Sustainable E-waste supply chain management with price/sustainability-sensitive demand and government intervention

Abstract

The impact of appropriate E-waste management practices on the environment, human health, and natural resources has made E-waste management an interesting research topic in recent decades. Research also shows that government intervention is an important factor in controlling the emission volume produced by waste management systems. This study considers a base E-waste supply chain in which a collection center is responsible for collecting E-waste and delivering it to a recycling center. The recycling center recovers valuable material and sells it to electronic device manufacturing companies using a price/sustainability-sensitive demand. E-waste material recovery generates emissions that are undesirable for manufacturing companies. Two extended cases regarding the base supply chain are studied, as well: (1) the recycling center is also active in E-waste collection. (2) There are two active recycling centers. Although the sustainability-sensitive demand is a controlling factor for material recovery emissions, government interferences through tariff and emission penalties make sure that sustainable issues are considered in the material recovery process. Each plant in this study makes a marginal profit by processing E-waste; therefore, it is important to know which plant is the primary decision-maker when it comes to price. Because of its capability in terms of solving interactive decision-making problems, game theory is used to model different scenarios in our problem. Equilibrium values are derived, and a numerical example with parameter sensitivity analysis is provided to show the applicability of the proposed models. The results show that the E-waste supply chain makes more profit and selects a higher level of material recovery sustainability if the plants work under a centralized decision-making framework. Moreover, it is more profitable for the entire E-waste supply chain if the recycling center undertakes a portion of the E-waste collection activity.

Introduction

In recent decades, changing human lifestyles and technology developments have imported more electronic devices into our life; many needs that 1 day were satisfied manually or by mechanical devices are satisfied by electronic devices today. The short life span of electronic devices soon led to a plethora of older devices that were considered “out-of-date” and, thus, were determined to be waste. Waste Electrical and Electronic Equipment (WEEE), or E-waste, includes electrical and electronic equipment that is at the end of its product life cycle [1]. E-waste can be generated at the end of life (EOL) of electrical household gadgets, IT and telecommunication gadgets, illumination gadgets, medical devices, toys, monitoring and control instruments, automatic dispensers, etc. E-waste may contain both hazardous and non-hazardous materials, including plastic, glass, wood and plywood, concrete and ceramics, and other materials besides metal components [2].

Growing E-waste volume has been a global challenge, especially in developing countries. Between 2001 and 2013, just cell phones that had been imported into Nigeria resulted in 8920 t of copper, 120 t of lead, 720 t of nickel, 40 t of chromium, and 1310 t of bromine waste for this country [3]. Other statistics show that in 2014, Brazil produced 7 kg of E-waste per capita; this amount is even larger for other countries, with 8.7 kg for Russia and 20.1 kg for Australia recorded for the same year [4]. It was predicted that in 2017, about 72 million tons of E-waste was generated globally [2].

Besides simply the rapid growth of E-waste generation in recent years, a noticeable problem that also arose was the lack of appropriate recovery technology, as well as the risk of releasing toxic and hazardous material if E-waste was not discarded properly [5]. Whether it is because of the lack of available information, the prevalence of inappropriate infrastructures and inefficient regulations, or a combination, there are not many efficient E-waste management systems in most parts of the world. Regulations can affect the E-waste management practices in a country, but they may not do all the work themselves. Supervisory intensity and costs, penalties, waste disposal costs, and revenue from illegal dumping are the factors that affect contractors and government departments’ decisions in an E-waste management system [6].

Bhaskar and Turaga [7] studied the impacts of E-waste management and handling rules in Ahmedabad, a city in India. They showed that—even after running the above-mentioned rules—only between 5 and 15% of the total E-waste in Ahmedabad was recycled by formal processing facilities. Some other studies have shown that working on public education regarding sustainability can significantly improve participation in recycling and accentuate the importance of green production strategies [8,9,10].

A large portion of generated E-waste is still landfilled with household waste or recycled by the informal sector. A lack of appropriate E-waste recycling diminishes valuable and, in some cases, non-renewable resources. It also generates environmental and human health risks. For example, E-waste landfill releases contaminated leachates into groundwater. Melting computer chips with acid acidifies soil and contaminates the water [11], and E-waste incineration contributes to air pollution by releasing toxic fumes. On the other hand, the valuable metal concentration in E-waste is significantly more than in corresponding natural ores, and that makes this sector an interesting business for both formal and informal recycling channels. In recent years, many private-sector and research organizations have entered the E-waste management sector, because the presence of valuable metals is a good resource of income [12].

Recently, many researchers have studied various E-waste management practices in different frameworks and in different parts of the world. A case study in Turkey was developed by [13] to analyze the familiarity of Turkey’s population with technological products, as well as predict the potential of generating E-waste in the following years [14] developed an integrated E-waste management approach for improving the environmental and human health conditions during E-waste disposal activities. Research on the subject has been mostly restricted to reviewing the current practices and managerial insights to improve the E-waste management systems, but studying the E-waste management problem in a modeling format has rarely been looked at thus far. Studying the current E-waste management condition in China and providing improvement solutions [15], investigating the E-waste management system in France in terms of recycling and resource recovery practices [16], and a review of E-waste sources and recognizing the major challenges of E-waste generation, global trade, and management strategies [17] are among these rare studies.

An appropriate E-waste management system may include E-waste generation, collection centers, dismantling, separation, transportation, treatment, storage, distribution, recovery, and disposal processes. Therefore, we can model E-waste management as a strategic supply chain problem [18, 19]. In this field, [4] proposed a WEEE collection system with different collection locations and a pre-treatment phase to establish the complete recycling of all relevant components in Brazil [19] applied mixed-integer linear programming to optimize the logistics, production, and distribution processes in a waste supply chain model. In another study, a model was developed by [20] specifically for laptop disposal plants to reduce the E-waste generation in the supply chain of a laptop manufacturer.

Regarding a supply chain view of E-waste management, there is always the possibility of collaboration or competition between the stages, such as between the E-waste source, collection parties, logistic providers, disposal facilities, government, etc. Recently, game theory [21] was applied in some waste management studies to optimize the decisions of the waste management supply chain’s players [22]. A game theory approach was applied by [23] to model the E-waste-generating behaviors of the electronic device consumers. This study shows that the convenience of services, a consumer’s desire to overvalue life-end products, the usage time of products, a consumer’s overview of product obsolescence, and the re-marketability of refurbished product all affect the decisions of E-waste supply chain players [24] modeled a two-echelon reverse supply chain and assumed that recycling publicity was a factor that stimulated the recovery demand. They applied game theory to find the optimal values for the recovery cost and volume in an E-waste recycling system. In another study, Chen et al. [6] used game theory to find the optimal penalty and supervision level by the government in a construction waste disposal system.

Government intervention can be an important player regarding the defining of game models in an E-waste supply chain. Government intervention in making green products has been studied in the literature [25, 26], but government presence in E-waste supply chain models has not been investigated in any study so far. Therefore, here we begin to study an E-waste supply chain problem that includes three stages: government, E-waste recycler, and E-waste collector. Our work is inspired by [24, 27] in terms of using game theory to find the equilibrium values in a waste management supply chain.

We start off by assuming that this supply chain is working to recover valuable materials out of collected E-waste and sell this recovered material to electronic device manufacturing plants. There should be a government presence when it comes to collecting emission penalties, according to international conventions and widely accepted social norms. On the other hand, manufacturers are sustainable, sensitive, and prefer advanced processes that recover high-quality material using lower emissions. E-waste management and recycling systems are responsible for incurring the investment in a sustainable material recovery process. The collection center and recycling center can be in different in decision-making power sequence in regards to the size and reputation of the firms, but the government would always be the most powerful decision-making entity in the supply chain.

This study defines three cases based on the number and role of the supply chain members. Each case is studied under different scenarios, which are defined based on the decision-making sequence of the supply chain members. A sensitive demand for selling E-waste recovered material is applied, and game models are developed to maximize the profit for each scenario. Sensitive demand refers to the demand that depends on a supply chain’s decisions—price and sustainability level in particular, when it comes to this study. More explicitly, the manufacturers’ demand for recycled material increases at lower prices and higher sustainability levels. The novelty of this work is fivefold:

• The financial relationship between the firms in an E-waste supply chain is studied according to their decision-making dominance

• The impact of government intervention on the financial and sustainable performance of an E-waste supply chain is studied

• The financial impact of a recycling center that also fulfills a portion of the E-waste collection activities is studied

• The impact of centralization between the collection center and recycling center is studied

• The performance of an E-waste supply chain with two recycling centers is compared to that of an E-waste supply chain with one recycling center

In the remainder of this paper, the section “Problem description” defines the notations and problem descriptions for all cases and scenarios. Game models are developed and equilibrium values are obtained in the section “Model formulation”. The section “Numerical example” shows the applicability of the proposed model with a numerical example and some sensitivity analysis. A discussion and the managerial implications of this study are provided in the section “Discussion and managerial implications”. Finally, the section “Conclusions and future research” concludes the study and mentions some possible directions for future research.

Problem description

This study investigates the practice of E-waste management with government intervention. We consider three cases of a supply chain each with three stages of participants. In the first case, a collection center collects the E-waste to transport it to the only recycling center in the system. The recycling center is responsible for recovering the valuable material out of the E-waste and selling it to the electronic device manufacturers. The recycling center has the technology and license to recover the valuable material out of the E-waste. The material recovery process generates emissions that are harmful both to human health and to the environment. Consumer companies prefer recovered material that is high-quality and generates low emissions during the recycling process. The E-waste management system is responsible for providing the required technology that generates lower emissions and higher quality recovered material.

Material recovery emission

We assume that the recovery process for each pound of E-waste generates $a - b\theta > 0$ units of emission. $\theta$ is the level of sustainable improvement in E-waste material recovery processes compared to a benchmark condition. It is notable that any recovery process has a nonnegative amount of emission released; therefore, $0 < \theta < \frac{a}{b}$. Since E-waste recovery has negative environmental impacts, and the higher quality of recovered material saves more resources, the government uses its authority to improve the recovery process. The government collects emission penalties and recovery taxes from the recycling center. This tax is used by the government to support the sustainable practices of waste management and/or other sustainability concerns. Therefore, the money asked for from the manufacturers includes the material price and service tax. The government itself might incur some emission cost, according to the social responsibility norms and international conventions.

It is notable that in the literature, governments are logically assumed to be interested in reducing the negative environmental impacts and improving the social welfare [28, 29]. However, the required investments for improving the social welfare are secured through the collected tax. Since the government is a non-profit organization, all collected money is returned to the community at the end. As with some other articles in the literature [25, 26, 30], we develop a profit maximization function for the government to secure fund availability for green investments and control material recovery emissions.

Waste collection and recycling frameworks

In Case 1, the collection center decides on its own asking price from the recycling center for each pound of E-waste. The recycling center then decides on the recovered material price and the level of sustainability improvement. The recycling center is responsible for paying for the sustainability improvement. Finally, the government decides on the tax per pound of recycled E-waste. Figure 1 shows Case 1 schematically.

Fig. 1

Process flowchart for Case 1

Three scenarios are studied in Case 1 regarding the decision-making sequence between the players: the three-level leader–follower game, the collection center–recycling center centralized model, and the collection center–recycling center Nash model [31]. All three scenarios are explained in further detail in “Model formulation”.

In Case 2, the recycling center starts to directly interact with the waste source. Therefore, it collects half of its required E-waste for the recovery process directly from the source, and the other half from the collection center. The demand function parameters do not change from using a dual-channel collection method in this model. Other assumptions are like those of Case 1. Figure 2 shows the supply chain configuration in Case 2.

Fig. 2

Process flowchart for Case 2

Two scenarios are studied for Case 2: a three-level leader–follower game as well as a collection center–recycling center Nash model. Both scenarios are better explained in the section “Model formulation”. It is worth noting that the outcomes of the centralized alliance between the collection center and the recycling center are exactly like Case 1; therefore, the centralized alliance scenario is not studied for Cases 2 and 3.

Case 3 studies the appearance of a new recycling center that serves the same customer market. We assume that, in Case 3, the government incurs the investment and decides on the level of sustainability improvement. Other assumptions are like those in Case 1. Figure 3 shows the supply chain configuration in Case 3.

Fig. 3

Process flowchart for Case 3

We now will study two scenarios for case three: a three-level leader–follower game where both recycling centers make their decisions simultaneously (Nash equilibrium), and a collection center–recycling centers Nash model.

Methodology

An E-waste supply chain includes units that are working for profit. According to our logical assumption, demand is a function of price, which means that the supply chain’s demand lessens if the price increases. Price-sensitive demand shows that each supply chain member cannot increase its own marginal profit in perpetuity, and each member should ask for a marginal profit that guarantees a gainful level of demand. In such a situation, it would be crucial for supply chain members to have the authority in the decision-making process. Game theory is the methodology that has been used to model interactive decision-making problems, such as the problem in this study [32].

In regular decision-making problems, a plant decides to optimize its own objective function without considering the reactions of the partners or competitors. However, in game decision-making problems, each plant knows that its decision elicits a reaction from the other plant, which in turn affects its own objective function. In the current study, the final price of the recovered material is calculated based on the price that the collection center asks of the recycling center, as well as the recycling center’s marginal profit, emission penalty, and tax. Therefore, the supply chain member who is dominant in the decision-making sequence (according to reputation, size, market share, financial strength, etc.) has more power to make decisions that maximizes their own profit. More explicitly, it makes difference in regards to profit if a member is the first decision-maker, the second, or the third.

Game theory is the methodology that is applied to many pricing studies with sensitive demand [28, 31, 33]. In a game between two members, it is possible that a member is the leader or follower of the decision-making sequence (Stackelberg game); additionally, it is possible that both members have the same decision-making power and thus make decisions simultaneously (Nash equilibrium). In a Stackelberg game, the collection center, recycling center, and government interact with each other as followers and leaders. Each member makes a decision to optimize its own objective function. A Stackelberg game occurs when a member is in such an advantageous position where it can make decisions first. This member is the Stackelberg game leader, and a member who makes decisions after the leader is called a follower [34]. A Nash equilibrium provides a solution for noncooperative games, including those with two or more members. If all members have each chosen their decision in regards to the equilibrium decisions of the other members, and no member can change their decision, then these decisions and their payoffs would establish a Nash equilibrium [32].

Notation

We use the following notation in our modeling:

Index
${\text{RC}}_{z}$	Recycling center z$, z = 1,2$
${\text{CC}}$	Collection center
${\text{Gov}}$	Government
${\text{SC}}$	Total supply chain
Parameters
$D$	Potential demand for recovered electronic material (pound) $D > 0$
$d$	Demand function for recovered material (pound) $d > 0$
$\alpha$	Demand function coefficient for sensitivity to price $\alpha > 0$
$\beta$	Demand function coefficient for sensitivity to quality and sustainability $\beta > 0$
$c$	E-waste collection and management cost for the collection center (monetary unit/pound)$c > 0$
$i$	Investment parameter to sustainably improve recovery technology at the recycling center $i > 0$
$a$	Fixed recovery emission per pound of E-waste, $a > 0$
$b$	Sustainability improvement effect on reducing the emission, $b > 0$
$e$	Unit emission trading price $e > 0$
$\lambda$	Unit emission social responsibility cost for the government $\lambda > 0$
Decision variable
$w$	E-waste selling price to the recycling center (monetary unit/pound)
$m$	Recycling center marginal profit (monetary unit/per pound)
$t$	Tariff imposed by the government (monetary unit/per pound)
$\theta$	Level of quality and sustainability improvement (compared to a benchmark)
Dependent variable
$P$	Material price asked for from the manufacturer (per pound) ($P = w + m$)
$\Pi_{j}$	Profit function of player $j, j = {\text{CC, RC}}_{ 1} , {\text{ RC}}_{ 2} , {\text{Gov, SC}}$

Decision-making power

The participant that is dominant makes their decisions first. However, in game theory, we must first find the optimal equilibrium value of the follower. This is because the decision-making leader would make better decisions if they have knowledge about the follower’s decision. This trade-off is like playing chess; the white player plays before the black player, but, on the other hand, the white player can make better moves if he knows the black player’s moves beforehand. To avoid any misunderstanding, we show the decision-making and problem-solving sequences schematically in Fig. 4.

Fig. 4

Decision sequences for studied scenarios

Model formulation

Here, we consider three players in the E-waste management system—the collection company, recycling facility, and government. This section studies different cases based on the process flow between the supply chain members. It is assumed that all parameters are deterministic and known by the supply chain members.

Case 1: one collection center, one recycling center, government intervention

In the simplest form of an E-waste management system, a collection center delivers E-waste to the recycling center. The collection center determines the price of E-waste per pound $w$. The recycling center processes E-waste to both recover the valuable material and dispose of the scrap. The recycling center determines the level of sustainability regarding the recovery process $\theta$, as well as the marginal profit of selling the recovered material $m$ (indirectly $P$). The government adjusts the material recovery sustainability via laying down taxes and emission penalties.

Recovered material demand is a function of price according to the demand functions discussed in the literature [33, 35]. The sustainability level $\theta$ also increases the manufacturers’ desire to buy recovered material. Therefore, manufacturer demand for recovered material increases as the levels of $\theta$ get higher. We define the demand in the form of function (1):

$$ d = D - \alpha (P + t) + \beta \theta . $$

(1)

The following scenarios find the equilibrium values based on which decision-making sequence is being used.

Scenario 1: three-level leader–follower game

In this scenario, the government is the most powerful decision-making entity. The recycling center follows the government, and the collection center in turn follows the recycling center in decision-making. According to the decentralized leader–follower structure, we should first solve the collection center profit function for $w$ [36]. The collection center’s profit function is as follows:

$$ \Pi_{\text{CC}} = ( {w - c} ) ( {D - \alpha ( {m + w + t} ) + \beta \theta } ) . $$

(2)

Lemma 1

According to the validity of$\frac{{\partial^{2} \Pi_{\text{CC}} }}{{\partial w^{2} }} = - 2\alpha < 0$, the collection center profit function (2) is concave (has a maximum value) with respect to the E-waste selling price to the recycle center ($w$).

Proposition 1

Given the recycling center marginal profit ($m$), the tariff imposed by the government ($t$), and the level of sustainability improvement ($\theta$), the maximum value of the collection center’s profit function in Eq. (2) is achieved when the E-waste selling price to the recycling center is$w = \frac{{D + c\alpha - ( {m + t} )\alpha + \beta \theta }}{2\alpha }$.

Proof

Presented in Appendix A.

The value for $w$ in proposition 1 is a function of $m$ and $\theta$. We use a backward induction technique [30] to find the value of $w$ based on the known parameters of the problem. The recycling center profit function should be solved to find the values for $m$ and $\theta$. The recycling center’s profit function is as follows:

$$ \Pi_{\text{RC}} = ( {m - e ( {a - b\theta } )} ) ( {D - \alpha ( {m + w + t} ) + \beta \theta } ) - i\theta^{2} . $$

(3)

Lemma 2

For$8i\alpha > ( {be\alpha + \beta } )^{2}$and$w = \frac{{D + c\alpha - ( {m + t} )\alpha + \beta \theta }}{2\alpha }$, profit function (3) is joint concave in regards to both the recycling center marginal profit ($m$) and the level of sustainability improvement ($\theta$).

Proof

Presented in Appendix B.

Proposition 2

There exists a unique value for the recycling center marginal profit ($m$), and the level of sustainability improvement ($\theta$) that maximizes the recycling center’s profit (${{\Pi }}_{\text{RC}}$).

Proof

Presented in Appendix A.

The government’s profit function is solved by finding the optimal value of tax parameter $t$. It is worth noting that the unit emission trading price ($e$) in function (4) is the constant fine that the government receives from the recycling center if a unit of emission is released because of the material recovery process. The government’s profit function is as follows ($w, m,$ and $\theta$ values should substitute):

$$ \Pi_{\text{Gov}} = ( { ( {e - \lambda } ) ( {a - b\theta } ) + t} ) ( {D - \alpha ( {m + w + t} ) + \beta \theta } ). $$

(4)

Lemma 3

If Lemma 2 holds true, and the unit emission trading price is greater than unit emission social responsibility cost for the government ($e > \lambda$), government’s profit function (${{\Pi }}_{\text{Gov}}$) is concave (has a maximum value) with respect to its imposed tariff ($t$).

Proof

Presented in Appendix B.

Proposition 3

Assuming that the concavity conditions are satisfied, the equilibrium values (optimal decision values) of Case 1 for the three-level leader–follower game are as noted in Table 1.

Table 1 Equilibrium values for Case 1

Proof

Presented in Appendix A.

Scenario 2: government Stackelberg with collection center–recycling center centralized model

It is a logical possibility that the collection center and the recycling center form an alliance and cooperate in a centralized model. In this case, the government is still the leader in the decision-making sequence, and the collection center–recycling center alliance follows the government. However, in this case, the collection center would not ask for a separate profit, and all profit for the alliance comes from the recycling center marginal profit. We solve profit function (5) to find the values for $m$ and $\theta$.

$$ \Pi_{\text{CC - RC}} = ( {m - e ( {a - b\theta } )} ) ( {D - \alpha ( {m + {\text{c}} + t} ) + \beta \theta } ) - i\theta^{2} . $$

(5)

Lemma 4

For$4i\alpha > ( {be\alpha + \beta } )^{2}$, profit function (5) is joint concave in both the recycling center marginal profit ($m$) and the level of sustainability improvement ($\theta$).

Proof

Presented in Appendix B.

Proposition 4

There exists a unique value for both the recycling center marginal profit ($m$) and the level of sustainability improvement ($\theta$) that maximizes the profit function of collection center–recycling center alliance (${{\Pi }}_{\text{CC - RC}}$).

Proof

Presented in Appendix A.

The government profit function is:

$$ \Pi_{\text{Gov}} = ( { ( {e - \lambda } ) ( {a - b\theta } ) + t} ) ( {D - \alpha ( {m + c + t} ) + \beta \theta } ) . $$

(6)

Lemma 5

If Lemma 4 holds true and the unit emission trading price is greater than the unit emission social responsibility cost for the government ($e > \lambda$), the government’s profit function is concave with respect to its imposed tariff ($t$).

Proof

Same as Lemma 3.

Proposition 5

When satisfying the concavity conditions, the equilibrium values (optimal decision values) of Case 1 for the collection center–recycling center centralized model are as noted in Table 1.

Proof

Presented in Appendix A.

Scenario 3: government Stackelberg with collection center–recycling center Nash model

Here again, government is the most powerful member in the decision-making sequence. However, here, we must consider the Nash structure between the collection center and the recycling center. In the decision-making sequence, it is possible that the collection center and the recycling center make decisions simultaneously, but not in a centralized model. More explicitly, each player tries to maximize its own profit regardless of the other player’s profit, but no player is simply the leader or follower. The concavity conditions should follow the assumptions in “Scenario 1: Three-level leader–follower game”.

Proposition 6

If the concavity conditions in Lemmas 1 and 2 hold true, there exist unique values for the E-waste selling price for the recycling center ($w$), the recycling center marginal profit ($m$), and the level of sustainability improvement ($\theta$) that simultaneously maximizes collection center and recycling center profit functions [Eqs. (2), (3)].

Proof

Presented in Appendix A.

Proposition 7

Satisfying Lemma 3 and substituting the obtained values of$w$, $m$, and$\theta$from Proposition 6 into the profit functions (2) and (3), the equilibrium values of Case 1 for the collection center–recycling center Nash model are as noted in Table 1.

Proof

Presented in Appendix A.

Corollary 1

For$4i\alpha > ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )$, the relation${\text{t}}^{{{\text{Scen}}1}} < {\text{t}}^{{{\text{Scen}}3}} < {\text{t}}^{{{\text{Scen}}2}}$, $\theta^{{{\text{Scen}}1}} < \theta^{{{\text{Scen}}3}} < \theta^{\text{Scen2}}$, $\Pi_{\text{Gov}}^{\text{Scen1}} < \Pi_{\text{Gov}}^{\text{Scen3}} < \Pi_{\text{Gov}}^{\text{Scen2}}$, $\Pi_{\text{SC}}^{\text{Scen1}} < \Pi_{\text{SC}}^{\text{Scen3}} < \Pi_{\text{SC}}^{\text{Scen2}}$holds true.

Proof

Through algebraic reasoning.

In the second and third scenarios, decision-making power is divided between the collection center and the recycling center; therefore, the authority of the recycling center when it comes to selecting the value for $\theta$ is reduced. In Case 1, it is assumed that the recycling center is the only player that incurs the cost for sustainability investment. Therefore, in the first scenario, when the recycling center takes priority over the collection center, it prefers a lower level of quality and sustainability improvement, which consequently reduces the demand and total supply chain profit. On the other hand, the best profit and sustainability achievements occur when the collection center and the recycling center work in a centralized model. This removes double marginalization, and the recycling center can invest more in sustainability improvement. This stimulates demand and improves the supply chain profit.

The government’s profit is affected by taxes, and these taxes interact with price, with both of them negatively impacting the price-sensitive demand. The price in the first scenario is at its highest level because of double marginalization between the collection center and the recycling center; therefore, the government should reduce the tax value to keep the demand at a reasonable level. It reduces the government’s income, as well. Corollary 1 shows that if the power of recycling center is lessened, the government can obtain more profit.

Corollary 2

For$4i\alpha > ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )$, the relation$\Pi_{\text{CC}}^{\text{Scen1}} + \Pi_{\text{RC}}^{\text{Scen1}} < \Pi_{\text{CC}}^{\text{Scen3}} + \Pi_{\text{RC}}^{\text{Scen3}} < \Pi_{\text{CC - RC}}^{\text{Scen2}}$holds true.

Proof

Through algebraic reasoning.

The alliance of the collection center and the recycling center makes more profit in a centralized model when compared to the combined profits of these two same players when they are not working in a centralized model. Corollary 2 shows that joining the alliance is more profitable if the players are currently working under the three-level leader–follower model, compared to when the collection center and the recycling center are working under the Nash structure.

Case 2: one collection center, one recycling and collection center, government intervention

In this second case, we study the situation in which the recycling center collects E-waste as well. This might be due to financial interests, or to reduce the negative consequences of disrupting the main collection center’s activity. We assume that the E-waste collection demand splits 50%–50% between the collection center and the recycling center. Managing the E-waste collection activity for the recycling center has the same cost as for collection center ($c$ per pound).

Concavity of the profit functions in Case 2 is provable using the same proofs as Case 1.

Scenario 1: three-level leader–follower game

In this scenario, government is the leader in the decision-making sequence. The recycling center follows the government, and the collection center follows the recycling center. Therefore, the first step in a backward induction solution is to solve the collection center profit function to obtain $w$.

$$ \Pi_{\text{CC}} = ( {w - c} )\frac{1}{2} ( {D - \alpha ( {m + w + t} ) + \beta \theta } ). $$

(7)

Proposition 8

Given the recycling center marginal profit ($m$), the tariff imposed by the government ($t$), and the level of sustainability improvement ($\theta$), the maximum value of the collection center’s profit function in Eq. (7) is achieved when the E-waste selling price to the recycling center is$w = \frac{{D + c\alpha - ( {m + t} )\alpha + \beta \theta }}{2\alpha }$.

The profit function of the recycling center in Case 2 is:

$$ \Pi_{\text{RC}} = ( {m - e ( {a - b\theta } )} )\frac{1}{2} ( {D - \alpha ( {m + w + t} ) + \beta \theta } ) + ( {m + w - c - e ( {a - b\theta } )} )\frac{1}{2} ( {D - \alpha ( {m + w + t} ) + \beta \theta } ) - i\theta^{2} . $$

(8)

Proposition 9

There exists a unique value for the recycling center marginal profit ($m$) and the level of sustainability improvement ($\theta$) that maximizes the recycling center’s profit function in Eq. (8).

Proof

Presented in Appendix A.

The government profit function is the same as function (4). Therefore, by substituting the values of $w, m$, and $\theta$ in function (4), it is possible to find optimal $t$.

Proposition 10

The equilibrium values (optimal decision values) of Case 2 for the three-level leader–follower model are as noted in Table 2.

Table 2 Equilibrium values for Case 2

Scenario 2: government Stackelberg with collection center–recycling center Nash model

It is logically possible that the collection center is on the same decision-making level as the recycling center. The recycling center still works in the collection market, but the collection center does not follow the recycling center in the decision-making sequence. Just as in “Scenario 3: government Stackelberg with collection center–recycling center Nash model”, we model a Nash game between the collection center and the recycling center. Government is still the decision-making leader. The profit functions of the Nash players follow Eqs. (7) and (8).

Proposition 11

There exist unique values for the E-waste selling price to the recycling center ($w$), the recycling center marginal profit ($m$), and the level of sustainability improvement ($\theta$) that simultaneously maximizes the collection center and recycling center profit functions [Eqs. (7), (8)].

Proof

Presented in Appendix A.

Proposition 12

The equilibrium values (optimal decision values) of Case 2 for the collection center–recycling center Nash model are as noted in Table 2.

Proof

Presented in Appendix A.

Corollary 3

For both the three-level leader–follower game and collection center–recycling center Nash scenarios, if$5i\alpha > ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )$holds true, Case 1 and Case 2 profit equilibrium values follow${{\Pi }}_{\text{Gov}}^{\text{Case 1}} < {{\Pi }}_{\text{Gov}}^{\text{Case 2}}$, and${{\Pi }}_{\text{SC}}^{\text{Case 1}} < {{\Pi }}_{\text{SC}}^{\text{Case 2}}$.

Proof

Through algebraic reasoning.

The relationship between the total profit for the collection center and the recycling center depends on the required sustainability investment. Generally, however, double marginalization is reduced when the recycling center performs both the collection and recovery processes. This stimulates demand and opens the room for the government to ask for more tax revenue. Therefore, both the government and the total supply chain achieve a larger profit if the recycling center participates in E-waste collection.

Case 3: one collection center, two recycling centers, government intervention

This case is expanding upon case one, in which the recycling center is not active in E-waste collection. In this scenario, a new recycling center obtains the license and requirements to enter the E-waste material recovery business. Customer companies that recover material keep the same sensitivity level to price and sustainability improvements, but the collection center can ask a different price from the new recycling center based on its required E-waste characteristics. Therefore, the recovered material price for the new recycling center can be changed, as well. In this case, we assume that the government incurs the required investment for improving the material recovery according to previous sustainability commitments, and that improving the sustainability level is the government’s decision. We also assume that the new recycling center did a marketing campaign and is positioned to serve half of the potential market.

Scenario 1: three-level leader–follower game (recycling centers play Nash)

In this scenario, the collection center has a different profit function because of the different E-waste selling prices at the different recycling centers. Considering subscript 1 for recycling center 1 and subscript 2 for recycling center 2, Eq. (9) defines the collection center’s profit function:

$$ \begin{aligned} \Pi_{\text{CC}} &= ( {w_{1} - c_{1} } )\frac{1}{2} ( {D - \alpha ( {m_{1} + w_{1} + t} ) + \beta \theta } ) \\&\quad+ ( {w_{2} - c_{2} } )\frac{1}{2} ( {D - \alpha ( {m_{2} + w_{2} + t} ) + \beta \theta } ). \end{aligned} $$

(9)

The collection center’s profit function is concave, as proven in Case 1.

Proposition 13

Given each recycling center marginal profit ($m_{z}$), the tariff imposed by the government ($t$), and the level of sustainability improvement ($\theta$), the maximum value of the collection center’s profit function is achieved when the E-waste selling price to each recycling center is:

$$ w_{z} = \frac{{D - t\alpha + \beta \theta + \alpha c_{z} - \alpha m_{z} }}{2\alpha } , \quad z = 1, 2. $$

Proof

Same as proposition 1.

Recycling centers have the following profit functions:

$$ \Pi_{{{\text{RC}}_{z} }} = ( {m_{z} - e ( {a - b\theta } )} )\frac{1}{2} ( {D - \alpha ( {m_{z} + w_{z} + t} ) + \beta \theta } ) ,\quad z = 1, 2. $$

(10)

Lemma 6

According to the validity of$\frac{{\partial^{2} \Pi_{{{\text{RC}}_{z} }} }}{{\partial m_{z}^{2} }} = - \frac{\alpha }{2} < 0$, each recycling center’s profit function (10) is concave with respect to its own marginal profit ($m_{z}$).

Proposition 14

Given the tariff imposed by the government ($t$) and the level of sustainability improvement ($\theta$), the maximum value of each recycling center’s profit function is achieved when its marginal profit ($m_{z}$) is$m_{z} = \frac{{D + ae\alpha - t\alpha - be\alpha \theta + \beta \theta - \alpha c_{z} }}{2\alpha } , \quad z = 1, 2$.

Proof

Same as Proposition 1.

In Case 3, the government profit function should consider the sustainability investment, as well. Therefore, the government profit function is:

$$ \Pi_{\text{Gov}} = \mathop \sum \limits_{z = 1}^{2} ( { ( {e - \lambda } ) ( {a - b\theta } ) + t} )\frac{1}{2} ( {D - \alpha ( {m_{z} + w_{z} + t} ) + \beta \theta } ) - i\theta^{2} . $$

(11)

Lemma 7

For$16i\alpha > ( {\beta + b\alpha \lambda } )^{2}$, the government’s profit function in Eq. (11) is joint concave in its imposed tariff ($t$) and the level of sustainability improvement ($\theta$).

Proof

Presented in Appendix B.

Proposition 15

Equilibrium values (optimal decision values) for the three-level leader–follower model in Case 3 are as noted in Table 3.

Table 3 Equilibrium values for Case 3

Scenario 2: government Stackelberg with collection center–recycling centers Nash model

Starting the material recovery business by building a new recycling center can modify the power structure in the supply chain. This modification might make the collection center have the same decision-making power as the recycling centers. This occurred in the section “Scenario 3: government Stackelberg with collection center–recycling center Nash model”, where we modeled a Nash game between the collection center and the recycling centers. The government is still the decision-making leader. The profit functions of the Nash players follow Eqs. (9) and (10).

Proposition 16

There exist unique values for the E-waste selling price for each recycling center ($w_{z}$) and each recycling centers’ marginal profit ($m_{z}$) that simultaneously maximizes collection center’s and recycling center’s profit functions [Eqs. (9), (10)].

Proof

Presented in Appendix A.

Proposition 17

Equilibrium values (optimal decision values) for the collection center–recycling centers Nash model in Case 3 are as noted in Table 3.

Corollary 4

The effect of the sustainable sensitivity parameter on equilibrium values is as noted in Table 4.

Table 4 Sustainable-sensitivity variation impacts

Proof

Through partial derivatives and algebraic reasoning.

Table 4 shows that customers that are more willing to engage in sustainable material recovery increase the profitability of the total supply chain. It is seen that the level of variations in a supply chain’s profit versus $\beta$ variations is larger when the collection center and recycling centers work in a Nash model. This goes back to the level of marginalization in the collection center and recycling centers. In the leader–follower model, the profit of the chain is dominated by the pricing strategy of the recycling centers, which mitigates the marginal profit of the collection center. However, in the Nash structure, where demand is stimulated by a larger level of dedication to sustainability improvement, the collection center and the recycling centers can determine their own marginal profit at the same decision-making power level. This increases the collection center profit, and consequently, the total supply chain profit.

Corollary 5

The effect of recovery fixed emission parameter ($a$) on equilibrium values are as noted in Table 5.

Table 5 Impact of recovery fixed emission variation

Proof

Through partial derivatives and algebraic reasoning.

All the equilibrium values in Table 5 decrease over the recycling fixed emissions. If the E-waste is at a level where the recovering process generates considerable emissions, then this enforces a large penalty on the waste management system and decreases the supply chain profit. On the other hand, the variations of $\theta$ versus $a$ show that when the recovery of material produces considerable emissions, the investment in sustainable processes does not significantly benefit the system, and the government prefers to incur the penalty. In this case, it is better for the collection center to collect the E-waste in a manner that is more environmentally friendly and basically generates less emissions.

Numerical example

The numerical example in this section shows the applicability of the proposed game models in an E-waste management supply chain. A subsequent parameter sensitivity analysis provides insights for real-life applications of the discussed models. We assume that an E-waste supply chain with collection center “A” is responsible for collecting E-waste and delivering it to recycling center “B”. The collection center has some ongoing expenses such as transportation cost, labor cost, etc. Historical data show that the collection center incurs $c = 10$ monetary units to collect and deliver for each pound of E-waste taken to the recycling center. The collection center asks for $w$ monetary units per pound from the recycling center to compensate for cost $c$ and make some marginal profit.

Recycling center “B” is a for-profit company that recovers valuable material out of E-waste and sells it to electronic device manufacturers. The material recovery from each pound of E-waste generates $10 - 0.12\theta > 0$ units of emission ($a = 10$, $b = 0.12$), where $\theta$ is the level of sustainability improvement in the material recovery process. The emission is reduced when we increase $\theta$, but it cannot become a negative value. The government charges tariff and emission trading prices on each pound of recovered material to source a portion of its own expenses for ongoing social welfare and sustainability projects. The unit emission trading price paid from the recycling center to the government is $e = 0.7$ monetary units, and, to be socially responsible, the government invests $\lambda = 0.5$ monetary units to reduce the environmental impacts of the recycling emissions.

Potential demand for the recycling center’s recovered material is 1000 lb, but this demand decreases for higher prices with a sensitivity coefficient of $\alpha = 44$. On the other hand, electronic device manufacturers are interested in recovered material that is produced through a more environmentally friendly recycling process. Therefore, the manufacturer’s demand increases in higher levels of recycling sustainability with a sensitivity coefficient of $\beta = 35$. The money that the electronic device manufacturers pay for each pound of recovered material includes the recycling center’s asked price $P$ and the government’s tariff per pound $t$. Thus, the demand function for the recovered material is $d = 1000 - 44 \times (P + t )+ 35 \times \theta$.

Increasing the sustainability level of the recycling process requires investment in facilities, training, etc. The investment parameter $i$ in this example uses a value greater than 9 to ensure that the results pass the profit functions’ concavity requirements. The following numerical analyses show the trade-off between the parameters and decision variables under different cases and scenarios.

Impact of various i on Case 1 equilibrium values

An investment in sustainability potentially increases the supply chain’s cost. On the other hand, we believe that a more sustainable recovery process is a desirable feature for the recovered material customer. Table 6 shows the effect of varying investment parameters on the equilibrium values of Case 1.

Table 6 Case 1: equilibrium values analysis

Table 6 results are in the same direction as Corollaries 1 and 2. An interesting numerical outcome shows that the government profit is greater than that of the collection center or the recycling center. The government is making profit off of both the emission penalty and the taxes, and in all three scenarios, the government is the leader and most powerful entity in the supply chain. Because the government is not incurring any production or service cost, it can lead all scenarios in a way that leads to a significant profit. However, it is notable that government should not keep this profit and return it back to the community by incurring long-term sustainable investments.

The other observation regards the largest supply chain profit in the centralized collection center–recycling center scenario. Government power is constant in all scenarios; therefore, the relationship between the collection center and the recycling center determines the supply chain profit. In agreement with many former studies (i.e., [31, 35]), this centralized relationship can produce more profit when compared to uneven decision-making power structures. Moreover, when the collection center and the recycling center both play Nash, the supply chain produces a higher profit compared to the case in which the recycling center is the leader and the collection center is a follower. However, the recycling center loses an observable profit in Nash structure because of higher level of optimal sustainability level $\theta$. This, however, stimulates the demand, which positively affects the collection center’s profit. The total trade-off leads to a larger supply chain profit in the Nash structure compared to the leader–follower structure.

Increasing the required investment for improving the material recovery sustainability negatively affects the profits and level of sustainability improvement, which is economically logical. However, there is a noticeable trend in the recycling center’s profit when it plays the Nash game with the collection center. In the Nash scenario, the recycling center makes more profit when the investment cost is increased. This may not look logical, because the recycling center is responsible for the investment, but it is worth noting that the higher sustainability investment is mirrored in the service price and, thus, reduces the demand. Therefore, even in the Nash scenario, both players agree to reduce the level of sustainability improvement when the required investment goes up. This lower value of $\theta$ saves some sustainability investment money for the recycling center, and thus, the recycling center’s profit goes up, although there is a larger investment parameter.

Impact of unit emission social responsibility cost for the government (${{\lambda }}$) on sustainability improvement (${{\theta }}$)

The government incurs emission costs based on international conventions and/or a social commitment to reduce emissions. The government social monetary commitment might affect the level of sustainability for material recovery process.

Figure 5 uses $i = 20$ to obtain numerical results for the Case 1 leader–follower scenario. When the government social responsibility cost increases, the level of $\theta$ goes down. Logically, it is to be expected that the level of sustainability will increase when there is higher cost parameter for emissions. However, it is worth noting that there is a fixed emission ($a$) for recovering each pound of E-waste, and this value is multiplied by $\lambda$ as well. Higher $\theta$ values increase demand, which is equal to the larger emission cost for the government. Therefore, at the current parameter values, it is more profitable for the government (which is leading the decision-making process) to have lower levels of $\theta$ when $\lambda$ increases. Comparing Fig. 5a and b shows that, at lower fixed emission levels ($a$), increasing the government social responsibility cost does not affect $\theta$ so much.

Fig. 5

Sustainability improvement level versus government emission cost

Impact of sustainability investment cost parameter (i) on level of sustainability improvement ($\theta$) and E-waste selling price (w) in Case 2

In Case 2, the recycling center both collects a portion of E-waste and manages waste transportation to the recycling center facility. Figure 6 shows the variations of $\theta$ and $w$ compared to various values of the sustainability investment parameter for both studied scenarios.

Fig. 6

Impact of sustainability investment parameter variations on $\theta$ and $w$

Increasing the sustainability investment parameter mitigates the desire to improve the level of sustainability. More explicitly, supply chain units must pay more money to improve their sustainability level. Since these plants are for-profit companies, they decide to invest less in sustainability improvement when the required investment becomes larger. Figure 6a shows that when the collection center and recycling center are in the Nash power structure, they both adopt a larger level of sustainable improvement. In the leader–follower model, the recycling center prefers smaller $\theta$ values, because it is the only member responsible for paying the required investment. However, the collection center stimulates larger $\theta$ values when it receives equal decision-making power in the Nash model. This is the reason why, in Fig. 6a, the $\theta$ value for Scenario 2 is consistently larger than that in Scenario 1. A larger $\theta$ value is preferred by the collection center, as it stimulates the demand and profit, but imposes no additional cost on the collection center.

A larger required investment for sustainability improvement is considered an excessive cost for the recycling center. Since both the recycling center and the collection center benefit from higher demand, they both decide on a lower E-waste selling price ($w$) to prevent an increase in the recovered material price. This mitigates the demand reduction because of the price increase. According to Fig. 6b, the Nash model gives the collection center more authority to ask for larger w values from the recycling center.

Impact of second recycling center’s E-waste collection cost (c₂ ) on recovered material price (P₁ , P₂)

The second recycling center may have different requirements regarding their received E-waste. These different requirements cause different waste collection and transportation costs for the collection center. Figure 7 shows the effect of the sustainability investment parameter on the prices for recovered materials when it comes to different E-waste collection costs in a leader–follower scenario.

Fig. 7

Impact of sustainability investment parameter variations on $P_{1}$ and $P_{2}$

The recovered material price asked by each recycling center is a function of the cost of purchasing E-waste. Figure 7 shows that, since these two recycling centers are not competing on price, when the E-waste buying price for the second recycling center is more than the E-waste buying price for the first recycling center, it requires a higher selling price for the recovered material and vice versa. On the other hand, Fig. 7 shows that increasing $i$ decreases nonlinearly the recovered material selling price. A larger investment parameter reduces the government profit in this case. Therefore, the government—which is the decision-making leader—selects lower values of $\theta$ that mitigates demand for the recovered material. In turn, both recycling centers thus reduce the selling price to sell more recovered material and counteract the demand reduction of $\theta$ in response to a larger sustainability investment parameter.

Impact of second recycling center’s E-waste collection and management cost (c₂) on supply chain total profit

In terms of making profit, it is worthwhile for the supply chain manager to notice how the emergence of a new recycling center may affect the profitability of the total supply chain.

Figure 8 shows the total supply chain profit for the leader–follower scenario in Cases 1 and 3. It is assumed that $i = 20, c_{1} = 10$, and $c_{2}$ varies from 5 to 15. If the second recycling center requires cheaper E-waste than the first recycling center, then in most of the $c_{2}$ values, the supply chain makes a larger profit when it works with two recycling centers compared to only one. However, it is worth noting that a break-even point does not happen when $c_{1} = c_{2} = 10$, but only when $c_{2}$ is slightly less than $c_{1}$. In fact, when $c_{1} = c_{2} = 10$, Case 1 is more beneficial than Case 3. This is because the player that is responsible for sustainability improvement cost ($i\theta^{2}$) has changed. Figure 8 shows that, at the E-waste collection cost value for Case 3 (which creates the same supply chain structure as Case 1), the supply chain profit is reduced because the government is responsible for incurring the sustainability improvement cost ($i$). Therefore, from the point of view of the supply chain, it is more profitable if the recycling center incurs the sustainability improvement cost, rather than the government.

Fig. 8

Impact of second recycling center’s E-waste collection cost on supply chain profit

Discussion and managerial implications

Our study considers three different cases for an E-waste supply chain with government intervention. A numerical example is provided for each case, and sensitivity analysis has been shown for the variation in each important parameter. The equilibrium and optimal decision values are obtained with some consideration for algebraic constraints. These constraints are imposed to ensure that the solution is logically reasonable. For instance, without imposing these constraints, it is possible to find a negative value for the profit function, and we know that supply chain members are for-profit plants. Therefore, constraints should be set to make sure that optimal decisions lead to positive profit values.

This study considers the interaction between a collection center and a recycling center (or two recycling centers, in Case 3). Algebraic reasoning and numerical examples prove that the supply chain achieves a higher profit margin if the collection center and recycling center work under centralization. In centralized management, one unit makes integrated decisions for both the collection center and the recycling center. Centralized management reduces the uncertainties between the supply chain members and removes double marginalization in pricing. The result of centralization is the same as in many former studies [37], which shows that the profit for the government and the total supply chain increases under integrated decision-making, compared to when each member makes decisions individually. A finding relevant to managers is that members must try to reach an agreement to work within an integrated framework. Since the government profit increases in a centralized framework as well, members can use this as a bargaining tool in discussions with the government.

Our numerical results show that when the collection center and recycling center are both centralized, even the E-waste recovery process becomes more sustainable. This can be a bargaining tool when the supply chain manager wants to deal with decision-makers who are more interested in environmental issues, such as sustainable E-waste recovery processes. Moreover, a government authority may find it very practical to increase their sustainability level and profit just by convincing the collection center and recycling center to work within a centralized framework. In fact, supply chain members avoid integrated decision-making, because their individual profits may decrease compared to when they make these decisions individually. However, the profit of the total supply chain increases when integration measures are used. Therefore, the supply chain manager must ensure each member that he is dividing any extra supply chain profit between the members to compensate for their individual profit losses. This integration requires a great level of trust between all members; therefore, the government can support the dividend of integrated profit sharing to motivate members to enter a centralized framework.

Results show that the government loses profit when the recycling center is the leading decision-maker when working with the collection center. This is because of the recycling center’s dominance in making decisions on both sustainability and the recovered material price. This decision-making dominance enables the recycling center to in turn affect government decisions. Therefore, the government should work with the collection center to empower it to be on an even footing with the recycling center in decision-making.

We also studied a case in which the recycling center performs a portion of the total E-waste collection activity. This is a policy which can be used to counter E-waste supply disruption. More explicitly, if the collection center is not able to collect the waste, the recycling center still has some materials to work with. Financially, this makes sense, since the supply chain members are working for profit. However, when the recycling center does collection activity, as well, it is likely that double marginalization has been removed for a portion of the recovered material. This is similar to a case in which members work under a centralized framework. Therefore, the total supply chain profit improves if the recycling center does a portion of the collection process. However, the collection center will lose some of its business, but the recycling center can share a portion of its excessive profit to keep the collection center financially satisfied. Moreover, it is not so bad for the collection center, since it will then find some idle capacity that can be assigned to other waste collection activities.

We also considered the sustainability-sensitive demand in our study. The results show that a larger sustainability level is recommended for customers who are more sensitive to these kinds of issues. Once the demand increases for more sustainable E-waste material recovery, a larger optimal value of sustainability can be selected. In fact, larger $\theta$ requires more investment, but also leads to increased profit along with increased demand, which compensates this investment. Therefore, if the government is willing to have more sustainable E-waste recovery processes, it should invest in training the community about the importance of environmental issues. When the demand becomes more sensitive to material recovery sustainability, supply chain members should then select more sustainable recovery methods.

It is worth noting that considering the environment when recovering materials is not a cost-free business decision. Obtaining new technologies and devices, training personnel, recruiting skilled personnel, waste preprocessing, etc. impose costs on a sustainable E-waste recovery process. Managers might be interested in finding out how sustainability investment variations affect the level of recovery process sustainability. Our study shows that when the required investment for improving the recovery of E-waste material increases, the supply chain members are less motivated to participate in improving sustainability. A small $\theta$ value is environmentally an undesirable circumstance, but supply chain members may select a small $\theta$ because of financial reasons. In this situation, the government can increase the financial desirability of selecting larger $\theta$ values by providing loans and financial support to the supply chain members.

Conclusions and future research

This study investigated the application of game theory by modeling an E-waste supply chain with three players: the government, a recycling center, and a collection center. It was assumed that the government influences E-waste recycling by imposing tariffs and emission penalties. Three different cases were defined based on the operational condition of each stage, and game scenarios were developed for each case. The possibility of centralized decision-making between the collection center and the recycling center in Case 1 showed that the E-waste supply chain profit is at its highest level when these two parties work in a centralized manner.

When the recycling center and collection center cannot work under a centralized decision-making structure, the supply chain total profit when the two parties are at the same level in the decision-making sequence was larger than when the collection center followed the recycling center in the decision-making sequence. Moreover, when the recycling center entered the E-waste collection market and did a portion of the total E-waste collection activity, the supply chain profit was greater than when all E-waste was gathered by the collection center.

Dealing with customers who are more sensitive to material recovery sustainability increased the total profit of the supply chain. A larger sensitivity to the level of sustainability increased the demand for recovered material at the same level of sustainability investment; therefore, the total profit increased as well. This shows that the supply chain should apply ways to increase the recovered material buyers’ desire for sustainable recovery processes, since this is positively mirrored in the total profit of the supply chain.

Another interesting outcome of this study looked at the fixed emissions released by the recovery process. It was shown that when the fixed released emissions increased, improving the material recovery process may not increase the profit. This shows the importance of sorting and dismantling E-waste before starting the recovery process. If the high-emission-releasing material was removed from the E-waste before starting the recovery process, that would increase the total supply chain profit. This study considered the effect of transferring the responsibility of sustainability investment between the government and the recycling center as well. It was shown that, all else being equal, it was more profitable for the E-waste supply chain if the recycling center (instead of government) incurred the sustainability investment.

For future studies, the other demand functions—such as nonlinear demand functions or functions that consider price competition between different players—can be studied as an extension to the current research. Looking into advertising sustainability practices and their effect on demand functions could be studied as well. In many real E-waste management cases, the informal sector runs all the waste management processes and sells off the recovered material to the market. Therefore, developing a model that studies both formal and informal E-waste collection channels would be an interesting research topic. Finally, we assumed that all the parameters were known, but considering uncertainty would make our models more useful in practice.

Appendices

Appendix A

Proof of Proposition 1

The value of $w$ maximizes the collection center’s profit function if it solves the first derivative of the profit function equal to zero:

$$ \begin{aligned} \frac{{\partial \Pi_{\text{CC}} }}{\partial w} &= D + c\alpha - (m + t + 2w)\alpha + \beta \theta = 0 \to w \\ &= \frac{D + c\alpha - (m + t)\alpha + \beta \theta }{2\alpha }. \end{aligned} $$

(12)

Proof of Proposition 2

If we simultaneously solve the first derivative of ${{\Pi }}_{\text{RC}}$ with respect to $m$ and $\theta$, we find the following optimal values:

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 8i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

(13)

$$ m = \frac{{\alpha ( { - 4i ( {c - ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {4i - be ( {be\alpha + \beta } )} )}}{{8i\alpha - ( {be\alpha + \beta } )^{2} }}. $$

(14)

Proof of Proposition 3

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (4), we can find the value for $t$. We can substitute (15) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 8i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 8i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {8i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + a ( { - \alpha ( {8i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}}. $$

(15)

Proof of Proposition 4

If we simultaneously solve the first derivative of ${{\Pi }}_{\text{CC - RC}}$ with respect to $m$ and $\theta$, we find the following optimal values:

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 4i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

(16)

$$ m = \frac{{\alpha ( { - 2i ( {c - ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {2i - be ( {be\alpha + \beta } )} )}}{{4i\alpha - ( {be\alpha + \beta } )^{2} }}. $$

(17)

Proof of Proposition 5

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (6), we can find the value for $t$. We can then substitute (18) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 4i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 4i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {4i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + a ( { - \alpha ( {4i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 4i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}} . $$

(18)

Proof of Proposition 6

To find the optimal values for $m$, $w$, and $\theta$, we should simultaneously solve $\frac{{\partial \Pi_{\text{CC}} }}{\partial w} = 0$, $\frac{{\partial \Pi_{\text{RC}} }}{\partial \theta } = 0$, and $\frac{{\partial {{\Pi }}_{\text{RC}} }}{\partial m} = 0$. Therefore:

$$ w = \frac{{ - 2Di + 2i ( {ae + t} )\alpha + c ( { - 4i\alpha + ( {be\alpha + \beta } )^{2} } )}}{{ - 6i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

(19)

$$ m = \frac{{\alpha ( { - 2i ( {c - 2ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {2i - be ( {be\alpha + \beta } )} )}}{{6i\alpha - ( {be\alpha + \beta } )^{2} }}, $$

(20)

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 6i\alpha + ( {be\alpha + \beta } )^{2} }}. $$

(21)

Proof of Proposition 7

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (4), we can find the value for $t$. We can then substitute (22) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 6i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 6i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {6i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + a ( { - \alpha ( {6i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 6i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}} . $$

(22)

Proof of Proposition 9

If we simultaneously solve the first derivative of ${{\Pi }}_{\text{RC}}$ with respect to $m$ and $\theta$, we find the following optimal values:

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 6i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

(23)

$$ m = \frac{{\alpha ( { - 2i ( {c - 2ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {2i - be ( {be\alpha + \beta } )} )}}{{6i\alpha - ( {be\alpha + \beta } )^{2} }}. $$

(24)

Proof of Proposition 11

To find the optimal values for $m$, $w$, and $\theta$, we should simultaneously solve $\frac{{\partial \Pi_{\text{CC}} }}{\partial w} = 0$, $\frac{{\partial \Pi_{\text{RC}} }}{\partial \theta } = 0$, and $\frac{{\partial {{\Pi }}_{\text{RC}} }}{\partial m} = 0$ from Eqs. (7) and (8). Therefore:

$$ w = \frac{{ - 2Di + 2i ( {ae + t} )\alpha + c ( { - 3i\alpha + ( {be\alpha + \beta } )^{2} } )}}{{ - 5i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

(25)

$$ m = \frac{{\alpha ( { - i ( {c - 4ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {i - be ( {be\alpha + \beta } )} )}}{{5i\alpha - ( {be\alpha + \beta } )^{2} }}, $$

(26)

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 5i\alpha + ( {be\alpha + \beta } )^{2} }}. $$

(27)

Proof of Proposition 12

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (4), we can find the value for $t$. We can then substitute (28) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 5i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 5i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {5i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + + a ( { - \alpha ( {5i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 5i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}}. $$

(28)

Proof of Proposition 16

To find the optimal values for $w_{z}$ and $m_{z}$, we should simultaneously solve $\frac{{\partial {{\Pi }}_{\text{CC}} }}{{\partial w_{z} }} = 0$, $\frac{{\partial {{\Pi }}_{{{\text{RC}}_{z} }} }}{{\partial m_{z} }} = 0$ from Eqs. (9) and (10). Therefore:

$$ w_{z} = \frac{{ - 12i ( {D - a\alpha \lambda } ) + ( { - 54i\alpha + 5 ( {\beta + b\alpha \lambda } )^{2} } )c_{z} + ( { - 6i\alpha + ( {\beta + b\alpha \lambda } )^{2} } )c_{3 - z} }}{{6 ( { - 12i\alpha + ( {\beta + b\alpha \lambda } )^{2} } )}}, $$

(29)

$$ m_{z} = \frac{{\begin{array}{*{20}c} {6 ( {D ( { - 2i + be ( {\beta + b\alpha \lambda } )} ) + a ( {2i\alpha \lambda + e ( { - 12i\alpha + \beta^{2} + b\alpha \beta \lambda } )} )} )} \\ { + ( {18i\alpha - ( {\beta + b\alpha \lambda } ) ( {\beta + b\alpha ( {3e + \lambda } )} )} )c_{1} + ( { - 6i\alpha + ( {\beta + b\alpha \lambda } ) ( {\beta + b\alpha ( { - 3e + \lambda } )} )} )c_{2} } \\ \end{array} }}{{6 ( { - 12i\alpha + ( {\beta + b\alpha \lambda } )^{2} } )}}. $$

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Appendix B

Proof of Lemma 2

We should find the sign for the principal minors of a Hessian matrix:

$$ H = \left ( {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{{\partial m^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{\partial m\partial \theta }} \\ {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{\partial \theta \partial m}} & {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{{\partial \theta^{2} }}} \\ \end{array} } \right ) = \left ( {\begin{array}{*{20}c} { - \alpha } & {\frac{1}{2} ( { - be\alpha + \beta } )} \\ {\frac{1}{2} ( { - be\alpha + \beta } )} & { - 2i + be\beta } \\ \end{array} } \right ) . $$

(31)

The first principal minor of $H$ is $H^{1} = - {{\alpha }} < 0$. The objective function is joint concave in $m$ and $\theta$ if $2i\alpha - \frac{1}{4} ( {be\alpha + \beta } )^{2} > 0$.

Proof of Lemma 3

If we substitute the values of $m$ and $\theta$ from proposition (2) into profit function (4), and take the second derivative with respect to $t$, we have:

$$ \frac{{\partial {{\Pi }}_{\text{Gov}} }}{{\partial t^{2} }} = \frac{{4i\alpha^{2} ( { - 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}}{{ ( { - 8i\alpha + ( {be\alpha + \beta } )^{2} } )^{2} }} . $$

(32)

The government profit function is concave in $t$ if $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{{\partial t^{2} }} < 0$. This is only true if $- 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } ) < 0$. If Lemma 2 holds true, $8i\alpha > ( {be\alpha + \beta } )^{2}$. Therefore, $- 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } ) < 0$ holds true if and only if $e > \lambda$.

Proof of Lemma 4

We should find the sign for the principal minors of a Hessian matrix:

$$ H = \left ( {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{{\partial m^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{\partial m\partial \theta }} \\ {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{\partial \theta \partial m}} & {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{{\partial \theta^{2} }}} \\ \end{array} } \right ) = \left ( {\begin{array}{*{20}c} { - 2\alpha } & { - be\alpha + \beta } \\ { - be\alpha + \beta } & { - 2i + 2be\beta } \\ \end{array} }\right ). $$

(33)

The first principal minor of $H$ is $H^{1} = - 2{{\alpha }} < 0$. The objective function is joint concave in $m$ and $\theta$ if $4i\alpha - ( {be\alpha + \beta } )^{2} > 0$.

Proof of Lemma 7

We should find the sign for the principal minors of a Hessian matrix:

$$ H = \left ( {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{{\partial t^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{\partial t\partial \theta }} \\ {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{\partial \theta \partial t}} & {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{{\partial \theta^{2} }}} \\ \end{array} } \right ) = \left( {\begin{array}{*{20}c} { - \frac{\alpha }{2}} & {\frac{1}{4} ( {\beta + b\alpha ( {2e - \lambda } )} )} \\ {\frac{1}{4} ( {\beta + b\alpha ( {2e - \lambda } )} )} & {\frac{1}{2} ( { - 4i + b ( {be\alpha + \beta } ) ( { - e + \lambda } )} )} \\ \end{array} } \right ) . $$

(34)

The first principal minor of $H$ is $H^{1} = - \frac{\alpha }{2} < 0$. The objective function is joint concave in $t$ and $\theta$ if $\frac{1}{16} ( {16i\alpha - ( {\beta + b\alpha \lambda } )^{2} } ) > 0$.