Effects of governmental supportive policies on waste management for two substitutable products made of virgin and waste materials: A game-theoretic approach

Abstract

In recent years, the concept of sustainability has attracted a great amount of attention, due to increasing energy resources scarceness. Waste recycling is known as an efficient approach to improve sustainability and save energy. In this view, a sustainable supply chain is established in the current study to investigate the effects of waste recycling on sustainable development. The considered supply chain consists of the government, two manufacturers, a supplier, a waste depot, and a recycler. Under this structure, two substitutable products are made of the virgin and recyclable waste materials. The supplier provides the virgin materials for the first product. The waste depot collects the non-recycled waste, whereas the recycler recycles it and supplies the recycled waste for the second product. Also, the government supports the second product to provide an incentive for its members to collect and recycle more waste. Then, the game theory is applied to make decisions under the considered structure. Finally, the results are revealed and some managerial insights are provided. It is derived that the governmental supportive policies play a significant role in resources conservation and energy storage. Moreover, increasing the quality of the product made of the recyclable waste improves the government’s utility.

Keywords

Waste management sustainable development environmental sustainability recycling governmental supportive policies game theory

Introduction

Motivation

With growing energy resources scarceness, the concept of sustainable development has attracted a significant attention in recent years (Jafari, 2023a; Khan et al., 2023). Environmental, economic and social aspects are known as the fundamental pillars of the sustainable development (Jafari, 2023b; Tanveer et al., 2022). Waste recycling has a considerable effect on the environmental sustainability (Bulach et al., 2018; Dias et al., 2022). It reduces the air pollution and the greenhouse gases emissions as well as conserves the natural resources and the renewable and nonrenewable energies (Fu et al., 2022; Jafari, 2023c; Nketiah et al., 2022). Using the recyclables to manufacture a new product decreases the need for the virgin materials and this saves the natural resources and energy by reducing the need for mining or extracting the virgin materials (Callesen et al., 2022).

Governments play a major role for achieving the sustainable development goals (ElMassah and Mohieldin, 2020; Hassan et al., 2019). In this point of view, providing an incentive for the economic enterprises by implementing some supportive policies like reducing the tax rates or paying a subsidy can be considered as the common approaches in order to reach these goals (Mohammadzadeh et al., 2022; Naserabad et al., 2021).

The current study considers a supply chain structure consisting of the government, two manufacturers, a supplier, a waste depot, and a recycler for manufacturing two substitutable products in a competitive market. One of these products is manufactured from the virgin materials by the first manufacturer, whereas another one is produced using a recyclable waste by the second manufacturer. The supplier provides the virgin materials for the first manufacturer. Moreover, the waste depot collects the required non-recycled waste and then the recycler processes it (including separating, clearing, and washing operations) for the second manufacturer. In addition, the government supports the second product by stimulating the members involved in its supply chain to collect and use more waste. Then, the game-theoretic frameworks are applied in order to make decisions under the considered structure. The game theory is an efficient mathematical approach to analyze the complicated relationships established among some rational agents in order to make their decisions under competitive and cooperative situations (Jafari, 2021; Jafari, 2023d; Jafari, 2024; Jafari and Safarzadeh, 2023; Jafari et al., 2020).

Literature review

The current study aims to establish a supply chain structure where two substitutable products are manufactured using the virgin and waste materials. In this setting, the game-theoretic framework is applied to make decisions. To our knowledge, researches on how to investigate the recycling process under a supply chain structure using the game theory are still limited that most of them have been addressed as follows:

Yi et al. (2016) established a dual-recycling channel in a retailer-oriented closed-loop supply chain. Using the game theory, they implemented a two-period approach in order to set prices in each recycling channel. Li et al. (2019) studied the end-of-life products under a closed-loop supply chain with an option to open the online recycling channel. Xu et al. (2020) considered the capacity constraints in the recycling channels under a closed-loop supply chain including one manufacturer and one retailer, where the manufacturer sells his products to customers and the retailer acts as an interface. Matsui (2022) analyzed the competition established between a recycling company and a general third-party collector under a dual-channel closed-loop supply chain applying the game theory. Yu et al. (2021) used a game model to analyze the competition between the legal and illegal recyclers considering the quality level of the end-of-life vehicles. Moreover, Zhang et al. (2021) studied a closed-loop supply chain by considering the governmental reward-penalty mechanisms.

Guo et al. (2018) provided some directions for the waste recycling companies under a supply chain including one processor and one recycler by considering different scenarios. Jafari et al. (2017) investigated the waste recycling process on a three-echelon supply chain consisting of the collector, recycler, and manufacturer, where a product is manufactured from a recyclable waste. Furthermore, Parsaeifar et al. (2019) developed a game-theoretic model in order to calculate a gap between the green and non-green recyclables under a competitive three-echelon supply chain.

Jafari (2019) considered a supply chain structure including two collectors and three firms, where the collectors collect the plastic and metal bottles and the firms clean and fragment them. Then, the recycled materials are used in textile industry as an appropriate substitutable for cotton. Moreover, Jafari (2022) discussed the economic and environmental perspectives of sustainability under a supply chain structure where the PET plastic bottles are recycled and reused in textile industry.

Liu et al. (2020) discussed the dynamic relations between a waste producer and a recycler in waste recycling process using different market scenarios. Jin et al. (2020) proposed three waste recycling methods including internal recycling in a firm, intrafirm collaborative processing, and centralized processing in an industrial environment by providing a basis for the governmental supportive policies. He and Yuan (2020) analyzed the competitive relationships between the building materials enterprises and the waste recycling companies under a construction waste recycling market. Yang et al. (2021) established a multi-agent waste recycling system including the governments, individuals, and firms in order to find an efficient recycling model for the considered problem. Moreover, Cohen et al. (2021) developed a model for the recycling process by considering the social and economic aspects of sustainability.

Research contributions

In the current study, a novel sustainable supply chain is established in which two substitutable products are manufactured and then sold to customers. One of them is produced using the virgin materials, while another one is procured from a recyclable waste. Under the considered structure, the government provides some supportive policies for motivating the members who produce the second product to collect and utilize more waste. For this reason, the government pays a portion of the collecting, recycling, and producing costs of the second product. Moreover, the game theory is applied in order to make the decisions under different situations established among the members. To our knowledge, this study is the first one that deals with two substitutable products manufactured from the virgin and waste materials by considering the governmental supportive strategies to improve the environmental sustainability.

The remainder of the paper is organized as follows: The research problem is described in Section ‘Problem description’. The game-theoretic models are developed in Section ‘Game-theoretic approach’. In Section ‘A real example’, a real example is provided. Section ‘ Results and discussion’ deals with the results and managerial insights. Ultimately, the conclusions are presented in Section ‘ Summary and conclusions’.

Problem description

Consider two substitutable products where one of them is produced from the virgin materials and another one is made of the waste. Suppose the products 1 and 2 are produced from the virgin and waste materials, respectively. A supplier provides the virgin materials required to produce the product 1. In addition, to produce the product 2, its manufacturer purchases the required recycled materials from a recycler. The recycler also provides the non-recycled waste (waste that the recycling operations like separating, clearing, and washing have not been implemented on it) from a waste depot. As a matter of fact, the waste depot collects the non-recycled waste and then the recycler recycles it. Ultimately, the manufacturer uses the recycled waste (waste obtained after the recycling operations) to produce the product 2.

In the current study, a supply chain is considered consisting of the government, two manufacturers, a supplier, a waste depot, and a recycler. Under the considered supply chain, two substitutable products entitled as the product 1 (made of the virgin materials) and the product 2 (made of the waste materials) are manufactured and then sold to customers. The manufacturers 1 and 2 produce the products 1 and 2, respectively. The supplier provides the required virgin materials for the manufacturer 1. The waste depot collects the non-recycled waste, while the recycler recycles it and purveys the recycled waste to the manufacturer 2. Moreover, to motivate the members involved in the product 2’s supply chain, the government applies some supportive policies. In fact, the government supports the waste depot, recycler, and manufacturer 2 to collect and use more waste materials. In this view, the government undertakes to pay a portion of the producing cost of the product 2 as well as the collecting and recycling costs of its required waste. This leads to decreasing the final price of the product 2 as well as extending its demand. Clearly, the waste required to manufacture the product 2 would increase in this situation.

Under the considered structure, the supplier determines the price of the virgin materials for the manufacturer 1. The waste depot specifies the price of the non-recycled waste for the recycler. The recycler sets the price of the recycled waste for the manufacturer 2. Moreover, the manufacturers 1 and 2 declare the prices of the products 1 and 2 for customers, respectively. Then, based on the prices set for both products and their qualities, the customers’ demands for the products as well as the demands for the virgin and waste materials would be specified.

The notations used in this study are defined as follows:

Parameters
$C_{s}$	Supplying cost per ton of the virgin materials for the supplier
$C_{w}$	Collecting cost per ton of the non-recycled waste for the waste depot
$C_{r}$	Recycling cost per ton of the non-recycled waste for the recycler
$C_{1}$	Producing cost per ton of the product 1 for the manufacturer 1
$C_{2}$	Producing cost per ton of the product 2 for the manufacturer 2
$q$	Quality level of the product 2 with respect to the product 1 ( $0 < q < 1$ )
$τ_{s}$	Virgin materials required to produce one ton of the product 1 ( $τ_{s} > 1$ )
$τ_{r}$	Recycled waste required to produce one ton of the product 2 ( $τ_{r} > 1$ )
$τ_{w}$	Non-recycled waste required to procure one ton of the recycled materials ( $τ_{w} > 1$ )
$β$	Government’s subsidy rate paid to the members involved in the product 2’s supply chain ( $0 < β < 1$ )
$V_{1}$	Maximum possible demand for the product 1 in the market
$V_{2}$	Maximum possible demand for the product 2 in the market
$δ_{1}$	Self-price elasticity of the demands (change of the demand for a product as its price increases by one unit)
$δ_{2}$	Cross-price elasticity of the demands (change of the demand for a product as the price of another one increases by one unit) ( $δ_{1} > δ_{2}$ )
$γ_{1}$	Elasticity of the demand for the product 1 with respect to the quality level of the product 2
$γ_{2}$	Elasticity of the demand for the product 2 with respect to its quality level ( $γ_{2} > γ_{1}$ )
Independent variables
$P_{s}$	Price per ton of the virgin materials set by the supplier to the manufacturer 1
$P_{w}$	Price per ton of the non-recycled waste set by the waste depot to the recycler
$P_{r}$	Price per ton of the recycled waste set by the recycler to the manufacturer 2
$P_{1}$	Price per ton of the product 1 set by the manufacturer 1 to customers
$P_{2}$	Price per ton of the product 2 set by the manufacturer 2 to customers
Dependent variables
$Q_{s}$	Demand of the virgin materials for the supplier
$Q_{w}$	Demand of the non-recycled waste for the waste depot
$Q_{r}$	Demand of the recycled waste for the recycler
$Q_{1}$	Demand of the product 1 for the manufacturer 1
$Q_{2}$	Demand of the product 2 for the manufacturer 2
$U_{s}$	Supplier’s profit
$U_{w}$	Waste depot’s profit
$U_{r}$	Recycler’s profit
$U_{1}$	Manufacturer 1’s profit
$U_{2}$	Manufacturer 2’s profit
$U S_{1}$	Total profit of the product 1’s supply chain
$U S_{2}$	Total profit of the product 2’s supply chain
$U_{g}$	Government’s utility

The demands for the final products, the virgin materials, and the recycled and non-recycled wastes are formulated as follows:

Q_{1} = V_{1} - δ_{1} P_{1} + δ_{2} P_{2} - γ_{1} q

(1)

Q_{2} = V_{2} - δ_{1} P_{2} + δ_{2} P_{1} + γ_{2} q

(2)

Q_{s} = τ_{s} Q_{1}

(3)

Q_{r} = τ_{r} Q_{2}

(4)

Q_{w} = τ_{w} Q_{r} = τ_{w} τ_{r} Q_{2}

(5)

Evidently, the demands of the virgin and waste materials are obtained based on the demands for the final products.

The members’ profits as well as the government’s utility are also formulated as follows:

U_{s} = P_{s} Q_{s} - C_{s} Q_{s} = τ_{s} (P_{s} - C_{s}) Q_{1}

(6)

U_{w} = P_{w} Q_{w} - (1 - β) C_{w} Q_{w} = τ_{w} τ_{r} (P_{w} - (1 - β) C_{w}) Q_{2}

(7)

U_{r} = P_{r} Q_{r} - P_{w} Q_{w} - (1 - β) C_{r} Q_{w} = τ_{r} (P_{r} - τ_{w} P_{w} - τ_{w} (1 - β) C_{r}) Q_{2}

(8)

U_{1} = P_{1} Q_{1} - P_{s} Q_{s} - C_{1} Q_{1} = (P_{1} - τ_{s} P_{s} - C_{1}) Q_{1}

(9)

U_{2} = P_{2} Q_{2} - P_{r} Q_{r} - (1 - β) C_{2} Q_{2} = (P_{2} - τ_{r} P_{r} - (1 - β) C_{2}) Q_{2}

(10)

U S_{1} = U_{s} + U_{1} = (P_{1} - τ_{s} C_{s} - C_{1}) Q_{1}

(11)

U S_{2} = U_{w} + U_{r} + U_{2} = (P_{2} - (1 - β) (τ_{w} τ_{r} C_{w} + τ_{w} τ_{r} C_{r} + C_{2})) Q_{2}

(12)

U_{g} = Q_{w} = τ_{w} τ_{r} Q_{2}

(13)

As mentioned earlier, the government is trying to motivate the members to collect and utilize more waste. Thus, the total amount of the collected waste has been assumed as the government’s utility. The structure of the problem has been depicted in Figure 1.

To ensure that the decisions are made feasibly and rationally, the following constraints are taken into consideration:

{\begin{matrix} V_{1} \geq δ_{1} P_{1} - δ_{2} P_{2} + γ_{1} q \\ \begin{matrix} V_{2} \geq δ_{1} P_{2} - δ_{2} P_{1} - γ_{2} q \\ \begin{matrix} P_{s} \geq C_{s} \\ \begin{matrix} P_{w} \geq (1 - β) C_{w} \\ P_{r} \geq τ_{w} P_{w} + τ_{w} (1 - β) C_{r} \\ P_{1} \geq τ_{s} P_{s} + C_{1} \end{matrix} \\ P_{2} \geq τ_{r} P_{r} + (1 - β) C_{2} \end{matrix} \\ \begin{matrix} P_{2} \leq P_{1} \\ P_{r} \leq P_{s} \end{matrix} \end{matrix} \end{matrix}

(14)

In relation (14), the first two constraints ensure that the demands for the final products and the demands for the required materials are non-negative. The next five constraints are incorporated into the problem to allocate non-negative profits to the members. The price of the product 2 must be less than the price of another one. Otherwise, customers will not have an incentive to buy the product 2 with lower quality and higher price. Furthermore, the last constraint guarantees that the price set for the recycled waste is lower than the price of the virgin materials. Otherwise, the manufacturer 2 has no motivation for utilizing the waste to produce his products and therefore he uses the virgin materials with higher quality and lower price, doubtlessly.

Obviously, when the manufacturers sell their products to customers without any profit, the demands faced by them will be non-negative. In fact, in this situation, it is expected that customers are stimulated to purchase both products. According to relation (14), we have: $P_{1} \geq τ_{s} P_{s} + C_{1} \geq τ_{s} C_{s} + C_{1}$ and $P_{2} \geq τ_{r} P_{r} + (1 - β) C_{2} \geq τ_{r} (τ_{w} P_{w} + τ_{w} (1 - β) C_{r}) + (1 - β) C_{2} \geq τ_{r} (τ_{w} (1 - β) C_{w} + τ_{w} (1 - β) C_{r})$ $+ (1 - β) C_{2} = (1 - β) (τ_{w} τ_{r} C_{w} + τ_{w} τ_{r} C_{r} + C_{2})$ . Regarding these explanations, the demands for both products would be non-negative by considering relations $P_{1} = τ_{s} C_{s} + C_{1}$ and $P_{2} = (1 - β) (τ_{w} τ_{r} C_{w} + τ_{w} τ_{r} C_{r} + C_{2})$ . Therefore, the following relation should be considered into the problem as an obvious and logical assumption:

{\begin{matrix} V_{1} \geq δ_{1} (τ_{s} C_{s} + C_{1}) - δ_{2} (1 - β) (τ_{w} τ_{r} C_{w} + τ_{w} τ_{r} C_{r} + C_{2}) + γ_{1} q \\ V_{2} \geq δ_{1} (1 - β) (τ_{w} τ_{r} C_{w} + τ_{w} τ_{r} C_{r} + C_{2}) - δ_{2} (τ_{s} C_{s} + C_{1}) - γ_{2} q \end{matrix}

(15)

In the next section, the decisions will be made applying the game-theoretic approach.

Game-theoretic approach

In this section, the members (players) make their decisions using the game-theoretic approach.

Nash game

Nash game refers to a common scenario in which the players have similar decision powers. In this setting, they make their decisions independently and simultaneously. Under the considered problem, this game is formulated as follows:

{\begin{matrix} \begin{matrix} \underset{P_{1}}{m a x} U_{1} = (P_{1} - τ_{s} P_{s} - C_{1}) Q_{1} \\ \begin{matrix} \underset{P_{2}}{m a x} U_{2} = (P_{2} - τ_{r} P_{r} - (1 - β) C_{2}) Q_{2} \\ \begin{matrix} \underset{P_{s}}{m a x} U_{s} = τ_{s} (P_{s} - C_{s}) Q_{1} \\ \underset{P_{w}}{m a x} U_{w} = τ_{w} τ_{r} (P_{w} - (1 - β) C_{w}) Q_{2} \end{matrix} \end{matrix} \\ \underset{P_{r}}{m a x} U_{r} = τ_{r} (P_{r} - τ_{w} P_{w} - τ_{w} (1 - β) C_{r}) Q_{2} \end{matrix} \\ B y c o n s i d e r i n g c o n s t r a i n t (14) \end{matrix}

(16)

Now, the members try to determine the decisions by maximizing their own profits, independently and simultaneously. We have:

{\begin{cases} \frac{d U_{1}}{d P_{1}} = V_{1} - δ_{1} (2 P_{1} - τ_{s} P_{s} - C_{1}) + δ_{2} P_{2} - γ_{1} q, \frac{d^{2} U_{1}}{d {P_{1}}^{2}} = - 2 δ_{1} < 0 \\ \frac{d U_{2}}{d P_{2}} = V_{2} - δ_{1} (2 P_{2} - τ_{r} P_{r} - (1 - β) C_{2}) + δ_{2} P_{1} + γ_{2} q, \frac{d^{2} U_{2}}{d {P_{2}}^{2}} = - 2 δ_{1} < 0 \\ \frac{d U_{s}}{d P_{s}} = τ_{s} (V_{1} - δ_{1} P_{1} + δ_{2} P_{2} - γ_{1} q) = Q_{s} \geq 0, \frac{d^{2} U_{s}}{d {P_{s}}^{2}} = 0 \\ \frac{d U_{w}}{d P_{w}} = τ_{w} τ_{r} (V_{2} - δ_{1} P_{2} + δ_{2} P_{1} + γ_{2} q) = Q_{w} \geq 0, \frac{d^{2} U_{w}}{d {P_{w}}^{2}} = 0 \\ \frac{d U_{r}}{d P_{r}} = τ_{r} (V_{2} - δ_{1} P_{2} + δ_{2} P_{1} + γ_{2} q) = Q_{r} \geq 0, \frac{d^{2} U_{r}}{d {P_{r}}^{2}} = 0 \end{cases}

(17)

Profit functions $U_{1}$ and $U_{2}$ are strictly concave to $P_{1}$ and $P_{2}$ respectively. Consequently, the responses obtained from solving the first-order derivatives of these functions can generate an equilibrium in Nash game, if they are feasible.

Functions $U_{s}$ , $U_{w}$ , and $U_{r}$ are increasing in line with respect to $P_{s}$ , $P_{w}$ , and $P_{r}$ , respectively. Therefore, the supplier, waste depot, and recycler prefer to set the maximum possible values for these decisions. Regarding constraint (14), the maximum possible values for prices $P_{s}$ , $P_{w}$ , and $P_{r}$ are equal to $(P_{1} - C_{1}) / τ_{s}$ , $(P_{r} - τ_{w} (1 - β) C_{r}) / τ_{w}$ and $(P_{2} - (1 - β) C_{2}) / τ_{r}$ , respectively. In fact, it is reasonable that the supplier, waste depot, and recycler set their prices equal to $P_{s} = (P_{1} - C_{1}) / τ_{s}$ , $P_{w} = (P_{r} - τ_{w} (1 - β) C_{r}) / τ_{w}$ , and $P_{r} = (P_{2} - (1 - β) C_{2}) / τ_{r}$ , respectively. However, in this situation, the profits allocated to both manufacturers would be equal to zero and therefore they have no motivation to participate into the business. For stimulating the manufacturers to take part into the business under Nash game, it is assumed that the profits assigned to them in each supply chain are not less than the profits allocated to the other members in that chain, that is, relations $U_{1} \geq U_{s}$ $U_{2} \geq U_{w}$ and $U_{2} \geq U_{r}$ are considered in this game. Note that the members have similar decision powers under Nash game. Thus, it can be expected that the manufacturers’ profits would be at least as much as the values allocated to the others. By simplifying these relations, the maximum possible values for $P_{s}$ , $P_{w}$ , and $P_{r}$ would be equal to $(P_{1} + τ_{s} C_{s} - C_{1}) / 2 τ_{s}$ $(P_{2} - (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} - 2 τ_{w} τ_{r} C_{w})) / 3 τ_{w} τ_{r}$ and $(2 P_{2} - (1 - β) (2 C_{2} - τ_{w} τ_{r} C_{r} - τ_{w} τ_{r} C_{w}) / 3 τ_{r}$ , respectively. Now, substituting these prices into the first-order derivatives $d U_{1} / d P_{1}$ and $d U_{2} / d P_{2}$ and solving them simultaneously, the best responses of $P_{1}$ and $P_{2}$ can be achieved. Finally, by replacing the obtained $P_{1}$ and $P_{2}$ into relations $P_{s} = (P_{1} + τ_{s} C_{s} - C_{1}) / 2 τ_{s}$ $P_{w} = (P_{2} - (1 - β) (C_{2} + C_{r} - 2 C_{w}) τ_{w} τ_{r}) / 3 τ_{w} τ_{r}$ , and $P_{r} = (2 P_{2} - (1 - β) (2 C_{2} - τ_{w} τ_{r} C_{r} - τ_{w} τ_{r} C_{w}) / 3 τ_{r}$ , the other prices would also be calculated.

Proposition 1. The equilibrium decisions adopted by the members under Nash game are:

{\begin{matrix} {P_{1}}^{N} = \frac{4 δ_{1} (V_{1} - γ_{1} q) + 3 δ_{2} (V_{2} + γ_{2} q) + 2 {δ_{1}}^{2} (C_{1} + τ_{s} C_{s}) + δ_{1} δ_{2} (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})}{6 δ_{1}^{2} - 3 δ_{2}^{2}} \\ \begin{matrix} {P_{2}}^{N} = \frac{3 δ_{1} (V_{2} + γ_{2} q) + 2 δ_{2} (V_{1} - γ_{1} q) + δ_{1} δ_{2} (C_{1} + τ_{s} C_{s}) + {δ_{1}}^{2} (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})}{4 δ_{1}^{2} - 2 δ_{2}^{2}} \\ \begin{matrix} {P_{s}}^{N} = \frac{{P_{1}}^{N} + τ_{s} C_{s} - C_{1}}{2 τ_{s}} \\ {P_{w}}^{N} = \frac{{P_{2}}^{N} - (1 - β) (C_{2} + C_{r} - 2 C_{w}) τ_{w} τ_{r}}{3 τ_{w} τ_{r}} \end{matrix} \\ {P_{r}}^{N} = \frac{2 {P_{2}}^{N} - (1 - β) (2 C_{2} - τ_{w} τ_{r} C_{r} - τ_{w} τ_{r} C_{w}}{3 τ_{r}} \end{matrix} \end{matrix}

(18)

According to relation (15), it is easily proved that the obtained prices meet relation (14). Therefore, solution (18) is feasible and generates an equilibrium in Nash game.

Note that the considered demands, the allocated profits, and the government’s utility are also obtained by replacing the calculated prices into relations (1)–(13).

Stackelberg game

Under Stackelberg game, the manufacturers’ decision powers are assumed to be higher than the others. In fact, the manufacturer 1 imposes his decisions on the supplier, whereas the manufacturer 2 imposes his decisions on the waste depot and recycler. In this setting, initially, the manufacturers as the leaders make their decisions independently and simultaneously. Then, the supplier, waste depot, and recycler as the followers determine their prices independently and simultaneously. Stackelberg game model is as follows:

{\begin{matrix} \begin{matrix} \begin{matrix} L e v e l 1 : L e a d e r s ’ d e c i s i o n s \\ \begin{matrix} \underset{P_{1}}{m a x} U_{1} = (P_{1} - τ_{s} P_{s} - C_{1}) Q_{1} \\ \underset{P_{2}}{m a x} U_{2} = (P_{2} - τ_{r} P_{r} - (1 - β) C_{2}) Q_{2} \end{matrix} \end{matrix} \\ L e v e l 2 : F o l l o w e r s^{"} d e c i s i o n s \\ \begin{matrix} \underset{P_{s}}{m a x} U_{s} = τ_{s} (P_{s} - C_{s}) Q_{1} \\ \underset{P_{w}}{m a x} U_{w} = τ_{w} τ_{r} (P_{w} - (1 - β) C_{w}) Q_{2} \\ \underset{P_{r}}{m a x} U_{r} = τ_{r} (P_{r} - τ_{w} P_{w} - τ_{w} (1 - β) C_{r}) Q_{2} \end{matrix} \end{matrix} \\ B y c o n s i d e r i n g c o n s t r a i n t (14) \end{matrix}

(19)

First, based on the manufacturers’ decisions, the other members specify their prices independently and simultaneously.

Regarding relation (17), profit functions $U_{s}$ , $U_{w}$ , and $U_{r}$ are increasing in line to $P_{s}$ , $P_{w}$ , and $P_{r}$ , respectively. As a result, it is preferable for the supplier, waste depot, and recycler to set their prices as much as possible. Similar to what was previously explained in Nash game, in this situation, the manufacturers’ profits are equal to zero and they are not motivated to participate into the trade. In Stackelberg game, the manufacturers’ decision powers are higher than the others. Let $θ_{1}$ as the decision power of the manufacturer 1 in comparison to the supplier and $θ_{2}$ as the decision power of the manufacturer 2 compared to the waste depot and recycler ( $θ_{1}, θ_{2} > 1$ ). Under Stackelberg game, it is assumed that the manufacturer in each supply chain contributes to the business, if the profit assigned to him would be greater than to the other members in that chain with respect to their decision powers. As a matter of fact, relations $U_{1} \geq θ_{1} U_{s}$ , $U_{2} \geq θ_{2} U_{w}$ , and $U_{2} \geq θ_{2} U_{r}$ are considered in this game to provide an encouragement for the manufacturers. According to these relations, the maximum feasible values for prices $P_{s}$ , $P_{w}$ , and $P_{r}$ would be equal to $(P_{1} + τ_{s} θ_{1} C_{s} - C_{1}) / (1 + θ_{1}) τ_{s}$ , $(P_{2} - (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} - τ_{w} τ_{r} (1 + θ_{2}) C_{w})) / (2 + θ_{2}) τ_{w} τ_{r}$ , and $(2 P_{2} - (1 - β) (2 C_{2} - τ_{w} τ_{r} θ_{2} C_{r} - τ_{w} τ_{r} θ_{2} C_{w}) / (2 + θ_{2}) τ_{r}$ , respectively. Functions $U_{1}^{'}$ and $U_{2}^{'}$ are obtained by substituting these prices into profit functions $U_{1}$ and $U_{2}$ . Now, the manufacturers determine their prices by maximizing these functions, independently and simultaneously. We have:

{\begin{matrix} \frac{d U_{1}^{"}}{d P_{1}} = \frac{θ_{1} (V_{1} - δ_{1} (2 P_{1} - τ_{s} C_{s} - C_{1}) + δ_{2} P_{2} - γ_{1} q)}{1 + θ_{1}}, \frac{d^{2} U_{1}^{"}}{d {P_{1}}^{2}} = - \frac{2 δ_{1} θ_{1}}{1 + θ_{1}} < 0 \\ \frac{d U_{2}^{"}}{d P_{2}} = \frac{θ_{2} (V_{2} - δ_{1} (2 P_{2} - (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})) + δ_{2} P_{1} + γ_{2} q)}{2 + θ_{2}}, \frac{d^{2} U_{2}^{"}}{d {P_{2}}^{2}} = - \frac{2 δ_{1} θ_{2}}{2 + θ_{2}} \end{matrix}

(20)

Clearly, functions $U_{1}^{'}$ and $U_{2}^{'}$ are strictly concave with regards to $P_{1}$ and $P_{2}$ , respectively. By simultaneously solving the first-order derivatives (20), prices $P_{1}$ and $P_{2}$ are calculated. Eventually, the solution under Stackelberg game can be given by replacing these values into the relations obtained for the other prices.

Proposition 2. The equilibrium decisions in Stackelberg game are determined as follows:

{\begin{matrix} {P_{1}}^{S} = \frac{2 δ_{1} (V_{1} - γ_{1} q) + δ_{2} (V_{2} + γ_{2} q) + 2 {δ_{1}}^{2} (C_{1} + τ_{s} C_{s}) + δ_{1} δ_{2} (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})}{4 δ_{1}^{2} - δ_{2}^{2}} \\ \begin{matrix} {P_{2}}^{S} = \frac{2 δ_{1} (V_{2} + γ_{2} q) + δ_{2} (V_{1} - γ_{1} q) + δ_{1} δ_{2} (C_{1} + τ_{s} C_{s}) + 2 {δ_{1}}^{2} (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})}{4 δ_{1}^{2} - δ_{2}^{2}} \\ \begin{matrix} {P_{s}}^{S} = \frac{{P_{1}}^{S} + τ_{s} θ_{1} C_{s} - C_{1}}{(1 + θ_{1}) τ_{s}} \\ {P_{w}}^{S} = \frac{{P_{2}}^{S} - (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} - τ_{w} τ_{r} (1 + θ_{2}) C_{w})}{(2 + θ_{2}) τ_{w} τ_{r}} \end{matrix} \\ {P_{r}}^{S} = \frac{2 {P_{2}}^{S} - (1 - β) (2 C_{2} - τ_{w} τ_{r} θ_{2} C_{r} - τ_{w} τ_{r} θ_{2} C_{w}}{(2 + θ_{2}) τ_{r}} \end{matrix} \end{matrix}

(21)

Regarding relation (15), the feasibility of solution (21) would be inferable.

Chain-to-chain game

In Chain-to-chain game, the members involved in each supply chain agree to act in union and make their decisions in an integrated manner. As a matter of fact, in this game, the members in each chain determine their decisions by maximizing the total profit of that chain. It can be stated that the decisions are obtained under a competition established between both supply chains. Chain-to-chain game is formulated as follows:

{\begin{matrix} \begin{matrix} \underset{P_{1}}{m a x} U S_{1} = (P_{1} - τ_{s} C_{s} - C_{1}) Q_{1} \\ \underset{P_{2}}{m a x} U S_{2} = (P_{2} - (1 - β) (τ_{w} τ_{r} C_{w} + τ_{w} τ_{r} C_{r} + C_{2})) Q_{2} \end{matrix} \\ B y c o n s i d e r i n g c o n s t r a i n t (14) \end{matrix}

(22)

We have:

{\begin{matrix} \frac{d U S_{1}}{d P_{1}} = V_{1} - δ_{1} (2 P_{1} - τ_{s} C_{s} - C_{1}) + δ_{2} P_{2} - γ_{1} q, \frac{d^{2} U S_{1}}{d {P_{1}}^{2}} = - 2 δ_{1} < 0 \\ \frac{d U S_{2}}{d P_{2}} = V_{2} - δ_{1} (2 P_{2} - (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})) + δ_{2} P_{1} + γ_{2} q, \frac{d^{2} U S_{2}}{d {P_{2}}^{2}} = - 2 δ_{1} < 0 \end{matrix}

Obviously, $U S_{1}$ and $U S_{2}$ are strictly concave to $P_{1}$ and $P_{2}$ , respectively. Thus, the solution calculated by solving their first-order derivatives generates the equilibrium in Chain-to-chain game, if it would be feasible. $P_{1}$ and $P_{2}$ are obtained by solving the first-order derivatives (23).

Proposition 3. The equilibrium decisions made by the manufacturers in Chain-to-chain game are as follows:

{\begin{matrix} {P_{1}}^{C} = \frac{2 δ_{1} (V_{1} - γ_{1} q) + δ_{2} (V_{2} + γ_{2} q) + 2 {δ_{1}}^{2} (C_{1} + τ_{s} C_{s}) + δ_{1} δ_{2} (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})}{4 δ_{1}^{2} - δ_{2}^{2}} \\ {P_{2}}^{C} = \frac{2 δ_{1} (V_{2} + γ_{2} q) + δ_{2} (V_{1} - γ_{1} q) + δ_{1} δ_{2} (C_{1} + τ_{s} C_{s}) + 2 {δ_{1}}^{2} (1 - β) (C_{2} + τ_{w} τ_{r} C_{r} + τ_{w} τ_{r} C_{w})}{4 δ_{1}^{2} - δ_{2}^{2}} \end{matrix}

(24)

It is easily proved that solution (24) is feasible and thus it can be considered as the equilibrium under Chain-to-chain game.

However, due to ignoring the intra-chain decisions in Chain-to-chain game, it is not possible to determine the profits assigned to the members. For this reason, it is assumed that the profit of each chain is shared among its members based on their decision powers. In fact, if $θ_{1}$ is the decision power of the manufacturer 1 compared to the supplier and $θ_{2}$ is the decision power of the manufacturer 2 in comparison to the waste depot and recycler, then the profits allocated to the members will be calculated as follows.

Proposition 4. The profits assigned to the members in Chain-to-chain game are equal to:

{\begin{matrix} {U_{s}}^{C} = \frac{1}{1 + θ_{1}} U {S_{1}}^{C} \\ \begin{matrix} {U_{w}}^{C} = \frac{1}{2 + θ_{2}} U {S_{2}}^{C} \\ {U_{r}}^{C} = \frac{1}{2 + θ_{2}} U {S_{2}}^{C} \\ {U_{1}}^{C} = \frac{θ_{1}}{1 + θ_{1}} U {S_{1}}^{C} \end{matrix} \\ {U_{2}}^{C} = \frac{θ_{2}}{2 + θ_{2}} U {S_{2}}^{C} \end{matrix}

(25)

where, $U {S_{1}}^{C}$ and $U {S_{2}}^{C}$ respectively are the profits determined for the supply chains of the products 1 and 2 in Chain-to-chain game.

Obviously, if the decision powers of the members would be similar, then relation $θ_{1} = θ_{2} = 1$ is considered to determine their profits. In this situation, the same profits are assigned to the members included in each chain.

Eventually, it is adequate to apply relations (6)–(8) in order to set the intra-chain prices $P_{s}$ , $P_{w}$ , and $P_{r}$ , respectively.

A real example

Consider the disposable plastic cups market where two competing manufacturers produce their products and sell them to customers. The manufacturer 1 uses the virgin materials to produce the product 1, whereas and the manufacturer 2 utilizes the plastic waste to produce the product 2. The manufacturer 1 provides the required virgin plastic from a supplier. Moreover, to manufacture the product 2, initially, a waste depot collects the required waste, and subsequently, a recycler performs the recycling operations on it. Finally, the manufacturer 2 uses the recycled waste to produce this product.

Suppose the supplying cost per ton of the virgin plastic is equal to $C_{s} = 300$ $, the collecting cost per ton of the plastic waste is $C_{w} = 70$ $, and the recycling cost per ton of the collected waste is equal to $C_{r} = 30$ $. Moreover, the producing costs per ton of the disposable plastic cups for the manufacturers 1 and 2 are respectively equal to $C_{1} = 250$ $ and $C_{2} = 200$ $. The amount of the virgin plastic required to produce one ton of the product 1, the amount of the collected waste required to prepare 1 ton of the recycled waste, and the amount of the recycled plastic waste required to produce 1 ton of the product 2 are $τ_{s} = 2$ , $τ_{w} = 2$ , and $τ_{r} = 3$ tons, respectively. The quality level of the product 2 with respect to the product 1 is equal to $q = 0.6$ . The maximum possible demands for the products 1 and 2 in the market have been respectively estimated to be $V_{1} = 30, 000$ and $V_{2} = 20, 000$ tons. The self-price and cross-price elasticities of the customers’ demands respectively are $δ_{1} = 20$ and $δ_{2} = 5$ . Also, the elasticities of the demands for the products 1 and 2 with respect to the quality level of the product 2 are equal to $γ_{1} = 4000$ and $γ_{2} = 10, 000$ , respectively. Note that the government undertakes a portion of the producing cost of the product 2 as well as the collecting and recycling costs of its required plastic waste in order to create an incentive for companies to use the plastic waste in producing. The government’s subsidy rate to support the members involved in the product 2’s supply chain is $β = 0.3$ . The results obtained from the investigated games for this instance are summarized in Table 1.

Table 1.

The results given by the developed games for the provided real example.

Games	Prices
Games	$P_{1}$	$P_{2}$	$P_{s}$	$P_{w}$	$P_{r}$
Nash	1433.98	1383.87	445.99	94.77	323.08
Stackelberg	1250.79	1086.35	366.80	66.54	210.18
Chain-to-chain	1250.79	1086.35	366.80	66.54	210.18
	Demands
	$Q_{1}$	$Q_{2}$	$Q_{s}$	$Q_{w}$	$Q_{r}$
Nash	5839.78	5492.47	11,679.57	32,954.84	16,477.42
Stackelberg	8015.87	10,526.98	16,031.75	63,161.90	31,580.95
Chain-to-chain	8015.87	10,526.98	16,031.75	63,161.90	31,580.95
	Profits
	$U_{1}$	$U_{2}$	$U_{s}$	$U_{w}$
Nash	1,705,154.41	1,508,363.05	1,705,154.41	1,508,363.05
Stackelberg	2,141,807.34	3,324,521.84	1,070,903.67	1,108,173.95
Chain-to-chain	2,141,807.34	3,324,521.84	1,070,903.67	1,108,173.95
	Profits			Government’s utility
	$U_{r}$	$U S_{1}$	$U S_{2}$	$U_{g}$
Nash	1,508,363.05	3,410,308.82	4,525,089.14	32,954.84
Stackelberg	1,108,173.95	3,212,711.01	5,540,869.74	63,161.90
Chain-to-chain	1,108,173.95	3,212,711.01	5,540,869.74	63,161.90

In Stackelberg game, the manufacturers’ decision powers exceed the others. In this setting, the decision power of the manufacturer 1 in comparison to the supplier and the decision power of the manufacturer 2 compared to the waste depot and recycler have been considered to be $θ_{1} = 2$ and $θ_{2} = 3$ , respectively. In fact, it is assumed that the decision power of the manufacturer 1 is doubled compared to the supplier, while the decision power of the manufacturer 2 is trebled in comparison to the waste depot and recycler.

Regarding the results provided in Table 1, one can derive that the members’ profits as well as the government’s utilities are equal under Stackelberg and Chain-to-chain games. The manufacturers’ profits in Stackelberg and Chain-to-chain games are higher than in Nash game. This is concluded in reverse for the profits assigned to the supplier, waste depot, and recycler. Also, the government’s utility under Stackelberg and Chain-to-chain models are higher than under Nash model.

Results and discussion

In this section, the results obtained from the developed models will be presented.

Comparisons of the members’ profits and the government’s utility

In this section, the decisions are analyzed under different governmental scenarios as well as the investigated games.

Comparisons under the governmental scenarios

First, the members’ profits and the government’s utility are evaluated under the governmental strategies. For this reason, the following scenarios are considered:

Scenario $S_{1}$ : The government does not support the members who are included in the product 2’s supply chain ( $β = 0$ )

Scenario $S_{2}$ : The government supports the members involved in the product 2’s supply chain ( $β > 0$ )

Result 1. Under different governmental scenarios, the following relations are met:

{\begin{matrix} {U_{1}}^{S_{1}} \geq {U_{1}}^{S_{2}} \\ \begin{matrix} {U_{2}}^{S_{1}} \leq {U_{2}}^{S_{2}} \\ {U_{s}}^{S_{1}} \geq {U_{s}}^{S_{2}} \\ {U_{w}}^{S_{1}} \leq {U_{w}}^{S_{2}} \end{matrix} \\ \begin{matrix} {U_{r}}^{S_{1}} \leq {U_{r}}^{S_{2}} \\ {U_{g}}^{S_{1}} \leq {U_{g}}^{S_{2}} \end{matrix} \end{matrix}

(26)

Managerial insight 1

The governmental supportive policies from the members involved in the product 2’s supply chain lead to a growth in their profits as well as a decline in the members’ profits in the product 1’s supply chain. Also, this motivational policy enhances the customers’ demand for the product 2 as well as the amount of the waste materials needed for its production, which will improve the government’s utility.

It can be stated that by implementing the governmental supportive policy, the producing cost of the product 2 in addition to the collecting and recycling costs of its required waste will be reduced and this results in a decline in its price as well as a rise in its demand. In this situation, the profits allocated to the members who are included in the product 2’s supply chain and the government’s utility will also increase. Moreover, it can be concluded that some customers for product 1 switch to buy product 2 and this leads to decline in demands and the profits gain by the members involved in the product 1’s supply chain.

Comparisons under the developed games

Here, the effects of the considered games are investigated on the decisions. For this purpose, the members’ profits and the government’s utility are compared under Nash, Stackelberg, and Chain-to-chain games.

Result 2. Under the investigated games, the following relations can be derived:

{\begin{matrix} {U_{1}}^{N} \leq {U_{1}}^{S} = {U_{1}}^{C} \\ \begin{matrix} \begin{matrix} {U_{2}}^{N} \leq {U_{2}}^{S} = {U_{2}}^{C} \\ {U_{s}}^{N} \geq {U_{s}}^{S} = {U_{s}}^{C} \end{matrix} \\ {U_{w}}^{N} \geq {U_{w}}^{S} = {U_{w}}^{C} \\ {U_{r}}^{N} \geq {U_{r}}^{S} = {U_{r}}^{C} \end{matrix} \\ {U_{g}}^{N} \leq {U_{g}}^{S} = {U_{g}}^{C} i f A_{1} \geq 0 \end{matrix}

(27)

where, we have:

\begin{array}{l} A_{1} = 4 (V_{1} - q γ_{1}) (δ_{1}^{2} δ_{2} - δ_{2}^{3}) + 3 (V_{2} - q γ_{2}) (4 δ_{1}^{3} - 3 δ_{1} δ_{2}^{2}) + (C_{1} + τ_{s} C_{s}) (8 δ_{1}^{3} δ_{2} - 5 δ_{1} δ_{2}^{3}) \\ - (1 - β) (12 δ_{1}^{4} - 13 δ_{1}^{2} δ_{2}^{2} + 4 δ_{2}^{4}) (C_{2} + τ_{r} τ_{w} C_{r} + τ_{r} τ_{w} C_{w}) \end{array}

Managerial insight 2

In a situation in which the decision powers of the manufacturers exceed the others, the cooperation or competition among the members do not affect their profits and the government’s utility.

Managerial insight 3

By considering a dominant decision power for the manufacturers, higher profits would be allocated to them compared to a situation in which all the members own similar decision powers. This inference can be derived in reverse for the supplier, waste depot, and recycler. Furthermore, by considering a threshold condition, when the members have different decision powers, the government’s utility is more than a situation in which they have similar decision powers.

Sensitivity analysis of the members’ profits and the government’s utility

In this section, a sensitivity analysis is implemented in order to determine the effects of the parametric changes on the decisions made by the members.

The effects of changing the quality level of the product 2 in comparison to the product 1 on the decisions are illustrated in Figure 2.

Managerial insight 4

Improving the quality level of the product 2 leads to a decrease in the profits of the members involved in the product 1’s supply chain, an increase in the profits allocated in the product 2’s supply chain, and an improvement in the government’s utility.

The sensitivity analysis of the government’s subsidy rate paid to the members included in the product 2’s supply chain is demonstrated in Figure 3.

Managerial insight 5

By supporting the members included in the product 2’s supply chain, their profits and the government’s utility improve, whereas the members’ profits in the product 1’s supply chain worsen.

In the following, the effects of changing the self-price and cross-price elasticities of the customers’ demands are investigated on the decisions. The analysis is depicted in Figure 4.

Managerial insight 6

Decreasing the self-price elasticity as well as increasing the cross-price elasticity of the customers’ demands improve the members’ profits and the government’s utility.

Ultimately, the effects of changing the elasticities of the demands with respect to the quality level of the product 2 are analyzed. These changes are shown in Figure 5.

Managerial insight 7

Considering the quality level of the product 2, less the elasticity of the demand for the product 1 as well as more the elasticity of the demand for the product 2 improve the members’ profits and the government’s utility.

Summary and conclusions

In this study, the effects of the waste recycling were investigated on the sustainable development and the energy storage concepts. For this reason, a supply chain including the government, two manufacturers, a supplier, a waste depot, and a recycler was considered where two substitutable products are manufactured and delivered to customers. The manufacturers 1 and 2 produce the products 1 and 2 using the virgin and waste materials, respectively. The supplier procures the virgin materials for the manufacturer 1. The waste depot collects the waste and the recycler recycles it for the manufacturer 2. Furthermore, the government applies some supportive policies for motivating the members included in the product 2’s supply chain to collect and utilize more waste.

Then, the game theory was applied to make the decisions under the considered problem. In this setting, the game-theoretic models including Nash, Stackelberg, and Chain-to-chain were developed under various situations established among the members. In Nash game, the members have the same decision powers and they determine their decisions independently and simultaneously. Under Stackelberg game, the manufacturers have dominant decision powers over the others. Moreover, in Chain-to-chain game, the members included to manufacture each product agree to make their decisions in an integrated manner.

Subsequently, the models were investigated and some managerial insights were revealed. The main results indicate that supporting the members involved in the product 2’s supply chain leads to a growth in their profits and a decline in the profits allocated to the others. This motivational policy would also improve the government’s utility. Moreover, when the manufacturers have higher decision powers than the others, competition or cooperation established among the members do not affect their profits and the government’s utility.

Eventually, a sensitivity analysis was done to determine the effects of changing the considered parameters on the members’ profits and the government’s utility.

Some possible suggestions for future studies are: In this research, the government’s subsidy rate was considered as a constant value. In future studies, it can be specified as the government’s decision variable. Other motivational policies like the tax reduction can be taken into account to provide an incentive for the members. In this study, a linear and deterministic function was considered to determine the customers’ demands. A nonlinear or stochastic demand function can be used in future researches.

Footnotes

Correction (Dec 2025):

Online article updated to correct affiliation of S Safarzadeh to Department of Industrial Engineering.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Hamed Jafari

Soroush Safarzadeh

References

Bulach

Schüler

Sellin

, et al. (2018) Electric vehicle recycling 2020: Key component power electronics. Waste Management & Research 36: 311–320.

Callesen

Pedersen

Carolus

, et al. (2022) Recycling nutrients and reducing carbon emissions in the baltic sea region—Sustainable or economically infeasible? Environmental Management 69: 213–225.

Cohen

Halfon

Schwartz

(2021) Trust between municipality and residents: A game-theory model for municipal solid-waste recycling efficiency. Waste Management 127: 30–36.

Dias

Cenci

Bernardes

, et al. (2022) What drives WEEE recycling? A comparative study concerning legislation, collection and recycling. Waste Management & Research 40: 1527–1538.

ElMassah

Mohieldin

(2020) Digital transformation and localizing the sustainable development goals (SDGs). Ecological Economics 169: 106490.

Niu

, et al. (2022) Exploration of public stereotypes of supply-and-demand characteristics of recycled water infrastructure-Evidence from an event-related potential experiment in Xi’an, China. Journal of Environmental Management 322: 116103.

Guo

Tseng

M-L

, et al. (2018) Two-echelon reverse supply chain in collecting waste electrical and electronic equipment: A game theory model. Computers & Industrial Engineering 126: 187–195.

Hassan

Lee

Mokhtar

(2019) Streamlining non-governmental organizations’ programs towards achieving the sustainable development goals: A conceptual framework. Sustainable Development 27: 401–408.

Yuan

(2020) Investigation of construction waste recycling decisions by considering consumers’ quality perceptions. Journal of Cleaner Production 259: 120928.

10.

Jafari

(2019) Sustainable development by reusing of recyclables in a textile industry including two collectors and three firms: A game-theoretic approach for pricing decisions. Journal of Cleaner Production 229: 598–610.

11.

Jafari

(2021) Nurse scheduling problem by considering total number of required nurses as well as nurses’ preferences for working shifts: An algorithmic game-theoretic approach. Scientia Iranica (Inpress).

12.

Jafari

(2022) Investigating environmental and economic aspects of sustainability by recycling PET plastic bottles: A game-theoretic approach. Clean Technologies and Environmental Policy 24: 829–842.

13.

Jafari

(2023a) Pricing and ordering decisions of recyclables under a sustainable supply chain management: A game-theoretic approach. Scientia Iranica (Inpress).

14.

Jafari

(2023b) A game-theoretic approach to set prices concerning a product manufactured from plastic waste. Journal of Decisions and Operations Research 8: 463–477.

15.

Jafari

(2023c) A game-theoretic approach to select a channel for supplying required materials in producing a product manufactured from recyclables. Journal of Decisions and Operations Research 8: 844–860.

16.

Jafari

(2023d) Energy storage by improving energy-efficiency of electricity home appliances under governmental supporting policies: A game-theoretic approach. Journal of Energy Storage 63: 106972.

17.

Jafari

(2024) Manufacturer’s decisions about establishing online selling channel under a two-echelon supply chain: A game-theoretic approach. Journal of Industrial and Management Optimization 20: 2062–2088.

18.

Jafari

Hejazi

Rasti-Barzoki

(2017) Sustainable development by waste recycling under a three-echelon supply chain: A game-theoretic approach. Journal of Cleaner Production 142: 2252–2261.

19.

Jafari

Hejazi

Rasti-Barzoki

(2020) Game theoretical approach to price a product under two-echelon supply chain containing e-tail selling channel. International Journal of Services and Operations Management 36: 131–160.

20.

Jafari

Safarzadeh

(2023) Producing two substitutable products under a supply chain including two manufacturers and one retailer: A game-theoretic approach. Journal of Industrial and Management Optimization 19: 3650–3670.

21.

Jin

Tang

Zhou

, et al. (2020) A government value compensation model of waste recycling in an industrial park: A game theory approach. Journal of Cleaner Production 275: 122976.

22.

Khan

SAR

Tabish

(2023) Mapping and visualizing of research output on waste management and green technology: A bibliometric review of literature. Waste Management & Research: 0734242X221149329

23.

Feng

Luo

(2019) Strategic introduction of an online recycling channel in the reverse supply chain with a random demand. Journal of Cleaner Production 236: 117683.

24.

Liu

Nie

Yuan

(2020) Interactive decisions of the waste producer and the recycler in construction waste recycling. Journal of Cleaner Production 256: 120403.

25.

Matsui

(2022) Optimal timing of acquisition price announcement for used products in a dual-recycling channel reverse supply chain. European Journal of Operational Research 300: 615–632.

26.

Mohammadzadeh

Zegordi

Kashan

, et al. (2022) Optimal government policy-making for the electric vehicle adoption using the total cost of ownership under the budget constraint. Sustainable Production and Consumption 33: 477–507.

27.

Naserabad

Rafee

Saedodin

, et al. (2021) Multi-objective optimization of a building integrated energy system and assessing the effectiveness of supportive energy policies in Iran. Sustainable Energy Technologies and Assessments 47: 101343.

28.

Nketiah

Song

Cai

, et al. (2022) Predicting citizens’ recycling intention: Incorporating natural bonding and place identity into the extended norm activation model. Journal of Cleaner Production 377: 134425.

29.

Parsaeifar

Bozorgi-Amiri

Naimi-Sadigh

, et al. (2019) A game theoretical for coordination of pricing, recycling, and green product decisions in the supply chain. Journal of Cleaner Production 226: 37–49.

30.

Tanveer

Khan

SAR

Umar

, et al. (2022) Waste management and green technology: Future trends in circular economy leading towards environmental sustainability. Environmental Science and Pollution Research 29(53): 80161–80178.

31.

Shi

Chen

(2020) Pricing and collection rate for remanufacturing industry considering capacity constraint in recycling channels. Complexity 2020: 1–13.

32.

Yang

Long

Chen

, et al. (2021) A comparative analysis of express packaging waste recycling models based on the differential game theory. Resources, Conservation and Recycling 168: 105449.

33.

Huang

Guo

, et al. (2016) Dual recycling channel decision in retailer oriented closed-loop supply chain for construction machinery remanufacturing. Journal of Cleaner Production 137: 1393–1405.

34.

Tianshan

Khan

SAR

(2021) Investigating the effect of government subsidies on end-of-life vehicle recycling. Waste Management & Research 39(6): 860–870.

35.

Zhang

Tang

Bunn

, et al. (2021) Comparative evaluation and policy analysis for recycling retired EV batteries with different collection modes. Applied Energy 303: 117614.