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Computers & Industrial Engineering 128 (2019) 807-830 ELSEVIER Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie Modeling carbon regulation policies in inventory decisions of a multi-stage r® green supply chain: A game theory approach Kourosh Halat, Ashkan Hafezalkotob* College of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran ARTICLE INFO ABSTRACT Keywords: Green supply chain Inventory cost Carbon emission Government regulation Stackelberg game Governments and policymakers around the world put their best efforts to control the pollutions and climate change. Thus, they set various regulations to reduce greenhouse gases and carbon footprints. It is expected that firms should follow these regulations while maintaining their profitability. In this regard, firms can manage their carbon emissions across their supply chain (SC) by inventory management, since operational adjustments may affect the amount of carbon emissions, such as changing the production quantity and the frequency of trans¬ portations. This study applies a Stackelberg game between the government and a multi-stage green supply chain (GSC), in which the government’s goal is to maximize social welfare and that of the GSC is to minimize its cost. First, we formulate the inventory cost and carbon emission of a multi-stage GSC under two decision-making structures: non-coordinated and coordinated GSCs. Second, we develop eight bi-level mathematical program¬ ming models considering the structure of the GSC and the regulations of the government including carbon cap, carbon tax, carbon trade, and carbon offset. Then two solution approaches will present for solving problems based on the type of the GSC structure. This study examines the effect of coordination and carbon regulations on inventory cost, carbon emission and the objective function of the government. 1. Introduction The concurrent development of industries and environment has become a new challenge in the world. The emission of greenhouse gases (GHGs), especially carbon, from industrial activities is one of the main issues for the environment and is the primary cause of global warming (Stern, 2006). In recent years, the regulators and governments have paid significant attention to this problem. Consequently, many policies and regulations have formulated. For instance, the Kyoto Protocol had ratified in an international treaty, and as a result, a cap-and-trade system had been established. According to this system, a firm may sell its excess emission from the assigned amount units through the emis¬ sion trading market. An emission trading system (ETS) is an incentive policy instrument for managing the emission of GHGs (Du, Ma, Fu, Zhu, & Zhang, 2015). Moreover, policymakers have developed other plans to reduce the emission of carbon. For example, the US EPA established carbon emission allowance for power plants at the national level (Kuo, Hong, & Lin, 2016), and some countries such as Australia implemented the carbon taxation scheme (Zakeri, Dehghanian, Fahimnia, & Sarkis, 2015). In order to meet the requirements of governments, companies re¬ quire managing their environmental impacts and wastes among all stages of their supply chain (SC) while maintaining the profitability. In this area, firms need to integrate decisions in their strategic plan and evaluate them on a continuous basis, as various methods have devel¬ oped in the literature to determine performance rate of organization strategies (see Sobhanallahi, Gharaei, & Pilbala, 2016a, 2016b). How¬ ever, the companies seem more interested in increasing their profits rather than investing in carbon footprint reduction, and thus reducing the emission of carbon has become a critical challenge for companies and governments (Zhao, Liu, Zhang, & Huang, 2017). The green supply chain management (GSCM) can help companies to save energy, reduce pollution and continuously do their business by considering environ¬ mental impact and resource efficiency (Hu & Li, 2011). Therefore, in¬ tegrating green policies with inventory and production systems is vital for business success (Gharaei, Karimi, & Shekarabi, 2018; Gharaei, Pasandideh, & Akhavan Niaki, 2018). A firm needs to manage its carbon footprint across the SC to meet the governments’ regulations. Various methods can deploy in the GSCM to reduce carbon footprints such as redesigning the product or packaging, using new technologies in manufacturing, and using efficient vehicles for product delivery. Meanwhile, many studies showed that inventory management could be very effective for the reduction of carbon emission (see Bouchery, Ghaffari, Jemai, & Dallery, 2012; Chen & Monahan, 2010; Hovelaque & "Address: College of Industrial Engineering, Islamic Azad University, South Tehran Branch, Entezari Ally, Oskoui St, Choobi Brg, Tehran 1151863411, Iran. E-mail addresses: st khalat@azad.ac.ir (K. Halat), ajiafez@azad.ac.ir, hafezalkotob@iust.ac.ir (A. Hafezalkotob). https://doi.Org/10.1016/j.cie.2019.01.009 Received 25 June 2018; Received in revised form 27 December 2018; Accepted 3 January 2019 Available online 06 January 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved. K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Bironneau, 2015). The inventory policy of a firm determines variables which may affect the emission of carbon such as a number of deliveries, order or production quantities and storage amount. As a result, a firm can reduce its emission by operational adjustment and defining en¬ vironmental inventory policies. The carbon regulations and the need for profitability of companies will make the interaction between the government and SCs. Therefore, in this study, we formulate this interaction by game theory approach. The government imposes a regulation to control the emission of carbon and a green supply chain (GSC) requires to minimize its chain-wide costs and comply with the requirements of the government. Hence, there is a trade-off between the total cost of the GSC and the amount of the emission of carbon under governmental regulations. In the real world, a producer may use several players in its SC to release a product to market. An efficient management should consider the complete end-to-end process such as design, procurement, pro¬ duction planning, distribution, and fulfillment (Gharaei & Pasandideh, 2016). In this regard, we consider a multi-stage GSC that there is a manufacturer in its center and contains a number of suppliers, dis¬ tributors, and retailers. These members incur ordering, transportation, production, and inventory holding costs. Two different structures are applied to the GSC. In the first structure, the members use a co¬ ordination mechanism to manage the flow of the products as a cen¬ tralized network, which is called the coordinated GSC. In the second structure, the members make their decisions separately, which is called the non-coordinated GSC. On the other hand, in this study, on the reduction of carbon emis¬ sion, we examine four different regulations, namely carbon cap, carbon tax, carbon trade, and carbon offset, which the government sets them. In the carbon cap framework, the amount of carbon that emitted by a GSC should be less than a certain cap. In a carbon tax scheme, the government determines tax per unit of carbon emission as a penalty for the GSC. According to the carbon trade or the cap-and-trade policy, each carbon emitter can gain an allowance, that is, if firms produce carbon more than their allowance, they should purchase emission from the market as a penalty, but if firms generate carbon less than their allowance, they can sell their surplus emission and gain revenue. Therefore, it is an incentive system to encourage companies in en¬ vironmental efforts (Zakeri et al., 2015). The last scheme is the carbon offset or the cap-and-offset, in which specific caps are imposed on an emitter and just excess emission is penalized by purchasing emission credits (Schapiro, 2010). Each of these regulations can influence the optimal solution of the GSC in a different way. We propose different mathematical models to examine how firms managing their inventories under each regulation. By applying the above-mentioned regulations and the structure of the GSC to the problem, the Stackelberg game has found to be a useful tool for formulating interaction between the government and the GSC. The fundamental research questions are as follows: 1. How can the inventory cost and carbon emission of the multi-stage GSC be formulated? 2. How can different carbon emission regulations be modeled and can influence the optimal strategy between members? 3. What should be the Stackelberg equilibrium for these problems? This paper is organized as follows. The literature is reviewed in Section 2. The assumptions of the problem are discussed in Section 3. The mathematical models of the problem are presented in Section 4. The solution approach described in Section 5. A numerical example for solving the models, exploring different methods, is presented in Section 6. Finally, the conclusion of the study is presented in Section 7. 2. Literature review The literature is categorized into three parts. The first part reviews the researches that consider carbon footprints in inventory manage¬ ment. The second part analyzes the existing literature on games be¬ tween SCs and governments and the third part presents the research gap and contributions of this study. 2.1. Inventory management considering the emission of carbon In the past few years, the literature on carbon footprints in in¬ ventory management has increased considerably. In this field, one of the leading studies is reported by Benjaafar, Li, and Daskin (2013). They developed models that consider both inventory cost and carbon emission. They investigated the effect of different regulations, such as emission caps, taxes on emission, cap-and-offset, and cap-and-trade, on the basic inventory models. Similarly, a model that incorporates carbon emission parameters into the classical EOQ model developed by Hua, Cheng, and Wang (2011) and Chen, Benjaafar, and Elomri (2013). They found the optimal ordering quantity and emission under different carbon policies. Bouchery et al. (2012) proposed a multi-objective problem that integrates carbon emission criteria into an inventory model for a two-echelon SC. They proved that the operational adjust¬ ment is an effective way of reducing environmental impacts. Toptal, Ozlii, and Konur (2014) studied the classical EOQ model under three different carbon regulations, where a retailer has the option to invest in carbon reduction. They showed that the cap-and-trade policy gives more motivation for green investment. Schaefer and Konur (2014) analyzed a multi-item inventory model under the carbon-cap regula¬ tion. They used a genetic algorithm to solve this problem. Bazan, Jaber, and Zanoni (2015) developed two models for coordination of a two- echelon SC with considering GHG emission under tax policy. They showed that vendor-managed inventory (VMI) model has more benefit for the SC economically than classical coordination model. Also, Jiang, Li, Qu, and Cheng (2016) presented a VMI model for a SC consist of one manufacturer and a supplier under a carbon trade scheme. They showed that the optimal solution of the manufacturer depends on the carbon emission parameter related to transportation. Pasandideh, Niaki, and Gharaei (2015) developed VMI problem for the single vendor-buyer problem and used sequential quadratic programming approach to find the optimal production quantity. Zakeri et al. (2015) examined the effect of the carbon trade and carbon tax schemes on the cost of an Australian SC. They showed that the carbon trade scheme is a better option to measure the performance of a SC. A carbon trade me¬ chanism motivates companies to apply GSCM with financial incentives. Likewise, Ding, Zhao, An, and Tang (2016) showed that an incentive mechanism is more effective for reducing the pollutants that produced by a SC. Du et al. (2015) evaluated the optimal solution for the mem¬ bers of a two-echelon SC in a carbon trade system considering the Stackelberg game. Dye and Yang (2015) established an inventory re¬ plenishment model considering the deteriorating product and the trade credit condition under carbon trade and carbon offset policies. Garcia- Alvarado, Paquet, Chaabane, and Amodeo (2017) proposed a model for optimal replenishment inventory with remanufacturing under the cap- and-trade scheme. They used a Markov decision process to solve this model. Ma, Ji, Ho, and Yang (2018) presented a mathematical model for determining the optimal production quantity and supplier selection in a SC consisting of a manufacturer and suppliers under the carbon tax scheme. Gurtu, Jaber, and Searcy (2015) examined the impact of in¬ creasing carbon tax on the total cost of a SC. They found that tax can reduce environmental impacts but changes optimal cost too. Chen, Wang, Kumar, and Kumar (2016) developed a newsvendor model under carbon trade scheme and analyzed the trade-off between costs and the carbon emission reduction by investing in the green technology. In another study, a model to manage order quantity and green investment simultaneously were proposed by Nematollahi, Hosseini-Motlagh, and Heydari (2017). Ghosh, Jha, and Sarmah (2017) considered a two- echelon SC that aims to minimize inventory cost under carbon cap policy. Bouchery, Ghaffari, Jemai, and Tan (2017) suggested an 808 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 inventory model for a two-echelon SC with carbon emission. They showed that coordination among SC members can reduce both the cost and emission. In another study, two coordination mechanisms studied for a two-echelon closed loop SC with considering GHG emission and energy usage by Bazan, Jaber, and Zanoni (2017). Tsao, Lee, Chen, and Liao (2017) studied a newsvendor problem considering carbon emis¬ sion, trade credit, and product recycling under three different regula¬ tions including carbon cap, carbon tax, and carbon trade. Some studies examined other criteria along with the inventory problem. Yi, Xue, and Zhimin (2014) applied a pricing variable into the problem and presented a model for ordering and pricing strategies in a two-echelon SC with a stochastic demand under the carbon-cap scheme. Similarly, Hovelaque and Bironneau (2015) investigated a joint in¬ ventory and pricing problem in a firm under the carbon tax and carbon trade regulations. Zheng, Liao, and Yang (2016) further developed this problem for an SC with a stochastic demand under carbon trade reg¬ ulation. Moreover, Miao, Mao, Fu, and Wang (2018) formulated a pricing and production quantity problem of a manufacturer with re¬ manufacturing under the carbon tax and carbon trade regulations. Some studies, considered carbon emission parameters in a joint SC design and inventory planning problem and developed a multi-objec¬ tive optimization problem (Alhaj, Svetinovic, & Diabat, 2016; Jamshidi, Ghomi, & Karimi, 2012; Kannegiesser & Gunther, 2014; Mallidis, Vlachos, Iakovou, & Dekker, 2014). Given that in a real world, a SC may consist of more than two levels and should manage environmental aspects in its all stages. Gharaei and Pasandideh (2016) developed inventory model for multi-product and multi-stage production SC. Moreover, Gharaei and Pasandideh (2017a) developed inventory cost model for four-echelon SC and used sequen¬ tial quadratic programming to find solutions. In another similar study, Gharaei, Pasandideh, et al. (2018) examined the inventory model of a multi-echelon SC with imperfect products. Hoseini Shekarabi, Gharaei, and Karimi (2018) developed an optimal lot-sizing policy for a multi¬ product, multi-stage, centralized SC and used a decomposition method for solving the problem. Although many other research works, such as Ben-Daya, As’ad, and Seliaman (2013), Gharaei and Pasandideh (2017b), Gharaei, Pasandideh, and Arshadi Khamseh (2017), Nikandish, Eshghi, and Torabi (2009), Zhang (2013) and Zhao, Wu, and Yuan (2016), studied inventory control in a multi-stage SC, hut a few studies considered carbon emission parameters into the inventory model of a multi-stage GSC. Sarkar, Ganguly, Sarkar, and Pareek (2016) applied a coordination mechanism to a three-echelon SC and found the optimum inventory, production, and carbon emission cost. Huang, Wang, Zhang, and Pang (2016) established a Stackelberg game model in a three-echelon SC for optimizing inventory planning, product line design, supplier selection, and pricing with considering GHG emissions. They applied a genetic algorithm to find the optimal solution. Gharaei, Karimi, et al. (2018) developed a MINLP model for integrated multi¬ product and multi-buyer SC with considering quality inspection, VMI, and green policies under tax cost of GHG emissions. They proposed an OA/ER/AP algorithm to find the optimal solution of large-scale pro¬ blems. The studies in this field are summarized and categorized in Table 1. 2.2. Game between the government and the supply chain The regulators and governments all around the world impose dif¬ ferent policies and instruments on firms in order to protect the en¬ vironment and control global warming. These policies imply an inter¬ active relationship between governments and companies. In this regard, game theory has found as a useful approach for modeling and analyzing this relationship. In this regard, some researchers developed a two- person game between the government and a firm with discrete strate¬ gies, in which the government may choose to inspect or penalize firms and the firms accordingly decide to implement green strategies (see Dayi & Jianwei, 2011; Kim, 2015; Shutao & Jiangao, 2011; Zhao, Neighbour, McGuire, & Deutz, 2013). One of the first research studies that analyzed a game between the government and a SC at an operational planning level conducted by Sheu (2011). In which, a Nash bargaining model was presented to analyze competition between two SCs with reverse logistics under government financial intervention. According to that model, SCs aim to maximize their profit by determining the production quantity and price, and the government aims to maximize social welfare by de¬ termining tax and subsidy. Jin, Wang, and Mei (2011) developed a game model between the government and a SC and found the optimal solution for the production quantity, product price, and subsidy by maximizing social welfare. Sheu and Chen (2012) analyzed the effect of government intervention on two rival green SCs. They suggested that the combined use of the green taxation and subsidy strategy is more effective for the government rather than pure taxation scheme. Sheu and Gao (2014) examined the cooperation mechanism between the members of reverse SCs under tax and subsidy strategies of the gov¬ ernment. They showed that cooperation could increase the profit of the firms and utility of the government. In practice, the government usually is more powerful than SCs and therefore the Stackelberg game can he more suitable to characterize the interaction between them. Hafezalkotob (2015) developed a production and pricing game between the government and two regular and green SCs, which determines optimal pricing and tariffs under the revenue seeking and environmental protection policies of the government. In another similar study, Madani and Rasti-Barzoki (2017) presented a model for pricing, greening strategies, and tariffs in both centralized and decentralized GSCs. Hafezalkotob (2017) defined four policies, such as energy saving, revenue seeking, social welfare, and sustainable development, for the Stackelberg leader government. He evaluated the effect of each policy on a GSC with three different structures comprising competition, coopetition, and cooperation. Hong, Chu, Zhang, and Yu (2017) developed a Stackelberg model between a government and multiple firms under carbon trade scheme where each firm determines its production quantity and government determines the carbon cap and a target for carbon reduction. In addition, there are other studies in the field of carbon emission regulations that can be referred. Zhao, Neighbour, Han, McGuire, and Deutz (2012) adopted the game theory approach to analyze the optimal strategies for reducing material risk and carbon emission in a SC under governmental sanctions and in¬ centives policies. Kuo et al. (2016) studied a Nash game between the government and a firm, in which the government aims to maximize social welfare by imposing carbon tax regulation and the firm aims to maximize its profit. Wu, Liu, and Xu (2017) established an evolutionary game for the reduction of carbon emission and adopted a simulation approach for solving the game model. Hafezalkotob (2018) studied a Stackelberg game between a government and two rival GSCs under different intervention policies for energy saving. Table 2 lists and ca¬ tegorizes the researches that developed game theory models for inter¬ action between government and SC(s). 2.3. Research gap and contributions Although several studies developed inventory models in a GSC, the review of the literature shows that only a few applied an inventory model in a multi-stage GSC with considering carbon emission para¬ meters. On the other hand, to the best of our knowledge, no study has considered a game between the government and a multi-stage GSC aiming to optimize inventory cost and carbon emission policies. Thus, in this study, two main contributions are presented to fill this research gap. First, we consider carbon emission parameters in the in¬ ventory model of a multi-stage GSC, and also apply the coordinated and non-coordinated structures to the GSC. Second, we formulate the ob¬ jective of the government under four different regulations, namely carbon cap, carbon tax, carbon trade, and carbon offset, and we ex¬ amine the effect of each regulation on the optimal decisions of the GSC. 809 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Table 1 List of related studies in the inventory management under carbon regulations. References Model Echelons SC structure Decision variables of the SC Regulation Centralized Decentralized Carbon cap Carbon tax Carbon Carbon trade offset Bazan et al. (2015) MINLP Two-echelon X • Production quantity • Replenishment cycle X Bazan et al. (2017) MINLP Two-echelon X • Production quantity X Benjaafar et al. (2013) MILP Multi- X X • Order quantity X X X X echelon • Backorder quantity Bouchery et al. (2012) EOQ Two-echelon X • Order quantity X X Chen et al. (2013) EOQ Single¬ echelon • Order quantity X X X X Chen et al. (2016) NLP Two-echelon X • Order quantity • Price X Ding et al. (2016) NLP Two-echelon X X • Production quantity X Du et al. (2015) NLP Two-echelon X • Order quantity • Price X Garcia-Alvarado et al. (2017) NLP Two-echelon X • Production quantity X Ghosh et al. (2017) NLP Two-echelon X • Order quantity X Gurtu et al. (2015) MINLP Two-echelon X • Production quantity X Hovelaque and Bironneau EOQ Single- • Order quantity X X (2015) echelon • Price Hua et al. (2011) EOQ Single¬ echelon • Order quantity X Jiang et al. (2016) EPQ Two-echelon X • Production quantity X Miao et al. (2018) NLP Single- • Production quantity X X echelon • Price Toptal et al. (2014) EOQ Single¬ echelon • Order quantity X X X Tsao et al. (2017) Newsvendor Single- • Order quantity X X X echelon • Price • Credit period Zakeri et al. (2015) MILP Multi- X • Order quantity X X echelon • Warehouse selection Gharaei, Karimi, et al. (2018) MINLP Two-echelon X • Production quantity X This paper MINLP Multi¬ echelon X X • Replenishment cycles X X X X As a result, we develop eight scenarios according to the structure of the GSC and government regulations. We comprehensively compare the effects of different governmental carbon regulations on coordinated and non-coordinated GSC. 3. Problem description In this study, we consider a multi-stage GSC that operates under the carbon regulations of the government and delivers one type of product to the market. As shown in Fig. 1, at the first stage of the GSC, there are S suppliers, who supply raw materials and send them to a manufacturer. At the next stage, the manufacturer uses the raw materials at a constant rate to produce the finished products. Then, the finished products are delivered to distribution centers that are denoted by D, who transport products to R retailers. At the last stage of the GSC, each retailer re¬ ceives the products from only one distributor and fulfills the demand of a customer. This study assumes a constant demand rate with zero lead- time, finite production rates, and the shortage is not allowed. Each member of the GSC incurs the ordering, inventory holding, and transportation costs. In addition, the manufacturer incurs the produc¬ tion cost too. The objective of the GSC is to minimize its cost by de¬ termining the optimal inventory decisions. There are two approaches for the chain members to make their decisions. First, they can operate individually regardless of others performances (i.e., a decentralized SC). Second, they can use a coordination mechanism to manage the flow of products and production processes more efficiently and expect to re¬ duce their costs (i.e., a centralized SC). Three methods for inventory coordination in a centralized three-echelon SC were presented by Khouja (2003). We used one of these mechanisms, in which the re¬ plenishment cycle at each stage is an integer multiplier of the replenishment cycle of the subsequent stage. This coordination me¬ chanism used by several authors such as Fluang, Huang, and Newman (2011), Zhang (2013) and Zhao et al. (2016). These two approaches in the decision making of chain members are called the non-coordinated and coordinated GSC. Moreover, each firm in the GSC emits carbon according to its in¬ ventory management processes. For instance, each ordering cycle can generate carbon emission due to the weight of the empty vehicles used for order delivery and holding inventory generate carbon due to warehousing activities and electricity consumption of the warehouse equipment. In addition, delivering products and production can emit significant carbon mostly because of fuel consumption of the vehicles and manufacturing activities. This study assumes that the amount of carbon emission is linearly related to per ordering cycle, held in the inventory per unit time, per transportation cycle, and per unit pro¬ duced. This approach to modeling the carbon emission is used in many studies (see Bouchery et al., 2017; Hovelaque & Bironneau, 2015; Kannegiesser & Gunther, 2014; Schaefer & Konur, 2014; Toptal et al., 2014). In addition, the government imposes one of the four carbon reg¬ ulations, namely, carbon cap, carbon tax, carbon trade, and carbon offset, for managing the carbon footprints of the GSC. The government aims to maximize social welfare (SW) that incorporates the trade-off between the consumer surplus, the profit of the firms, government net revenue and environmental benefits (see Hafezalkotob, 2018; Hong et al., 2017; Sheu & Chen, 2012; Sheu & Gao, 2014). The interaction between the government and the GSC will evaluate by Stackelberg game theory, where, the government is a leader because of its authority and the GSC is a follower in decision making. We formulate the problem in eight scenarios based on the structure of the GSC and the type of 810 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 X P3 H 43 3 2 S' d •P s }_ J_ — CU ^ »—I E — 1 E—' E—' E — 1 C—? PJ UwUwh fl rP 42 X) X) 42 42 ‘G 42 X XI X s- 3 3 3 3 3 Ctf 43 ’ET p to iu iu CO E—' co E—' co E—< E—• £ tZ) U U U U ^ ^ ^ tc s s 811 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Government's regulation Decision making structure of the GSC Non-coordinated GSC Coordinated GSC o o Carbon cap Carbon tax Carbon trade Carbon offset Scenario 1 Scenario 3 Scenario 2 Scenario 6 Scenario 3 Scenario 7 Scenario 4 Scenario S Fig. 2. The government’s regulations versus decision-making structures of the GSC. government regulation that shown in Fig. 2. In the next section, the mathematical models of inventory and carbon emission will discuss. Each scenario will develop as a bi-level programming problem. Then the solutions to achieve Stackelberg equilibrium will present. there is no coordination between them, retailer r will order finished products from distributor d every cycle time, Based on the standard method for computing an economic order quantity, the total cost of the / h retailer is: *Cy = Kjdj T 4J 2 (1) The inventory cost of all retailers in the non-coordinated GSC is: i=r 1 3ij ( 2 ) Under a coordination mechanism, all retailers order within a common cycle time T, hence the total cost of all retailers is: A4 ; h 4 .2)4 ; T id = v (-^ + J J 4 “ T 2 (3) As stated in Section 3, the carbon emission of each firm in the GSC linearly associated with the frequency of delivery, the frequency of transportation, the average storage amount, and the production rate. Therefore, the total carbon emission of all retailers in the non-co¬ ordinated GSC is given by: J 4j J=rt D 4 .iTi '4 J J 4J. (4) 4. The problem formulation In this section, we first formulate the cost and the amount of carbon emission at each stage of the GSC. Then, the objective function of the government will formulate and finally, eight mathematical models will develop. 4.1. Notations where the first term is the amount of emission because of the replen¬ ishment cycle and the second term is the amount of emission due to product holding that is obtained by multiplying the average inventory level and the related factor. Similarly, the carbon footprints of retailers in the coordinated GSC can be written as. . D 4 j T (5) The parameters and decision variables of the mathematical models are listed in Table 3. We use the superscript index c to indicate co¬ ordinated GSC and its members (and related parameters) and n to refer non-coordinated ones. 4.2. The cost and carbon emission of retailers (Stage 4) Retailers receive products from distributors and sell them in the market. It is evident that they incur ordering and holding costs. When 4.3. The cost and carbon emission of distributors (Stage 3) The demand of each distributor is the sum of the orders of retailers which are supplied by the distributor. It means that if we consider that R distributor dl supplies retailers rl to R then, D 3 ^ = 2 D 4J . When ]=rl distributors operate separately, the inventory cost of the distributor dl is expressed as: 812 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Table 3 The list of notations used to model the problem. Parameters of the model: i Index for the stage of the GSC, where i = {1, 2, 3, 4} indicates suppliers, manufacturer, distributors and retailers respectively r Index for retailers, where r = rl, r2, ...,R d Index for distributors, where d = dl, d2, ...,D m Index for the manufacturer s Index for suppliers, where s = si, s2, ...,S j Index for firms’ number in each stage, where j = {rl, dl, m, si, Dij Demand rate of firm’s j at stage i of the GSC Aij Fixed ordering cost per cycle time of firm’s j at stage i of the GSC hij Holding cost per unit of product per unit time of firm’s j at stage i of the GSC bn Fixed transportation cost per cycle time of firm’s j at stage i of the GSC S m Setup cost per cycle time of the manufacturer h w Raw materials’ holding cost per unit time of the manufacturer e°j Carbon emissions factor per replenishment's order of firm’s j at stage i e h Carbon emissions factor for holding inventory of firm’s j at stage i efj Carbon emissions factor per transportation cycle of firm’s j at stage i e P. Carbon emissions factor per production quantity of firm’s j at stage i P m Manufacturer’s production rate / 3 S The portion of raw material supplied by supplier s ICij Inventory cost of firm’s j at stage i of the GSC /C” Inventory cost of all firms at stage i of the non-coordinated GSC ICf Inventory cost of all firms at stage i of the coordinated GSC IC n Inventory cost of the non-coordinated GSC IC C Inventory cost of the coordinated GSC Zij Total cost of firm’s j at stage i of the GSC including inventory cost and carbon-related cost Zf Total cost of all firms at stage i of the non-coordinated GSC including inventory cost and carbon-related cost Zf Total cost of all firms at stage i of the coordinated GSC including inventory cost and carbon-related cost Z n Total cost of the non-coordinated GSC including inventory cost and carbon-related cost Z c Total cost of the coordinated GSC including inventory cost and carbon- related cost Ef Carbon emissions of all firms at stage i of the non-coordinated GSC E( Carbon emissions of all firms at stage i of the coordinated GSC E n Carbon emissions of the non-coordinated GSC E c Carbon emissions of the coordinated GSC GNR The government net revenue u.gov Utility function of the government Decision variables: Tij Replenishment cycle time of firm’s j at stage i in the non-coordinated GSC T Replenishment cycle time of retailers in the coordinated GSC kd Integer multiplier of the cycle time for distributors k m Integer multiplier of the cycle time for the manufacturer k w Integer multiplier of the cycle time for raw materials k s Integer multiplier of the cycle time for suppliers C Quantity of carbon for trading in carbon trade and carbon offset schemes r The government tax per unit of carbon emission p The carbon price per unit in carbon trade and carbon offset schemes cap The government carbon cap A 3 ji hxdlDxdlTi.dl . V u T n V ^dl IC Xdl = + -o- + 2j "3 .dlTtjDdj + 2_, 3 ’dl j=rl i=rl 3 -/ j=n ( 6 ) The first and second terms above are the fixed ordering and holding costs, respectively. To avoid shortages, a distributor requires having enough inventory to satisfy all demands that if the retailers order at the same time altogether. The largest possible cumulative ordering size is 2 TyQy, therefore the third term in Eq. (6) is the holding cost to an- j ticipate this largest order. The fourth term is the transportation cost per order. Therefore, the total cost of all distributors in the third stage of the non-coordinated GSC is: D IC + j=dl 3 j ,Tii + t + t r=rl r=rl A (7) In the coordinated GSC, all distributors have a common replenish¬ ment cycle that is an integer multiplier of T. This common replenish¬ ment cycle is k d T. The inventory level of a coordinated GSC at each stage is shown in Fig. 3. As can be seen, the order quantity of distributor d is D d k d T, and the maximum inventory level of distributor d is D d T(k d — 1). The average inventory of each distributor is calculated as: _ DyTjkd - 1)7 + D 3J T(k d - 2)T + D 3J T(k d -3 )T + ...+D 3J T 2 3J k d T _ D 3J T(k d - 1) 2 ( 8 ) Therefore, the total cost of all distributors in a coordinated GSC is: jqc _ ^ (—y. + ■ j=dl k d T 2 1) + —) T (9) Similar to the method that we used to calculate the carbon emission for retailers, we can calculate the emission produced by a distributor in both the coordinated and the non-coordinated GSC based on the or¬ dering cycle, average inventory, and transportation cycle, which is expressed as follows: d o nr K 1 bs= + j=dl r=rl 4 ’ r ( 10 ) e > , - 2 < s s+ j=dl DijT(k d 1 ) 1 i „ + e 3,f ) (ID The first term of the Eqs. (10) and (11) represents the emission due to the ordering cycle. The second term is the emission for holding the inventory and the last term is the emission because of the transportation cycle. 4.4. The cost and carbon emission of the manufacturer (Stage 2) In the second stage of the considered GSC, there is a manufacturer D who has to fulfill the demand of all distributors i.e., D 2 , m = 2 A/- The i=d 1 manufacturer receives raw materials from the suppliers and converts them into the finished product at a constant rate P m . Although an in¬ teger multiplier between replenishment cycles has not used in the non- coordinated structure, in the manufacturing stage the inventory levels of the raw materials and the final products are synchronized by k w . The total cost of the manufacturer in the non-coordinated GSC is: n _ S m ^ h 2 in D 2 .m Tl.m D 2 , (1 - + Yj th.m TyDy + 2 ^ Z r m j=dl j=dl T 3 j A 2 .n 1 , , V / + T T + K 2j (' K-uj l 2,m .s'=.si \T) 2 , m T 2 , m XCA.m/lm) + k w - 1) ( 12 ) The first term in the above equation is the setup cost of the manu¬ facturer per production cycle. The second term is the holding cost of the finished product. The average inventory is calculated based on the classical economic production quantity. The third term is the holding cost anticipating receiving orders of all distributors at the same time. The fourth term is the cost of transportation of the finished products to distributors. The fifth term is the fixed ordering cost for raw materials, and the sixth term is the holding cost for raw materials. In the coordination mechanism, the production cycle of the manu¬ facturer is an integer multiplier of the replenishment cycle of dis¬ tributors. Therefore, the production cycle is k m k d T. The production quantity of the manufacturer is D 2 m k m k d T per cycle and the production time is ( D 2jn /P m )k m k d T. As shown in Fig. 3, the number of the 813 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 M.MJ Fig. 3. Inventories behavior in each stage of the coordinated GSC where, k d = 3, k m = 4, k w = 1 and k s = 1. remaining products after the production time is D 2 , m kdT(k m — 1). Hence, the average inventory of the finished product can be calculated as follows. jc _ s(.(D2,mkmk d T)(P2 t mfPmkmk d T)/2)\ m ~ 1 k m k d T ' . / D 2 ,mk d T(km — 1 )k d T + D 2 ,mk d T(km — 2)k d T + ... + D 2 ,m(k d T + ^ k m k d T > _ ^(.D2,mk d T) ^k m D2,m^ ^02,m.k d T(k m — 1 )^ 2 Pm 2 = + (fe m - D) (13) inventory of raw materials is calculated by the same method that we used in Eq. (13). jc / (1^2,m / kd T )[3 S /J'i.m k w k m W ~ { 2 k„k m k d T + @s^2,mkmkdT(k w — l)k m kdT + fi s D2,mkmkdT (kw — 2)k m kdT + ... + j3 s D 2 ,m(kmkdT)^ ) -( (t^2,m / Pm)(kmk d T)(3 s D2,m 2 M kwkmk d T ^s^2,mkmk d T (k w — 1 )\ = ( kD2m 2 n,kdT y<iD 2 , m /p m ) + (k ~ d) (14) The manufacturer orders raw materials to supplier s with the amount of /3 s r> 2 ,m at every cycle time k w k m kdT and uses them with rate P m . The inventory level of raw materials depicted in Fig. 3. The average Considering the average inventory of the finished products and raw materials, the total cost of the manufacturer under the coordination mechanism can be expressed as: 814 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 IC C 2 = + kmkdT A2,m kwkmk-d T + k m — 1) + i ^ 2 ,n ' 9 vwn + h y* D2m !; mkdT )«D2,m/P,n ) + K b2 ,m km^dT -i) (15) = S ( j=Sl k s k w JCffi k d T h k s k w k m k c [Tfi s D2'r l + 7. + k m k d T ( 21 ) Carbon footprints of the manufacturer are formulated using the following equations in the non-coordinated and the coordinated GSC, respectively. T?n _ 2,m , h 2 ~ l 2 ,m , [(^=(1 - ^f)) + ( MD ^)(p2, m /P m ) + K - 1)] + e 2 ,m ljj =d l y + e 2 ,mP ) 2 ,mT 2 ,m (16) “2 ,m . ’-‘2 = 2T1^+ E$ = kmkdT im[( D2m 2 kdT { k m ^ + k m -l “m )) + ( Zs ^ D2 f mkdT )aD 2 , m /p m ) + k w + e 2,m k m k d T ^2,m^2,mk m k^T (17) In the above equations, the first term is the amount of carbon that is emitted per setup cycle. The second term is the carbon emission be¬ cause of inventory holding that is obtained by multiplying the average inventory of the finished products and the raw materials with its coefficient. Carbon emission rates may vary from one warehouse to another based on equipment and technology used. However, we assume that carbon emission rates are equal for the finished products and raw materials. The third term is the amount of emission related to trans¬ portation, and the final term is the amount of carbon that is emitted during the production time. 4.5. The cost and carbon emission of suppliers (Stage 1) 4.6. The model of a supply chain When there is no coordination, the total cost of the multi-stage GSC is the summation of the costs of all members, which means the sum of Eqs. (2), (7), (12), and (18), and as a result, we have: IC n = IC% + ICf + ICf + ICf (22) The total carbon emission of the non-coordinated GSC is: E" = E'f + Ef + Ef + Ef (23) It is easy to see, that the total cost of the GSC in the coordinated scheme is the sum of Eqs. (3), (9), (15) and (19), which is given by: IC C = ICl + ICl + IC% + ICf (24) Moreover, the total carbon emission of the coordinated GSC is: E c = El + Ef + E c 2 + Ef (25) 4.7. Modeling the regulations of the government In the previous sections, we formulated both cost and carbon emission of the GSC. Now we define the objective function of the government in line with the social welfare (SW) maximization. Then, the problem under each regulation will develop. The SW can be defined as a trade-off between utility functions of the producers and environ¬ ments. The first element of the SW is producer surplus, which is the summation of all firms’ profit. Given that, as we have not considered the GSC income, instead of the GSC profit maximization, the cost minimization has been applied in the model of the government. The second element of the SW is the government revenue (GNR) that ob¬ tained through carbon taxes and the last element is environmental benefits. In this paper, the reduction in the total carbon emission of the GSC has considered as environmental benefits. Hence, the objective function of the government can be written as follow. Each supplier supplies different kinds of raw materials required for the manufacturer. Thus, the quantity of the raw material that is sup¬ plied by supplier s per unit time is P s D 2 , m . Each supplier incurs the ordering cost, inventory holding cost and transportation cost for deli¬ vering raw materials to the manufacturer. In the non-coordinated structure, the total cost of the suppliers is: lc n _ y , -Aij + kij'Zij P s P 2 ,m + j= s i Tl J 2 k w T 2<m Qgj In the coordination structure, the replenishment cycle of all sup¬ pliers is a multiple integer of the manufacturer’s replenishment cycle, k s k w k m k,iT. From the inventory level of the suppliers that depicted in Fig. 3, it is evident that the average inventory of the suppliers is, k s k w k m k d TY,fi s T> 2 ,ml2 and the ordering cycle is, 1 /k s k w k m k d T. Thus, S the cost of suppliers under the coordination mechanism is: j.^ c _ y 1 r -^i j /zij k s k w k m k d Tfi s D2 <n: 1 k s k„k m k d T + 2 bu + ---) kw k m k d T (19) Similar to previous stages, the carbon emission of suppliers in each structure obtained by multiplying the ordering cycle, average in¬ ventory, and transportation cycle in the relevant emission factors which are given by: j=sl w Tl,j^s^ 2 ,n + e 1 k Ts •'•Mi 1 7 I ( 20 ) Min u gov = (1 - Q)Z - T1GNR + pE (26) where the first term is producer surplus that increases by decreasing the total cost of the GSC. The second term is the revenue of the government that needs to be maximized. In the above equation, O is the trade-off between the cost of the GSC and the GNR that determined by the government. The range of this parameter can be 0 < Cl < 1. The last term in Eq. (26) shows environmental benefits, where p ^ 0 is the en¬ vironmental importance factor from the point of view of the govern¬ ment. The low value of p indicates that the government considers economic objectives more than environmental objective and when p approaches to oo, the government does not consider the benefit of the GSC and his revenue. As described in Section 3, the government (Stackelberg leader) sets a regulation to control his concerns related to carbon emission and social welfare. As mentioned, four regulations are considered: carbon cap, carbon tax, carbon trade, and carbon offset. Each of these reg¬ ulations is briefly described as follows. A. In the carbon cap policy, the carbon emission of the GSC should not exceed a certain cap, which is determined by the government. B. Under the carbon tax policy, the government penalizes the GSC according to the amount of emission by setting tax. This study as¬ sumes that tax is linearly associated per unit carbon. C. In the carbon trade scheme that is also called cap-and-trade or cap- and-price, the government penalizes for emission that exceeds the specified cap. Moreover, in this scheme, the government encourages the firms that emit less than their cap and accordingly reward them. 815 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Table 4 Bi-level mathematical models of the government and the GSC’s for each scenario. Regulation Non-Coordinated GSC Coordinated GSC Carbon cap Min Ug 0V (cap) = (1 — Q)Z n + pE n Min Z n = IC n (Tij, k w ) s.t E n ^ cap Z n < M z cap > 0 Tij>0Vi,j k w is integer number Min Ug 0V (cap) = (1 - H)Z C + pE c Min Z c = IC C (T, k d , k m , k w , k s ) s.t E c ^ cap Z c ^M z cap > 0 T> 0 kd, k m , k w , k s are integer numbers Carbon tax Min u gov (t) = (1 - D.)Z n - QzE n + pE n Min Z n = IC n (Tij, k w ) + z. E n Z n ^M z r > 0 Tij > 0 V i, j k w is integer number Min u gov (r) = (1 - n)Z c - D.zE c + pE c Min Z c = IC C (T, k d , k m , k w , k s ) + z. E c Z c ^M z z > 0 T> 0 k d , k m , k w , k s are integer numbers Carbon trade Min u gov (p, cap) = (1 - d)Z n + pE n Min Z n = IC n (Tij, k w ) + p. C s.t E n — C = cap c^C^c Z n ^ M z cap > 0 p > 0 Tq > 0 V i, j k w is integer number Min Ug 0V (p, cap) = (1 - Cl)Z c + pE c Min Z c = IC C (T, k d , k m , k w , k s ) + p. C s.t E c - C = cap c^C ^c Z c ^M z cap > 0 p > 0 T> 0 k d , k m , k w , k s are integer numbers Carbon offset Min u gov (p, cap) = (1 — D.)Z n + pE n Min Z n = IC n (Tij, k w ) + p. C + s.t E n - C + = cap Z n ^M z cap > 0 p > 0 Ttj > 0 V i, j k w is integer number Min Ug 0V (p, cap) = (1 - H)Z C + pE c Min Z c = 7C C (T, k d , k m , k w , k s ) + p. C+ s.t E c - C + = cap Z c ^M z cap > 0 p > 0 T> 0 k d , k m , k w , k s are integer numbers This scheme is implemented through a carbon trading market, where a firm can buy carbon if its amount of emission exceeds the carbon cap, or else a firm can sell its extra carbon credit and gain revenue. D. Carbon offset or cap-and-offset is a policy where the government imposes the tax only on emission that exceeds a certain cap. In this setting, the government sets an emission cap but allows the GSC to reduce its emission by purchasing emission offsets through third parties. Under the above regulations, the GSC (Stackelberg follower) makes its decisions about replenishment cycles. Based on the structure of the GSC and type of regulation, eight scenarios are developed. The math¬ ematical models of each scenario are presented in Table 4. Generally, the Stackelberg problems show a hierarchical structure similar to a bi¬ level programming problem (BLPP). Each scenario is a bi-level mixed integer nonlinear problem (MINLP). Note, we consider that the gov¬ ernment variables such as cap and tax are imposed on each firm of the GSC distinctly. As can be seen in Table 4, in the carbon cap regulation (the first row of Table 4), the government determines cap at the top level of the model and at the lower level, the GSC determines replenishment cycles at each stage and also has a constraint for the amount of its carbon emission. In all scenarios, we define a constraint Z < M z as an in¬ dividual rationality, where indicates that the GSC will continue its business if obtain a minimum utility. In the models related to the carbon tax scheme (the second row of Table 4), the government at the top level problem determines tax per unit carbon, which will add to the GSC’s costs at the lower level. In this scheme, the GNR obtained from the tax that multiplied by the amount of emission. In the carbon trade and carbon offset schemes, the government determines a cap and carbon price and the GSC trade its emission at price p. The carbon price can consider either exogenous or endogenous. However, because the carbon trade market is more powerful than the companies, we assume that the carbon price is exogenous and de¬ termined by the government. At the top level problem of the carbon trade models (the third row of Table 4), the government determines cap and carbon price. Then at the lower level problem, the GSC transfer the carbon quantity C to the market at price p. The positive value C in¬ dicates the amount of carbon purchased from the trading market and the negative value C indicates the amount of carbon sold in the market. In this scheme the constraint E — C = cap shows the carbon balance, if E > cap the GSC must buy C units of carbon from the trading market, also if E ^ cap, the value of C is negative and the GSC can sell C units of carbon on the trading market and compensate part of its cost. Besides, the transferable carbon quantity C has a lower bound c and an upper bound c. These boundaries determined by the amount of supply and demand for the carbon trading market. In the models related to the last regulation (the fourth row of Table 4), the term p. C will add to the GSC cost only when C is positive, i.e., E > cap. 5. Solution approach We confront a hierarchical decision-making problem where a leader first makes his decisions at the top level and then the followers make their decisions according to the decision at the top level. In this section, we investigate the solution for each scenario distinctively. First, the non-coordinated scenarios are investigated and then the coordinated ones. We used different approaches to solve non-coordinated and co¬ ordinated scenarios. In this regard, we seek to prove convexity of the 816 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Table 5 Single-level mathematical models for Scenarios 1-4. Scenario 1: Min u gov (cap) = (1 — Q )Z n + pE n s.t E n ^ cap 2 (A4J + b 3,dl + A 4 ,j e 4j + A 4j e 3,di> I d 4J ( b 4j + 2 h 3t dl + Ujejfj) _ 1 2 ( A 3,j + b 2,m + A 3J e 3j + A 3J e 2, m ) 7 ]j D 3j(h3J + 2h2,m + A3je^j) To* 2 (Sm + A 2,m e 2 t m + ^2,m + b\j + A2 ,m e {j) / k\v) I 02,m(^2,m(l-+ + few — 1) + ^2,m e 2 mi 1 -^ U On „ On ’ On *- D ’ m +kw- 1)] + 2 A 2 ,m^ m ) „ on r = ( 2 ^U+4j e i°j) lj ^ Ps D 2 ,m( h lj + 4 j) Z n ^M z cap, Tij ^ 0, is integer number Aij ^ 0V i, j Scenario 2: Min Ugov (?) = (1 — Q)Z” - D,rE n + pE n S.t Ta; — _ J 2 ( A 4j + ^3,dZ + Tie^j + e| >di )) D 4j(h 4 j + 2h 3j dl + _ |2(A 3 J + l>2, m + r(e^+e| im )) ^ n 3j <tl 3j + 2h2 >m + Tejfj) 2(Sm + (A2,m + &l d + TC ij) ! ^w) i D2,m( h 2,mV ~ + <*» +1» - 1) + [1 - V on s on ’ on -Eft(-^ + feiv-l)] + 2T C f m ) - on r = 204^+Tefj) lj ^ & D 2,m(hlj + wfj) Z n ^M z r, Tq > 0, fc w is integer number Scenario 3: Min Ugovip, cap ) = (1 - Q)Z n + pE n s. t Z n = IC n (Tij, k w ) + p.C T h = 2(A 4 j + but + 01,4j - A2,4j)(e|j + e| d p) n/= J D 4 j(h 4 j + 2h 3j dl + 01,4j ~ A 2,4j)e 4 j) 1 2 ( A 3j + b 2,m + 01,3J - A2,3j)( e 3j + e 2,m^ ^ D 3j( h 3j + 2h2, m + 01,3 j ~ *2,3j) e 3j) 2 (Sm + Te 2,m + ^2,m + ^1 j + 01,2,m - 42,2 ,m) e [j) / ^w) In /i ^2,m. , , „ „ ,t>2,m ^2,m02,m(l--—) + Ziw Z&(—^— on on h k w - 1) + 01,2,m - A2,2,m)c2,m i 1 ~ n ’ m + - 1)] + 201,2,m — %2,2,m) e £ m ) - on Tfi = 2(Alj + 01,ij - A2,ij)efj) 1,j ^ Ps D 2,m(hj + 01,1 j -^2,1 E n - C = cap c^C^c Ai ,ij(cap -E n + C) = 0 h,ij(-cap +E n -C) = 0 Z n ^M z cap, p, Tij > 0, k w is integer number Ai,ij, fa,ij ^ 0V i, j Scenario 4: Min Ug 0 v {p, cap) = (1 — Q,)Z n + pE n s.t Z n = IC n {Tij, k w ) + p. C + j 2(A 4 j + b 3y dl + 01,4 j ~ ^2,4 j)( e 4 j + c| d p) D 4 j(h 4 j + 2h 3 dl + 01,4j - A2,4j) e 4j) j 2 (A 3 j + b 2 , m + 01,3 j ~* 2 , 3 j)(.e 3 j + e 2 ,m^ ' )j D 3 j( h 3 J + 2 Zi 2 , m + 01 , 3 J - ^ 2 , 3 j) e 3 j) (continued on next page) 817 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Table 5 ( continued ) I 2(Sm + + (A 2 ,m + t>lj + (h,2,m ~ 2-2,2,m) e [j) / kw) D 2 ,m(li2,m(l - + *»-!) + Ul,2, m - A 2 ,2, m )e| m [l - Em s rm ’ rm 1 + ZHs 1 (- - fcn 1 - 1)] + 2 Wl,2,m —2-2,2,m)^ m ) ] s Pr " T , _ I 2(Aij + (4,ij-,12.1J)«i°j) ~ ) ft n 2,m(h\ j + Ui j j 2 2 ,i j)e(U) E n - C+ = cap *,y(cap - B» + C) = 0 h,ij(-cap + E" - C) = 0 Z"^M- cap, p, Tij > 0, k w is integer number 7l,y, A 2 ,y > 0 V i, j objective function of the non-coordinated GSC as follows. Proposition 1.. The inventory cost junction of the non-coordinated GSC , Eq. (22) is convex for over 7y > 0 and fc w ^ 1. See Appendix A for proof. Proposition 2.. The total carbon emission function of the non-coordinated GSC, Eq. (23) is convex for over 7y > 0 and k w ^ 1. See Appendix B for proof. According to Colson, Marcotte, and Savard (2007) when the pro¬ blem of the follower at the lower level is convex and regular, it can be reformulated to a single level by replacing the Karush-Kuhn-Tucker (KKT) conditions of the lower level problem. Propositions 1 and 2 showed that the non-coordinated GSC problem is convex. Therefore, in the following, the KKT conditions of the GSC problem in Scenarios 1-4 are replaced and as a result, each scenario is converted into the equivalent single-level problem. dIC n , dE"_ mj + + hj( -Ay trijDjj + + hlj lt D + e h D y = 0 „ _ j 1 2(A 3j - + b 2 ,m + hjCij + A3je2, m ) J ]j D 3 j(h 3 j + 2 h 2 . m + A. 3 je 3 j) dIC n A, dE " —Sm , h 2 ,mD2,m f i P>2,m\ A 2 ,m dT 2 , m + - m dT 2 , m 1 2,m + 2 l 1 Pm J kwT 2,m , KD S Ps D 2,m / D2,m A ( ~ e 2,m 2 \ P m * J k w T2 m i A2,m l T lm + e 2 . m (M 1 - P>2,m Pm )) + \(D 2 , m )\ Pm + k w - lj + e 2,m D 2,m e lj jy T lm )=° (33) (34) (35) 5.1. The non-coordinated GSC under carbon cap regulation (Scenario 1) In this scenario, the non-coordinated GSC at the lower level problem aims to minimize his inventory cost and has a constraint where, E n < cap. The KKT conditions of this problem are as follows. dIC n dE n - + A; j - 3T,j dTij = 0 v i,j (27) 2(S m + ^ 2 ,m e 2 ,m + (A 2 , m + & 1 J + ^ 2 ,m e lj)/^w) d 2 ,Ahi.mii -°^ L ) + + few - i) + AmCl.mi 1 - tr 1 + YiPsi-jr 1 + ~ ^ + ^A-.mCi.m) (36) Eij < cap V i, j (28) Ay (Eij - cap) = 0 Vi, j (29) cap, Tij > 0, Ay ^ 0 (30) where Ay is a multiplier associated with the KKT conditions and may consider as the shadow price for each constraint of the problem. By using Eq. (27), the best response strategies of the chain members can be obtained at each stage of the GSC as follows. SIC" dT 4j i.dl ~ ii.dl (31) dIC n + /ti dE n 'a7b ~ A ij , hijP s D 2 , w , T 2 + Aiji—y^ + ei,; -) = o (37) T » _ | 2 iAj + h je°j) J ]j P s D 2 ,m(hij + Aij-ey) (38) We can rewrite the Scenario 1 to a single-level MINLP problem by replacing the optimal values of the replenishment cycles of the GSC into the upper-level problem. We list the single-level problems of each scenario in Table 5. 5.2. The non-coordinated GSC under carbon tax regulation (Scenario 2) 2 (A 4 j + b 3 ji + A 4 je4j + Ajcldi) (hnj + 2 h 3 ji + A 4 je 4j ) (32) The above equation shows the optimal value for the retailers’ re¬ plenishment cycle. The optimal solution for other stages of the GSC is computed as follows. Under the carbon tax regulation, the non-coordinated GSC face an unconstrained problem where its total cost is Z n = TC n + r. E". To obtain the best response strategies of the GSC, the first order derivatives of Z n with respect to each decision variable 7y, must satisfy the con- dition —— = 0. Thus, replenishment cycles of the GSC members re- dT ij garding carbon tax can be found as follows. 818 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 dZ n —Ad h*jD 4J 3 ,dt + Dijh-xil - ~2~ + + e \li—r ~ e Ldi~^r) - 0 '4J 1 *j C 4j |2(A 4 j + b 3<d i + r(e4j + e^^)) T4 n j D 4 j(h 4 j + 2h 3d i + re 4 j) 3Z n -A 3 j h 3J D 3 j b 2 , m 2 + + h 2 , m D 3 j 2 J 3J Z J 3J + = 0 (39) (40) (41) I 2 (A 3 j + fe 2 ,m + ^(e 3 j + C 2 , m )) \ Dy(h 3 j + 2h 2 , m + te 3 j) (42) 3Z” 3T2, m -5m 2 ,m /-m _ ^ ,m \ ^2 ,n T 2,m 2 Pm hw X H\ D'i.m - + T( Ti„ + eU^l k w T( »2. i^ + K- l) )) + / D 2,n + i)] + e 2,m D 2,m ~ e lj kiT 2 ) ~ 0 (43) 5.4. The non-coordinated GSC under carbon offset regulation (Scenario 4) Under the carbon offset scheme, if the carbon emission is more than a cap, i.e., E n > cap, the C units of carbon transfer to the third party at price p, in this situation the problem mathematically can formulate similar to the carbon trade problem. Otherwise when E n < cap, the value of C must equal zero, this situation mathematically can formulate similar to the carbon cap problem. Therefore, this scenario can be re¬ formulated into a single-level MINLP as shown in Table 5. To solve Scenarios 1—4, each problem from Table 5 is coded in GAMS software and solved by Baron solver. Now, we present a solution for the coordinated GSC under each carbon regulation. The lower level problems of the Scenarios 5-8 contains five decision variables that one of them is continues (T) and four of them are integer (fed, k m , k w , fc s ). To solve these problems, we propose an algebraic method due to the difficulties related to the con¬ vexity of inventory cost function. This method used by several re¬ searchers such as Ben-Daya et al. (2013), Leung (2009), and Sarkar et al. (2016), that is developed in this paper to find the best response strategies of the GSC. 5.5. The coordinated GSC under carbon cap regulation (Scenario 5) The objective function of the coordinated GSC in Scenario 5 is to minimize IC C , that can be expressed as: T* ™ — 2 (S m + re 2 m + (A 2m + b 4 j + re/j)/fc„,) | D 2 , m (h 2 , m (l - D ^) + h w ZP s ( D ^ + k w - 1) on ^ r m I + rel n [l - D f^ + 2/3 s (^ + k w - 1)] + 2r efj dZ " _ -Au hjf s D 2:n _ 1 ' dT, + ) = 0 V T i*J = 2(Aij + re°j) P s D 2 ,m(hij + zefj) (44) (45) (46) Hence, the problem of Scenario 2 can be rewritten as presented in Table 5. r d s r 1C‘= Z Z Z f kd | — ^2 ,m) + k m | —— 1 -1" ^2,m^2,n j=rl j=dl j=sl + - lj + K&D 2 , m k w + hjKk^Di, + D 2 , m (h 4 j — h 2 j) 1 if. 1L , ^2,m k d [ 3J+ k m [ m+ 2 ’ m+ K + 6w + Aj) w k s k w J +A tJ + b 3J By changing variables, Eq. (54) can be simplified to: IC C = YT+ — (54) (55) 5.3. The non-coordinated GSC under carbon trade regulation (Scenario 3) The objective function of the non-coordinated GSC in Scenario 3 is the minimization of Z" = TC" + p. C. This problem has a constraint, E" — C = cap that replaced by E n — C < cap and - E n + C ^ —cap. As a result, the KKT conditions of the GSC problem in this scenario can be express as: dZ" d(E n - C) d(-E n + C ) - + Ai i i - + A? i i - 3T,J UJ aiy 2 ' ,J 3Iy = 0 v i,j (47) E n - C ^ cap (48) - E n + C ^ -cap (49) di,ij(cap — E n + C) = 0 (50) hij(.-cap + E" - C) = 0 (51) c^C ^c (52) cap, p, Ttj > 0, A lf y, X 2 ,ij ^ 0 V i, j (53) The Eq. (47) is converted to 3Z"/37y + (Ai,ij — Az,iJ)dE n /dTij = 0 and based on that we can compute each replenishment cycle similar to the previous section. As a result, the optimal values of the lower level problem are calculated and replaced into the government’s problem as shown in Table 5. See Appendix C for all values. According to Eq. (55) the optimal replenishment cycle of the re¬ tailers and then optimal inventory cost can easily find as: (56) IC C * = 2 JYX (57) By expanding the terms of the Eq. (57), we can obtain each decision variable iteratively. The IC C * rewrite as: IC C * = V2 j(k d co + ctiX-Zu + ct 2 ) V k d (58) By using the perfect squares method, the above equation is trans¬ formed to: IC c * = V2 - fva(] 2 + [fuv + fa.ia 2 } 2 } 112 k d (59) The optimal value of the k d can be obtained when the other vari¬ ables are fixed and the first term in above equation equal zero, i.e., [k d fa><x 2 — ] 2 = 0. As a result, we have: The only term in Eq. (59) that contains the integer variable k m is Hsu. We have: 819 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 = J(/ 3 + k m (r 2 + k»H))(S,4y + + ^Pi)) = J(r 2 + Kyi)(v 2 + ^-<Pi) + Jn'LjAij (61) Similar the way that we computed kd, the optimal value of k m is obtained by fixing values of other variables and setting the term [fc m l(r 2 + Krd'Z^j - <J(<P 2 + i-ydv?] 1 = o in the Eq. (61). As a re¬ sult, we have: k * = fa + jvOh (y 2 + Kn) YjAy J (62) The optimal value for the variable k w can be calculated by setting the term j(y 2 + k w y 1 )(cp 2 + fc ' tp,) = 0 from Eq. (61). Therefore, we have: (/ 2 + fc wft )(<P 2 + t—Pi) = - JmI 2 + + V^pI1 2 } 1/2 = ° lr* — K-u) — Mi 7lP2 (64) The term y jy l cp l in the Eq. (63), contains the integer variable k s . Thus, the last integer variable of the GSC problem fc s , can be obtained as follows. •Jy iPi — /Z (.(.hwP s D 2 , m + hijk s p s D 2 ,m)(A 2tm + b 2 j + -^)) = z — [fe s y h[ jfi y D 2 m (A 2 m + hjj) — ^Aijh„P s D 2 m ] J I +[^hijf$ s D 2 ,mAij + ^jh„l3 s D 2 m (A 2 m + bi j) |- Aijh„ 7 (A 2m + b 2 j) (65) ( 66 ) In the following subsections, we will calculate the best response strategies of the GSC in Scenarios 6-8 by using the method that de¬ scribed in this section. In the end, a comprehensive approach for sol¬ ving problems of the coordinated GSC under each carbon regulation will present. 5.6. The coordinated GSC under carbon tax regulation (Scenario 6) The total cost of the coordinated GSC under carbon tax regulation is Z c = IC C + tE c . Similar to Section 5.5, we can simplify the inventory x cost to IC C = YT H-. Also, we can rewrite total emission as follows. T E c = Z Z Z ^[D 2 , m eij - Dje^j j=sl j=dl j=rl + kd(P2,m e 3J ~ (^2,m^2,m (~ir^ + 1) v r m + e 2 ,m^ s E > 2 ,m(-jr^ — 1) + 2e£ m D 2im + e 2m P s D 2<m k„ + eijk s k w ^ s D 2 ,m))] + yl e 4j + e 3j + r d K + ^l m + ei m + e i + St))] The above equation can be simplified to: E c = YT + — T (67) ( 68 ) See Appendix D for all values. The objective function of the GSC in this scenario rearrange as: Z c = IC C + t. E c = YT + - + r. (YT + —) T T (69) We can find each decision variable of the coordinated GSC with regarding the government decision variable, i.e., carbon tax r itera¬ tively. By placing the first derivative of the Eq. (69) equal to zero, the optimal retailers’ replenishment cycle and total cost are: T* = j X+ tx V Y+ tY Z c * = Y.l’YfA + A + t. (Y.jAAEL + ±_ ) IY+tY j X+ & VY+tY X+ tx v Y+ tY V Y+ tY X = 2 i](Y+ t?)(X + tX) (70) (71) The Eq. (71) expand as follows to calculate the optimal value of the integer variable kj. Z" = 2ij(Y+ tY)(X+ rX) = 2 y j(( k ^t) + r( k ^))((f d + a 2 ) + r(l d + *)) = sf 2 j(k d (co + rco) + cci + zaiX'AbAL + a 2 + ra 2 ) (72) By using the perfect squares method, the above equation is con¬ verted to: (X[k dy l(ai + tco)(cc 2 + zcc 2 ) - J(v + Tu)(ai + rai)] 2 l Z" = V2 ) kd [ +[V ( tt1 + r “)(t> + to) + V(“i + r “i)(“2 + ta 2 )] 2 (73) The optimal value of kj will obtain by fixing other variables and setting k d2 j(co + rcd)(cc 2 + ra 2 ) - i](v + Tu)(ai + rcii) = 0. As a result, we have: k d = (v + rfiXai + rai) (a) + rfi)(a 2 + ra 2 ) (74) The only term that has variable k m in the Eq. (73) is t/(co + rcd)(v + w) that can be expanded as follows. y/(cO + TCd)(v + Tv) = (k m (y 2 + Kyi + r(f 2 + KyJ) + y 2 + rf 3 ) Ctt -(?>2 + + + + T, A 3j K m K w j 1+ reSj) K I(y 2 + Kn + t(% + Kn))CZ A 2 j + re°j) -J(<P2 + Z-fl + r($2 + t~ 9 i ))(ft + r%) V % K-W (72 + fc w7i + r(f 2 + Kn))(f 2 + -rfi + r ($2 + irP i3) + JG3 + l ?0< + re 3j) (75) By setting the first term of the above equation equal zero, the op¬ timal value of k m is: Table 6 Input parameters for retailers. j D 4,j h 4,j ^4j e 4J ej}. 4j rl 7500 30 110 0.7 0.2 r2 12,000 25 95 0.7 0.3 r3 10,000 35 90 0.75 0.3 r4 5500 30 100 0.65 0.2 820 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Table 7 Input parameters for distributors. j hsj A 3,j b 3,j * e y dl 20 85 15 0.6 0.4 1.7 d2 24 90 14 0.6 0.4 1.8 Table 8 Input parameters for suppliers. j hij A hi bij •& <- e U si 7 70 18 0.5 0.4 1.6 s2 8 80 25 0.5 0.3 1.6 s3 8 75 22 0.6 0.3 1.4 kZ = + t @2 + }fo))(a + m) (.72 + fc w7i + r(f 2 + k^XEAsj + re°j) j (76) The third term of the Eq. (75) contains k„, therefore we transform it algebraically and set it equal zero as follows. + + r P 2 ) - + + ty 2 )] 2 I +Ly'(n + tfi)(¥>i + *&) + yl(r 2 + tfiKvi + r fe)] 2 = o (77) (gi + Wi Xr 2 + *%) (Yi + Wi)(p 2 + T Pi) (78) To obtain optimal value of the last integer variable k s , we set the term ^(ft + v{)(Vi + T Pi) = 0 from Eq. (77). •J(r1 + ifi)(?>i + r@i) = . S (.k s ^ s D 2im (h\j + TElj) + fi s D 2,mhw + rP s D 2 .rn^mK—^~ + ^ 2 - m + + Ze \j) 1 k s ^(fi s D 2 ,mhij + T&D 2 , m e}j)(A 2 , m + bij + re}j) 2 z fcs + ™°j)(P s D 2 ,mhw + Tp s D 2 , m e£ m ) j + + t[3 s D 2 + rafj) +p s D 2%m h w + T^D 2m e 2m )(A 2-m + b\j + Te(j) (79) ' (^lj + «lj)(A 2| „, (80) 5.7. The coordinated GSC under carbon trade regulation (Scenario 7) In this scenario, the coordinated GSC seeks to minimize Z c = IC C + p. C at the lower level problem. According to a constraint of this problem, where E c — C = cap, the total cost function of the GSC can be written as Z c = IC C + p. (E c — cap). Similar the technique that we used in the previous section, the total cost function can be simplified to Z c = YT + j + p. (YT + jr) — P- cap. Thus, it is easy to calculate the following equations. m ix + px ~ \ Y+ pY (81) Z c * = 2^(Y + pY)(X + pX) + p. cap (82) The optimal values of integer variables can be found by the same method that used in the previous section. As a result, the multipliers of the retailers’ replenishment cycle are: Table 9 The computational results for each scenario of the non-coordinated GSC. Decision Variables Scenario number 1 2 3 4 T 4,rl 0.022 0.022 0.054 0.024 T4,r2 0.017 0.017 0.036 0.018 T 4,r3 0.016 0.016 0.04 0.018 T4,r4 0.023 0.023 0.064 0.025 T 3 ,dl 0.015 0.015 0.023 0.016 T3,d2 0.016 0.016 0.026 0.017 T2,m 0.018 0.025 0.003 0.018 Ti, s i 0.017 0.016 0.006 0.014 Ti, s2 0.017 0.017 0.007 0.015 Ti, s3 0.023 0.023 0.011 0.020 k w 1 1 4 1 ics 30,367 30,268 32,226 29,476* /C 3 n 34,439* 34,508 52,451 35,242 ics 55,159 52,938* 175,930 55,238 ICf 24,230* 24,235 34,796 24,576 IC n 144,195 141,948 295,403* 144,531 ZS 30,367 30,499 -1,810,859* 29,476 z" 34,439 35,061 -15,178* 35,242 zs 55,159* 55,527 2,302,522 55,238 z( 24,230 24,847 -535,545* 24,576 z n 144,195 145,935 -59,061* 144,531 ES 232 232 261 227* ES 556 553 371* 525 ES 1918 2589 793* 1940 E( 621 612 425* 552 E n 3328 3986 1850* 3244 T - 1 - - P - - 3554 11 cap 1918 - 195 158,545 GNR - 3986 - - UgOV 103,156 105,898 -26,185* 102,939 Table 10 The computational results for each scenario of the coordinated GSC. Decision Variables Scenario number 5 6 7 8 T 0.017 0.016 0.004 0.016 kd 1 1 1 1 km 1 1 1 1 k w 1 1 5 1 k s 1 1 1 1 IC% 32,075* 33,008 100,830 33,008 ics 1709* 1815 7251 1815 ics 22,926 22,077 21,925* 22,077 ics 11,310 10,645* 13,305 10,645* IC C 68,020 67,544* 143,311 67,544* zs 32,075* 33,256 100,614 33,008 zS 1709* 2034 7658 1815 zS 22,926 23,687 22,514 22,077* zS 11,310 11,121 13,198 10,645* Z c 68,020 70,098 143,985 67,544* ES 243* 249 718 249 ES 206* 219 875 219 ES 1711 1610 823* 1610 Ef 506 476* 595 476* E c 2665 2553* 3011 2553* X - 1 - - P - - 1 1 cap 1711 - 234 12,657 GNR - 2553 - - Ugov 54,139 53,805* 101,446 53,294* 821 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 U GO O CD rg tM o 3,50,000 3,00,000 2,50,000 2,00,000 1,50,000 1,00,000 50,000 Effect of p on the IC Fig. 4. The sensitivity of the optimal values of the inventory cost to the parameter p. Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7 Scenario 8 (u + pg)fa + pa i) (oj + pcc)(a 2 + P&i) (83) kZ = (<p 2 + ir<Pi + P(v 2 + r-Pi))(/3 + P© K W /C W (r 2 + k w y i + p (?2 + + pefj) j (84) k* — (Pi + Pp!)(/ 3 + pg) (n + pn)('P 2 + m) (85) (K + P e 2,m)(AlJ + P e lj) (^1J + P e lj)(^2,m + blj + P e lj) ( 86 ) 5.8. The coordinated GSC under carbon offset regulation (Scenario 8) The total cost of the GSC in this scenario is Z c = IC C + p. (E c — cap) + . It means, if the carbon emission of the GSC is more than a certain cap, E c > cap, the total cost will be Z c = IC C + p. (cap — E c ) + . And if the carbon emission is less than a cap, E c < cap, the total cost will be Z c = IC C . The optimal values of the GSC in this scenario can be obtained in two cases by using the proposed algebraic method, when E c > cap the decision variables of the lower level problem can formulate as the carbon trade problem and when E c ^ cap the variables can formulate as the carbon cap problem. Consequently, the optimal values of the GSC in Scenario 8 are as fol¬ lows. ^/f EC < ca P E c > cap y y+py (87) fcrf = (u + pff)(gl + p8i) (ty + piS)(a2 + pa 2 ) E c ^ cap E c > cap kl = (<P2 +i-<Pi)r3 (Y2 + kwYpAsj (<P2 + +P(V 2 + r-9i))(X3 + PY: 3 ) _KW_MV_ t/2 + Wl + P(?2+ k»fl)X2l3 j + pe°j) E c < cap E c > cap E c < cap (fi + pfiX/2+ pn) pc v rnn (ri+mX?>2+P92) " ( 88 ) (89) (90) fc* = A\jh w hlj(A2,m + blj) (Ji„, +pc| m )(Aij +pe 1 ° J .) (hi j + pcfj)(A 2 .m + bij +pe(j) E c ^ cap E c > cap (91) Based on the best response strategies of the models, we propose the following solution method for coordinated problems (Scenarios 5-8). - Step 1. For a given value of governments’ parameters and variables, calculate a continues value of k*. - Use Eq. (66) for the Scenario 5. - Use Eq. (80) for the Scenario 6. - Use Eq. (86) for the Scenario 7. - Use Eq. (91) for the Scenario 8. - Step 2. In each scenario calculate a continues values of k*„ kZ , kd And compute replenishment cycle T*. - Use Eqs. (56),(60),(62),(64) for the Scenario 5. - Use Eqs. (70),(74),(76),(78) for the Scenario 6. - Use Eqs. (81)-(85) for the Scenario 7. - Use Eqs. (87)-(91) for the Scenario 8. 2,30,000 U 1,80,000 o oj 1,30,000 ■5 o 80,000 § 30,000 73 o -20,000 <D H -70,000 -1,20,000 Effect of p on the Z Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7 Scenario 8 Fig. 5. The sensitivity of the optimal values of the total cost to the parameter p. 822 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Effect of p on the E 4500 4000 —•— Scenario 1 —B— Scenario 2 -=0 — 4 — Scenario 3 ' -—— . — a —*— Scenario 5 *■———^ --w -W- — 4 — Scenario 6 4 - 4 - - ~ 4 . —•— Scenario 7 p=0 p=10 p=20 p=30 B ^cenanoS Fig. 6. The sensitivity of the optimal values of the carbon emission to the parameter p. - Step 3 . In each scenario, put the values of k s = ffc s l, k s = [fc s J, k w = | k w |, k w = [fc w J, k m = | k m |, k m = \k m j, k^ = f k ^|, k^ = k t i in the GSC problem and compute Z c . - Step 4 . In each scenario, let kk *„ fc*„ k% be the integer values that leading to a minimum Z c . - Step 5. Put the optimal values of the coordinated GSC in the gov¬ ernment problem and use Baron solver through GAMS software to find the decision variables of the government and u* m . 6. Numerical example We illustrate the applicability of the mathematical models devel¬ oped previously through the following numerical example. In practice, the input data for modeling the problem can obtain through several sources. Usually, the values of the parameters of the inventory model can be derived from the firms’ formal financial statements. These statements record the firm’s costs and revenues clearly and are avail¬ able to the government to be audited. Besides, the carbon emissions of firms are measurable by using standards and tools that developed by several organizations such as GHG protocol, EPA, Carbon Trust and ISO 14064 (Benjaafar et al., 2013). There are also third parties that provide services for measurement of GHG emissions such as testing capabilities of a product, online emission measurements, emissions monitoring, evaporative emission testing, data acquisition and reporting, etc. The emission data needs to be reported to the government in order to meet the requirements. For example, the European firms that trade their carbon credit under the ETS system, are obligated to document and report their emissions (Ellerman, Convery, & De Perthuis, 2010). Therefore, the firms’ financial and environmental information can be used by both GSC and government, to build an effective model to manage costs and carbon emissions. In this example, we consider a GSC consists of four retailers, two distributors, one manufacturer, and three suppliers. The input parameters for the retailers and distributors are presented in Tables 6 and 7, respectively. In this example, the values of the manufacturer parameters are S m = 450, h m = 15, P m = 36,000, b 2 , m = 20, A m = 70 and h w = 10 and the carbon emission factors of the manufacturer are e% m = 0.4, e 2 , m = 0-3, = 1-5 and = 2. Moreover, each unit of the product requires three types of raw materials that are supplied by three different suppliers. The consumption ratio of raw materials in a product is /J sl = 2, /3 s2 = 2 and /3 s3 = 1. The other input parameters of suppliers are presented in Table 8. Finally, the parameters of the government are p = 5 and H = 0.4. This example is formulated by the methods that described in Section 5 and was programmed in GAMS v24.1.2 software. The computational results of the discussed problem are presented in Tables 9 and 10 under non-coordinated and coordinated structures of the GSC. From the results, we can compare the cost of the GSC among dif¬ ferent scenarios. As it is evident, the inventory cost of the GSC under the coordinated structures is less than that in the non-coordinated ones. In Scenario 3 (non-coordinated GSC under the carbon trade scheme), the value of the GSC’s total cost becomes negative. It means the GSC earn revenue more than its inventory cost by selling carbon credits. However, in the coordinated GSC the least total cost obtained at Scenario 8. Also, reviewing the cost of the GSC reveals that each reg¬ ulation gives advantages for some members and disadvantages for some others. The lowest cost of the retailers, distributors, and suppliers of the non-coordinated GSC are obtained under the carbon trade scheme but the manufacturer’s lowest cost is obtained under the carbon cap scheme. Moreover, in the coordinated GSC, retailers, and distributors reach least cost under the carbon cap scheme but the manufacturer and suppliers can reach their least cost under the carbon offset scheme. From the environmental benefits perspective, the results show that the lowest carbon emission is obtained in Scenario 3 due to the financial incentives of the carbon trade system for carbon footprint reduction. Effect of p on u 2,30,000 <D a • Scenario 1 <5 > o “ 1,30,000 £ B Scenario 2 4 Scenario 3 C 80,000 o - -jk . -- —4— Scenario 4 ■ X Scenario 5 ▼ LJUWiaiiu u (D -70,000 p=0 p=lo/ p=20 p=30 —B— Scenario 8 Fig. 7. The sensitivity of the optimal values of the utility function of the government to the parameter p. 823 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Effect of £2 on the IC g 3,00,000 6 a " 2,50,000 O tJ) u 2,00,000 o C/3 8 1,50,000 >3 o S 1,00,000 .2 fg 50,000 ■-■-i-• 0=0 Q=0.3 0=0.6 0=1 -Scenario 1 - Scenario 2 - Scenario 3 - Scenario 4 - Scenario 5 - Scenario 6 - Scenario 7 - Scenario 8 Fig. 8. The sensitivity of the optimal values of the inventory cost to the parameter £2. Also, the minimum amount of carbon emission in the coordinated scenarios is obtained under the carbon tax and carbon offset regula¬ tions. It can also be seen that in the most scenarios the carbon emission of the coordinated GSC is less than that in the non-coordinated GSC as well as cost. However, by the assumptions of this paper, the carbon trade regulation can be the most desirable for the government due to the minimum value of the u go „ in Scenario 3. For further analysis, the sensitivity of the inventory cost, total cost, carbon emission, and utility function to the government’s parameter p has been investigated. Figs. 4-6 show that by increasing p, the in¬ ventory cost and total cost of the GSC slightly increase in all scenarios except for Scenario 3. The cost of the GSC in Scenario 3 is more sen¬ sitive to the parameter p. Also, when the government’s parameter p increases, the carbon emission of the GSC will decrease. It means when the government is more concerned about the environment than the economic aspects, he puts more pressure on the GSC for carbon re¬ duction and as a result, the GSC incurs more cost. In addition, from Fig. 7, we can see that the utility function of the government is very sensitive to the parameter p. Thus, the government with right choice can achieve the desirable social welfare. Moreover, the sensitivity analysis of the cost and emission of the GSC, and utility function of the government in accordance to the parameter £2 shown in Figs. 8-11. The results show that by increasing £2; i.e., the government pays more attention to his revenue than GSC costs; the utility function of the government decreases. The cost func¬ tion of the GSC is more sensitive to parameter £2 under the carbon tax scheme due to the existence of GNR on the government’s objective function. By increasing £2 to its maximum level, the total cost of the GSC reaches its maximum acceptable value M,, under Scenarios 2 and 6. In contrary, when £2 increases the carbon emissions decrease. However, the GSC’s cost and emission are not sensitive to £2 under Scenarios 5, 7 and 8. The effect of government’s parameters on the replenishment cycle of the manufacturer is important because changes in this cycle will change the production quantity of the GSC. The sensitivity of the manu¬ facturer’s replenishment cycle T 2 , m in accordance with government’s parameters p and £2 are shown in Figs. 12 and 13. It can be seen that both p and £2 have inverse relationship with the manufacturer’s re¬ plenishment cycle in the non-coordinated GSC. However, these para¬ meters have no effect on the manufacturer’s replenishment cycle in the coordinated GSC. We now investigate the sensitivity of inventory input parameters to the results. The demand, holding cost and ordering cost change from + 25% to +100% and their effect on the cost and emission of the GSC and the government’s utility function are determined as presented in Tables 11 and 12. We find that the cost and the carbon emission in¬ crease simultaneously in all scenarios when the market demand rises and consequently the government’s utility function also increases. Furthermore, when the holding costs increase, the total cost of the GSC increases as expected and the utility function of the government in¬ creases too, but the emission of the GSC decreases. Also, increasing the ordering costs will increase the total cost, the utility function of the government and the carbon emission. From the numerical example, the following managerial insights can be inferred. • The results from different scenarios show that inventory decisions have an impact on the carbon emission of the GSC. Thus, a firm can reduce his carbon emission through operational adjustment instead of other costly methods. • The members of the GSC can reduce their both inventory cost and their carbon emission by implementing a coordination mechanism for managing product flows. Consequently, cooperation helps both the government and the GSC to achieve their goals. However, the Effect of £2 on the Z U 10,06,000 cn ’ 5 O jg 8,06,000 o tS 6,06,000 o o 3 4,06,000 o <D £ 2,06,000 6,000 0=0 0=0.3 0 = 0.6 0=1 Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7 Scenario 8 Fig. 9. The sensitivity of the optimal values of the total cost to the parameter Q. 824 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Effect of Q on the E 4,000 U GO O —■- —•— Scenario 4h O g 3,000 —■— Scenario ♦ Scenario —±— Scenario £ X / - \ 4 x ^ m <u 2,500 G O G ▼ ■ V- x 1 Scenario ♦ Scenario q z,uuu <0 G ♦ Scenario H 0=0 0=0.3 0=0.6 0=1 Fig. 10. The sensitivity of the optimal values of the carbon emission to the parameter £2. coordinated scheme may be difficult to implement in practice be¬ cause it requires a decision maker who has information of all members. • When the government increases the value of environmental im¬ portance factor p, the cost of the GSC will increase and the carbon emission will decrease. Thus the cost and carbon emission have an inverse relationship. Also, when the government increases the value of his revenue importance factor £2, the GSC cost increase and as a result the emission decrease. Moreover, each regulation affects the cost and the carbon emission of each stage of the GSC differently. • Among the four regulations that examined in this study, the carbon trade scheme can be more appropriate from the social welfare point of view. This result shows that governments may achieve their goals by setting incentive policies. • By changing the government’s policies, the appropriate regulation can be different. If only the environmental benefits are considered by the government, the schemes such as carbon tax and carbon offset may be effective as well as carbon trade scheme. But the governments should consider that these schemes are kind of a command-and-control policy and deviate firms from their optimal solutions and increase their cost and dissatisfaction. Besides, if the government only considers the GNR, the carbon tax policy may be more effective and the optimal solution may be different. • The sensitivity analysis of the model to the government’ environ¬ mental importance factor shows that the total cost of the GSC, the carbon emission and social welfare are affected by government regulations and decisions. Thus, the government can achieve the optimal trade-off between the carbon emission and the profit of firms by applying an appropriate regulation and adjusting the op¬ timal parameters and variables. • The sensitivity analysis of the model to the inventory parameters shows that the carbon emission of the GSC has a direct relation with demand and ordering cost but has an inverse relationship with the inventory holding cost. In addition, the utility function of the gov¬ ernment has a direct relationship with parameters of the inventory model. 7. Conclusion The purpose of this study was to contribute to the field of inventory management of a multi-stage GSC under carbon emission regulations. This paper used to determine the importance of the inventory decisions for compliance the GSC with carbon regulations and also analyzing the effect of these regulations on the cost and carbon emission. Therefore, a mathematical model is developed for inventory cost and carbon emis¬ sion of a four-echelon GSC based on the non-coordinated and co¬ ordinated decision-making structures. In addition, four different reg¬ ulations associated with controlling carbon emission have examined in the model of the GSC. Thus, eight scenarios are developed based on these regulations and the structure of the GSC. Each scenario modeled as a Stackelberg game, where at first the government (leader) as a more powerful player, set a regulation and based on that, determines a cer¬ tain cap, carbon tax or carbon price to maximize SW. Second, the GSC (follower) determines the replenishment cycle of each stage and the production quantity. This game expressed as a bi-level MINLP. Then the problems of the non-coordinated GSC reformulated to single-level problems by replacing the KKT condition of the lower level problems. In addition, an algebraic method presented for solving the problems of the coordinated GSC. The findings of this study from the numerical example highlight that inventory decisions have an influence on the carbon emission of the GSC and consequently, operational adjustment can be an effective way for carbon reduction. Moreover, the firms in a GSC can reduce both their cost and carbon emission by using a coordination mechanism in Effect of £2 on u gov o3 80,000 i e g -1,20,000 o 00 H -3,20,000 o o -5,20,000 O G B -7,20,000 & § -9,20,000 <D G -11,20,000 Scenario Scenario Scenario Scenario Scenario Scenario Scenario Scenario 1 2 3 4 5 6 7 8 Fig. 11. The sensitivity of the optimal values of the utility function of the government to the parameter Q. 825 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Effect of p on T 2 m 0.030 (D 0.000 p=0 p—10 p—20 p—30 - Scenario - Scenario - Scenario - Scenario - Scenario * Scenario - Scenario - Scenario 1 2 3 4 5 6 7 8 Fig. 12. The sensitivity of the manufacturer’s replenishment cycle to the parameter p. 0.030 <D re¬ 's 0.025 <L> >. S 0.020 O Vh G § g 0.015 I § '3 § 0.010 t <D H 0.000 Effect of £2 on T 2 m 0=0 Q=0.3 0=0.6 0=1 - Scenario - Scenario - Scenario - Scenario - Scenario - Scenario - Scenario - Scenario 1 2 3 4 5 6 7 8 Fig. 13. The sensitivity of the manufacturer’s replenishment cycle to the parameter 12. Table 11 The sensitivity of the results to the input parameters in the non-coordinated GSC. Changes of Dij Scenario 1 Scenario 2 Scenario 3 Scenario 4 Z n E n Ugov Z" E n Ugov Z n E n Ugov Z" E n Ugov + 25% 161,215 3720 115,331 163,160 4457 118,397 160,341 4457 118,489 161,591 3627 115,089 + 50% 180,243 4160 128,944 182,418 4983 132,372 179,423 4983 132,568 180,663 4055 128,674 + 75% 201,518 4651 144,164 203,949 5571 147,996 200,758 5571 148,310 201,987 4534 143,861 + 100% 225,303 5200 161,180 228,021 6229 165,464 224,612 6229 165,910 225,829 5069 160,842 Changes of hij Z n E n Ugov Z n E n Ugov Z n E n Ugov Z n E n Ugov + 25% 160,486 3256 112,569 162,496 3787 114,918 -86,245 1851 -42,490 160,796 3182 112,385 + 50% 178,828 3201 123,301 181,077 3626 125,326 - 54,845 1840 -23,705 179,132 3131 123,136 + 75% 199,450 3166 135,499 201,911 3503 137,262 -19,098 1840 -2,256 199,768 3097 135,346 + 100% 222,605 3153 149,329 225,259 3419 150,884 222,786 3419 150,768 222,956 3082 149,185 Changes of Aq Z" E n Ugov Z n E n Ugov Z n E n Ugov Z" E n Ugov + 25% 155,429 3356 110,037 157,164 4021 112,797 154,490 4021 112,801 155,765 3272 109,818 + 50% 168,188 3406 117,945 169,933 4080 120,727 -78,710 1852 -37,966 168,539 3318 117,714 + 75% 182,659 3482 127,004 184,430 4165 129,815 -57,625 1852 -25,313 183,039 3385 126,751 + 100% 199,051 3584 137,352 200,866 4279 140,202 -31,419 1842 -9,642 199,473 3477 137,069 inventory decisions. In addition, the inventory cost and the carbon emission of the GSC have inverse relations and the government can trade-off the costs and carbon emissions by setting appropriate reg¬ ulation and parameters. Among the different regulations that examined, the carbon trade scheme can be more effective from the point of view of both GSC and SW. However, the results may differ by changing the government policies. For instance, if the government seeks to maximize his revenue the carbon tax policy may be more effective. Finally, we propose that future studies can consider complex assumptions of inventory management such as deteriorating products and delayed payment during demand uncertainty. In addition, the developed models can be expanded by engaging the concepts of rewards-driven systems and maintenance scheduling with GSCM (see Duan, Deng, Gharaei, Wu, & Wang, 2018; Gharaei, Naderi, & Mohammadi, 2015). Besides, this study considered a Stackelberg game between the GSC and the government. One may consider other games such as cooperation for further study. Besides, the impact of other regulations of the government on the inventory cost is an interesting future research area. 826 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 Table 12 The sensitivity of the results to the input parameters in the coordinated GSC. Changes of Dy Scenario 5 Scenario 6 Scenario 7 Scenario 8 Z c E c Ugov Z c E c Ugov Z c E c Ugov Z c E c Ugov + 25% 76,074 2466 57,973 78,586 2459 58,465 137,877 3694 101,197 76,074 2466 57,973 + 50% 85,051 2757 64,814 87,860 2750 65,364 154,307 4130 113,235 85,051 2757 64,814 + 75% 95,087 3082 72,463 98,228 3074 73,078 172,675 4618 126,693 95,087 3082 72,463 + 100% 106,309 3446 81,014 109,820 3437 81,703 193,212 5163 141,741 106,309 3446 81,014 Changes of hij Z c E c Ugov Z c E c Ugov Z c E c Ugov Z c E c Ugov + 25% 75,735 2118 56,031 77,881 2114 56,455 137,213 3202 98,336 75,735 2118 56,031 + 50% 84,425 2039 60,852 86,483 2037 61,259 153,035 3118 107,410 84,425 2039 60,852 + 75% 94,217 1971 66,386 96,199 1970 66,780 170,835 3056 117,779 94,217 1971 66,386 + 100% 105,225 1916 72,715 107,147 1915 73,098 190,825 3018 129,586 105,225 1916 72,715 Changes of Aij Z c E c Ugov Z c E c Ugov Z c E c Ugov Z c E c Ugov + 25% 75,737 2332 57,104 78,123 2325 57,570 136,968 3447 99,418 75,737 2332 57,104 + 50% 84,380 2485 63,055 86,931 2477 63,552 152,487 3627 109,628 84,380 2485 63,055 + 75% 94,080 2667 69,785 96,826 2657 70,317 169,932 3847 121,197 94,080 2667 69,785 + 100% 104,957 2881 77,379 107,932 2868 77,954 189,520 4112 134,272 104,957 2881 77,379 Acknowledgement This paper has been accomplished on the basis of a Ph.D. disserta¬ tion by Kourosh Halat supervised by Prof. Ashkan Hafezalkotob at Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran. The authors would like to appreciate the reviewers and editor for their insightful comments. Appendix A The IC n is convex when the inventory cost of each stage of the GSC be convex. The second derivation of the inventory cost function of retailers is calculated as follows: d 2 IC" ^ J=rl (A.l) The second derivation of the inventory cost function of the retailers is more than zero, as a result /C" is convex. The second derivation of the inventory cost function of distributors is calculated as follows: 52/C "= t^ L >° j=dl J 3 J dTi 3 J (A. 2) The above equation shows IC" is convex. The inventory cost function of the manufacture contains two decision variables (T 2 m , k„). The hessian matrix of /C," is calculated as follows: H,rf = 2Sm _j_ 2+2,m T 2,m Kwl 2 ,m A2,m k 2 y2 kwT 7 yy S Axm +KJ] + h w Z (^) 2^2 ,m k&Tz,* > 0 The inventory cost function of the manufacturer is convex only if: s / 2S m 2A 2 m 2A 2 m A2 m ''7^3 T ^l,3m ) V ,2 rp2 *2,m *2,m ^w^2,m + K 2 (^)) 2 > 0 s=sl The second derivation of the inventory cost function of suppliers is calculated as follows: ^>0 tk T h Given that the inventory cost of each stage of the GSC is convex, as a result, the Eq. (22) is convex. Appendix B (A. 3) (A.4) (A. 5) The E n is convex when the carbon emission of each stage of the GSC be convex. The second derivation of the carbon emission function of retailers is calculated as follows: a 2 g" y r 2 Zj R 2 p° Ze 4j i 4 J T 3 . j=rl 4 J > 0 (B.l) 827 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 The second derivation of the inventory cost function of the retailers is more than zero, as a result E£ is convex. The second derivation of the carbon emission function of distributors is calculated as follows: D 2e? ^=^>0 s'y & % The Eq. (B.2) shows fs 3 " is convex. The carbon emission function of the manufacture is convex due to the following equation. 3 2 E1 2el„ T 3 1 2,m > 0 The second derivation of the carbon emission function of suppliers is calculated as follows: d -^=i 2 4>o Given that the carbon emission of each stage of the GSC is convex, as a result, the Eq. (23) is convex. Appendix C (B.2) (B.3) (B.4) Y (K& D m + hjks&Dm) = Y U3 s D m (h w + hijks )) j j (C.l) Y ( h + h 2m D m + KP s D m { D ™ 1)) j *m (C.2) (^3 J ^2,m) j (C.3) / 3 + k m (y 2 + k w y 2 ) (C.4) An Y ~ M j (C.5) k d cc + cti 2 (C.6) : Y + bij + j s (C.7) : Sm + ^2 ,m (C.8) Y Ai j + + i) j K-m K-w (C.9) 2 (a 4J + fa 3j ) j (C.10) V — +a 2 kd (C.ll) Appendix D j (D.l) ?2 = e^ m D m ( D p m + 1) + e^D m (° p m - 1) + 2 ef m D m (D.2) % = An 2 <C V ~ e 2,m ) J (D.3) s = f 3 + k m (y 2 + k w %) (D.4) “l = An Y ( e *J ~ e 3j) j (D.5) - _ k d co + a 3 2 (D.6) 828 K. Halat and A. Hafezalkotob Computers & Industrial Engineering 128 (2019) 807-830 = Z «/ + e p j s (D.7) V 2 = e 2.m + (D.8) = Z e 3J + + T-Vd j K-m K-w (D.9) “2 = Z K + e 3j) j (D.10) X = -- \~ CC2 k d (D.ll) References Alhaj, M. A., Svetinovic, D., & Diabat, A. (2016). A carbon-sensitive two-echelon-in- ventory supply chain model with stochastic demand. Resources, Conservation and Recycling, 108, 82-87. Bazan, E., Jaber, M. 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