Sustainable multi-channel supply chain design: an intuitive fuzzy game theory approach to deal with uncertain business environment

Abstract

By introducing the concept of sustainable development, managers and policymakers in many industries have been encouraged to consider environmental and social issues in addition to economic objectives in their planning. Following this concept, sustainable supply chain management has become the main concern of many studies. Among all the strategies to achieve sustainability targets in a supply chain, cooperating with third-party logistics companies has attracted lots of attention. By providing more sustainable and efficient transportation services, 3PLs can help all types of regular, closed-loop, and circular SCs achieve more profit, while they are still sustainable, at least in distribution and collection/recycling stages. This study investigates the sustainable multi-channel SC design problem in the presence of the government and 3PLs. To bring the present study closer to the real-world situation, the problem is modeled using an intuitionistic fuzzy uncertainty approach. Considering the government as the leader of the SC in two centralized and decentralized decision structures, game theory has been applied to model the game between players and obtain optimal decision values. For the first time in the literature, public awareness toward green activities of the players, emission reduction, uncertainty, and delivery time have been considered in this study. The results show the presence of a 3PL will reduce the delivery time and the amount of pollution. Also, the findings confirm that governments can control the players' activities and encourage them to apply green strategies using financial tools.

1 Introduction

Emissions and environmental issues have turned into one of the most critical human concerns in the new millennium all around the world. Producers as well as the transportation sectors are the main releasers of greenhouse gases (GHGs). Hence, many studies in recent decades have been done on supply chain management (SCM) and green supply chain management (GSCM) in a wide variety of industries in order to find sustainable solutions to this problem (You et al., 2021). A regular supply chain (SC) includes activities related to production and to delivery of the final product from suppliers to customers (Lummus & Vokurka, 1999), while a circular SC will also have an agent to collect the used product in order to recycle it (De Angelis et al., 2018; Guo et al., 2018). A SC can include several suppliers, producers, distributors, central depots, recycling centers, and, finally, customers (Amin & Zhang, 2013). The goal of GSCM is to achieve a stable balance between the profits of chain members and environmental policy (Mahmoudi et al., 2021), while sustainable supply chain management (SSCM) considers social dimensions of activities in addition to environmental and economic aspects (Feng et al., 2021; Mahmoudi et al., 2019). Indeed, the main goal of the SSCM is to strike a suitable balance between economic development, environmental monitoring, and social assets (Sikdar, 2003).

Research over the past decade has shown a relationship between Consumer Awareness (CAs) and designed Green Products (GPs) (Jayaram & Avittathur, 2015). GPs are designed and produced to reduce excessive consumption of energy and natural resources, as well as reduce or eliminate released toxic substances that are harmful to both consumers and the environment (Zhang et al., 2021). Many companies see green product production as a key policy and strategy to improve competition in a competitive market, creating a green image of the company in the consumers’ minds and moving forward to sustainable development (Pakseresht et al., 2020). On the other hand, the cost of green activities and green production and the competition among members of the SCs and SCs make managers hesitant to apply green policies without any precise market analysis (Beamon, 1999; Mahmoudi & Rasti-Barzoki, 2018).

In addition to the pollution caused by production activities, the product distribution and collection activities from the producer to the consumers, and from consumers to producers, are playing a major role in spreading the pollution in both regular and circular SCs. The use of modern and environmentally friendly transportation systems/vehicles can be a great help in reducing pollution and increasing consumer satisfaction. Based on this, the issue of sustainable activities for third-party logistics (3PLs) companies has become a significant and challenging problem (Doherty & Hoyle, 2009; Stekelorum et al., 2021). 3PLs are companies that by arranging contracts that are usually long term, provide outsourced logistics services to producers. These companies provide logistics services such as material management, product distribution, product collection for recycling, etc. (Jiang et al., 2014; Ngo, 2021).

Many big companies, such as Apple and Samsung in the technology industry, Mercedes-Benz and Volkswagen in the automotive industry, and Fonterra and Arla in the dairy industry that offer similar products and have a large market use various strategies to control the price of products and other related items. As the core of a SC, logistics has a significant impact on the efficiency and internal costs of a SC (Li et al., 2012). On the other hand, the delivery of the ordered products to the customers in the fastest possible time is one of the most critical factors that can lead to a significant increase in the loyalty of the consumer toward the producer and distribution channels (Mahmoudi et al., 2021). By outsourcing logistics activities, companies can decrease their operational costs and their financial risk (Aktas & Ulengin, 2005). Therefore, in addition to the environmental debate, the goal of major companies in cooperating with 3PLs is to reduce the costs and improve services (Pamucar et al., 2018). According to increasing global attention toward sustainable development and sustainability, the 3PLs have found that environmental sustainability will soon be a critical criterion for their consumers (Genovese et al., 2013). As a result, recently, most of 3PL companies are trying to provide environmentally friendly services (Zailani et al., 2011).

Uncertainty in real-world problems is one of the main reasons that highly complicates the planning and decision-making process. For almost all industries, the demand value and product price can be considered the most important factors in any decision-making process that depends on some uncertain parameters/values. While pricing problems are always considered as one of the most complicated decision-making processes in supply chain management, uncertainty in demand or any other parameter makes this process even more complicated. Considering that product prices will seriously affect the market share of any company, there are a large number of studies that have applied different methodologies to investigate the pricing problem, especially in an uncertain business environment (Mahmoudi et al., 2020; Tian et al., 2014). However, studies in the field of uncertainty have shown that it is difficult to select a suitable probabilistic distribution function (Xie et al., 2006). A good strategy to deal with this issue is to consider the experts’ and managers’ opinions using fuzzy theory.

To consider the uncertainty in different problems, fuzzy sets have been introduced by Zadeh (1965). This theory includes two important concepts: degree of membership and degree of non-membership, so that the degree of non-membership is considered as a complement to membership of one. The values of these parameters are usually determined based on the decision-makers’ opinion. Since there is no consensus about an objective process to obtain such critical information, it seems reasonable to consider the degree values of membership as an interval instead of an exact value (Jafarian et al., 2018). On the other hand, managers may not consider it as a complement to a membership grade of one in determining the degree of non-membership degree. In this case, they are hesitant about allocating amounts to parameters. To overcome this problem, Atanassov (1986) developed the theory of classical fuzzy sets and introduced the concept of intuitive fuzzy sets. In intuitive fuzzy sets, three functions are used to express the concepts of degree of membership, degree of non-membership, and degree of hesitation. In intuitive fuzzy theory, the expert can mention his/her own opinion better than in classic fuzzy. Intuitive fuzzy sets are an excellent tool to deal with vague, inexact decision information and uncertainty and ambiguity in the decision-making process (Atanassov, 1986). According to the characteristics of the fuzzy theory, it has many advantages over the probability theory (Zimmermann, 2000).

Governments play a key role in how members of a SC make decisions. Studies show that the government’s financial interventions on the activities of different members of a SC can encourage them to engage in sustainable operations (Mahmoudi & Rasti-Barzoki, 2018). Therefore, considering the government as a strategic player in the SSCM problem can provide more reliable results. In this study, considering the issue of uncertainty and cooperation with 3PLs, the pricing problem for green and non-green SCs under the government’s financial intervention will be modeled using game theory. The general objective is to evaluate the financial effects of governmental tariffs on the players’ optimal strategies and the sustainability of the SCs, while they are facing uncertainty in the information. Also, for the first time in the literature, the problem of cooperation or non-cooperation with 3PLs in a fuzzy environment will be analyzed and the impacts of cooperation on the sustainability of SCs will be examined.

In particular, game theory has been applied in this study to address the following research questions:

• (a) How do the governmental tariffs affect the optimal and equilibrium values, players’ strategies and sustainability of the SCs in different scenarios?

• (b) Is cooperation with 3PLs an efficient strategy to achieve sustainable development targets?

• (c) What is the relationship between the sustainability of the supply chain and public awareness of green production and cooperation with 3PLs?

The rest of this article is set up as follows: In Sect. 2, a review of previous studies and research gaps has been presented. In Sect. 3, the concepts, theorems, and relations proposed in intuitive fuzzy theory are briefly explained. Sections 4 and 5 present the problem definition and proposed models, respectively. Numerical examples, sensitivity analysis, and managerial insight are discussed in Sect. 6. Finally, Sect. 7 presents concluding remarks and directions for future studies.

2 Literature review

In this section, the literature related to different aspects of this study has been reviewed.

2.1 Sustainable/green supply chain management

There have been a growing number of studies conducted on SCM during the last decades. Product delivery time, the amount of greenness of the product and the sustainability of SC, selecting appropriate distribution centers, pricing, etc., are the main themes of the conducted studies. In particular, since introducing the concept of sustainable development in 1987 (Brundtland et al., 1987), the subject of green and sustainable supply chains has attracted lots of researchers’ and policymakers’ attention. In addition to the concepts, different tools have also been applied to conduct existing research, such as operations research methods and statistical methods. In particular, game theory has turned into a popular method among researchers to evaluate different policies and find equilibrium strategies in different issues of SCM. For example, Fleischmann et al. (1997) showed that the production of a green product increases the demand for that product. That is why producers are moving to make their products greener to take advantage of the competition that has been created in the target market. Yan et al. (2011) analyzed the effects of coordination and different strategies on sharing profits in the multi-channel SC. In this structure, the manufacturers can sell their products through both online and traditional channels. Results showed that creating an online channel in the SC could increase the competition among members. Swami and Shah (2011) examined a SC consisting of a producer and a retailer. In this study, the effect of green production on increasing demand has been considered. Using game theory and considering Stackelberg competition between members, they concluded that the optimal green effect rate for the producer and retailer was equal to the green cost rate. Basiri and Heydari (2017) focused on the coordination of the green channel in a two-stage SC, in order to determine the equilibrium prices for the green and non-green products, as well as the greening degree of the products. This study showed that cooperation in the SC will increase degree of the greenness of the product. Hajiaghaei-Keshteli and Fard (2019) investigated a sustainable closed-loop supply chain network design problem. They applied a new multi-objective mixed integer nonlinear programming model and a discount feature for the transportation system in their study. Furthermore, they implicated a new heuristic method to solve the problem. The result showed that their method had better performance than other traditional methods.

Mahmoudi and Rasti-Barzoki (2018) developed a two-population evolutionary game model to evaluate the long-term behavior of producers and retailers under the government’s financial intervention. They assumed producers could apply green or non-green strategies and considered three different decision-making scenarios for the government as the leader of the game. As an important result, they showed that governmental tariffs are very effective tools to push the SCs toward sustainable production in both the short- and long term. Considering a population of governments, suppliers and producers, Xu et al. (2020) applied evolutionary game theory to investigate the impacts of government presence on the GSCM problem. They found that a green strategy by one of the members of the SC can affect the behavior of the opposing player in order to implement the green policies. Also, if the government seeks to make the greening policies more effective, it should put more financial pressure on the producers. In another similar study, Hadi et al. (2020) modeled the pricing problem in a green SC in the presence of the government. They considered two types of green and non-green products for the supply chain and used the evolutionary game theory approach to study the problem and evaluate the long- and short-time strategies of the players. The results show that the implementation of conservative environmental policies can improve the utility of the government and SC’s members. Sana (2020) studied the pricing problem in a SC when both green and non-green products are produced. They assumed that government could affect SC’s activities through taxes or subsidies. Also, Liu et al. (2020) considered a two-echelon SC and explored the problem of decision making and coordination between members by taking social responsibility into account. They used a game theory approach to analyze the problem and showed that considering a penalty can have a negative effect on producer performance. Meng et al. (2021) examined the impacts of financial interventions by governments on the performance of supply chain members. To this end, they considered a dual-channel green supply chain and analyzed the performance of the members in the case of receiving or not receiving subsidies. The results showed that the allocation of subsidies reduces the price of green products and increases the producer’s profit.

2.2 Cooperation with 3PL companies

Outsourcing, especially in logistical activities, was always a practical strategy in order to reduce the related costs. There are some studies that have evaluated the effects of cooperation with 3PL companies on the SC’s performance. Lai et al. (2009) developed a game theoretic approach for controlling greenhouse gas emissions in an energy SC. In their model, the government seeks to determine the desired tax rate for greenhouse gas emissions. Furthermore, each member of the SC seeks to maximize its profits. Xu et al. (2012) studied a network of independent SCs, using a cooperative game model. Their results showed that cooperation between SCs can lead to a reduction in transportation costs and also in greenhouse gas emissions. Ozsakalli et al. (2014) investigated the importance of planning for logistics companies. The results of the study show that, in addition to cooperation with 3PL companies, comprehensive planning should be designed to achieve significant cost savings. Jiang et al. (2014) examined the decision-making and coordination problems in a SC, including one producer and two retailers, while a 3PL company was considered for the logistics services. They analyzed the margin effects of the 3PL company’s profit on the SC’s activities and developed three different contracts among the players in a decentralized decision-making model. Mahmoudi et al. (2021) used game theory to study a green supply chain, considering three different models for products sales. They assumed that the products were delivered to customers through the retail channel or direct channel. In addition, it was assumed that products were transported from producers to some retailers by a 3PL company, and the rest of the retailer received products from producers directly. Their study indicated that cooperating with 3PL company has a great impact on increasing supply chain sustainability. For a closed-loop supply chain, Lang et al. (2016) and Qi et al. (2017) applied game theory in order to investigate the pricing and greenhouse gas emissions problems in the SC.

2.3 Applications of the fuzzy approach on the supply chain

Uncertainty in SCM is another subject that a group of studies have focused on. Wei and Zhao (2011) investigated the optimal pricing problem in a closed-loop SC with a producer and two competitive retailers in fuzzy conditions. As a result, they found that the amount of profit obtained in the centralized model is much higher than in the decentralize model. Also, the retail price in the centralized model is lower than in the decentralized one. Zhao et al. (2012) studied the pricing problem for a SC with alternative products in a fuzzy environment and indicated that the highest profit will be provided when the members cooperate with each other. Considering Nash competition (NC) between different channels, Wu (2013) used game theory to study two competitive SCs in which demand is a function of retail and wholesale prices. Zhao et al. (2013) addressed the issue of pricing and manufacturing services in a two-level SC consisting of two producers and a common retailer, and considered some parameters such as customer demand, production costs, and producer service costs in coefficients fuzzy form. They concluded that if the market is stable, decision makers can have acceptable predictions of the future of the market. Zhao and Wei (2014) used a game theory approach to study a SC with fuzzy demand that was a function of retail prices and sales effort. They considered both centralized and decentralized decision-making patterns and modeled the game between the players in both patterns. The results showed that in symmetric and asymmetric information conditions, the coordinated channel provides a better result than the traditional channel. Soleimani (2016) focused on the pricing problem in a two-channel SC including a producer and a retailer operating under fuzzy uncertainty. In this study, it was shown that when the producer is the leader of the game, an increase in the cross-price elasticity to the other channel’s price will reduce the producer’s profit. Lin et al. (2018) developed the fuzzy DEMATEL method to study the sustainable SC management problem and showed that compared to the previous studies, their method provides more reliable results. Jafarian et al. (2019) studied the impact of different power structure strategies on the optimal decisions of SC members in pricing. They considered two rival SCs and solved the problem in an intuitive fuzzy environment with a game theory approach. Considering a three-person game, Chavoshlou et al. (2019) studied the issue of GSCM in order to find the best optimal solutions for the players under different scenarios. To obtain more realistic results, they investigated the problem under conditions of uncertainty and used fuzzy theory. Sarkar et al. (2020) investigated the pricing problem in the SC under conditions of fuzzy uncertainty and according to the concept of cooperation in advertising, when the players are producers, retailers, and customers. The results showed that increasing advertising cooperation will increase the profitability of the entire supply chain. Taleizadeh et al. (2020) examined a closed-loop supply chain considering the features of fuzzy theory. Pricing and advertisement problems were investigated in their research and the result showed that the profit of members in a centralized approach is higher than in a decentralized one.

2.4 Research gaps and contributions

Listing the key features of the current study, Table 1 compares this study with existing ones in the literature. Also, according to this table, it can be easily found which new features have been covered in this study.

Table 1 Key differences among current study and current studies

While cooperating with 3PLs seems like a crucial strategy to achieve sustainable development goals in the SCs, considering the uncertainty in the modeling makes the results more reliable, and modeling both centralized and decentralized decision structures provides more comprehensive findings and views. Due to Table 1, and to the best of the authors’ knowledge, there is not any study in the literature that considers all these issues together. According to this research gap, as its main contribution, this research has focused on the problem of sustainable supply chain design considering the government’s financial intervention and cooperation strategies with 3PLs in the presence of uncertainty. To this end, two SCs, including two producers (green and non-green), two retailers, and a 3PL, are being considered. Alternative products are produced by producers, and it is assumed that the first and second producers produce green and non-green products, respectively. The 3PL has modern warehousing and transportation facilities and provides services to both producers. Cooperation with 3PLs is aimed at reducing greenhouse gas emissions due to its special features. The government has been considered as leader of the market who monitors players’ activities and imposes tariffs based on its strategies and players’ profits. Due to the inability to predict the exact values of the parameters for various reasons in the real world (such as demand fluctuations or price fluctuations), for better simulation, the problem is modeled in the intuitive fuzzy environment. In order to present a comprehensive decision structure, the problem has been modeled for both centralized and decentralized decision patterns. Using game theory, the optimal decision values, optimal prices and reactions of the players to the government’s tariffs have been obtained.

3 Revisit of intuitive fuzzy set theory

Each element of a set has a membership value in the interval [0, 1] in the classical fuzzy set theory. Suppose Y is a universal set and y is an element in this set. R is a fuzzy set in Y that has been characterized by a membership function as $\mu_{R} :Y \to [0,1]$. The $\mu_{R} (y)$ value for any $y \in Y$ represents the degree of membership of y in R. When the value of $\mu_{R} (y)$ is 1, it shows that the element y is completely in set R and when $\mu_{R} (y) = 0$, it means y is not entirely in set R. However, assigning values between 0 and 1 for $\mu_{R} (y)$ indicates the degree of partial membership. The intuitive fuzzy theory has some differences with the classical mode. So, in order to model the understudy problem, important definitions and theories related to the intuitive fuzzy theory are briefly explained.

Definition 1

Assume Ѱ is as a non-empty set, P(Ѱ) is the power set of Ѱ and Cr is a credibility measure. For each event $\tilde{Y}$ in P(Ѱ), the credibility measure, ${\text{Cr}}\left\{ {\tilde{Y}} \right\}$, indicates the credibility that $\tilde{Y}$ will occur. The credibility space can be defined as a triplet (Ѱ, P(Ѱ), Cr). The credibility measure Cr: Ѱ → [0, 1] is a set function satisfying following axioms (Liu, 2009):

Axiom 1

${\text{Cr}}\left\{ \Psi \right\} = 1$.

Axiom 2

If $\tilde{Y} \subseteq \tilde{Z}$, then ${\text{Cr}}\left\{ {\tilde{Y}} \right\} \le {\text{Cr}}\left\{ {\tilde{Z}} \right\}$.

Axiom 3

For any event $\tilde{Y}$, ${\text{Cr}}\left\{ {\tilde{Y}} \right\} + {\text{Cr}}\left\{ {\tilde{Y}^{c} } \right\} = 1$.

Axiom 4

For each event $\tilde{Y}_{i}$ with $\sup_{i} {\text{Cr}}\left\{ {\tilde{Y}_{i} } \right\} < 0.5,{\text{Cr}}\left\{ {\bigcup\nolimits_{i} {\tilde{Y}_{i} } } \right\} = \sup_{i} {\text{Cr}}\left\{ {\tilde{Y}_{i} } \right\}$ (Note. Sup is the abbreviation of supremum).

Definition 2

Let $\left( {\Psi_{t} ,P(\Psi_{t} ),{\text{Cr}}_{t} } \right),t = 1,2,...,n$, be credibility spaces, $\Psi = \prod\nolimits_{t} {\Psi_{t} }$ and ${\text{Cr}} = \wedge_{t} {\text{Cr}}_{t}$. The product credibility space can be defined by $(\Psi ,P(\Psi ),{\text{Cr}})$, where ${\text{Cr}}\left\{ {\tilde{Y}} \right\}{ (}\forall \tilde{Y} \in P(\psi ){)}$ is determined as follows (Liu, 2007):

$$ {\text{Cr}}\left\{ {\tilde{Y}} \right\} = \left\{{\begin{array}{*{20}c} {\mathop{\mathrm{sup}}\nolimits_{{(\Psi_{1},\Psi_{2} ,\Psi_{3} ,...,\Psi_{n} ) \in \tilde{Y}}}\mathop{\mathrm{min}}\nolimits_{1 \le t \le n} {\text{Cr}}_{t}\left\{ {\theta_{t} } \right\},} &{{\text{if}}\;\mathop{\mathrm{sup}}\nolimits_{{(\Psi_{1} ,\Psi_{2},\Psi_{3} ,...,\Psi_{n} ) \in \tilde{Y}}}\mathop{\mathrm{min}}\nolimits_{1 \le t \le n} {\text{Cr}}_{t}\left\{ {\Psi_{t} } \right\} \le 0.5}\\ {1 - \mathop{\mathrm{sup}}\nolimits_{{(\Psi_{1} ,\Psi_{2} ,\Psi_{3},...,\Psi_{n} ) \in \tilde{Y}^{c} }}\mathop{\mathrm{min}}\nolimits_{1 \le t \le n} {\text{Cr}}_{t}\left\{ {\theta_{t} } \right\},} &{{\text{if}}\;\mathop{\mathrm{sup}}\nolimits_{{(\Psi_{1} ,\Psi_{2},\Psi_{3} ,...,\Psi_{n} ) \in \tilde{Y}}}\mathop{\mathrm{min}}\nolimits_{1 \le t \le n}{\text{Cr}}_{t} \left\{ {\Psi_{t} } \right\} > 0.5} \\\end{array} } \right. $$

(1)

Definition 3

$\tilde{\varepsilon }$ is a fuzzy variable which is defined as a measurable function of $(\Psi ,P(\Psi ),{\text{Cr}})$ to the ${\mathbb{R}}$ numbers. The membership function of $\tilde{\varepsilon }$ can be obtained by Liu (2009):

$$ \mu_{{\tilde{\varepsilon }}} (x) = \left( {2{\text{Cr}}\left\{ {\tilde{\varepsilon } = x} \right\}} \right) \wedge 1,x \in {\mathbb{R}} $$

(2)

Lemma 1

For any $\beta \subseteq {\mathbb{R}}$:

$$ {\text{Cr}}\left\{ {\tilde{\varepsilon } \in \beta } \right\} = \frac{1}{2}\left( {\mathop {\sup }\limits_{x \in \beta } \mu_{{\tilde{\varepsilon }}} (x) + 1 - \mathop {\sup }\limits_{{x \in \beta^{c} }} \mu_{{\tilde{\varepsilon }}} (x)} \right) $$

(3)

Proof

Please see Liu (2009). □

Definition 4

Assume $\tilde{\varepsilon }_{1} ,\tilde{\varepsilon }_{2} ,...,\tilde{\varepsilon }_{n}$ are the fuzzy variables defined on $\left( {\Psi_{t} ,P(\psi_{t} ),{\text{Cr}}_{t} } \right),t = 1,2,...,n$ and $f:{\mathbb{R}}^{n} \to {\mathbb{R}}$ is a function. In this case, $\tilde{\varepsilon } = f(\tilde{\varepsilon }_{1} ,\tilde{\varepsilon }_{2} ,...,\tilde{\varepsilon }_{n} )$ will be a fuzzy variable defined on the product credibility space $\left( {\prod\nolimits_{t} {\Psi_{t} } ,P(\prod\nolimits_{t} {\Psi_{t} } ), \wedge_{t} {\text{Cr}}_{t} } \right)$ (Liu, 2002).

Definition 5

Considering M as a non-empty universal set, the intuitionistic fuzzy set $\tilde{Y}$ in M can be defined as (Atanassov, 1986):

$$ \tilde{Y} = \left\{ {\left\langle {x,\mu_{{\tilde{Y}}} (x),\upsilon_{{\tilde{Y}}} (x)} \right\rangle |x \in M} \right\}\;\;\;{\text{where}}\;\;\;\left\{ {\begin{array}{*{20}l} {\mu_{{\tilde{Y}}} (x):M \to [0,1]} \hfill \\ {\upsilon_{{\tilde{Y}}} (x):M \to [0,1]} \hfill \\ {0 \le \mu_{{\tilde{Y}}} (x) + \upsilon_{{\tilde{Y}}} (x) \le 1\quad \forall x \in M} \hfill \\ \end{array} } \right. $$

(4)

$\mu_{{\tilde{Y}}} (x)$ and $\upsilon_{{\tilde{Y}}} (x)$ show the membership and non-membership degrees of M to $\tilde{Y}$, respectively.

The hesitation or indeterminacy function is another function that can be defined for the intuitionistic fuzzy set $\tilde{Y}$ as follows:

$$ \pi_{{\tilde{Y}}} (x) = 1 - \mu_{{\tilde{Y}}} (x) - \upsilon_{{\tilde{Y}}} (x) $$

(5)

The indeterminacy function for an element such as x in the universal set determines the uncertainty value about the membership situation of x (belonging or not belonging to $\tilde{Y}$). If the value of $\pi_{{\tilde{Y}}} (x)$ is small, the knowledge about the variable x is more definite. If the relation $\mu_{{\tilde{Y}}} (x) = 1 - \upsilon_{{\tilde{Y}}} (x)$ is established for all elements, the intuitionistic fuzzy set will be equal to the ordinary fuzzy set.

Definition 6

Each intuitionistic fuzzy variable, $\tilde{\varepsilon }$, consists of two fuzzy variables such as $\tilde{\varepsilon }^{ + }$ and $\tilde{\varepsilon }^{ - }$ which have been defined in (Ѱ, P(Ѱ), Cr), where $\mu_{{\tilde{\varepsilon }^{ + } }} (x) \le \mu_{{\tilde{\varepsilon }^{ - } }} (x) \, \forall x \in {\mathbb{R}}$. Equations (6) and (7) can be used as follows to obtain the membership and non-membership functions of $\tilde{\varepsilon }$ (Jafarian et al., 2019):

$$ \mu_{{\tilde{\varepsilon }}} (x) = \mu_{{\tilde{\varepsilon }^{ + } }} (x) = \left( {2{\text{Cr}}\left\{ {\tilde{\varepsilon }^{ + } = x} \right\}} \right) \wedge 1,\;\;\;x \in {\mathbb{R}} $$

(6)

$$ \mu_{{\tilde{\varepsilon }}} (x) = 1 - \mu_{{\tilde{\varepsilon }^{ - } }} (x) = 1 - \left( {2{\text{Cr}}\left\{ {\tilde{\varepsilon }^{ - } = x} \right\}} \right) \wedge 1,\;\;\;x \in {\mathbb{R}} $$

(7)

Definition 7

$\tilde{\varepsilon }$ is nonnegative if ${\text{Cr}}\left\{ {\tilde{\varepsilon }^{ - } < 0} \right\} = 0$.

Definition 8

The α-optimistic and the α-pessimistic values ($\alpha \in [0,1]$) of $\tilde{\varepsilon }$ are as follows:

$$ \tilde{\varepsilon }_{\alpha O}^{ + } = \sup \left\{ {x|{\text{Cr}}\left( {\left\{ {\tilde{\varepsilon }^{ + } \ge x} \right\}} \right) \ge \alpha } \right\} $$

(8)

$$ \tilde{\varepsilon }_{\alpha P}^{ + } = \inf \left\{ {x|{\text{Cr}}\left( {\left\{ {\tilde{\varepsilon }^{ + } \le x} \right\}} \right) \ge \alpha } \right\} $$

(9)

$$ \tilde{\varepsilon }_{\alpha O}^{ - } = \sup \left\{ {x|{\text{Cr}}\left( {\left\{ {\tilde{\varepsilon }^{ - } \ge x} \right\}} \right) \ge \alpha } \right\} $$

(10)

$$ \tilde{\varepsilon }_{\alpha P}^{ - } = \inf \left\{ {x|{\text{Cr}}\left( {\left\{ {\tilde{\varepsilon }^{ - } \le x} \right\}} \right) \ge \alpha } \right\} $$

(11)

Lemma 2

For two nonnegative intuitionistic fuzzy variables, such as $\tilde{\varepsilon }$ and $\tilde{\zeta }$, and any $\alpha \in [0,1]$, the following relationships are established:

$$ (\tilde{\varepsilon } + \tilde{\zeta })_{\alpha P} = \varepsilon_{\alpha p} + \zeta_{\alpha P} = (\tilde{\varepsilon }^{ + }_{\alpha P} + \tilde{\zeta }^{ + }_{\alpha P} ;\tilde{\varepsilon }^{ - }_{\alpha P} + \tilde{\zeta }^{ - }_{\alpha P} ) $$

(12)

$$ (\tilde{\varepsilon } + \tilde{\zeta })_{\alpha O} = \varepsilon_{\alpha O} + \zeta_{\alpha O} = (\tilde{\varepsilon }^{ + }_{\alpha O} + \tilde{\zeta }^{ + }_{\alpha O} ;\tilde{\varepsilon }^{ - }_{\alpha O} + \tilde{\zeta }^{ - }_{\alpha O} ) $$

(13)

$$ (\tilde{\varepsilon } - \tilde{\zeta })_{\alpha P} = \varepsilon_{\alpha p} + \zeta_{\alpha O} = (\tilde{\varepsilon }^{ + }_{\alpha P} - \tilde{\zeta }^{ + }_{\alpha O} ;\tilde{\varepsilon }^{ - }_{\alpha P} - \tilde{\zeta }^{ - }_{\alpha O} ) $$

(14)

$$ (\tilde{\varepsilon } - \tilde{\zeta })_{\alpha O} = \varepsilon_{\alpha O} + \zeta_{\alpha P} = (\tilde{\varepsilon }^{ + }_{\alpha O} - \tilde{\zeta }^{ + }_{\alpha P} ;\tilde{\varepsilon }^{ - }_{\alpha O} - \tilde{\zeta }^{ - }_{\alpha P} ) $$

(15)

$$ (\tilde{\varepsilon }.\tilde{\zeta })_{\alpha P} = \varepsilon_{\alpha P} .\zeta_{\alpha P} = (\tilde{\varepsilon }^{ + }_{\alpha P} .\tilde{\zeta }^{ + }_{\alpha P} ;\tilde{\varepsilon }^{ - }_{\alpha P} .\tilde{\zeta }^{ - }_{\alpha P} ) $$

(16)

$$ (\tilde{\varepsilon }.\tilde{\zeta })_{\alpha O} = \varepsilon_{\alpha O} .\zeta_{\alpha O} = (\tilde{\varepsilon }^{ + }_{\alpha O} .\tilde{\zeta }^{ + }_{\alpha O} ;\tilde{\varepsilon }^{ - }_{\alpha O} .\tilde{\zeta }^{ - }_{\alpha O} ) $$

(17)

Proof

Please see Jafarian et al. (2019). □

Definition 9

The expected value of the intuitionistic fuzzy variable, $\tilde{\varepsilon }$, can be obtained as:

$$ \begin{aligned} E[\tilde{\varepsilon }] & = \tau \left( {\int_{0}^{ + \infty } {{\text{Cr}}\left\{ {\tilde{\varepsilon }^{ + } \ge x} \right\}{\text{d}}x} - \int_{ - \infty }^{0} {{\text{Cr}}\left\{ {\tilde{\varepsilon }^{ + } < x} \right\}{\text{d}}x} } \right) \\ & \quad + (1 - \tau )\left( {\int_{0}^{ + \infty } {{\text{Cr}}\left\{ {\tilde{\varepsilon }^{ - } \ge x} \right\}{\text{d}}x} - \int_{ - \infty }^{0} {{\text{Cr}}\left\{ {\tilde{\varepsilon }^{ - } < x} \right\}{\text{d}}x} } \right),\quad \tau \in [0,1] \\ \end{aligned} $$

(18)

$\tau$ is named the fuzzification parameter which shows the degree of emphasis on the membership function of $\tilde{\varepsilon }$.

Lemma 3

For any nonnegative intuitionistic fuzzy variable such as $\tilde{\varepsilon }$ that have finite expected value, Eq. (19) will be true:

$$ E[\tilde{\varepsilon }] = \frac{\tau }{2}\int\limits_{0}^{1} {(\tilde{\varepsilon }^{ + }_{L\alpha } + \tilde{\varepsilon }^{ + }_{U\alpha } ){\text{d}}\alpha } - \frac{1 - \tau }{2}\int\limits_{0}^{1} {(\tilde{\varepsilon }^{ - }_{L\alpha } + \tilde{\varepsilon }^{ - }_{U\alpha } ){\text{d}}\alpha } $$

(19)

Proof

Please see Ban (2008).□

Lemma 4

Assume $\tilde{\varepsilon }$ and $\tilde{\zeta }$ are intuitionistic fuzzy numbers, and $a,b \in {\mathbb{R}}$. Then,

$$ E[a\tilde{\varepsilon } + b\tilde{\zeta }] = aE[\tilde{\varepsilon }] + bE[\tilde{\zeta }] $$

(20)

Proof

Please see Zhou et al. (2008).□

4 Problem description

In this study, two SCs are considered. The problem involves two producers, two retailers, a 3PL, and a government assumed to be the leader of the game. The first producer produces the green products, and the second producer will produce the non-green products. Each SC can distribute its product through either the direct distribution channel or the indirect distribution channel. The 3PL jointly provides services for both SCs. The 3PL transfers the first producer’s products to the first retailer and the second producer’s products to the second retailer. In the direct channel, the producer pays the transportation costs of the products. The study also assumes that the cost of 3PL for shipping is paid jointly by the producer and retailer. In the indirect channel, the 3PL receives %r of the product transportation costs from retailers and %(1 − r) from the producers. As it has been already mentioned, the government monitors the SCs’ activities and tries to control them through financial intervention. Due to the amount of pollution released by each producer and their production volume, the government will either support the producers by paying subsidies or penalize them by taking taxes. In the current study, two scenarios are considered for the government. In the first scenario, the government’s goal is to reduce the amount of pollution released throughout the SCs’ activities (producing sector and transportation sector), while in the second scenario, the government’s goal is to increase its revenue through tax collection. This problem has been studied in both centralized and decentralized conditions. Also, this study seeks to reduce the delivery time of the product to the customer. Fleischmann et al. (1997) showed that delivery time affects customer demand. The schematic framework of the understudy problem is shown in Fig. 1. In order to consider the fact of uncertainty in real-world problems, some of the promoters have been considered in their fuzzy form. Please note that many major companies such as Apple, Samsung, Dell, and Mercedes-Benz have the same SC and market structure as considered in this study. Hence, the results of this study could be useful for managers of such companies.

The time order of the introduced game is as follows:

• (1) The government will obtain the optimal tariffs for each SC aiming to maximize its profit or minimize the total environmental impacts

• (2) Each producer will obtain optimal delivery times through direct channels. Also, the optimal greenness degree of the green producer will be obtained.

• (3) The 3PL company will determine its greenness degree and the optimal delivery time through indirect channels.

• (4) The retailers will obtain the optimal retail prices.

4.1 Notations

All symbols and notations used in this study are as follows (Note. Intuitionistic fuzzy variables are indicated by a Tilde symbol (~) at the top):

Indexes

I :

The set of producers, indexed by $i \in I = \left\{ {1,2,...,\left| I \right|} \right\}$.

J :

The set of retailers, indexed by $j \in J = \left\{ {1,2,...,\left| J \right|} \right\}$.

H :

The set of 3PLs, indexed by $h \in H = \left\{ {1,2,...,\left| H \right|} \right\}$.

Q :

The set of channels, indexed by $q \in Q = \left\{ {1,2,...,\left| Q \right|} \right\}$.

$Y$ :

The set of 3PL channels, indexed by $y \in Y \subseteq Q$.

U :

The set of pollutions types, indexed by $u \in U = \left\{ {1,2,...,\left| U \right|} \right\}$.

Parameters

$\tilde{\alpha }_{q}$ :

The market base in channel $q$.

$\tilde{\lambda }_{i}$ :

The greenness degree elasticity of demand ($\tilde{\lambda }_{i} \ge 0$).

$\tilde{\beta }$ :

The self-price elasticity of demand ($\tilde{\beta }_{i} \ge 0$).

$\tilde{\gamma }$ :

The cross-price elasticity of demand ($\tilde{\gamma } \ge 0$).

$\tilde{\eta }_{i}$ :

The cost coefficient of the greenness degree per unit for producer i, ($\tilde{\eta }_{i} \ge 0$).

$\tilde{\psi }$ :

The sensitivity coefficient of delivery time on demand, ($\tilde{\psi } \ge 0$).

$\varsigma$ :

The sensitivity coefficient of the green transportation on demand function, ($\varsigma \ge 0$).

$k$ :

The public awareness coefficient of the cooperation with the 3PL on the channel (the 3PL advantage factor), (k ≥ 0).

$\tilde{a}_{f}$ :

The advertising cost of cooperation with 3PL as an advantage in order to increase the public awareness about the importance of cooperation with 3PLs, ($\tilde{a}_{f} \ge 0$).

$\tilde{g}$ :

The fixed investment on reducing delivery time ($\tilde{g} \ge 0$).

$\tilde{u}$ :

The sensitivity coefficient of delivery time on the utility function.

$z_{b}$ :

The upper bound set by the government for the amount of pollution emitted by the 3PL.

$v$ :

The released emission reduction cost per unit.

r :

The rate of participation of SC’s members in payment of the freight costs by 3PL, ($r \ge 0$).

$\tilde{m}_{1}$ :

The payment of the first producer and first retailer to the 3PL company for transporting each unit of product, ($\tilde{m}_{1} \ge 0$).

$\tilde{m}_{2}$ :

The payment of the second producer and second retailer to the 3PL company for transporting each unit of product, ($\tilde{m}_{2} \ge 0$).

$\tilde{\chi }_{i}$ :

The cost of transporting each unit from the producer i to the customers in the direct channel, ($\tilde{\chi }_{i} > 0$).

$\tilde{c}_{t}$ :

The shipping cost per unit for 3PL providers, ($\tilde{c}_{t} > 0$).

$\tilde{c}_{i}$ :

The cost of production for producer i, ($\tilde{c}_{i} > 0$).

$s_{i}$ :

The selling price through the direct channel for producer i = 1, 2, ($s_{i} > 0$).

$w_{i}$ :

The wholesale price through the indirect channel for producer i = 1, 2, ($w_{i} > 0$).

$cp_{u}$ :

The cost of releasing destructive gas type u.

${\text{MPR}}_{j}$ :

The minimum amount of the profit that the retailer j has considered for itself (${\text{MPR}}_{j} \ge 0$).

${\text{MPI}}_{i}$ :

The minimum amount of the profit that the producer i has considered for itself (${\text{MPI}}_{i} \ge 0$).

${\text{MPT}}_{h}$ :

The minimum amount of the profit that the 3PL h has considered for itself (${\text{MPT}}_{h} \ge 0$).

${\text{MPG}}$ :

The minimum amount of the profit that the government has considered for itself (${\text{MPG}} \ge 0$).

${\text{UPG}}$ :

The upper bound for the released pollution considered by the government (${\text{UPG}} \ge 0$).

Decision variables

$p_{j}$ :

The retail price (j = 1, 2).

$t_{1}$ :

The delivery time through the direct channel of the first producer.

$t_{2}$ :

The delivery time through the 3PL channels.

$t_{3}$ :

The delivery time through the direct channel of the second producer.

$z$ :

Amount of released emission by the 3PL.

$\theta$ :

The greenness degree of the first producer’s products.

$\varphi_{i}$ :

The tariff assigned to the producer i by the government, $\varphi_{i} > 0$ shows subsidies and $\varphi_{i} < 0$ indicates taxes.

Dependent variables

$D_{q}$ :

The demand through the channel $q$.

$\pi_{j}$ :

The profit function of the retailer j, j = 1, 2.

$\pi_{{3{\text{pl}}}}$ :

The profit function of the third-party logistic provider.

$\pi_{{m_{i} }}$ :

The profit function of the producer i, i = 1, 2.

$\pi_{G}$ :

The profit function of the government.

4.2 General assumptions

Here is a list of general assumptions considered through different parts of this study:

Assumption 1

The retail price (p_i) is higher than the wholesale price (w_j). Also, the wholesale price for producers is higher than the total production costs ($\tilde{c}_{i}$) (Mahmoudi et al., 2021).

Assumption 2

The government is the leader of the game in all considered decision structures.

Assumption 3

Demand in each channel is a linear function of the price in the self-channel and other channels, as well as the greenness degree of the product (Jamali et al., 2021).

Assumption 4

The initial market demand for both producers is a large enough value.

Assumption 5

Changing the price of a product through a related channel has a greater impact on demand than changing the price of a product through an opposite channel (Huang & Wang, 2017) ($\tilde{\beta } > \tilde{\gamma } > 0$).

Assumption 6

The number of the customers for green products is higher than that of non-green products (Li et al., 2016) ($\tilde{\lambda }_{1} > \tilde{\lambda }_{2} > 0$).

Assumption 7

The wholesale price (w) and selling price of products in the direct channel (s) are considered as parameters. This assumption has been used in previous studies, such as Zhang and Liu (2013) and Basiri and Heydari (2017).

5 Model development

Using game theory and fuzzy theory, the described problems have been modeled in this section.

5.1 Demand functions

Based on the problem structure, the demand functions for different channels are as follows:

$$ D_{1} = \tilde{\alpha }_{1} - \tilde{\beta }s_{1} + \tilde{\gamma }(p_{1} + p_{2} + s_{2} ) + \tilde{\lambda }_{1} \theta + \tilde{\psi }(t_{2} + t_{3} - t_{1} ) $$

(21)

$$ D_{2} = \tilde{\alpha }_{2} - \tilde{\beta }p_{1} + \tilde{\gamma }(s_{1} + p_{2} + s_{2} ) + \tilde{\lambda }_{1} \theta + \tilde{\psi }(t_{1} + t_{3} - t_{2} ) + k\tilde{a}_{f} + \varsigma (z_{b} - z) $$

(22)

$$ D_{3} = \tilde{\alpha }_{3} - \tilde{\beta }p_{2} + \tilde{\gamma }(p_{1} + s_{1} + s_{2} ) - \tilde{\lambda }_{2} \theta + \tilde{\psi }(t_{1} + t_{3} - t_{2} ) + k\tilde{a}_{f} + \varsigma (z_{b} - z) $$

(23)

$$ D_{4} = \tilde{\alpha }_{4} - \tilde{\beta }s_{2} + \tilde{\gamma }(p_{1} + p_{2} + s_{1} ) - \tilde{\lambda }_{2} \theta + \tilde{\psi }(t_{1} + t_{2} - t_{3} ) $$

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The presence of 3PL in the channel is an advantage for that channel. To consider this advantage, the term $k\tilde{a}_{f}$ has been added to the demand functions for the second and third channels (Mahmoudi & Rasti-Barzoki, 2018).

5.2 Decentralized model (DCM)

5.2.1 Profit functions

In the decentralized model, none of the members of the supply chain are willing to cooperate with each other; they are only seeking to maximize their own profit. The profit functions of the SC members in this scenario are as follows:

$$ \pi_{1} = D_{2} (p_{1} - w_{1} - r_{1} \tilde{m}_{1} ) $$

(25)

$$ \pi_{2} = D_{3} (p_{2} - w_{2} - r_{2} \tilde{m}_{2} ) $$

(26)

$$ \pi_{{{\text{3pl}}}} = (\tilde{m}_{1} + \tilde{m}_{2} - \tilde{c}_{t} )(D_{2} + D_{3} ) - (\tilde{g} - \tilde{u}t_{2} )^{2} - v(z_{b} - z)^{2} $$

(27)

$$ \pi_{{m_{1} }} = D_{1} (s_{1} - \tilde{c}_{1} - \tilde{\chi }_{1} + \varphi_{1} ) + D_{2} (p_{1} - \tilde{c}_{1} - r_{3} \tilde{m}_{1} + \varphi_{1} ) - \tilde{\eta }\frac{{\theta^{2} }}{2} - (\tilde{g} - \tilde{u}t_{1} )^{2} $$

(28)

$$ \pi_{{m_{2} }} = D_{3} (p_{2} - \tilde{c}_{2} - r_{4} \tilde{m}_{2} + \varphi_{2} ) + D_{4} (s_{2} - \tilde{c}_{2} - \tilde{\chi }_{2} + \varphi_{2} ) - (\tilde{g} - \tilde{u}t_{3} )^{2} $$

(29)

Here, the profit function of the 3PL company is a function of the costs that retailers and producers pay to ship their products ($\tilde{m}_{1} ,\tilde{m}_{2}$). $\tilde{m}_{1}$ is the summation of the payments of the first retailer and producer, and $\tilde{m}_{2}$ is the summation of the payments of the second retailer and producer. As mentioned before, the 3PL receives %r₁ of $\tilde{m}_{1}$ from the first retailer and ($\% r_{3} = \% (1 - r_{1} )$) percentage from the first producer. The same rule is true for the second retailer and producer ($r_{4} = 1 - r_{2}$). On the other hand, 3PL should also reduce product delivery time to retailers. If the delivery time increases, the 3PL will have to spend an additional fee to increase the efficiency of its distribution system and develop the service level. According to Savaskan and Van Wassenhove (2006), this cost is equal to $(\tilde{g} - \tilde{u}t_{2} )^{2}$. Also, the relationship $\tilde{m}_{1} + \tilde{m}_{2} > \tilde{c}_{t}$ must be established. The cost of the emission reduction and greenness has been considered in the 3PL’s profit function by $v(z_{b} - z)^{2}$. As it is clear, the more 3PL reduces emissions, the more cost has to be paid. In order to produce green products, the first producer has to pay an extra cost to increase the greenness degree of the products. This cost is equal to $\tilde{\eta }\frac{{\theta^{2} }}{2}$.

5.2.2 Problem solution

In this section, considering the decentralized decision-making structure, game theory has been used to model the problem and obtain the equilibrium values of the decision variables at each level of the SC. A Stackelberg game model is considered to model the problem, where the government is the leader of the game, and the other players are followers. Each member of the SC wants to maximize their profits. Generally, the mathematical structure of the game is as follows:

$$ \left\{ {\begin{array}{*{20}l} {E[\pi_{G} ]} \hfill \\ {\quad \left\{ {\begin{array}{*{20}l} {E[\pi_{{m_{1} }} ] = E\left[ {D_{1} (s_{1} - \tilde{c}_{1} - \tilde{\chi }_{1} + \varphi_{1} ) + D_{2} (p_{1} - \tilde{c}_{1} - r_{3} \tilde{m}_{1} + \varphi_{1} ) - \tilde{\eta }\frac{{\theta^{2} }}{2} - (\tilde{g} - \tilde{u}t_{1} )^{2} } \right]} \hfill \\ {E[\pi_{{m_{2} }} ] = E[D_{3} (p_{2} - \tilde{c}_{2} - r_{4} \tilde{m}_{2} + \varphi_{2} ) + D_{4} (s_{2} - \tilde{c}_{2} - \tilde{\chi }_{2} + \varphi_{2} ) - (\tilde{g} - \tilde{u}t_{3} )^{2} ]} \hfill \\ {E{[}\pi_{{{\text{3pl}}}} ] = E[(\tilde{m}_{1} + \tilde{m}_{2} - \tilde{c}_{t} )(D_{2} + D_{3} ) - (\tilde{g} - \tilde{u}t_{2} )^{2} - v(z_{b} - z)^{2} ]} \hfill \\ {\quad \left\{ {\begin{array}{*{20}c} {E[\pi_{1} ] = E[D_{2} (p_{1} - w_{1} - r_{1} \tilde{m}_{1} )]} \\ {E[\pi_{2} ] = E[D_{3} (p_{2} - w_{2} - r_{2} \tilde{m}_{2} )]} \\ \end{array} } \right.} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right. $$

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Since some of the problem parameters are considered intuitive fuzzy, the profit function of each retailer must be changed to its definite form. To this end, the expected value method can be used to convert functions to a certain state (Jafarian et al., 2019; Soleimani, 2016). Using this method, the expected profit function of the first and second retailers, the 3PL, and the first and second producers can be presented as follows, respectively:

$$ \begin{aligned} E[\pi_{1} ] & = E\left[ {\varsigma w_{1} } \right]z - kA_{6} - A_{8} - A_{9} - A_{12} - A_{14} - \theta A_{16} - E\left[ \varsigma \right]p_{1} z \\ & \quad + \left( {kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\beta w_{1} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{2} } \right]} \right)p_{1} \\ & \quad + E\left[ {\lambda_{1} } \right]p_{1} \theta - E\left[ \beta \right]p_{1}^{2} - A_{7} p_{2} + E\left[ \gamma \right]p_{1} p_{2} + E\left[ {\varsigma m_{1} } \right]r_{1} z \\ & \quad - \left( {kA_{1} + A_{3} + A_{4} + A_{11} + A_{13} } \right)r_{1} - \theta A_{15} r_{1} + E\left[ {\beta m_{1} } \right]p_{1} r_{1} \\ & \quad - A_{2} p_{2} r_{1} - A_{10} t_{1} + E\left[ \psi \right]p_{1} t_{1} - A_{5} r_{1} t_{1} - E\left[ \psi \right]p_{1} t_{2} + E\left[ {\psi m_{1} } \right]r_{1} t_{2} \\ & \quad + E\left[ {\psi w_{1} } \right]\left( {t_{2} - t_{3} } \right) + \left( {E\left[ \psi \right]p_{1} - A_{5} r_{1} } \right)t_{3} \\ \end{aligned} $$

(31)

$$ \begin{aligned} E[\pi_{2} ] & = E\left[ {\varsigma w_{2} } \right]z + E\left[ {w_{2} \lambda_{2} } \right]\theta - kA_{17} - A_{19} - A_{21} - A_{23} - A_{24} - A_{18} p_{1} \\ & \quad + \left( { - E\left[ \varsigma \right]z + kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\beta w_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{3} } \right]} \right)p_{2} \\ & \quad - E\left[ {\lambda_{2} } \right]p_{2} \theta + E\left[ \gamma \right]p_{1} p_{2} - E\left[ \beta \right]p_{2}^{2} + E\left[ {\varsigma m_{2} } \right]r_{2} z + E\left[ {m_{2} \lambda_{2} } \right]r_{2} \theta - kN_{3} r_{2} \\ & \quad - N_{6} r_{2} - N_{8} r_{2} - N_{12} r_{2} - N_{14} r_{2} - N_{4} p_{1} r_{2} + E\left[ {\beta m_{2} } \right]p_{2} r_{2} + E\left[ \psi \right]p_{2} t_{1} - L_{10} r_{2} t_{1} \\ & \quad + E\left[ {\psi w_{2} } \right]t_{2} - E\left[ \psi \right]p_{2} t_{2} + E\left[ {\psi m_{2} } \right]r_{2} t_{2} + \left( {E\left[ \psi \right]p_{2} - L_{10} r_{2} } \right)t_{3} - A_{20} \left( {t_{1} + t_{3} } \right) \\ \end{aligned} $$

(32)

$$ \begin{aligned} E[\pi_{{{\text{3pl}}}} ] & = - E\left[ g \right]^{2} - z^{2} E\left[ v \right] + 2E\left[ {\gamma m_{1} s_{1} } \right] + 2E\left[ {\gamma m_{2} s_{1} } \right] + 2E\left[ {\gamma m_{1} s_{2} } \right] + 2E\left[ {\gamma m_{2} s_{2} } \right] \\ & \quad + 2E\left[ {\varsigma m_{1} z_{b} } \right] + 2E\left[ {\varsigma m_{2} z_{b} } \right] + E\left[ {m_{1} \alpha_{2} } \right] + E\left[ {m_{2} \alpha_{2} } \right] + E\left[ {m_{1} \alpha_{3} } \right] + E\left[ {m_{2} \alpha_{3} } \right] \\ & \quad + 2k\left( {E\left[ {a_{f} m_{1} } \right] + E\left[ {a_{f} m_{2} } \right] - L_{1} } \right) + 2z\left( {E\left[ {\varsigma c_{t} } \right] + E\left[ {vz_{b} } \right] - L_{2} - L_{3} } \right) - 2L_{7} \\ & \quad - 2L_{8} - 2L_{11} - L_{12} - L_{13} - L_{14} + \theta \left( {E\left[ {m_{1} \lambda_{1} } \right] + E\left[ {m_{2} \lambda_{1} } \right] + E\left[ {c_{t} \lambda_{2} } \right] - L_{15} - L_{16} - L_{17} } \right) \\ & \quad + \left( {E\left[ {\beta c_{t} } \right] + E\left[ {\gamma m_{1} } \right] + E\left[ {\gamma m_{2} } \right] - L_{4} - L_{5} - L_{6} } \right)\left( {p_{1}^{*} + p_{2}^{*} } \right) \\ & \quad - t_{2} \left( { - 2E\left[ {gu} \right] - 2E\left[ {\psi c_{t} } \right] + 2\left( {A_{5} + L_{10} } \right) + B\left[ u \right]^{2} t_{2} } \right) \\ & \quad + 2\left( {E\left[ {\psi m_{1} } \right] + E\left[ {\psi m_{2} } \right] - L_{9} } \right)\left( {t_{1} + t_{3} } \right) \\ \end{aligned} $$

(33)

$$ \begin{aligned} E[\pi_{{m_{1} }} ] & = - E\left[ g \right]^{2} - \frac{1}{2}\theta^{2} E\left[ \eta \right] + z^{*} E\left[ {\varsigma c_{1} } \right] + E\left[ {\beta c_{1} s_{1} } \right] + E\left[ {\gamma s_{1} s_{2} } \right] + E\left[ {s_{1} \alpha_{1} } \right] + E\left[ {\beta s_{1} \chi_{1} } \right] - kM_{1} \\ & \quad - M_{3} - M_{4} - 2M_{5} - M_{8} - M_{9} - M_{10} - M_{16} - M_{18} - E\left[ \beta \right](p_{1}^{*} )^{2} + z^{*} B\left[ {\varsigma m_{1} } \right]r_{3} - kA_{1} r_{3} \\ & \quad - A_{3} r_{3} - A_{4} r_{3} - A_{11} r_{3} - M_{11} r_{3} + p_{1}^{*} ( - z^{*} E\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + E\left[ {\beta c_{1} } \right] + 2E\left[ {\gamma s_{1} } \right] \\ & \quad + E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{2} } \right] - M_{2} - M_{14} + E\left[ \gamma \right]p_{2}^{*} + E\left[ {\beta m_{1} } \right]r_{3} ) \\ & \quad + p_{2}^{*} \left( {E\left[ {\gamma s_{1} } \right] - 2M_{2} - M_{14} - A_{2} r_{3} } \right) + \theta \left( {E\left[ {s_{1} \lambda_{1} } \right] - 2M_{12} - M_{19} + E\left[ {\lambda_{1} } \right]p_{1}^{*} - M_{15} r_{3} } \right) \\ & \quad + \left( {2E\left[ {gu} \right] + E\left[ {\psi \chi_{1} } \right] - M_{6} + E\left[ \psi \right]p_{1}^{*} - A_{5} r_{3} } \right)t_{1} - E\left[ u \right]^{2} t_{1}^{2} \\ & \quad + \left( {E\left[ {\psi s_{1} } \right] - M_{17} - E\left[ \psi \right]p_{1}^{*} + E\left[ {\psi m_{1} } \right]r_{3} } \right)t_{2}^{*} \\ & \quad + \left( {E\left[ {\psi s_{1} } \right] - 2M_{7} - M_{17} + E\left[ \psi \right]p_{1}^{*} - A_{5} r_{3} } \right)t_{3} \\ & \quad + \left( { - z^{*} E\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + 2E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{1} } \right] + E\left[ {\alpha_{2} } \right]} \right. \\ & \quad \left. { + 2\theta E\left[ {\lambda_{1} } \right] - M_{13} + \left( { - E\left[ \beta \right] + E\left[ \gamma \right]} \right)p_{1}^{*} + 2E\left[ \gamma \right]p_{2}^{*} + 2E\left[ \psi \right]t_{3} )\varphi_{1} } \right) \\ \end{aligned} $$

(34)

$$ \begin{aligned} E[\pi_{{m_{2} }} ] & = - E\left[ g \right]^{2} + E\left[ {\varsigma c_{2} } \right]z^{*} + E\left[ {\beta c_{2} s_{2} } \right] + E\left[ {\gamma s_{1} s_{2} } \right] + E\left[ {s_{2} \alpha_{4} } \right] + 2E\left[ {c_{2} \lambda_{2} } \right]\theta + E\left[ {\beta s_{2} \chi_{2} } \right] \\ & \quad + E\left[ {\lambda_{2} \chi_{2} } \right]\theta - kN_{1} - 2N_{5} - N_{7} - N_{11} - N_{13} - N_{15} - \theta N_{16} - N_{19} - N_{21} \\ & \quad - N_{22} + \left( {E\left[ {\gamma s_{2} } \right] - 2N_{2} - N_{18} } \right)p_{1}^{*} \\ & \quad + \left( {kE\left[ {a_{f} } \right] + E\left[ {\beta c_{2} } \right] + E\left[ {\gamma s_{1} } \right] + 2E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{3} } \right]} \right)p_{2}^{*} \\ & \quad - E\left[ {\lambda_{2} } \right]\theta p_{2}^{*} - N_{2} p_{2}^{*} - N_{18} p_{2}^{*} + E\left[ \gamma \right]p_{1}^{*} p_{2}^{*} - E\left[ \beta \right](p_{2}^{*} )^{2} + E\left[ {\varsigma m_{2} } \right]z^{*} r_{4} \\ & \quad + E\left[ {m_{2} \lambda_{2} } \right]\theta r_{4} - kN_{3} r_{4} - N_{6} r_{4} - N_{8} r_{4} - N_{12} r_{4} - N_{14} r_{4} - N_{4} p_{1}^{*} r_{4} \\ & \quad + E\left[ {\beta m_{2} } \right]p_{2}^{*} r_{4} - \left( {2N_{9} + N_{20} } \right)t_{1} + E\left[ \psi \right]p_{2}^{*} t_{1} - L_{10} r_{4} t_{1} - N_{20} t_{2}^{*} - E\left[ \psi \right]p_{2}^{*} t_{2}^{*} \\ & \quad + E\left[ {\psi m_{2} } \right]r_{4} t_{2}^{*} + E\left[ {\psi s_{2} } \right]\left( {t_{1} + t_{2}^{*} } \right) + \left( {2E\left[ {gu} \right] + E\left[ {\psi \chi_{2} } \right] - N_{10} } \right)t_{3} \\ & \quad + E\left[ \psi \right]p_{2}^{*} t_{3} - L_{10} r_{4} t_{3} - E\left[ u \right]^{2} t_{3}^{2} - E\left[ \varsigma \right]z^{*} \left( {p_{2}^{*} + \varphi_{2} } \right) \\ & \quad + \left( {\begin{array}{*{20}l} {kE\left[ {a_{f} } \right] + 2E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{3} } \right] + E\left[ {\alpha_{4} } \right] - 2E\left[ {\lambda_{2} } \right]\theta - N_{17} } \hfill \\ { - E\left[ \beta \right]p_{2}^{*} + E\left[ \gamma \right]\left( {2p_{1}^{*} + p_{2}^{*} } \right) + 2E\left[ \psi \right]t_{1} } \hfill \\ \end{array} } \right)\varphi_{2} \\ \end{aligned} $$

(35)

Note A_i, N_i, L_i, M_i and H_i are the ith change of the variables, which are given in the supplementary file.

Lemma 1

The equilibrium values of p₁ and p₂ are:

$$ p_{1}^{*} = - \frac{{\left( {\begin{array}{*{20}l} {2E\left[ \beta \right]\left( { - E\left[ \varsigma \right]z + E\left[ {\lambda_{1} } \right]\theta + H_{1} + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \right)} \hfill \\ {\quad + E\left[ \gamma \right]\left( { - E\left[ \varsigma \right]z - B\left[ {\lambda_{2} } \right]\theta + H_{2} + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \right)} \hfill \\ \end{array} } \right)}}{{ - 4E\left[ \beta \right]^{2} + E\left[ \gamma \right]^{2} }} $$

(36)

$$ p_{2}^{*} = \frac{{\left( {\begin{array}{*{20}l} {E\left[ \gamma \right]\left( { - E\left[ \varsigma \right]z + \theta E\left[ {\lambda_{1} } \right] + H_{1} + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \right)} \hfill \\ {\quad + 2E\left[ \beta \right]\left( { - E\left[ \varsigma \right]z - \theta E\left[ {\lambda_{2} } \right] + H_{2} + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \right)} \hfill \\ \end{array} } \right)}}{{4E\left[ \beta \right]^{2} - E\left[ \gamma \right]^{2} }} $$

(37)

Proof

See “Appendix 1.”□

Lemma 2

The equilibrium values of the 3PL decision variables are:

$$ t_{2}^{*} = \frac{{E\left[ {gu} \right] + E\left[ {\psi c_{t} } \right] - A_{5} + H_{4} - L_{10} }}{{E\left[ u \right]^{2} }} $$

(38)

$$ z^{*} = \frac{{E\left[ {z_{b} v} \right] + E\left[ {\varsigma c_{t} } \right] + H_{3} - L_{2} - L_{3} }}{E\left[ v \right]} $$

(39)

Proof

See “Appendix 1.”□

Lemma 3

The equilibrium values of $\theta$, $t_{1}$ and $t_{3}$ are as follows:

$$ \theta^{*} = \frac{{\left( {H_{10} + H_{11} } \right)\left( {H_{8} + H_{6} \varphi_{1} } \right) - H_{7} \left( {H_{12} + H_{14} + H_{9} \left( {\varphi_{1} + \varphi_{2} } \right)} \right)}}{{ - H_{5} \left( {H_{10} + H_{11} } \right) + H_{7} \left( {H_{7} + H_{13} } \right)}} $$

(40)

$$ t_{1}^{*} = \frac{{\left( {\begin{array}{*{20}l} {H_{10} H_{13} \left( {H_{8} + H_{6} \varphi_{1} } \right) - H_{7} \left( {H_{8} H_{11} + H_{6} H_{11} \varphi_{1} + H_{13} \left( {H_{12} + H_{9} \varphi_{1} } \right)} \right)} \hfill \\ {\quad + H_{7}^{2} \left( {H_{14} + H_{9} \varphi_{2} } \right) + H_{5} \left( {H_{11} \left( {H_{12} + H_{9} \varphi_{1} } \right) - H_{10} \left( {H_{14} + H_{9} \varphi_{2} } \right)} \right)} \hfill \\ \end{array} } \right)}}{{\left( {H_{10} - H_{11} } \right)\left( {H_{5} \left( {H_{10} + H_{11} } \right) - H_{7} \left( {H_{7} + H_{13} } \right)} \right)}} $$

(41)

$$ t_{3}^{*} = \frac{{\left( {\begin{array}{*{20}l} { - H_{11} H_{13} \left( {H_{8} + H_{6} \varphi_{1} } \right) + H_{7} \left( {H_{8} H_{10} + H_{6} H_{10} \varphi_{1} + H_{13} \left( {H_{12} + H_{9} \varphi_{1} } \right)} \right)} \hfill \\ {\quad - H_{7}^{2} \left( {H_{14} + H_{9} \varphi_{2} } \right) + H_{5} \left( { - H_{10} \left( {H_{12} + H_{9} \varphi_{1} } \right) + H_{11} \left( {H_{14} + H_{9} \varphi_{2} } \right)} \right)} \hfill \\ \end{array} } \right)}}{{\left( {H_{10} - H_{11} } \right)\left( {H_{5} \left( {H_{10} + H_{11} } \right) - H_{7} \left( {H_{7} + H_{13} } \right)} \right)}} $$

(42)

Proof

See “Appendix 1.”□

5.3 Centralized model (CM)

5.3.1 Profit functions

In the centralized decision-making structure, it is assumed that the players at the same level cooperate to achieve the maximum amount of profit. In this case, the producers are at the same level and the retailers are assumed to be at the same level, too. Therefore, according to the explanations, the profit functions at each level of the SC can be written as follows:

$$ \begin{aligned} \pi_{{{\text{Retailers}}}} & = \pi_{1} + \pi_{2} \\ & = D_{2} \times (p_{1} - w_{1} - r_{1} \tilde{m}_{1} ) + D_{3} \times (p_{2} - w_{2} - r_{2} \tilde{m}_{2} ) \\ \end{aligned} $$

(43)

$$ \pi_{{{\text{3pl}}}} = (\tilde{m}_{1} + \tilde{m}_{2} - \tilde{c}_{t} )(D_{2} + D_{3} ) - (\tilde{g} - \tilde{u}t_{2} )^{2} - v(z_{b} - z)^{2} $$

(44)

$$ \begin{aligned} \pi_{{{\text{Manufactures}}}} & = \pi_{{m_{1} }} + \pi_{{m_{2} }} \\ & = (s_{1} - \tilde{c}_{1} - \tilde{\chi }_{1} + \varphi_{1} ) \times D_{1} + (p_{1} - \tilde{c}_{1} - r_{3} \tilde{m}_{1} + \varphi_{1} ) \times D_{2} - \tilde{\eta }\frac{{\theta^{2} }}{2} - (\tilde{g} - \tilde{u}t_{1} )^{2} \\ & \quad + (p_{2} - \tilde{c}_{2} - r_{4} \tilde{m}_{2} + \varphi_{2} ) \times D_{3} + (s_{2} - \tilde{c}_{2} - \tilde{\chi }_{2} + \varphi_{2} ) \times D_{4} - (\tilde{g} - \tilde{u}t_{3} )^{2} \\ \end{aligned} $$

(45)

5.3.2 Problem solution

Generally, the mathematical structure of the game in the centralized model is as follows:

$$ \left\{ {\begin{array}{*{20}l} {E[\pi_{G} ]} \hfill \\ {\quad \left\{ {\begin{array}{*{20}l} {E[\pi_{{m_{1} }} + \pi_{{m_{2} }} ] = E[(s_{1} - \tilde{c}_{1} - \tilde{\chi }_{1} + \varphi_{1} ) \times D_{1} + (p_{1} - \tilde{c}_{1} - r_{3} \tilde{m}_{1} + \varphi_{1} ) \times D_{2} - \tilde{\eta }\frac{{\theta^{2} }}{2} - (\tilde{g} - \tilde{u}t_{1} )^{2} } \hfill \\ {\quad + (p_{2} - \tilde{c}_{2} - r_{4} \tilde{m}_{2} + \varphi_{2} ) \times D_{3} + (s_{2} - \tilde{c}_{2} - \tilde{\chi }_{2} + \varphi_{2} ) \times D_{4} - (\tilde{g} - \tilde{u}t_{3} )^{2} ]} \hfill \\ {E{[}\pi_{{{\text{3pl}}}} ] = E[(\tilde{m}_{1} + \tilde{m}_{2} - \tilde{c}_{t} )(D_{2} + D_{3} ) - (\tilde{g} - \tilde{u}t_{2} )^{2} - v(z_{b} - z)^{2} ]} \hfill \\ {E[\pi_{{{\text{Retailers}}}} ] = E[\pi_{1} + \pi_{2} ] = E[D_{2} \times (p_{1} - w_{1} - r_{1} \tilde{m}_{1} ) + D_{3} \times (p_{2} - w_{2} - r_{2} \tilde{m}_{2} )]} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right. $$

(46)

Note A_i, N_i, L_i, M_i and Y_i are the ith change of the variables, which is given in the supplementary file.

Lemma 4

The equilibrium values of p₁ and p₂ are:

$$ p_{1}^{*} = \frac{{\left( {\begin{array}{*{20}l} {E\left[ \gamma \right]\left( { - zE\left[ \varsigma \right] - \theta E\left[ {\lambda_{2} } \right] + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right) + Y_{1} } \right)} \hfill \\ {\quad + E\left[ \beta \right]\left( { - zE\left[ \varsigma \right] + \theta E\left[ {\lambda_{1} } \right] + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right) + Y_{2} } \right)} \hfill \\ \end{array} } \right)}}{{2\left( {E\left[ \beta \right]^{2} - E\left[ \gamma \right]^{2} } \right)}} $$

(47)

$$ p_{2}^{*} = \frac{{\left( {\begin{array}{*{20}l} {E\left[ \beta \right]\left( { - zE\left[ \varsigma \right] - \theta E\left[ {\lambda_{2} } \right] + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right) + Y_{1} } \right)} \hfill \\ {\quad + E\left[ \gamma \right]\left( { - zE\left[ \varsigma \right] + \theta E\left[ {\lambda_{1} } \right] + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right) + Y_{2} } \right)} \hfill \\ \end{array} } \right)}}{{2\left( {E\left[ \beta \right]^{2} - E\left[ \gamma \right]^{2} } \right)}} $$

(48)

Proof

See “Appendix 2.”□

Lemma 5

The equilibrium values of t₂ and z are as follows:

$$ t_{2}^{*} = \frac{{2E\left[ {gu} \right] + Y_{3} + 2\left( {E\left[ {\psi c_{t} } \right] - A_{5} - L_{10} } \right)}}{{2E\left[ u \right]^{2} }} $$

(49)

$$ z^{*} = \frac{{2E\left[ {z_{b} v} \right] + 2\left( {E\left[ {\varsigma c_{t} } \right] - L_{2} - L_{3} } \right) + Y_{4} }}{2E\left[ v \right]} $$

(50)

Proof

See “Appendix 2.”□

Lemma 6

The equilibrium values of $\theta$, $t_{1}$ and $t_{3}$ are as follows:

$$ \theta^{*} = \frac{{\left( {Y_{12} + Y_{13} } \right)\left( {Y_{9} + Y_{8} \varphi_{1} + Y_{7} \varphi_{2} } \right) - Y_{6} \left( {Y_{14} + Y_{15} + \left( {Y_{10} + Y_{11} } \right)\left( {\varphi_{1} + \varphi_{2} } \right)} \right)}}{{2Y_{6}^{2} - Y_{5} \left( {Y_{12} + Y_{13} } \right)}} $$

(51)

$$ t_{1}^{*} = \frac{{\left( {\begin{array}{*{20}l} {Y_{6}^{2} \left( { - Y_{14} + Y_{15} - \left( {Y_{10} - Y_{11} } \right)\left( {\varphi_{1} - \varphi_{2} } \right)} \right) + Y_{6} \left( {Y_{12} - Y_{13} } \right)\left( {Y_{9} + Y_{8} \varphi_{1} + Y_{7} \varphi_{2} } \right)} \hfill \\ {\quad + Y_{5} \left( { - Y_{12} \left( {Y_{15} + Y_{11} \varphi_{1} + Y_{10} \varphi_{2} } \right) + Y_{13} \left( {Y_{14} + Y_{10} \varphi_{1} + Y_{11} \varphi_{2} } \right)} \right)} \hfill \\ \end{array} } \right)}}{{\left( {Y_{12} - Y_{13} } \right)\left( { - 2Y_{6}^{2} + Y_{5} \left( {Y_{12} + Y_{13} } \right)} \right)}} $$

(52)

$$ t_{3}^{*} = \frac{{\left( {\begin{array}{*{20}l} {Y_{6}^{2} \left( {Y_{14} - Y_{15} + \left( {Y_{10} - Y_{11} } \right)\left( {\varphi_{1} - \varphi_{2} } \right)} \right) + Y_{6} \left( {Y_{12} - Y_{13} } \right)\left( {Y_{9} + Y_{8} \varphi_{1} + Y_{7} \varphi_{2} } \right)} \hfill \\ {\quad + Y_{5} \left( {Y_{13} \left( {Y_{15} + Y_{11} \varphi_{1} + Y_{10} \varphi_{2} } \right) - Y_{12} \left( {Y_{14} + Y_{10} \varphi_{1} + Y_{11} \varphi_{2} } \right)} \right)} \hfill \\ \end{array} } \right)}}{{\left( {Y_{12} - Y_{13} } \right)\left( { - 2Y_{6}^{2} + Y_{5} \left( {Y_{12} + Y_{13} } \right)} \right)}} $$

(53)

Proof

See “Appendix 2.”□

5.4 The government’s model

As mentioned already, the government is the leader of the game in both decentralized and centralized structures. Tariffs are the government’s decision variables, and two different scenarios are considered to obtain the optimal values of these variables. In the first scenario, it is assumed the government wants to minimize the total amount of pollution released by the SC’s members. Equation (54) shows the government’s model in the first scenario:

$$ \begin{aligned} & {\text{min}}\;\;\pi_{{{\text{Gov}}}} = \sum\limits_{u = 1}^{\left| U \right|} {\sum\limits_{q = 1}^{\left| Q \right|} {cp_{u} D_{q} } } + \sum\limits_{q \in Q - Y} {\ell D_{q} } + \sum\limits_{y \in Y} {\partial \ell D_{y} } \\ & \quad \;\;\;{\text{S.t}} \\ & \quad \quad \quad \left\{ {\begin{array}{*{20}l} {\pi_{{m_{i} }} \ge {\text{MPI}}_{i} } \hfill & {\forall i} \hfill \\ {\pi_{{3{\text{pl}}_{h} }} \ge {\text{MPT}}_{h} } \hfill & {\forall h} \hfill \\ {\pi_{j} \ge {\text{MPR}}_{j} } \hfill & {\forall j} \hfill \\ {\sum\limits_{i = 1}^{\left| I \right|} {\left( { - \varphi_{i} } \right)D_{{m_{i} }} } \ge {\text{MPG}}} \hfill & {\forall i} \hfill \\ \end{array} } \right. \\ & \quad \quad \quad \quad \quad 0 \le \ell ,\partial \le 1 \\ \end{aligned} $$

(54)

where constraints 1–3 indicate the minimum satisfactory profit for the SC members (producers, 3PL, retailers). Based on these constraints, the SC members will stay in the game if and only if their profit is greater than or equal to the minimum considered profits. The fourth constraint is related to the government’s profits from the producers’ production activities, which must be greater than or equal to a minimum value considered by the government.

In the second scenario, it is assumed that the government tries to maximize its profits from the producers’ production activities. Also, it has set an upper bound for the total pollution released by the SC members. The proposed model for the second scenario is as follows:

$$ \begin{aligned} & {\text{max}}\;\;\pi_{{{\text{Gov}}}} = \sum\limits_{i = 1}^{\left| I \right|} {\left( { - \varphi_{i} } \right)D_{{m_{i} }} } \\ & \quad \;\;\;{\text{S.t}} \\ & \quad \quad \;\;\left\{ {\begin{array}{*{20}l} {\pi_{{m_{i} }} \ge {\text{MPI}}_{i} } \hfill & {\forall i} \hfill \\ {\pi_{{3pl_{h} }} \ge {\text{MPT}}_{h} } \hfill & {\forall h} \hfill \\ {\pi_{j} \ge {\text{MPR}}_{j} } \hfill & {\forall j} \hfill \\ {\sum\limits_{u = 1}^{\left| U \right|} {\sum\limits_{q = 1}^{\left| Q \right|} {cp_{u} D_{q} } } + \sum\limits_{q \in Q - Y} {\ell D_{q} } + \sum\limits_{y \in Y} {\partial \ell D_{y} } \le {\text{UPG}}} \hfill & {\forall i} \hfill \\ \end{array} } \right. \\ & \quad \quad \quad \quad \;\;0 \le \ell ,\partial \le 1 \\ \end{aligned} $$

(55)

where the constraints 1–3 are completely similar to the model in the first scenario. The objective function shows the total profit of the government, while the fourth constraint indicates that the total released pollution in the SC must be less than or equal to an upper bound considered by the government.

6 Computational results

6.1 An example

A numerical example is provided in this section, and the results have been analyzed in order to see how the proposed model is practical in dealing with the problems. Also, the effect of intuitive fuzzy parameters on the players’ equilibrium values has been investigated. Table 2 shows the considered values for intuitive fuzzy and definite parameters in the numerical example. The government’s models have been coded and solved using Gams 24.1 software after placing the equilibrium expressions and values obtained.

The obtained expected values for each intuitive fuzzy parameter are presented in Table 3.

Substituting the expected values of the parameters into the equilibrium equations obtained for the decision variables and solving the government model, the equilibrium values of the decision variables can be obtained. The equilibrium values for both centralized and decentralized problems are shown in Table 4.

The equilibrium values of the profit functions of the SC members are also given in Table 5.

According to Table 4, the retail prices in the centralized model in both scenarios are higher than the ones in the decentralized model. In the centralized model, the retail price for the first retailer is higher than the second retailer. The objective function in the first scenario of the government could be a reason for this. Since in the first scenario, the model aims to reduce the total released pollution, the producers have to spend more money in order to increase the greenness degree of the products or pay tax to the government. Therefore, to compensate for this extra cost, the selling price must be increased, which is quite evident in Table 4. This is also true for the decentralized model. Table 4 shows that the delivery time of the products in the 3PL channel (t₂) in both models is much lower than those on the other channels (t₁, t₃). This is an important finding that indicates why cooperating with 3PLs can be a critical strategy for the supply chains and improve their market brand. Also, in both centralized and decentralized models, the delivery time in the direct channel of the first producer (t₁) is less than the delivery time in the direct channel of the second producer (t₃). According to numerical results, as an interesting finding, the total pollution released in the decentralized model is 2.37% less than in the centralized model.

Corollary 1

The delivery time of products and the amount of pollution are directly related.

The greenness degree of the product produced by the first producer in the centralized model is much greater than in the decentralized model. While in the second scenario, the greenness degree of the product produced by the first producer in the centralized model is 690.6% higher than in the decentralized one, in the first scenario, this value is about 65.7%. Since the second producer produces a non-green product, it faces a government fine, and it is forced to pay taxes in both scenarios. However, the first producer has government support in the form of subsidies.

As shown in Table 5, in both models, the expected profit for the first retailer is much higher than the second retailer’s expected profit. The reason for this is that the price difference between the first and second retailers is meager, but since the product of the first producer is green, the demand for this product is higher than the one for non-green products, and as a result, the first retailer expects a higher profit. In both scenarios, the 3PL’s profit in the decentralized model is higher than the one in the centralized model. On the other hand, in the centralized model, the expected profit of players at the same level is greater than in the decentralized model; hence, it can be concluded that centralized decision structure is a better strategy for players in the same level. However, it should be noted that according to Table 4, this strategy increases the delivery time of the product to the customer and increases the level of pollution. If the goal of all members is to increase sustainability in the SC, producers and retailers should pay more attention to the environmental aspects and customer satisfaction than the economic aspect.

Corollary 2

Cooperating with the 3PL companies will reduce the delivery time and the amount of pollution, and also increase customer satisfaction and the sustainability of the SC.

Table 5 clearly shows that in both models the government’s revenue in the second scenario is significantly higher than in the one in the first scenario. Obviously, this predictable result is because of the nature of the models in different scenarios and the difference between their objective functions. However, the government’s profits in the decentralized model are higher than the centralized model. Therefore, the government, interestingly, will prefer to apply a strategy that leads to a non-cooperative decision structure among the players at the same level. Both the government’s profit and the sustainability of the SC will be better in the centralized model than in the decentralized model.

6.2 The effect of change in government tariffs $(\varphi_{1} ,\varphi_{2} )$ on the profits of the SC members

In this section, a sensitivity analysis of government’s variables (tariffs) has been conducted. Table 2 shows the changes in producers’ and retailers’ expected profit in the various strategies chosen by the government and various decision-making structures.

Figure 2i, ii depicts the changes in retailers’ profits versus tariffs imposed by the government. In the decentralized model (Fig. 2i), the increase and decrease in the profit of the first and second retailers is more dependent on the tariff imposed on the first producer. Figure 2i clearly shows that in the decentralized model, the expected profit of the first retailer increases as the subsidy allocated to the first producer (φ₁) increases, and the changes in φ₂ have no effect on it. Also, the expected profit of the second retailer decreases as the subsidy imposed on the first producer (φ₁) increases, and the changes in φ₂ have no effect on it, too.

In the centralized model (Fig. 2ii), by increasing the government’s support for the first producer and fining the second producer with taxation due to environmental issues, the first producer’s profit will be increased and the second retailer’s profit will be decreased. The same is true for the second retailer.

According to Fig. 2iii, in the decentralized model, the expected profit of the first producer is entirely dependent on the imposed tariff to it, though giving a high subsidy to the second producer can decrease the first producer’s profit insignificantly. As the amount of subsidy allocated to the first producer increases, its profit increases, too. In contrast, the profit of the second producer reacts to the tariff of the first producer in addition to its own tariff. The second producer’s expected profit is greater when the government pays the highest amount of subsidies to it and receives the highest tax from the first producer.

Figure 2iv has almost the same pattern as Fig. 2iii. As it can be deduced from this figure, although in the greater part of the space, the expected profit of the first producer is greater than the second producer’s profit, the difference between the retailers’ profit in the different tariff combinations is less than the one in the decentralized model. Even though it can be seen that in some cases, when the government gives subsidies to the first producer and takes tax from the second producer, interestingly, the second producer’s profit is still higher than the first producer’s. However, the maximum profit comes to the first producer when the government is indifferent to the second producer (zero tariff) and pays maximum subsidies to the first producer.

Corollary 3

The financial intervention of the government is a practical policy to increase the sustainability of the SCs.

6.3 The effect of change k on the selling price and greenness degree of the product

Using the sensitivity analysis of variables, in this section, the effects of changes in k on the prices and degree of greenness have been analyzed.

Figures 3 and 4 show the changes in retail prices of the first and second retailers against changes in k value, in both centralized and decentralized models and different governmental scenarios. In Figs. 3 and 4, the first and second scenarios of the government are applied, respectively. As it is clear from both figures, in both models and both scenarios, if public awareness about the importance of cooperation with 3PL in the channel (k) increases, the greenness degree and the selling price of products in retail channels will be increased, too.

6.4 The effect of changes in $\lambda_{1}$, $\lambda_{2}$, ς, and ψ values on the players’ profit

Tables 6, 7, 8, 9, 10 and 11 show the effect of changes in the λ₁ and λ₂ values in different scenarios on the expected profit of SC members. The numerical values in Tables 6 and 11 indicate that increasing λ₁ and λ₂ will not necessarily increase or decrease the expected profit of retailers. In fact, with increasing λ₁ and λ₂, the expected profit has increased in some cases and decreased in others.

Corollary 5

There isn’t any direct relationship between (λ₁, λ₂) and the expected profit of retailers.

According to Tables 6 and 11, in different λ₁ and λ₂ values, in all similar problems, the profit of the first retailer is higher than the second retailer’s. Also, the highest profit is obtained for the first retailer in the decentralized model in the second scenario and for the second retailer in the non-centralized model in the first scenario.

According to Tables 8 and 9, the changes in producers’ expected profit versus the changes in the λ₁ and λ₂ values are similar to changes in the retailers’ profit function. The only difference is that the highest profit for producers is achieved in the centralized model. Also, analyzing Tables 10 and 11 shows that the expected profit of 3PL in centralized model is higher than decentralized model. Also, it can be seen that the effects of changes in λ₁ and λ₂ on the profit of the 3PL company are in accordance with its effects on the producers’ profit.

According to Tables 12 and 13, the profit of the first retailer in each scenario is higher than the profit of the second retailer. It can also be seen that with the change of ς and ψ, the ratio of changes in the expected profit of the first retailer to its expected equilibrium profit is higher than the one for the second retailer. Furthermore, the retailers gain more profits in the decentralized model compared to the centralized one. Based on Tables 14 and 15, it can be claimed that the same is true for producers. Similar to λ₁ and λ₂, increasing ς and ψ does not necessarily increase the profits of producers and retailers. Therefore, how to adjust the intuitive fuzzy parameters (expert’s opinion), ultimately affects the outcome of the problem.

In general, by analyzing the results provided in Tables 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15, it can be clearly seen that the focus on increasing consumer awareness of the remarkable advantages and positive impacts of eco-friendly transportation systems and green products over the traditional non-green production systems can noticeably increase the sustainability of the supply chain. In other words, it can be a practical and effective strategy for governments and companies to use advertisements to increase public awareness in their society toward green production/green transportation in order to achieve both economic and environmental sustainability targets.

7 Conclusions and managerial insights

In this study, the problem of sustainable multi-channel supply chain design has been modeled for a case where there are two multi-channel SCs, including two producers, two retailers, and a 3PL company. Also, the government monitors the players’ activities, trying to achieve its personal goals by imposing tariffs on the producers. As for the decision variables, the proposed models are able to determine the equilibrium values of tariffs allocated to the producer, wholesale/retail prices in the different channels, the greenness degree of the products, and product delivery time in all channels. As an important contribution, the existing uncertainty in the real word is considered using intuitive fuzzy numbers to make the understudy problem more realistic.

Some of the important managerial insights obtained in this study can be expressed as follows:

• Examining the problems in an intuitive fuzzy environment makes the results closer to the actual behavior of the players in the real world. As a result, managers need to pay attention to the importance of considering the issue of uncertainty in the decision-making process. This insight has also been observed by other studies in the literature, such as Li (2014) and Chavoshlou et al. (2019).

• Cooperation with a 3PL and monitoring the SC’s members’ activities by the government will increase the sustainability of the supply chain over time.

• The government’s financial intervention is a practical toll in order to achieve sustainable development goals in the SC and encourage the players of the SC to apply more sustainable strategies. By setting a strategy to support green production, the government can reduce the delivery time of products to the customer, reduce the total amount of released pollution in the SC, increase customer satisfaction, and thus increase the sustainability of the SC.

• Increasing public awareness about the importance of cooperating with 3PL companies, despite the increase in the final price of the product, leads to increased sustainability in the SC.

Also, this research showed that the interests of producers and 3PL companies are mainly in the same direction. In this study, it was assumed that in the equilibrium strategy, the greater part of the products’ shipment costs will be paid by the producers. Therefore, it is logical for the producer and the 3PL company to be aligned.

There are some suggestions for future studies: first, the intuitive fuzzy theory is used in this study to consider the uncertainty, while future studies can apply other fuzzy theory methods and compare their results with this study. In addition to short-term behavior, it is necessary to analyze the long-term behavior of the players under any applied policy. Hence, the future studies can develop an evolutionary game model to analyze the dynamic behavior of the players. As the third suggestion, studying different SC structures by adding new players and considering different scenarios for the government can be an interesting but challenging research direction. Finally, the proposed approach can be applied to real-world data and the results can be discussed.

Appendices

Appendix 1: Proofs

1.1 Decentralized model

Proof of Lemma 1

To obtain the equilibrium values of p₁ and p₂, it is enough to solve the following system of equations:

$$ \left\{ \begin{gathered} \frac{{{\text{d}}E[\pi_{1} ]}}{{{\text{d}}p_{1} }} = 0 \Rightarrow \left( {\begin{array}{*{20}l} { - E\left[ \varsigma \right]z + kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\beta w_{1} } \right] + E\left[ {\varsigma z_{b} } \right]} \hfill \\ {\quad + E\left[ {\alpha_{2} } \right] + E\left[ {\lambda_{1} } \right]\theta - 2E\left[ \beta \right]p_{1} + E\left[ \gamma \right]p_{2} + E\left[ {\beta m_{1} } \right]r_{1} + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \hfill \\ \end{array} } \right) = 0 \hfill \\ \frac{{{\text{d}}E[\pi_{2} ]}}{{{\text{d}}p_{2} }} = 0 \Rightarrow \left( {\begin{array}{*{20}l} { - E\left[ \varsigma \right]z + kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\beta w_{2} } \right] + E\left[ {\varsigma z_{b} } \right]} \hfill \\ {\quad + E\left[ {\alpha_{3} } \right] - E\left[ {\lambda_{2} } \right]\theta + E\left[ \gamma \right]p_{1} - 2E\left[ \beta \right]p_{2} + E\left[ {\beta m_{2} } \right]r_{2} + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \hfill \\ \end{array} } \right) = 0 \hfill \\ \end{gathered} \right. $$

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Proof of Lemma 2

To obtain $t_{2}^{*}$ and $z^{*}$, it is enough to solve the following set of simultaneous equations:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}E[\pi_{{{\text{3pl}}}} ]}}{{{\text{d}}t_{2} }} = 0 \Rightarrow 2\left( {\begin{array}{*{20}l} {E\left[ {gu} \right] + E\left[ {\psi c_{t} } \right] - A_{5} - L_{10} - E\left[ u \right]^{2} t_{2} } \hfill \\ {\quad + \frac{{E\left[ \psi \right]\left( { - E\left[ {\beta c_{t} } \right] - E\left[ {\gamma m_{1} } \right] - E\left[ {\gamma m_{2} } \right] + L_{4} + L_{5} + L_{6} } \right)}}{2E\left[ \beta \right] - E\left[ \gamma \right]}} \hfill \\ \end{array} } \right) = 0} \hfill \\ {\frac{{{\text{d}}E[\pi_{{{\text{3pl}}}} ]}}{{{\text{d}}z}} = 0 \Rightarrow 2\left( {\begin{array}{*{20}l} { - zE\left[ v \right] + E\left[ {z_{b} v} \right] + E\left[ {\varsigma c_{t} } \right] - L_{2} - L_{3} } \hfill \\ {\quad + \frac{{E\left[ \varsigma \right]\left( { - E\left[ {\beta c_{t} } \right] - E\left[ {\gamma m_{1} } \right] - E\left[ {\gamma m_{2} } \right] + L_{4} + L_{5} + L_{6} } \right)}}{2E\left[ \beta \right] - E\left[ \gamma \right]}} \hfill \\ \end{array} } \right) = 0} \hfill \\ \end{array} } \right. $$

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Proof of Lemma 3

By substituting the equilibrium values in Eqs. (36), (37), (39) and (40) into Eqs. (34) and (35) and solving Eq. (58), the equilibrium values can be calculated:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}E[\pi_{{m_{1} }} ]}}{{{\text{d}}\theta }} = 0 \Rightarrow \theta H_{5} + H_{8} + H_{7} \left( {t_{1} + t_{3} } \right) + H_{6} \varphi_{1} = 0} \hfill \\ {\frac{{{\text{d}}E[\pi_{{m_{1} }} ]}}{{{\text{d}}t_{1} }} = 0 \Rightarrow \theta H_{7} + H_{12} + H_{11} t_{1} + H_{10} t_{3} + H_{9} \varphi_{1} = 0} \hfill \\ {\frac{{{\text{d}}E[\pi_{{m_{2} }} ]}}{{{\text{d}}t_{3} }} = 0 \Rightarrow \theta H_{13} + H_{14} + H_{10} t_{1} + H_{11} t_{3} + H_{9} \varphi_{2} = 0} \hfill \\ \end{array} } \right. $$

(58)

1.2 Centralized model

Proof of Lemma 4

To obtain the equilibrium values of p₁ and p₂, it is enough to solve the following system of equations:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}E[\pi_{{{\text{Retailers}}}} ]}}{{{\text{d}}p_{1} }} = 0 \Rightarrow \left( {\begin{array}{*{20}l} { - zE\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\beta w_{1} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{2} } \right]} \hfill \\ {\quad + \theta E\left[ {\lambda_{1} } \right] - A_{18} - 2E\left[ \beta \right]p_{1} + 2E\left[ \gamma \right]p_{2} + E\left[ {\beta m_{1} } \right]r_{1} - N_{4} r_{2} } \hfill \\ {\quad + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \hfill \\ \end{array} } \right) = 0} \hfill \\ {\frac{{{\text{d}}E[\pi_{{{\text{Retailers}}}} ]}}{{{\text{d}}p_{2} }} = 0 \Rightarrow \left( {\begin{array}{*{20}l} { - zE\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\beta w_{2} } \right] + E\left[ {\varsigma z_{b} } \right]} \hfill \\ {\quad + E\left[ {\alpha_{3} } \right] - \theta E\left[ {\lambda_{2} } \right] - A_{7} + 2E\left[ \gamma \right]p_{1} - 2E\left[ \beta \right]p_{2} - A_{2} r_{1} } \hfill \\ {\quad + E\left[ {\beta m_{2} } \right]r_{2} + E\left[ \psi \right]\left( {t_{1} - t_{2} + t_{3} } \right)} \hfill \\ \end{array} } \right) = 0} \hfill \\ \end{array} } \right. $$

(59)

Proof of Lemma 5

To obtain $t_{2}^{*}$ and $z^{*}$, it is enough to solve the following set of simultaneous equations:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}E[\pi_{{{\text{3pl}}}} ]}}{{{\text{d}}t_{2} }} = 0 \to \left( {\begin{array}{*{20}l} {\frac{{2E\left[ \psi \right]\left( { - E\left[ {\beta c_{t} } \right] - E\left[ {\gamma m_{1} } \right] - E\left[ {\gamma m_{2} } \right] + L_{4} + L_{5} + L_{6} } \right)}}{E\left[ \beta \right] - E\left[ \gamma \right]}} \hfill \\ {\quad - 4\left( { - E\left[ {\psi c_{t} } \right] + A_{5} + L_{10} + E\left[ u \right]^{2} t_{2} } \right) + 4E\left[ {gu} \right]} \hfill \\ \end{array} } \right) = 0} \hfill \\ {\frac{{{\text{d}}E[\pi_{{{\text{3pl}}}} ]}}{{{\text{d}}z}} = 0 \to \left( {\begin{array}{*{20}l} { - 4zE\left[ v \right] + 4\left( {E\left[ {z_{b} v} \right] + E\left[ {\varsigma c_{t} } \right] - L_{2} - L_{3} } \right)} \hfill \\ {\quad + \frac{{2E\left[ \varsigma \right]\left( { - E\left[ {\beta c_{t} } \right] - E\left[ {\gamma m_{1} } \right] - E\left[ {\gamma m_{2} } \right] + L_{4} + L_{5} + L_{6} } \right)}}{E\left[ \beta \right] - E\left[ \gamma \right]}} \hfill \\ \end{array} } \right) = 0} \hfill \\ \end{array} } \right. $$

(60)

After substituting the equilibrium values of the decision variables of the retailers and the 3PL into the producers’ profit functions, the expected profit for the producers can be presented as follows:

$$ \begin{aligned} E[\pi_{{{\text{Producers}}}} ] & = - 2E\left[ g \right]^{2} - \frac{1}{2}\theta^{2} E\left[ \eta \right] + E\left[ {\beta c_{1} s_{1} } \right] + E\left[ {\beta c_{2} s_{2} } \right] + 2E\left[ {\gamma s_{1} s_{2} } \right] + E\left[ {s_{1} \alpha_{1} } \right] + E\left[ {s_{2} \alpha_{4} } \right] \\ & \quad + E\left[ {\beta s_{1} \chi_{1} } \right] + E\left[ {\beta s_{2} \chi_{2} } \right] - kM_{1} - M_{3} - M_{4} - 2M_{5} - M_{8} - M_{9} - M_{10} - M_{16} \\ & \quad - M_{18} - kN_{1} - 2N_{5} - N_{7} - N_{11} - N_{13} - N_{15} - N_{19} - N_{21} - N_{22} \\ & \quad - E\left[ \beta \right](p_{1}^{*} )^{2} - E\left[ \beta \right](p_{2}^{*} )^{2} - \left( {kA_{1} + A_{3} + A_{4} + A_{11} + A_{13} } \right)r_{3} - kN_{3} r_{4} \\ & \quad - N_{6} r_{4} - N_{8} r_{4} - N_{12} r_{4} - N_{14} r_{4} + z^{*} \left( {E\left[ {\varsigma c_{1} } \right] + E\left[ {\varsigma c_{2} } \right] + E\left[ {\varsigma m_{1} } \right]r_{3} + E\left[ {\varsigma m_{2} } \right]r_{4} } \right) \\ & \quad + \theta \left( {E\left[ {s_{1} \lambda_{1} } \right] + 2E\left[ {c_{2} \lambda_{2} } \right] + E\left[ {\lambda_{2} \chi_{2} } \right] - 2M_{12} - M_{19} - N_{16} - M_{15} r_{3} + E\left[ {m_{2} \lambda_{2} } \right]r_{4} } \right) \\ & \quad + \left( {2E\left[ {gu} \right] + E\left[ {\psi s_{2} } \right] + E\left[ {\psi \chi_{1} } \right] - M_{6} - 2N_{9} - N_{20} - A_{5} r_{3} - L_{10} r_{4} } \right)t_{1} \\ & \quad - E\left[ u \right]^{2} t_{1}^{2} + \left( {E\left[ {\psi s_{1} } \right] + E\left[ {\psi s_{2} } \right] - M_{17} - N_{20} + E\left[ {\psi m_{1} } \right]r_{3} + E\left[ {\psi m_{2} } \right]r_{4} } \right)t_{2}^{*} \\ & \quad + \left( {2E\left[ {gu} \right] + E\left[ {\psi s_{1} } \right] + E\left[ {\psi \chi_{2} } \right] - 2M_{7} - M_{17} - N_{10} - A_{5} r_{3} - L_{10} r_{4} } \right)t_{3} - E\left[ u \right]^{2} t_{3}^{2} \\ & \quad + \left( {\begin{array}{*{20}l} { - z^{*} E\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + E\left[ {\gamma s_{1} } \right] + 2E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{1} } \right] + E\left[ {\alpha_{2} } \right]} \hfill \\ {\quad + 2\theta E\left[ {\lambda_{1} } \right] - \beta s_{1} + 2E\left[ \psi \right]t_{3} } \hfill \\ \end{array} } \right)\varphi_{1} \\ & \quad + \left( {\begin{array}{*{20}l} { - z^{*} E\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + 2E\left[ {\gamma s_{1} } \right] + E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{3} } \right] + E\left[ {\alpha_{4} } \right]} \hfill \\ {\quad - 2\theta E\left[ {\lambda_{2} } \right] - \beta s_{2} + 2E\left[ \psi \right]t_{1} } \hfill \\ \end{array} } \right)\varphi_{2} \\ & \quad + p_{1}^{*} ( - z^{*} E\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + E\left[ {\beta c_{1} } \right] + 2E\left[ {\gamma s_{1} } \right] + 2E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{2} } \right] \\ & \quad + \theta E\left[ {\lambda_{1} } \right] - M_{2} - M_{14} - 2N_{2} - N_{18} + 2E\left[ \gamma \right]p_{2}^{*} + E\left[ {\beta m_{1} } \right]r_{3} - N_{4} r_{4} \\ & \quad + E\left[ \psi \right]t_{1} - E\left[ \psi \right]t_{2}^{*} + E\left[ \psi \right]t_{3} + \left( { - E\left[ \beta \right] + E\left[ \gamma \right]} \right)\varphi_{1} + 2E\left[ \gamma \right]\varphi_{2} ) \\ & \quad + p_{2}^{*} ( - z^{*} E\left[ \varsigma \right] + kE\left[ {a_{f} } \right] + E\left[ {\beta c_{2} } \right] + 2E\left[ {\gamma s_{1} } \right] + 2E\left[ {\gamma s_{2} } \right] + E\left[ {\varsigma z_{b} } \right] + E\left[ {\alpha_{3} } \right] \\ & \quad - \theta E\left[ {\lambda_{2} } \right] - 2M_{2} - M_{14} - N_{2} - N_{18} - A_{2} r_{3} + E\left[ {\beta m_{2} } \right]r_{4} + E\left[ \psi \right]t_{1} \\ & \quad - E\left[ \psi \right]t_{2}^{*} + E\left[ \psi \right]t_{3} + 2E\left[ \gamma \right]\varphi_{1} + \left( { - E\left[ \beta \right] + E\left[ \gamma \right]} \right)\varphi_{2} ) \\ \end{aligned} $$

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Proof of Lemma 6

The following system of equations must be solved to obtain the equilibrium values of $\theta^{*}$, $t_{1}^{*}$, and $t_{3}^{*}$:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}E[\pi_{{{\text{Producers}}}} ]}}{{{\text{d}}\theta }} = 0 \to \theta Y_{5} + \left( {t_{1} + t_{3} } \right)Y_{6} + Y_{9} + Y_{8} \varphi_{1} + Y_{7} \varphi_{2} = 0} \hfill \\ {\frac{{{\text{d}}E[\pi_{{{\text{Producers}}}} ]}}{{{\text{d}}t_{1} }} = 0 \to \theta Y_{6} + t_{3} Y_{12} + t_{1} Y_{13} + Y_{14} + Y_{10} \varphi_{1} + Y_{11} \varphi_{2} = 0} \hfill \\ {\frac{{{\text{d}}E[\pi_{{{\text{Producers}}}} ]}}{{{\text{d}}t_{3} }} = 0 \to \theta Y_{6} + t_{1} Y_{12} + t_{3} Y_{13} + Y_{15} + Y_{11} \varphi_{1} + Y_{10} \varphi_{2} = 0} \hfill \\ \end{array} } \right. $$

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Appendix 2: Figures

See Figs. 1, 2, 3, and 4.

Fig. 1

A schematic framework of understudy problem

Fig. 2

Effect of changing government tariffs on the expected profit of the different players

Fig. 3

Effect of changing k on the price and greenness degree the products in the first scenario

Fig. 4

Effect of changing k on the price and greenness degree the products in the second scenario

Appendix 3: Tables

See Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15.

Table 2 Assigned values to intuitive fuzzy and definite parameters

Table 3 Expected values of intuitive fuzzy parameters

Table 4 Equilibrium values of decision variables

Table 5 Equilibrium values of SC members’ profits

Table 6 Optimal expected profit of retailers in different λ₁ values

Table 7 Optimal expected profit of retailers on different λ₂ values

Table 8 Optimal expected profit of producers in different λ₁ values

Table 9 Optimal expected profit of producers in different λ₂ values

Table 10 Optimal expected profit of 3PL in different λ₁ values

Table 11 Optimal expected profit of 3PL in different λ₂ values

Table 12 Optimal expected profit of retailers in different $\varsigma$ values

Table 13 Optimal expected profit of retailers in different $\psi$ values

Table 14 Optimal expected profit of producers in different $\varsigma$ values

Table 15 Optimal expected profit of producers in different $\psi$ values