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Computers & Industrial Engineering 128 (2019) 807-830 



ELSEVIER 


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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 


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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 


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P3 

H 


43 

3 

2 

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d 


•P s 


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42 X) X) 42 


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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: 


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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 


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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: 


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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. 


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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 


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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) 


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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) 


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