ2 = 0.45, and then select those parameters, θ, hr ,hm , e1 ,e2 , c p and C, corresponding to deterioration, inventory holding and carbon emission. Varying

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one of the parameters by +40%, +20%, −20% and −40% and keeping others unchanged, we perform the sensitive analysis and summarize the results in Table 3. From Table 3, we have the following observations: (i) In the RS contract, when the deterioration rate θ increases, the wholesale prices of the manufacturer, the green technology level, the selling prices and profits of two retailers, and the emission amount increase,

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whereas the manufacturer’s profit decreases. When θ is relatively high, production and storage of deteriorating product may cause more damage to the environment. The manufacturer as the main emitter adopts higher level of the green technology to reduce the carbon emissions and increases the wholesale prices so

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as to cut the technology cost. Increases in the wholesale prices lead to increases in the selling prices. Moreover, the RS contract urges the two competing retailers to adopt the same selling price for cooperation, and

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the market demands of the two retailers increase because the increase in the green technology level is higher than the changes in the selling prices. With VMI, increases in the selling prices and market demands of the two retailers lead to an increase in each retailer’s profit. On the other hand, for the manufacturer, the costs of inventory and buying the carbon emission permits increase when the market demands increase. With the increased investment cost on green technologies, the profit of the manufacturer decreases. As a result, the profit of the whole supply chain decreases because a decrease in the manufacturer’s profit is higher than the changes in the profits of the two retailers. These observations imply that a higher deterioration rate leads 24

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Table 3. Sensitivity analysis for several parameters in the RS contract with ρ1 = 0.5 and ρ2 = 0.45 Parameter

Percentage (Value)

rs (wrs 1 ;w2 )

srs

rs (prs 1 ;p2 )

θ

+40%(0.112) +20%(0.096) -20%(0.064) -40%(0.048)

(168.38;151.54) (168.25;151.42) (167.96;151.16) (167.80;151.02)

60.64 60.16 59.21 58.76

hr

+40%(4.2) +20%(3.6) -20%(2.4) -40%(1.8)

(168.37;151.53) (168.24; 151.41) (167.97; 151.18) (167.84;151.06)

hm

+40%(2.8) +20%(2.4) -20%(1.6) -40%(1.2)

(168.26;151.43) (168.18;151.36) (168.03;151.23) (167.95;151.16)

e1

+40%(70) +20%(60) -20%(40) -40%(30)

(191.03;171.93) (179.57;161.61) (156.64;140.98) (145.18;130.67)

e2

+40%(32.2) +20%(27.6) -20%(18.4) -40%(13.8)

cp

+40%(3.5) +20%(3) -20%(2) -40%(1.5)

C

+40%(14000) +20%(12000) -20%(8000) -40%(6000)

rs Em

(625.80;625.80) (625.45;625.45) (624.75; 624.75) (624.38;624.38)

(35,505;31,955;179,840) (35,487;31,938;180,460) (35,456;31,910;181,700) (35,443;31,899;182,300)

13,430 13,420 13,389 13,368

59.64 59.66 59.70 59.72

(625.43;625.43) (625.27; 625.27) (624.94;624.94) (624.77;624.77)

(35,422; 31,880;180,870) (35,446;31,902;180,980) (35,495;31,945;181,190) (35,519;31,967;181,300)

13,414 13,410 13,403 13,400

59.67 59.67 59.69 59.71

(625.29; 625.29) (625.20;625.20) (625.01; 625.01) (624.91;624.91)

(35,443;31,898;180,960) (35,457;31,911;181,020) (35,484;31,936;181,140) (35,498;31,949;181,210)

13,411 13,409 13,405 13,403

56.15 57.92 61.45 63.21

(653.86; 653.86) (639.48;639.48) (610.73; 610.73) (596.35; 586.35)

(31,397; 28,258; 16,3160) (33,403;30,063;171,990) (37,600;33,840;190,450) (39,792;35,813;200,100)

23,895 18,829 7,630.5 1,499.4

(177.52; 159.76) (172.81;155.53) (163.40;147.06) (158.70;142.83)

58.23 58.96 60.41 61.13

(636.91;636.91) (631.00; 631.00) (619.20;619.20) (613.30; 613.30)

(33,768;30,391;173,590) (34,614;31,153;177,320) (36,337;32,704;184,900) (37,215;33,493;18,8760)

17,884 15,676 11,078 8,690.1

(159.79; 143.81) (165.88;149.30) (167.14; 150.42) (163.40;147.06)

78.95 68.79 51.27 43.30

(620.90; 620.90) (625.345; 625.35) (620.93; 620.93) (613.28; 613.28)

(38,588;34,729;175,320) (36,632;32,969;178,030) (34,922;31,430;184,640) (34,881;31,393;188,830)

5,936.6 9,919.6 16,643 19,798

(168.10;151.30) (168.10;151.29) (168.10;151.29) (168.10;151.29)

59.68 59.68 59.68 59.68

(625.10;625.10) (625.10;625.10) (625.10;625.10) (625.10;625.10)

(35,470;31,923;191,080) (35,470;31,923;186,080) (35,470;31,923;176,080) (35,470;31,923;171,080)

13,407 13,407 13,407 13,407

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rs rs (Πrs r1 ;Πr2 ;Πm )

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to less profit and more carbon emissions of the whole supply chain when the decentralized supply chain is coordinated.

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(ii) In the RS contract, when the unit holding cost at each retailer’s side hr or the unit holding cost at the manufacturer’s side hm increases, the wholesale prices of the manufacturer, the selling prices, and the carbon emissions increase, whereas the green technology level, the profits of the two retailers, and the manufacturer’s profit decrease. With VMI, the manufacturer manages the inventory of the whole supply

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chain. A larger value of hr or hm urges the manufacturer to increase the wholesale prices to offset the

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higher holding cost. Increases in the wholesale prices lead to increases in the selling prices and the two competing retailers set the same selling prices when the supply chain is coordinated. The manufacturer also invests less in green technologies to offset the higher holding cost. A decrease in the green technology level and increases in the selling prices lower the market demands. Under this scenario, the profits of the two retailers decrease because the effect of the decreases in the market demands are higher than the effect of the

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increases in the selling prices. Moreover, a lower green technology level causes the manufacturer to emit more carbon emissions and to buy more emission permits when the carbon emissions are higher than the carbon cap. Increases in hr or hm and cost of buying carbon emission permits ultimately lower the profit

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of the manufacturer. These observations imply that a lower value of hr or hm benefits each member of the

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supply chain under VMI and plays a critical role in carbon emission reduction. (iii) In the RS contract, when the normal carbon emissions in the production stage e1 or that in the inventory holding stage e2 increases, the wholesale prices of the manufacturer, the selling prices, and the carbon emissions increase, whereas the green technology level, the profits of the two retailers, and the profit of the manufacturer decrease. When e1 or e2 is relatively small, the manufacturer is more willing to invest higher level of the green technology such that the carbon emissions are lower than the carbon cap, and the manufacturer generates more revenue by trading the carbon emission permits. Under this 26

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scenario, the manufacturer decreases the wholesale prices to incentive the two retailers for cooperation. On the other hand, the RS contract urges the two retailers to decrease the selling prices for stimulating the

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market demands and increasing the order quantities. Hence, when e1 or e2 decreases, the profits of the two retailers increase because the decreases in the selling prices are lower than the increases in the maker demands, and the manufacturer’s profit increases because of the increase in order quantities and trading the carbon emission permits.

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(iv) In the RS contract, when the unit trading price of carbon emission permit c p increases, the green

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technology level, the profits of two retailers increase, the profit of the manufacturer, and the carbon emissions decrease, whereas the wholesale prices of the manufacturer and the selling prices first increase and then decrease. When c p is relatively high, the manufacturer is more willing to invest the higher level of the green technology to reduce the carbon emissions so as to gain revenue or save cost. When the carbon emissions are higher than the carbon cap, a larger value of c p urges the manufacturer to increase the

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wholesale prices so as to cut the costs of the technology investment and buying carbon emission permits. When the carbon emission are less than the carbon cap, a larger value of c p urges the manufacturer to decrease the wholesale prices for stimulating the two retailers to increase the order quantities. Under this

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scenario, the changes of the selling prices are line with those of the wholesale prices. The increase in the

AC C

green technology level leads to an increase in the market demands and then the profits of the two retailers increase. However, the profit of the manufacturer decreases because increases in the green technology level and market demands lead to increases in the costs of technology investment and inventory holding. These observations imply that a relatively large value of c p benefits the two competing retailers while a relatively small value of c p benefits the manufacturer. Moreover, increasing the value of c p plays a positive role in reducing the carbon emissions. (v) In the RS contract, when the carbon cap C increases, the profit of the manufacturer increases, 27

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whereas the wholesale prices of the manufacturer, the selling prices, the green technology level, the profits of two retailers and the carbon emissions keep unchanged. These observations show that the carbon cap

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does not affect the optimal operational decisions of the two retailers and the manufacturer when the supply chain is coordinated by the RS contract. Moreover, a large value of the carbon cap benefits the primary emitter because the manufacturer has higher profit and emits the same carbon emissions when the carbon cap increases.

SC

We further conduct a robust sensitive analysis where multiple parameters vary simultaneously to study

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the effects on the supply chain coordination strategies. The Taguchi method is applied to test the robustness of the coordination system with respect to the main parameters θ, cm , hr , hm , hd , e1 , e2 , ρ1 , ρ2 , c p , and C. As discuss above, we select ρ1 = 0.5 and ρ2 = 0.45 to conduct the experimental design. The Taguchi method is widely used to study the impacts of varying multiple parameters simultaneously on various problems, and a detailed explanation of this method can be found in Taguchi (1962) and Taguchi (1987). In reality,

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the values of c p and C are usually set by the government; hence, we regard these two parameters as the uncontrolled factors and the others as the control factors when the Taguchi method is used to analyze the problem considered in this paper. Following the Taguchi method, we select the L12 (28 ) orthogonal table

EP

to assign eight control factors, including θ, cm , hr , hm , hd , e1 , e2 , ρ1 , and ρ2 , and select the L4 (22 ) orthogonal

AC C

table to assign the two uncontrolled factors, c p and C, in which levels 1 and 2 represent +20% and −20% of the initial values of the corresponding parameters, respectively. Based on the formula of the signal-to-noise (SN) ratio in the Taguchi method, we calculate the values of the SN corresponding to the manufacturer’s profit, both retailers’ profits, and the coordinated system’s carbon emissions. The experimental results are summarized in Table 4 and Figure 4. In addition, from the definition of the SN ratio proposed by Taguchi, we conclude that the larger the SN ratio is, the smaller the variance of each member’s profit is around the desired value. On the other hand, the smaller the SN ratio is, the smaller the variance of the system’s carbon 28

ACCEPTED MANUSCRIPT

*GZG 3KGTY

*GZG 3KGTY emissions is around the desired value. BZNKZG θ



IS cm

hNXr

NS h

cm IS

BZNKZG θ

NJ h

m

d

NX hr

NS hm

NJ hd







  





eK 1











ρ1 XNU

e2 K







ρ2 XNU





 





e1 K





 















(a) Main effects plot for the SN ratio of the manufacturer *GZG 3KGTY hNXr

hNS m





M ean of SN ratios



 



eK 1













ρ1 XNU

eK 2



ρ2 XNU

 





cm IS

BZNKZG θ

















(b) Main effects plot *GZG for 3KGTY the SN ratio of retailer 1

hNJ d







  



e1 K

hNXr









hNS m





ρ1 XNU

eK2

M AN U

M ean of SN ratios

cm IS

BZNKZG θ





9OMTGRZUTUOYK 2GXMKX OY HKZZKX

9OMTGRZUTUOYK 2GXMKX OY HKZZKX



ρ2 XNU

SC













ρ1 XNU

 



e2 K

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M ean of SN rat ios

M ean of SN ratios



NJ hd







ρ2 XNU

 









































9OMTGRZUTUOYK 9SGRRKX OY HKZZKX

9OMTGRZUTUOYK 2GXMKX OY HKZZKX

(c) Main effects plot for the SN ratio of retailer 2

(d) Main effects plot for the SN ratio of carbon emissions

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Figure 4: Main effects plot for the SN ratios of the manufacturer, retailer, and carbon emissions.

From Table 4, we can see that (i) when the levels of the parameters θ, cm , hr , hm , hd , e1 , e2 , ρ1 , and ρ2

EP

are 1, 1, 1, 1, 1, 2, 2, 2, and 2, respectively, the SN ratio corresponding to the profit of the manufacturer is the largest; (ii) when the levels of the parameters θ, cm , hr , hm , hd , e1 , e2 , ρ1 , and ρ2 are 1, 2, 2, 2, 1, 2, 2, 1,

AC C

and 2, respectively, the SN ratio corresponding to the profit of retailer 1 is the largest; (iii) when the levels of the parameters θ, cm , hr , hm , hd , e1 , e2 , ρ1 , and ρ2 are 2, 1, 2, 1, 2, 2, 2, 1, and 1, respectively, the SN ratio corresponding to the profit of retailer 2 is the largest; (iv) when the levels of the parameters θ, cm , hr , hm , hd , e1 , e2 , ρ1 , and ρ2 are 1, 1, 1, 1, 1, 1, 1, 1, 1, or 1, 1, 2, 2, 2, 1, 1, 1, 2, respectively, the SN ratio corresponding to the carbon emissions of the supply chain is the smallest. Figure 4 shows the following: (i) Based on the SN ratio, the optimal conditions for the manufacturer that may help her to get more 29

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Table 4. SN ratios of the profits and carbon emissions

hr

hm

hd

e1

e2

ρ1

ρ2

1

1

1

1

1

1

1

1

1

1

2

1

1

1

1

1

2

2

2

2

3

1

1

2

2

2

1

1

1

2

4

1

2

1

2

2

1

2

2

1

5

1

2

2

1

2

2

1

2

1

6

1

2

2

2

1

2

2

1

2

7

2

1

2

2

1

1

2

2

1

8

2

1

2

1

2

2

2

1

1

9

2

1

1

2

2

2

1

2

2

10

2

2

2

1

1

1

1

2

2

11

2

2

1

2

1

2

1

1

1

12

2

2

1

1

2

1

2

1

2

Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions Manufacturer Retailer 1 Retailer 2 Carbon emissions

EP

AC C

154,930 39,269 35,342 18,048 213,980 32,400 29,160 740 167,040 39,368 23,621 18,039 178,070 28,171 38,031 13,374 191,420 30,819 41,606 6,043 199,370 49,079 29,447 588 178,730 27,939 37,718 13,411 184,540 48,347 43,512 1,216 204,950 30,393 27,354 6,489 182,970 26,439 23,795 17,899 177,130 45,933 41,340 6,406 178,150 42,151 25,290 13,375

30

2 2

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cm

1 2 2 1 Response values 142,930 39,269 35,342 18,048 201,980 32,400 29,160 740 155,040 39,368 23,621 18,039 166,070 28,171 38,031 13,374 17,9420 30,819 41,606 6,043 187,370 49,079 29,447 588 166,730 27,939 37,718 13,411 172,540 48,347 43,512 1,216 192,950 30,393 27,354 6,489 170,970 26,439 23,795 17,899 165,130 45,933 41,340 6,406 166,150 42,151 25,290 13,375

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θ

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No.

1 1

164,490 39,015 35,114 24,085 212,220 29,635 26,672 8875 176,540 39,104 23,462 24,091 184,890 27,264 36,806 19,898 194,020 28,811 38,895 13,464 199,000 44,862 26,917 8815 185,130 27,150 36,653 19,583 185,130 44,472 40,024 8,898 206,470 28,579 25,721 13,439 192,020 26,316 23,684 23,726 180,410 43,168 38,851 13,414 184,620 40,940 24,564 19,583

SC

cp C

SN ratio

156,490 39,015 35,114 24,085 204,220 29,635 26,672 8875 168,540 39,104 23,462 24,091 176,890 27,264 36,806 19,898 186,020 28,811 38,895 13,464 191,000 44,862 26,917 8815 177,130 27,150 36,653 19,583 177,130 44,472 40,024 8,898 198,470 28,579 25,721 13,439 184,020 26,316 23,684 23,726 172,410 43,168 38,851 13,414 176,620 40,940 24,564 19,583

103.76 91.85 90.94 -86.56 106.36 89.81 88.89 -75.98 104.41 91.87 87.44 -86.56 104.91 88.85 91.46 -84.58 105.46 89.47 92.08 -80.37 105.76 93.41 88.97 -75.91 104.94 88.80 91.40 -84.50 105.09 93.31 92.39 -76.06 106.04 89.38 88.46 -80.47 105.20 88.42 87.51 -86.45 104.78 92.96 92.05 -80.43 104.91 92.37 87.93 -84.49

ACCEPTED MANUSCRIPT

profits are θ at level 2, cm at level 2, hr at level 2, hm at level 2, hd at level 1 or 2, e1 at level 2, e2 at level 2, ρ1 at level 2, and ρ2 at level 2. Parameters e1 , ρ1 , ρ2 , and e2 have a greater impact on the manufacturer’s

RI PT

profit, while parameters θ, cm , hr , hm , and hd have a lower impact on the manufacturer’s profit. Moreover, among these parameters, parameter e1 has the greatest impact on the manufacturer’s profit.

(ii) Based on the SN ratio, the optimal conditions for the retailer are θ at level 1, cm at level 2, hr at level 1 or 2, hm at level 2, hd at level 2, e1 at level 2, e2 at level 2, ρ1 at level 1, and ρ2 at level 2. Parameters

SC

ρ1 , e2 , and e1 have a greater impact on the profit of retailer 1, while parameters θ, cm , hr , hm , hd , and ρ2 have

impact on the profit of retailer 1.

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a lower impact on the profit of retailer 1. Moreover, among these parameters, parameter ρ1 has the greatest

(iii) Based on the SN ratio, the optimal conditions for the retailer are θ at level 1, cm at level 2, hr at level 2, hm at level 1 or 2, hd at level 1 or 2, e1 at level 2, e2 at level 2, ρ1 at level 2, and ρ2 at level 1. Parameters ρ2 and e1 have a greater impact on the profit of retailer 2 while parameters θ, cm , hr , hm , hd , e2 , and ρ1 have

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a lower impact on the profit of retailer 2. Moreover, among these parameters, parameter ρ2 has the greatest impact on the profit of retailer 2.

(iv) Based on the SN ratio, the optimal conditions for the supply chain that may help the supply chain

EP

have less carbon emissions are θ at level 2, cm at level 2, hr at level 1, hm at level 2, hd at level 2, e1 at level

AC C

1, e2 at level 1, ρ1 at level 2, and ρ2 at level 1. Parameters e1 and e2 have a greater impact on the supply chain’s carbon emissions, while parameters θ, cm , hr , hm , hd , ρ1 , and ρ2 have a lower impact on the supply chain’s carbon emissions.

In summary, from Figure 4, we can obtain the following management insights: When the supply chain is coordinated by the RS contract, the manufacturer should focus on how to reduce the value of e1 , and retailer i (i = 1, 2) should focus on negotiating the value of ρi (i = 1, 2), while the values of e1 and e2 play an key role in reducing carbon emissions for the supply chain. 31

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6. Concluding remarks The carbon cap-and-trade regulation and investment in green technologies are two effective methods to

RI PT

curb carbon emissions. This paper develops one-manufacturer and two-competing-retailer low-carbon supply chain models for deteriorating products under VMI. The manufacturer’s operational activities are the main sources of carbon emissions, and the manufacturer may invest in green technologies to curb carbon

SC

emissions under the cap-and-trade regulation. Considering that the demand of each retailer depends on the green technology level and the selling prices of both competing retailers, we formulate decentralized and

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centralized supply chain models. We also compare the profits and carbon emissions between decentralized and centralized models and propose the RS contract to coordinate the decentralized supply chain. Finally, we illustrate the theoretical results with a numerical example and use the Taguchi method to conduct sensitivity analysis on key parameters and study the impacts of carbon footprint on supply chain coordination.

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The results provide the following managerial insights. (i) When the manufacturer cooperates with the two competing retailers, there exists an upper bound on the profit penalty for decentralization and the carbon emissions of the system may be lower than those of the decentralized system. (ii) When the market demand

EP

rate is formulated as a linear function of the manufacturer’s green technology level and the two competing retailers’ selling prices, the low-carbon supply chain under VMI can be coordinated perfectly. (iii) When

AC C

the RS contract is acceptable for all three members of the supply chain, the supply chain benefits from a relatively smaller value of the unit trading price of carbon emission permit or a relatively larger value of the carbon cap. An increase in the unit trading price of carbon emission permit leads to a decrease in the carbon emissions and the carbon cap has no impact on the carbon emissions. (iv) When the RS contract is acceptable for all three members of the supply chain, an increase in one of the parameters θ, e1 , e2 , hr , and hm leads to a decrease in the profit of the whole supply chain and an increase in carbon emissions. In particular, the robust sensitivity analysis shows that among all parameters, e1 has the greatest impact on the 32

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manufacturer’s profit and carbon emissions, and ρi (i = 1, 2) has the greatest impact on the profit of retailer i.

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In this paper, we assume that the manufacturer and the retailer are risk-neutral. It would be interesting to investigate the impact of risk attitude on such low-carbon supply chain coordination with vendor-managed deteriorating product inventory in further research. We may extend such one-manufacturer and two-retailer supply chain to that with multiple manufacturers and multiple retailers for vendor-managed deteriorating

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Appendices

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Appendix A. Proof of Theorem 4.1

For any given w1 , w2 and s, from Eq. (16), we take the first partial derivative of Πdri(pi ) with respect to

EP

pi and have

∂Πdr1 (p1 ) = a + αs − 2βp1 + γp2 + βw1 ∂p1

AC C

(A.1)

and

∂Πdr2 (p2 ) = a + αs − 2βp2 + γp1 + βw2 . ∂p2

From ∂Πdr2 (p2 ) ∂p2

∂Πdr1 (p1 ) ∂p1

= 0, we have p∗1 (p2 ) =

= 0, we have

d p∗2 (p1 ) d p1

=

γ 2β .

a+αs+γp2 +βw1 2β

and further have

(A.2)

d p∗1 (p2 ) d p2

=

γ 2β .

Similarly, from

So p∗1 (p2 ) and p∗2 (p1 ) have one and only one intersection, which leads to 37

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a unique Nash equilibrium when 2β > γ. ∂Πdr1 (p1 ) ∂p1

= 0 and

∂Πdr2 (p2 ) ∂p2

= 0 yield

p1 =

(2β + γ)(a + αs) + 2β2 w1 + βγw2 4β2 − γ2

p2 =

(2β + γ)(a + αs) + 2β2 w2 + βγw1 4β2 − γ2

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SC

and

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Solving

Taking the second partial derivative of Πdri (pi ) with respect to pi , we have

∂2 Πdri (pi ) ∂p2i

(A.3)

(A.4)

= −2β < 0, which

means that Πdr1 (p1 ) and Πdr2 (p2 ) are concave in the selling price p1 and p2 , respectively. Substituting Eqs. (5), (6), (9) and Eq. (11) into Eq. (17), using Eqs. (10), (12), (13), (14), and Eq. (15)

Πdm (s, w1 , w2 ) =

2 X i=1

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to simplify it, we have

wi Di (s, p1 , p2 ) − [cm + c p (e1 − b1 s)]Q − [hr + θhd + c p (e2 − b2 s)]

2 X

Iri (A.5)

i=1

EP

1 −[hm + θhd + c p (e2 − b2 s)]Im − ηs2 + c p C 2 = (a + αs − βp1 + γp2 )(w1 − A + bc p φs) + (a + αs − βp2 + γp1 )(w2 − A + bc p φs)

AC C

1 − ηs2 + c pC. 2

Substituting Eqs. (A.3) and (A.4) into Eq. (A.5), we have

Πdm (s, w1 , w2 ) =

{(a + αs)β(2β + γ) + β[(γ2 − 2β2 )w1 + βγw2 ]} (w1 − A + bc p φs) (A.6) 4β2 − γ2 {(a + αs)β(2β + γ) + β[(γ2 − 2β2 )w2 + βγw1 ]} 1 + (w2 − A + bc p φs) − ηs2 + c p C. 2 2 2 4β − γ

38

ACCEPTED MANUSCRIPT

Taking the first partial derivatives of Πdm (s, w1 , w2 ) with respect to s, w1 and w2 yields

∂Πdm (s, w1 , w2 ) ∂s

bβc p φ [2(a + αs)(2β + γ) + (γ2 − 2β2 + βγ)(w1 + w2 )] − ηs 4β2 − γ2 αβ(2β + γ) + (w1 + w2 − 2A + 2bc p φs), 4β2 − γ2

(A.7)

RI PT

=

(A.8)

SC

∂Πdm (s, w1 , w2 ) β[(a + αs)(2β + γ) + (bc p φs − A)(γ2 − 2β2 + βγ)] + 2β[(γ2 − 2β2 )w1 + βγw2 ] = ∂w1 4β2 − γ2

M AN U

and

∂Πdm (s, w1 , w2 ) β[(a + αs)(2β + γ) + (bc p φs − A)(γ2 − 2β2 + βγ)] + 2β[βγw1 + (γ2 − 2β2 )w2 ] = . (A.9) ∂w2 4β2 − γ2

From Eqs. (A.7), (A.8) and (A.9), we further have =

∂2 Πdm (s,w1 ,w2 ) ∂s∂w2

=

β[bc p φ(γ−β)+α] , 2β−γ

TE D

2β(γ2 −2β2 ) ∂2 Πdm (s,w1 ,w2 ) , ∂s∂w1 4β2 −γ2

∂2 Πdm (s,w1 ,w2 ) ∂s2

and

=

4bc p φαβ ∂2 Πdm (s,w1 ,w2 ) 2β−γ −η, ∂w2

∂2 Πdm (s,w1 ,w2 ) ∂w1 ∂w2

=

1

2β2 γ . 4β2 −γ2

=

∂2 Πdm (s,w1 ,w2 ) ∂w22

=

Let ∇2 Πdm be the Hessian

matrix of Πdm (s, w1 , w2 ). We have

∂2 Πdm (s,w1 ,w2 ) ∂s2

∂2 Πdm (s,w1 ,w2 ) ∂s∂w1

∂2 Πdm (s,w1 ,w2 ) ∂s∂w2

∂2 Πdm (s,w1 ,w2 ) ∂w1 ∂s

∂2 Πdm (s,w1 ,w2 ) ∂w21

∂2 Πdm (s,w1 ,w2 ) ∂w1 ∂w2

∂2 Πdm (s,w1 ,w2 ) ∂w2 ∂s

∂2 Πdm (s,w1 ,w2 ) ∂w2 ∂w1

∂2 Πdm (s,w1 ,w2 ) ∂w22

AC C

EP

     2 d ∇ Πm =    

Using γ < β and η > −β[bc p φ(β−γ)−α]2 (2β−γ)(β−γ)

< 0,

[α+bc p φ(β−γ)]2 , β−γ

∂2 Πdm (s,w1 ,w2 ) ∂w21

< 0 and

we have η > ∂2 Πdm (s,w1 ,w2 ) ∂w22

β[α+bc p φ(β−γ)]2 (2β−γ)(β−γ) ,

         

(A.10)

and further have

< 0. Rearranging η >

∂2 Πdm (s,w1 ,w2 ) ∂s2

β[α+bc p φ(β−γ)]2 (2β−γ)(β−γ) ,

<

we obtain

β[α − bc p φ(β − γ)]2 < η(2β − γ)(β − γ) − 4bc p φαβ(β − γ). We further have ∂2 Πdd (s, w1 , w2 ) 2 ∂2 Πdm (s, w1 , w2 ) ∂2 Πdm (s, w1 , w2 ) −( ) · ∂s∂w1 ∂s2 ∂w21 39

(A.11)

ACCEPTED MANUSCRIPT

β{2(γ2 − 2β2 )[4bc p φαβ − η(2β − γ)] − β(2β + γ)[α − bc p φ(β − γ)]2 } (2β + γ)(2β − γ)2 β(γ2 − 2β2 − βγ)[4bc p φαβ − η(2β − γ)] > (2β + γ)(2β − γ)2 2 β (β + γ)[α − bc p φ(β − γ)]2 > 0. > (2β + γ)(2β − γ)(β − γ)

RI PT

=

From Eq. (A.10), we have

SC

4β2 (β + γ){(2β − γ)(β − γ)[4bc p φαβ − (2β + γ)η] + β(2β + γ)[α − bc p φ(β − γ)]2 } (A.12) (2β − γ)2 (2β + γ)2 −32β3 (β2 − γ2 )bc p φα < < 0, (2β − γ)2 (2β + γ)2

M AN U

|∇2 Πdm | =

which means that the Hessian matrix ∇2 Πdm is negative definite matrix. Hence, we have that Πdm (s, w1 , w2 ) is jointly concave in s, w1 and w2 . ∂Πdm (s,w1 ,w2 ) ∂s

= 0,

∂Πdm (s,w1 ,w2 ) ∂w1

have

∂Πdm (s,w1 ,w2 ) ∂w2

=

wd∗ 2

=

= 0, from Eqs. (A.7), (A.8), and (A.9), we

a + (β − γ)A + [α − bc p φ(β − γ)]s∗d

(A.13)

2(β − γ)

EP

wd∗ 1

= 0, and

TE D

Solving

AC C

and

s∗d =

β[a − (β − γ)A][α + bc p φ(β − γ)] , (2β − γ)(β − γ)η − β[α + bc p φ(β − γ)]2

(A.14)

respectively.

Substituting Eq. (A.13) into Eqs. (A.3) and (A.4), we have

d∗ pd∗ 1 = p2 =

a(3β − 2γ) + β(β − γ)A + [α(3β − 2γ) − bβc p φ(β − γ)]s∗d 2(β − γ)(2β − γ) 40

.

(A.15)

ACCEPTED MANUSCRIPT



From Eqs. (A.13), (A.14) and (A.15), we have

d∗ d∗ = a + αs∗d − βpd∗ a + αs∗d − βpd∗ 2 + γp1 1 + γp2

β{a − (β − γ)A + [α + bc p φ(β − γ)]s∗d } 2(2β − γ) (β −

γ)ηs∗d

M AN U

= =

2[α + bc p φ(β − γ)]

d∗ d∗ pd∗ = pd∗ 1 − w1 2 − w2

a − (β − γ)A + [α + bc p φ(β − γ)]s∗d 2(2β − γ) (β − γ)ηs∗d

2β[α + bc p φ(β − γ)]

EP

=

∗ ∗ d∗ wd∗ 1 − A + bc p φsd = w2 − A + bc p φsd

AC C

and

(B.2)

TE D

=

(B.1)

SC

= a + αs∗d + (γ − β)pd∗ 1

RI PT

Appendix B. Proof of Theorem 4.2

= =

(B.3)

a − (β − γ)A + [α + bc p φ(β − γ)]s∗d 2(β − γ) (2β − γ)ηs∗d . 2β[α + bc p φ(β − γ)]

Substituting Eqs. (B.1), (B.2) and (B.3) into Eq. (16) and Eq. (A.5), we have

d∗ d Πdr1 (pd∗ 1 ) = Πr2 (p2 ) =

(β − γ)2 η2 (s∗d )2 4β[α + bc p φ(β − γ)]2 41

(B.4)

ACCEPTED MANUSCRIPT

and

=

(2β − γ)(β − γ)η2 (s∗d )2 2β[α + bc p φ(β −

γ)]2

1 − η(s∗d )2 + c p C 2

η{(2β − γ)(β − γ)η − β[α + bc p φ(β − γ)]2 }(s∗d )2 2β[α + bc p φ(β − γ)]2

(B.5)

RI PT

d∗ Πdm (s∗d , wd∗ 1 , w2 ) =

+ c pC.

Substituting Eqs. (5), (6) and (9) into Eq. (11), after simplification with Eqs. (10) and (B.1), the total

SC

emissions amount is given by

∗ ∗ d ∗ Em (sd ) = 2[a + αs∗d + (γ − β)pd∗ 1 ][(e1 − bsd )φ1 + (e2 − bsd )(φ2 + φ3 )]

M AN U

(β − γ)η[e1 φ1 + e2 (φ2 + φ3 ) − bφs∗d ]s∗d α + bc p φ(β − γ)

.



TE D

=

(B.6)

Appendix C. Proof of Theorem 4.4

Substituting Eqs. (5), (6), (9) and Eq.(11) into Eq. (24), using Eqs. (10), (12), (13), (14) and Eq. (15)

2 X

pi Di (s, p1 , p2 ) − [cm + c p (e1 − b1 s)]Q − [hr + θhd + c p (e2 − b2 s)]

AC C

Πc (s, p1 , p2 ) =

EP

to simplify it, we have

i=1

2 X

Iri

(C.1)

i=1

1 −[hm + θhd + c p (e2 − b2 s)]Im − ηs2 + c pC 2

= (a + αs − βp1 + γp2 )(p1 − A + bc p φs) + (a + αs − βp2 + γp1 )(p2 − A + bc p φs) 1 − ηs2 + c p C 2

42

ACCEPTED MANUSCRIPT

∂Πc (s, p1 , p2 ) ∂s ∂Πc (s, p1 , p2 ) ∂p1

= α[p1 + p2 + 2(bc p φs − A)] + bc p φ[2(a + αs) + (γ − β)(p1 + p2 )] − ηs

(C.2)

RI PT

Taking the first partial derivatives of Πc (p, s) with respect to s, p1 and p2 yields

(C.3)

= a + αs + 2γp2 − 2βp1 + (γ − β)(bc p φs − A)

SC

and

M AN U

∂Πc (s, p1 , p2 ) = a + αs + 2γp1 − 2βp2 + (γ − β)(bc p φs − A). ∂p2

(C.4)

Taking the second partial derivatives of Πc (s, p1 , p2 ) with respect to s, p1 and p2 , we have 4αbc p φ−η, and

∂2 Πc (s,p1 ,p2 ) ∂s∂p1

∂2 Πc (s,p1 ,p2 ) ∂p22

= α+(γ−β)bc p φ,

∂2 Πc (s,p1 ,p2 ) ∂s∂p2

= −2β.

= α+(γ−β)bc p φ,

∂2 Πc (s,p1 ,p2 ) ∂p21

= −2β,

∂2 Πc (s,p1 ,p2 ) ∂s2

∂2 Πc (s,p1 ,p2 ) ∂p1 ∂p2

=

= 2γ,

TE D

Let ∇2 Πc be the Hessian matrix of Πc (s, p1 , p2 ). We have

∂2 Πc (s,p1 ,p2 ) ∂s2

∂2 Πc (s,p1 ,p2 ) ∂s∂p1

∂2 Πc (s,p1 ,p2 ) ∂s∂p2

∂2 Πc (s,p1 ,p2 ) ∂p1 ∂s

∂2 Πc (s,p1 ,p2 ) ∂p21

∂2 Πc (s,p1 ,p2 ) ∂p1 ∂p2

∂2 Πc (s,p1 ,p2 ) ∂p2 ∂s

∂2 Πc (s,p1 ,p2 ) ∂p2 ∂p1

∂2 Πc (s,p1 ,p2 ) ∂p22

EP

     ∇2 Πc =    

[α+(β−γ)bc p φ]2 β−γ

AC C

With β + γ > 0, we have ∂2 Πc (s,p1 ,p2 ) ∂s2

> 4bαc p φ +

[α−(β−γ)bc p φ]2 . 2β

         

From η >

(C.5)

[α+(β−γ)bc p φ]2 , β−γ

we have

= 4αbc p φ − η < 0 and

∂2 Πc (s, p1 , p2 ) ∂2 Πc (s, p1 , p2 ) ∂2 Πc (s, p1 , p2 ) 2 · − ( ) ∂s∂p1 ∂s2 ∂p21

= 2β(η − 4αbc p φ) − [α − (β − γ)bc p φ]2 > 0.

43

(C.6)

ACCEPTED MANUSCRIPT

We further have

which means that Πc (s, p1 , p2 ) is jointly concave in s, p1 and p2 . = 0,

∂Πc (s,p1 ,p2 ) ∂p1

s∗c =

= 0, and

∂Πc (s,p1 ,p2 ) ∂p2

= 0 yields

SC

∂Πc (s,p1 ,p2 ) ∂s

[a − (β − γ)A][α + (β − γ)bc p φ] (β − γ)η − [α + (β − γ)bc p φ]2

and

a + (β − γ)A + [α − (β − γ)bc p φ]s∗c . 2(β − γ)

TE D

c∗ pc∗ 1 = p2 =

M AN U

Solving

(C.7)

RI PT

|∇2 Πc | = 4(β + γ){[α + (β − γ)bc p φ]2 − (β − γ)η} < 0,

(C.8)

(C.9)



Appendix D. Proof of Theorem 4.5

EP

From Eqs. (C.8) and (C.9), we have

AC C

c∗ c∗ = a + αs∗c − βpc∗ a + αs∗c − βpc∗ 1 + γp2 2 + γp1

= =

(D.1)

a − (β − γ)A + [α + (β − γ)bc p φ]s∗c 2 ∗ η(β − γ)sc 2[α + (β − γ)bc p φ]

and

∗ ∗ c∗ pc∗ 1 − A + bc p φsc = p2 − A + bc p φsc

44

(D.2)

ACCEPTED MANUSCRIPT

=

a − (β − γ)A + [α + (β − γ)bc p φ]s∗c 2(β − γ) η(β − γ)s∗c . 2(β − γ)[α + (β − γ)bc p φ]

Substituting Eqs. (C.8), (D.1) and (D.2) into Eq. (C.1), we have

=

{a − (β − γ)A + [α + (β − γ)bc p φ]s∗c }2 1 ∗ 2 − η(sc ) + c p C 2(β − γ) 2 2 2 ∗ η{(β − γ)η − [α + (β − γ)bc p φ] }(sc ) + c p C. 2[α + (β − γ)bc p φ]2

(D.3)

SC

c∗ Πc (s∗c , pc∗ 1 , p2 ) =

RI PT

=

M AN U

Substituting Eqs. (5), (6) and (9) into Eq. (11), after simplification with Eqs. (10) and (D.1), the total emissions amount in the centralized system is given by

c ∗ ∗ ∗ Em (sc ) = 2[a + αs∗c + (γ − β)pc∗ 1 ][(e1 − bsc )φ1 + (e2 − bsc )(φ2 + φ3 )]

(D.4)

(β − γ)η[e1 φ1 + e2 (φ2 + φ3 ) − bφs∗c ]s∗c . α + bc p φ(β − γ)

TE D

=

EP



Appendix E. Proof of Theorem 4.8

AC C

From Eqs. (B.4) and (B.5), we have

2 X

d∗ d∗ d ∗ Πdri(pd∗ i ) + Πm (sd , w1 , w2 )

(E.1)

i=1

=

η{(β − γ)(3β − 2γ)η − β[α + bc p β(β − γ)]2 }(s∗d )2 2β[α + bc p φ(β − γ)]2

45

+ c p C.

ACCEPTED MANUSCRIPT

Comparing Eq. (D.3) with Eq. (E.1) and using

where ψ =

<

x y

with x > y > 0 and z > 0, we have

Πc (p∗c , s∗c ) < ψ, d∗ Πdr (p∗d ) + Πdm (s∗d , wd∗ 1 , w2 )

{(2β−γ)(β−γ)η−β[α+bc p φ(β−γ)]2 }2 . β{(β−γ)η−[α+bc p φ(β−γ)]2 }{(β−γ)(3β−2γ)η−β[α+bc p φ(β−γ)]2 }

Appendix F. Proof of Theorem 4.9

[e1 φ1 + e2 (φ2 + φ3 ) − bφs∗c ]s∗c . [e1 φ1 + e2 (φ2 + φ3 ) − bφs∗d ]s∗d

(E.3)

M AN U

d (s∗ ) Em d

=

SC

Comparing Eq. (B.4) with Eq. (D.4) yields

c (s∗ ) Em c

(E.2)

RI PT

1<

x+z y+z



TE D

Taking the first partial derivative of Πrs r (pi ) with respect to pi , i = 1, 2, from Eq. (27), we have

∂Πrs r (p1 ) = ρ1 (a + αs) − 2ρ1 βp1 + ρ1 γp2 + βw1 ∂p1

EP

and

(F.1)

AC C

∂Πrs r (p2 ) = ρ2 (a + αs) − 2ρ2 βp2 + ρ2 γp1 + βw2 . ∂p2

For i = 1, 2, using ∂Πrs r (p2 ) ∂p2

∂2 Πrs r (pi ) ∂p2i

= −2βρi < 0, we have that Πrs r (pi ) is concave in pi . From

(F.2)

∂Πrs r (p1 ) ∂p1

= 0 and

= 0, we have

prs 1 =

ρ1 ρ2 (2β + γ)(a + αs) + 2ρ2 β2 w1 + ρ1 βγw2 ρ1 ρ2 (4β2 − γ2 )

46

(F.3)

ACCEPTED MANUSCRIPT

and

ρ1 ρ2 (2β + γ)(a + αs) + 2ρ1 β2 w2 + ρ2 βγw1 ρ1 ρ2 (4β2 − γ2 )

(F.4)

RI PT

prs 2 =

∗ ∗ rs ∗ Substituting prs 1 = pc , p2 = pc and srs = sc into Eqs. (F.3) and (F.4), and using Eq. (C.9), we have

ρ1 {aγ + (β − γ)(2β − γ)A + [αγ − (2β − γ)(β − γ)bcP φ]s∗c } 2β(β − γ)

wrs 2 =

ρ2 {aγ + (β − γ)(2β − γ)A + [αγ − (2β − γ)(β − γ)bcP φ]s∗c } 2β(β − γ)

(F.5)

SC

wrs 1 =

TE D

From Eqs. (F.5) and (F.6), we have

M AN U

and

rs wrs 1 ρ2 = w2 ρ1 .

(F.6)

(F.7)

EP

When αγ ≥ (2β − γ)(β − γ)bcP φ, we have wrs i > 0, i = 1, 2 from Eqs. (F.5) and (F.6). On the other hand, when αγ < (2β − γ)(β − γ)bcP φ, we have

>

A bc p φ .

From Eq. (15), we have

> min{ be11 , be22 }. Using assumption s∗c < min{ be11 , be22 }, we further have wrs i > 0 for 0 < ρi < 1 and

AC C

A bc p φ

aγ+(β−γ)(2β−γ)A (2β−γ)(β−γ)bc p φ−αγ

i = 1, 2. According to the contract structure, the corresponding decision variables including selling price, green technology level, wholesale price and fraction parameter of each retailer’s revenue can be solved by Eqs. (F.5) and (F.6), implying that the supply chain is coordinated.

47

ACCEPTED MANUSCRIPT

Appendix G. Proof of Theorem 4.10

rs ρi prs i − wi =

ρi (β − γ)ηs∗c , i = 1, 2. 2β[α + (β − γ)bc p φ]

ρi η2 (β − γ)2 (s∗c )2 , i = 1, 2. 4β[α + (β − γ)bc p φ]2

M AN U

rs Πrs ri (pi ) =

SC

Substituting Eq. (G.1) into Eq. (27), we have

RI PT

From Eqs. (C.8), (C.9), (F.5) and (F.6), we have

We further have

2 X [2β − (β − γ)(ρ1 + ρ2 )]ηs∗c ∗ rs . [wrs − A + bc φs + (1 − ρ )p ] = p c i i i 2β[α + (β − γ)bc φ] p i=1

(G.1)

(G.2)

(G.3)

η{(β − γ)η[2β − (β − γ)(ρ1 + ρ2 )] − 2β[α + (β − γ)bc p φ]2 }(s∗c )2 + c p C. 4β[α + (β − γ)bc p φ]2

(G.4)

EP

rs rs Πrs m (srs , w1 , w2 ) =

TE D

Substituting Eqs. (G.3) and (D.1) into Eq. (30), we have

The RS contract is acceptable for each member of the supply chain, if and only if the profit of each

AC C

member in the RS contract is not less than that in the decentralized system. Comparing Eq. (B.4) with Eq. (G.2) yields

ρi ≥ {

β(β − γ)η − β[α + (β − γ)bc p φ]2 }2 , i = 1, 2. (2β − γ)(β − γ)η − β[α + (β − γ)bc p φ]2

48

(G.5)

ACCEPTED MANUSCRIPT

Comparing Eq. (B.5) with Eq. (G.4) yields

With β > γ and η >

[α+(β−γ)bc p φ]2 , β−γ

2β{(β − γ)η − [α + (β − γ)bc p φ]2 } . (2β − γ)(β − γ)η − β[α + (β − γ)bc p φ]2

RI PT

ρ1 + ρ2 ≤

(G.6)

we have that the values of ρ1 and ρ2 satisfying Eqs. (G.5) and (G.6) exist,

AC C

EP

TE D

M AN U

SC

which completes the proof of Theorem 4.10.

49