Nonlinear wave train in an inhomogeneous medium with the fractional theory in a plane self-focusing

Abbreviations

CE: Carbon emissions; CT: Cap-and-trad; GT: Green technology; PE: Promotional effort; SCnM: Supply chain model; EA: Environmental awareness; PR: Product recovery; $EA_e$: Environmental awareness effort; RP: Retail price; CLSC: Closed-loop supply chain; CS: Centralized system; DS: Decentralized system

1.   Introduction

In the $21^{ \rm{st}}$ century, humankind is being threatened by the current state of environmental degradation. Emissions of gases and waste generation from industries, overpopulation, and other types of environmental pollution created by human activity are mainly responsible for environmental degradation. People in the current scenario are increasingly attracted to green products due to numerous health issues from environmental pollution, environmental degradation, etc. Global warming is one of the most significant environmental issues caused by industrial carbon emissions (CE). The governments of many areas, including China, the United States, and Europe have taken initiatives and implemented effective policies to curb emissions and make the public aware of eco-friendly products. In response to government pressure and the growing desire for green products, supply chain managers invest in green innovation technologies to reduce emissions and find innovative ways to manufacture low-carbon products. The global computer producer Dell modeled a tool to assess environmental risk toward a smaller ecological footprint in the supply chain and identified energy efficiency improvement technologies to lower greenhouse gas emissions to meet sustainability requirements^*.

^* https://corporate.delltechnologies.com/en-in/social-impact/advancing-sustainability/sustainable-supply-chain/environment.htm

The cap-and-trade (CT) policy is a significant and effective emissions-controlling strategy compared with other regulations initiated by the government. In this policy, the government assigns an emissions cap to the manufacturing industries for a mentioned period and issues a quantity of emission allowances consistent with the cap. The government charges excess emissions costs from the production industries over the defined cap. The industries may sell or buy the allowances at a cost in an emissions trading market Xu et al. (2017). The more efficient companies, whose emissions are less than their allowances, can sell excess allowances to the other companies that cannot make reductions easily. The government of Gujarat, India, recently implemented an emissions cap to restrict air pollution and permitted industries to buy/sell ^†. Therefore, CT regulation is one of the most effective market-based mechanisms for manufacturing companies to curb CE.

^† https://indianexpress.com/article/business/in-a-first-gujarat-to-launch-trading-system-with-incentive-for-low-polluting-industries-5767563/

In a production-based supply chain, all the products are inevitably imperfect. A few of the produced items are imperfect due to factors such as production system unreliability, lack of skilled labor, weather conditions, low product quality, and others Mandal and Pal (2021). The majority of companies perform a rework process to reform the faulty products into their original versions. In some cases, imperfect products are recycled as raw materials and returned to the factory for further manufacturing. Sometimes, manufacturing companies sell imperfect products on a secondary market for a lower price.

In the recent era of the business environment, competitive actions among companies have hiked terribly. Market competition has prompted companies to enlarge sales volume to gain higher revenue and profit. The companies introduced several strategies, knowing the rivals' weaknesses to survive in the competitive market. In this regard, some manufacturers hired green technology (GT) to produce low-carbon products. The manufacturers/retailers also offered extended product warranty with a return/refund policy for the non-functioning product, while others designed proper strategies on selling price, product quality, and promotional effort (PE) to beat rivals Bai et al. (2019). Therefore, the supply chain model (SCnM), based on chain members' rivalry, has become very interesting to researchers.

Nowadays, product recycling has captured extreme attention from firms/industries due to shortages of resources and environmental problems Ranjbar et al. (2020). In the recycling process, collected used products are processed to manufacture new products at a lower production cost. Moreover, recycling alleviates environmental issues that originate from wastes/landfills in the environment. HP recycled 17000000 pounds of ocean-bound plastics to produce new HP products, viz., ink cartridges, monitors, laptops, etc. HP Elite Dragonfly is the first notebook manufactured by HP with ocean-bound plastic materials^‡. Recycling used products is a fruitful tactic to reduce emission levels and manufacturing costs. Therefore, product recycling in the SCnM has become a fascinating area in current research.

^‡ https://h20195.www2.hp.com/v2/GetDocument.aspx?docname=c06614535

The number of environmentally conscious customers is significantly increasing day after day. Customers' environmental awareness (EA) trends instigated firms/industries to modify basic production game plans Heydari et al. (2021). Regarding awareness, manufacturers exhibit an eager interest in low-carbon products. In a practical situation, complete recovery of used and waste products is nearly impossible; only a portion of the used product can be recovered. The rest is damaged, diminishing chain efficiency and negatively impacting the environment. To increase product recovery (PR) and protect the environment, chain members consider several positive measures such as green activities, promoting environment-related issues, spreading awareness about the benefit of recycling, etc. Therefore, chain members have executed environmental awareness efforts ($EA_e$) to make a greener globe.

The proposed article investigates answers to the following questions:

● How do the strategies on the greening level, retail price (RP), and PE instigate market demand? Which scenarios are beneficial to the players for individual and chain profit?

● Is GT effective in abating CE and which condition emits the minimum amount of carbon?

● How does $EA_e$ and buy-back price influence the recovery rate, and in which structure is the highest product recovery possible?

In these regards, our article aims to extend a competitive SCnM considering carbon abatement technology and PE in product recycling under carbon cap regulation. The rival retailers compete based on the RP and PE. Accordingly, we model a closed-loop supply chain (CLSC) comprising a manufacturer, two rival retailers, and a recycler with an $EA_e$ under a greening environment. In forward logistics, the manufacturer produces low-carbon products and satisfies the demand of retailers who fulfil green level, RP, and PE-influenced customers' demand. In reverse logistics, the recycler promotes $EA$ and offers an attractive buy-back price to the customers to recover more used products. The recycler supplies the converted raw materials of the recovered products to the manufacturer for the following production purposes. The proposed setup could be similar to an example: Canon, India, is a renowned company that manufactures various products and sells those items through different stores. The retail stores compete against each other for product prices to increase customer demand for better profit. Moreover, Canon, India has tied up with an authorized recycler who collects e-waste such as ink cartridges, toner cartridges, camera batteries, etc., and recycles the waste in an eco-friendly process^§. We analyze the behavior of the proposed model under a centralized and five decentralized scenarios: two manufacturer Stackelberg, retailer-recycler Stackelberg, and two Nash game structures. In each game-theoretic approach, we derive the optimal strategies of the chain members and compare the scenarios to determine which is better for individual profit. In connection to the example, Canon, India could play the Stackelberg game as a leader and find the optimal decisions to gain better individual profit. Again, retail shops and third-party recyclers could jointly participate in the Stackelberg game as leaders for higher joint profits. Moreover, the members could play the Nash game to derive optimal decisions individually for an individual profit maximization scenario.

^§ https://in.canon/en/consumer/web/e-waste

The primary novelties of this article are summarized as follows:

● Imperfect production in closed-loop supply chain: In real-life situations, this article considers a closed-loop supply chain with an unreliable production system producing some fraction of inferior quality items. Most authors (Bai et al. (2019), Huang et al. (2020), Pang et al. (2018), Xu et al. (2017)) focused on the supply chain with the production of perfect items only. Here, we include converting the produced imperfect items into raw components for use in the subsequent production.

● Green technology investment and CT policy: We consider GT investment done by the manufacturer to curtail CE during production. Moreover, we study the SCnM under the CT policy, where the manufacturer benefits from carbon allowances. The majority of current research (Gao et al. (2018), Parsaeifar et al. (2019), Rezaei and Maihami (2020), Xu et al. (2016)) paid attention to either green technology investment or CT regulation under gas emissions environment, whereas both are taken into consideration in this article.

● Rivalry in the closed-loop supply chain: In this article, we study the competitive behaviour between retailers. The rival retailers compete against each other for the retail price and product promotion. We consider that one retailer's market demand not only depends on its selling price and product promotion but is also sensitive to that of the rival. In the existing research (Bai et al. (2019), Modak et al. (2016), Mondal and Giri (2022), Parsaeifar et al. (2019)), only retail price-based rivalry is present but, jointly, the retail price and product promotion-based rivalry are incorporated in the present study.

● Variable product recovery rate: The recycler's variable product recovery rate is designed in this research. Here, the recycler offers an attractive buy-back price and yields environmental awareness efforts to motivate customers about product recycling and to increase the quantity of recovered items. To the best of our knowledge, the buy-back price and environmental awareness effort-dependent variable recovery rate have been considered only in the study of Mandal and Pal (2023).

The rest of the present study is framed as follows: Section 3 introduces a brief survey of related past literature. Section 4 interprets the problem statement with notations and assumptions applied to construct the model. Mathematical modelling of the CLSC with variable PR rates under a CE environment is designed, and the model's behavior under different decision-making systems is analyzed in Section 5. A numerical example with some observations is posted in Subsection 5.1. Again, a sensitivity analysis is performed to check the model's efficiency, and managerial insights with implications are outlined in Subsection 5.2. Finally, concluding remarks are drawn in Section 7.

2.   Literature Review

In this section, we briefly survey past research linked to our study. The current research mainly concentrates on the literature based on the following aspects: 1) supply chain with CE, green investment, and CT regulation; 2) competitive supply chain, variable market demand, and imperfect production; and 3) recycling in the supply chain.

2.1.   Supply chain with carbon emissions, green investment, and cap-and-trade regulation

Adnan et al. (2023) investigated pricing decisions in two competing supply chains, each consisting of one manufacturer and one retailer with green investment in the presence of consumers' green awareness. They studied three game-theoretical approaches to derive optimal decisions of the chain. Cao et al. (2020) developed a SCnM for two firms under remanufacturing subsidy and carbon tax policies to study optimal decisions on production and pricing. They investigated the two policies and analyzed which was better for the firms. Daryanto et al. (2019) investigated the optimal delivery quantity and size in an integrated three-phase SCnM of deteriorated products with carbon emission under emission reduction incentives. Gao et al. (2018) studied a two-layer SCnM including two members: single manufacturer, single retailer (SMSR) with cooperative emission reduction strategies under a carbon tax scheme. They analyzed the model under cooperation, non-cooperation, and emission abatement cost-sharing contracts. Haijie et al. (2024) investigated a CLSC under CT regulations with a dual recycling channel. Huang et al. (2020) examined the various carbon policies in a two-tier SCnM under green investment. They assumed that CE was processed during the product's production, storage, and transportation. Jauhari et al. (2020) developed a CLSC consisting of three members with green investment under a CT policy. They constructed the model under five scenarios, including one centralized and three Stackelberg game structures. Jamali and Rasti-Barzoki (2019) proposed a sustainable SCnM for two manufacturers and a single retailer to investigate the product's pricing and greening level under a centralized system (CS) and decentralized systems (DS). They included third-party logistics between manufacturers and retailers to curtail CE and lessen delivery time. Jiang et al. (2021) formulated a two-phase SCnM comprising SMSR with emission-influenced demand under carbon reduction investment. They studied the model under the coordination of cost-sharing contracts. Jianhui et al. (2023) examined decisions on price, green level, and recycling in a CLSC under governmental subsidies. Karim and Nakade (2021) investigated the optimal decisions on green investment and production for a SCnM comprising of SMSR with product quality disruption under CE restriction. Lin et al. (2019) examined how emission regulations affect SCnM decisions in GT investment. They considered two firms and investigated their individual and optimal joint strategies under CS and DS. Liu et al. (2018) presented a SCnM in a carbon abatement environment under CT regulation. Assuming emission-influenced demand, they studied the effect of the carbon price and customers' consciousness of the environment on the chain members' optimal decisions. Pang et al. (2018) investigated a SCnM coordination mechanism with the revenue-sharing contract under CT regulation. They considered customers' EA dependent on market demand and studied the influence of EA on CE in the chain. Taleizadeh et al. (2021) modeled a CLSC model comprising a manufacturer and a distributor with a quality improvement effort and carbon reduction strategy. They applied a cost-sharing contract and analyzed the model using the Nash and Stackelberg game approaches. Taleizadeh et al. (2021) examined a dual-channel green supply chain comprising a manufacturer and a retailer under cap and trade regulation. They investigated the impact of green investment in the curtailment of CE. Wang and Song (2020) constructed a direct-retail channel SCnM under a green environment to investigate pricing policies considering the price, sales effort, and green level dependent on market demand. They examined the proposed model under CS, DS, and collaborative manners. Xu et al. (2016) presented a sustainable two-layer SCnM considering CE under CT regulation with a coordination mechanism. They included sustainability level and selling price-influenced product demand and showed how emission trading price impacts the model's optimal strategies. Xu et al. (2019) constructed an SCnM to highlight pricing and emission-abating behaviour with environmental awareness to conscious customers about carbon emission under four different governmental subsidy strategies.

2.2.   Competitive supply chain, variable market demand, and imperfect production

Dolai et al. (2023) developed an imperfect production-based inventory model for green products under an advertisement-sensitive credit period. In their model, they considered variable screening rates sensitive to the learning effect of the workers and the number of cycles. Fadavi et al. (2022) studied a green supply chain consisting of two players, a manufacturer and a retailer, in a competitive environment. The players compete with each other for green and price-sensitive markets. Hosseini-Motlagh et al. (2021) presented a supply chain coordination problem for a manufacturer and two rival retailers with CE. Competition among retailers took place due to greening efforts. They analyzed the model under centralized, decentralized, and compensation-based contracts. Huang et al. (2016) studied a two-phase SCnM consisting of three players, viz., duopoly retailers and a manufacturer with pricing competition between the retailers. They analyzed the behaviour of chain members under six DS. Jafari et al. (2016) presented a SCnM under a dual-channel structure with a monopoly manufacturer and duopoly retailers. Their model analyzed pricing strategies for Collusion, Bertrand, and Stackelberg game approaches. Li et al. (2016) proposed a SCnM of green products under the pricing competition between the direct and retail channels. They investigated greening and pricing decisions under CS and DS. Mandal and Pal (2021) examined an imperfect production-based supply chain under a competitive trade credit financing environment. Considering selling price and PE-based rivalry between retailers, they analyzed the model under centralized and various decentralized game structures. Mondal and Giri (2020) constructed a two-period CLSCnM consisting of SMSR under a greening environment. They employed green level, marketing effort, and selling price-sensitive market demand in their model. Mondal and Giri (2022) examined a closed-loop green SCnM with retailers' competition and collection of used products under a carbon cap scheme. Their research included selling price and green level-sensitive linear demand patterns and analyzed the model under a CS and DS. Pal et al. (2015) investigated the optimal selling price and PE to maximize the profit of a two-echelon competitive SCnM by analyzing different coordination mechanisms. Pal et al. (2016) modelled a two-phase SCnM, including a supplier and two rival retailers, under a trade credit policy. Their study considered how selling price and credit period influenced competitive market demand and examined the model under integrated and Vertical Nash scenarios. Pal and Sarkar (2022) formulated a dual-channel competitive supply chain for two players under green investment. They analyzed the model using different decentralized structures from the Stackelberg and Nash games. Pal et al. (2021) constructed an imperfect production-based two-phase SCnM for deteriorated items under credit policy. They considered variable demand to be sensitive to product quality and promotional level. Panja et al. (2023) designed a joint offline and online retailer business by proposing a utility-based approach to reflect the choosing behaviour of the customers over the available alternatives. Parsaeifar et al. (2019) proposed a multi-product three-phase SCnM comprising one manufacturer, multiple suppliers, and retailers under the competition among the chain players with recycling of products. They assumed that RP and product greenness are variable linear demands of retailers.

2.3.   Recycling in supply chain

Asghari et al. (2022) studied a green CLSCnM consisting of a green manufacturer, a retailer, and a collector. They considered retail price and environmental efforts sensitive to variable market demand and analyzed the model under different decentralized scenarios. Behrooz et al. (2023) constructed a dual-channel CLSC with product recycling under a greening environment. Cao et al. (2022) investigated a CLSC with remanufacturing and product recycling. They considered various alliances: the original manufacturer, the remanufacturer, and the third-party recycling platform. Jiang and Zheng (2023) explored the pricing and remanufacturing decisions of two firms with product recycling in the presence of consumers' EA. The outcomes showed that firms trade between collection cost and profit when EA gets lower. A CLSCnM of duopolies retailers and one manufacturer was constructed by Modak et al. (2016) with product recycling. They considered sales price and recycling factors depending on end-customer demand and compared the Collusion and Cournot games model. Pal and Sarkar (2021) investigated a dual-channel supply chain in a green environment with product promotion and recycling of used items. Rezaei and Maihami (2020) modelled a multi-echelon SCnM comprising SMSR and a collector remanufacturing of collected products under carbon abatement strategies. They studied the model under Stackelberg, Nash game, and DS's bargaining structures and compared the resulting decentralized approaches with a CS. A CLSCnM with the returned product's remanufacturing under a technology license was formulated by Taleizadeh et al. (2019). They included technology investment under the CT policy to curtail CE and considered price, emission reduction, and quality effort-sensitive market demand. Tsao et al. (2018) designed a two-phase SCnM considering CE and remanufacturing returned products. After minimizing the network cost in the forward channel, they investigated remanufacturing centers' optimal replenishment cycle, number, and service areas. Wang and Wu (2020) investigated emission reduction and product collection strategies in a CLSCnM under the CT policy and explored the model under CS and DS. Wei et al. (2021) examined the effect of retailing and collecting channels strategies on optimal decisions and profit in a three-layer CLSC under a competitive collection environment. Zhang et al. (2020) designed a dual-channel CLSCnM to recycle inferior quality and waste products. They investigated pricing and quality decisions and proposed a sharing contract on revenue to stimulate retailers towards the collection of used products.

2.4.   Research gaps and contributions

The contribution of the current work concerning other closely related research is summarized in Tables 1 and 2. The following primary research gaps and contributions are introduced based on the existing literature connected to a CLSC system with variable recovery rates.

^¶ Green technology level

^|| Promotional effort

^** Environmental awareness effort

^†† Buy-back price

^‡‡ advertisement frequency

1. Several previous studies (Bai et al. (2019), Huang et al. (2020), Pang et al. (2018), Xu et al. (2017)) mainly focused on producing perfect quality items only. Here, we consider imperfect production by converting imperfect goods to raw materials for subsequent production.

2. Only a few researchers (Gao et al. (2018), Parsaeifar et al. (2019), Rezaei and Maihami (2020), Xu et al. (2016)) paid attention to one or two factors amongst carbon generation, green investment, and CT regulation, but we present the trio concurrently in the current study.

3. The RP-based rivalry usually exists in past literature (Bai et al. (2019), Modak et al. (2016), Mondal and Giri (2022), Parsaeifar et al. (2019)); however, RP and PE-based competition are jointly present in our article.

4. We assume GT level, RPs, and PEs sensitive linear market demand patterns, whereas it was absent in a more significant segment of former research (Jauhari et al. (2020), Rezaei and Maihami (2020), Taleizadeh et al. (2019), Wang and Wu (2020)).

5. As far as our knowledge, only the research of Mandal and Pal (2023) has considered the buy-back price and EA-influenced variable recovery rate of the used product under the environment discussed above.

Addressing the research gaps to conduct research, we explore a SCnM problem under the following aspects. 1. Construction of a CLSC, including one manufacturer and two rival retailers with used PR by a recycler. 2. Study of the emission abatement technology under CT regulation. 3. Investigation of the rivalry among the retailers on RPs and PEs. 4. Incorporation of a market demand influenced by RPs, retailers' PEs, and the manufacturer's green innovation level. 5. Introducing the buy-back price and $EA_e$-dependent PR rate.

Table 1 and 2 illustrate a brief comparative review of the present research with the existing literature.

3.   Problem description

A multi-layer SCnM consisting of a manufacturer, two rival retailers, and a single recycler is considered with recycling of used products in the presence of green investment and $EA_e$ under the CT policy. In forward logistics, the manufacturer produces green products with CE reduction incentives and wholesales the products to rival retailers. Here, some percentage of manufactured products are faulty due to the production system's unreliability. The $i$th retailer directly satisfies the end customer demand that is influenced by green innovation, RPs, and PEs. In backward logistics, the recycler collects used products from the end customers at an attractive buy-back price. The collected products go through an inspection process and are separated into two parts: in the first part, recyclable items are converted into raw components and delivered to the manufacturer; in the second part are disposable/landfill items.

The manufacturing system produces some faulty products due to labour and weather issues, deterioration of machine equipment, and a wide range of other controllable and uncontrollable factors (Pal et al. (2021)). The manufacturer spends money converting imperfect goods into raw materials that are reused for production in the future. Moreover, the manufacturer invests in GT to curtail CE during production under the CT policy (Bai et al. (2019)). Under the CT regulation, the government agency assigns firms specific carbon emission quotas (Xu et al. (2017)). An emission penalty is imposed against a firm that exceeds the pre-determined limit. In the chain, the competitive behavior between the retailers is investigated (Mandal and Pal (2021)). The retailers compete against each other for retail price and PE. Here, one retailer's demand is assumed to be dependent not only on its own retail price and PE but also on the rival's. The recycler recovers the used items from the customers for the purpose of recycling (Pal and Sarkar (2021)). The recycler offers attractive buy-back prices and promotes EA efforts to increase used product collection. Figure 1 indicates the graphical view of the problem.

3.1.   Notations

Throughout the article, the following notations are presented.

3.2.   Assumptions

The following assumptions are made to construct the model.

Assumption 1: The production system is unreliable; it produces some fraction of imperfect quality items, and upon bearing the cost of $C_{co}$ per unit, the items are converted into raw components.

Assumption 2: To curb CE and to increase product demand, the manufacturer invests in green innovation technology with associated unit cost $\frac{1}{2} \beta g^2$, where $g$ is the green innovation level, and $\beta (>0)$ is the investment cost coefficient. Similar to Bai et al. (2019), the cost function is taken in quadratic form.

Assumption 3: In the production time, CE per unit item is $(a-b g)$. Here, we assume $0\leq g < \frac{a}{b}$ to avoid negative emission. Greater values of $g$ imply lower CE. Bai et al. (2019)

Assumption 4: The rival retailers compete against each other for RP and product promotion. Here, one retailer's market demand depends on his and the other's RP and PE. Therefore, green innovation level, RP, and PE-sensitive end customers' linear demand pattern is considered and presented as:

$D_i(g, p_i, \rho_i)$ = $\; \gamma_i + \delta_i g - \zeta_i s_{ri} + \eta_i s_{rj} + \lambda_i \rho_i -\mu_i \rho_j$, $i, j = 1, 2$ and $i \neq j$, where $\gamma_i(>0)$ is the base market, $\delta_i(>0)$ measures the elasticity of demand regarding green innovation level, $\zeta_i(>0)$ measures the impact of RP on demand by the retailers, $\eta_i(>0)$ measures the effect of rival's RP on demand, $\lambda_i(>0)$ measures the influence of promotion on demand by the retailers, and $\mu_i(>0)$ measures the effect of rival's promotion on demand. Here, $\zeta_i > \eta_i$ and $\lambda_i > \mu_i$, as one retailer's demand is more sensitive to their RP and PE than the rival's Bai et al. (2019). To overcome mathematical complexity, we take the demand function as $D_i(g, p_i, \rho_i)$ = $\gamma_i + \delta g - \zeta s_{ri} + \eta s_{rj} + \lambda \rho_i -\mu \rho_j$, $i, j = 1, 2$ and $i \neq j$.

We consider that the end customers' demand for the individual retailer and each retailer's demand for the manufacturer are equal.

Assumption 5: The $i$th retailer spends per unit promotional cost $\frac{1}{2}\kappa_i$$\rho_i^2$ for promotion of product to increase market demand, where $\rho_i$ is the PE level and $\kappa_i(>0)$ is the promotional cost coefficient.

Assumption 6: The recycler recovers the used products from the end customers at a rate $R_r$. To motivate customers to recycle and to increase PR, the recycler offers the best buy-back price and promotes EA. As rising values of buy-back price and $EA_e$ positively impact product collection, the recovery rate is taken in the form: $R_r$ = $h(p_b+\psi)$. Moreover, the recycler expends cost $\frac{1}{2}$$l\psi^2$ for $EA_e$ to uprise PR, where $\psi$ is the awareness effort level and $l > 0$ is the cost coefficient.

4.   Mathematical modeling

In this CLSCnM, the manufacturing system generates perfect and imperfect quality products together. After receiving the manufacturer's environment-friendly perfect items, both retailers directly sell those items to the end customers. The system acquires perfect items at a rate (1-$\alpha$) times the production rate, where $0 < \alpha < 1$. The manufacturer funds GT, observing the end customers' tendency toward a green product. The manufacturer adopts the CT policy to control CE during production. From a rival's perspective, each retailer has the following options for sales increment: offering a lower selling price, extending PE, or applying both together. The recycler supervises the buy-back price of used products and the $EA_e$ level to increase EA among customers and acquire a good collection of used products. The collected products are inspected by the recycler and categorized into two parts. Figure 2 illustrates the supply chain workflow.

4.1.   Manufacturer's model

The manufacturing system manufactures products at $(D_1+D_2)/(1-\alpha)$, whereas the imperfect items are generated at $\alpha$ times the production rate. The faulty items are converted to raw components by the manufacturer for the next production. For production, the manufacturer accumulates raw materials/components from three sources, viz., supplier, self, and recycler. The manufacturer funds green innovation technology to meet end customers' need for greener products and curb CE. Meanwhile, by controlling emissions, the manufacturer obeys the CT regulation for a less polluted environment.

The manufacturer receives $x R_r (D_1+D_2)$ units recycled raw components per unit time from the recycler for production. Therefore, raw materials cost paid to recycler is $C_{mc} \; x R_r (D_1+D_2)$. Again, $\alpha \frac{D_1+D_2}{1-\alpha}$ units imperfect items are converted to raw components. So, converted raw materials cost is $C_{co}\; \alpha \frac{D_1+D_2}{1-\alpha}$. Supplier settles the remaining raw materials' requirement, hence raw components cost paid to supplier is $C_{ms}\left[\frac{D_1+D_2}{1-\alpha}- x R_r (D_1+D_2)-\alpha \frac{D_1+D_2}{1-\alpha}\right]$. The CE amount for the production of $\frac{D_1+D_2}{1-\alpha}$ units item is $(a-bg)\frac{D_1+D_2}{1-\alpha}$. Therefore, CE cost is $\left[C_e(a-bg)\frac{D_1+D_2}{1-\alpha}-C\right]$ and the associated GT cost is $\frac{1}{2}\frac{D_1+D_2}{1-\alpha}\beta g^2$. Sales revenue collected by the manufacturer from the two retailers is $s_{m1}D_1+s_{m2}D_2$.

The manufacturer's profit is denoted by $\Pi_M$ and presented as:

4.2.   Retailers' model

The end customers' demand for the retailers is influenced by each retailer's RP and PE, which proves the rivalry between the retailers. To survive in a rivalry environment, individual retailers desire to curtail the RP and augment the PE compared with rivals.

The $i$th retailer's buying price is $s_{mi} D_i$. Promotional cost for the $i$th retailer is $\frac{1}{2} D_i \kappa_i \rho_i^2$. Earned sales revenue of the $i$th retailer is $s_{ri} D_i$.

The underneath equation defines the $i$th retailer's profit.

4.3.   Recycler's model

The recycler's target is to collect as many used products from customers as possible. For this, the recycler offers the best buy-back price and awakens the public toward the environmental benefit of recycling. The gathered used products are inspected and divided into two parts. The first part is recyclable items to be converted into raw materials; the other is disposable/landfilled items. Only the $x$ fraction of collected products are converted into raw components and delivered to the manufacturer for the next production.

Buy-back cost of the recycler is $p_b R_r (D_1+D_2)$. Recycler's $EA_e$ cost is $\frac{1}{2} l \psi^2$. Recycler's product recyling (inspection and converting) cost is $C_s R_r (D_1+D_2)$. Recycler's collected revenue from the manufacturer for delivering raw materials is $C_{mc} x R_r (D_1+D_2)$.

The expression of the recycler's profit is given in the below equation.

Now, the following game theoretic models are considered:

● Centralized system (CS)

● Manufacturer-Stackelberg model 1 (MS1)

● Manufacturer-Stackelberg model 2 (MS2)

● Retailer-recycler Stackelberg model (RCS)

● Vertical Nash model 1 (VN1)

● Vertical Nash model 2 (VN2)

The determination of optimal decisions and, consequently, the profits of each player are discussed under all the game-theoretic approaches mentioned above.

4.4.   Centralized system (CS)

In the CS, the manufacturer, the retailers, and the recycler act as a team, and one centralized decision is taken to optimize the integrated profit of the chain. Here, the manufacturer makes a contract with the recycler in which the manufacturer will pay a fixed raw component cost to the recycler. Moreover, the manufacturer offers a deal to the retailers based on their selling prices, where $s_{m1} = z_1 s_{r1}$ and $s_{m2} = z_2 s_{r2}$, $z_1 \gtrless z_2$, whenever $s_{r1} \gtrless s_{r2}$ and $0 < z_1 < 1$, $0 < z_1 < 1$.

The integrated profit of the chain,

Now, the problem is to

Maximize $\Pi^{CS}(g, s_{ri}, \rho_i, p_b, \psi)$ subject to the constraints $\frac{a}{b} > g > 0$, $s_{ri} > 0$, $\rho_i > 0$, $p_b > 0$, $0 < (\psi+p_b) < \frac{1}{h}$.

Solution procedure: To optimize the profit function $\Pi^{CS}(g, s_{ri}, \rho_i, p_b, \psi)$, we derive the partial derivatives of $\Pi^{CS}(g, s_{ri}, \rho_i, p_b, \psi)$ concerning the decision variables up to second-order. Equating first-order derivatives equal to zero, the values of $g, s_{ri}, \rho_i, p_b$, and $\psi$ are determined. These values are optimal, i.e., $g = g^*, s_{ri} = s^*_{ri}, \rho_i = \rho^*_i, p_b = p^*_b$, and $\psi = \psi^*$ if the Hessian matrix ($HN_m$) corresponding to the profit function is negative definite, i.e., all eigenvalues of the $HN_m$ are negative. Where,

As all the second-order partial derivatives of $\Pi^{CS}(g, s_{ri}, \rho_i, p_b, \psi)$ are complicated, it is tough enough to find an analytical solution to the problem. We numerically test the above optimality condition using the well-known computer software Mathematica 11.1.1.

4.5.   Decentralized system (DS)

In DS, individual players can make their own decisions. Here, we undergo three Stackelberg and two Nash structures. In the Stackelberg approach, a game is played alternatively among the chain members by the leader-follower rule, where one member is the leader, and the rest are followers.

4.5.1.   Manufacturer-Stackelberg (MS1) model 1

In the Stackelberg game, all the supply chain members optimize their corresponding decisions sequentially according to the decision-making power. Here, the manufacturer leads the supply chain, and other members follow the manufacturer. According to the Stackelberg game principle, the optimal decisions of the followers are derived sequentially. Then, the leader uses the followers' findings in the profit function and derives optimal responses.

Here, the decision making power structures are:

Level 1: Manufacturer, max $\Pi^{ms1}_M(g, s_{m1}, s_{m2})$ subject to $g > 0$, $s_{m1} > s_{r1}$, and $s_{m2} > s_{r2}$

Level 2: Recycler, max $\Pi^{ms1}_{RC}(\psi, p_b)$ subject to $0 < (\psi+p_b) < \frac{1}{h}$, $p_b > 0$

Level 3: Two retailers (play individually), max $\Pi^{ms1}_{Ri}(s_{ri}, \rho_i)$ subject to $s_{ri} > 0$, $\rho_i > 0$

The two retailers derive optimal RPs and PEs independently to maximize their individual profits. Knowing the retailers' strategies, the recycler optimizes its own profit for the decision on $\psi$ and $p_b$. Meanwhile, observing the reactions on $\psi$, $p_b$ of the recycler and $s_{ri}$, and $\rho_i$ of the retailers, the manufacturer finds out the optimal decision on $g$, $s_{m1}$, and $s_{m2}$ to maximize own profit.

The objective of the $i$th retailer is to Maximize $\Pi^{ms1}_{Ri}$ subject to $s_{ri} > 0$, $\rho_i > 0$.

Proposition 4.1. $\Pi^{ms1}_{Ri}(s_{ri}, \rho_i)$ takes maximum value at $(s^{ms1}_{ri}, \rho^{ms1}_i)$ if the condition $2\zeta k_i-\lambda^2 > 0$ holds.

Proof. See the Appendix A

Now, replacing $s_{r1}$, $s_{r2}$, $\rho_1$, and $\rho_2$ by $s^{ms1}_{r1}$, $s^{ms1}_{r1}$, $\rho^{ms1}_1$, and $\rho^{ms1}_2$, respectively, we get profit of the recycler,

where $D^{\prime}_1$ and $D^{\prime}_1$ are obtained by substituting $s^{ms1}_{r1}$, $s^{ms1}_{r1}$, $\rho^{ms1}_1$, and $\rho^{ms1}_2$ in $D_1$ and $D_2$.

The target of the recycler is to Maximize $\Pi^{ms1}_{RC}(\psi, p_b)$ subject to $0 < (\psi+p_b) < \frac{1}{h}$, $p_b > 0$

Proposition 4.2. $\Pi^{ms1}_{RC}(\psi, p_b)$ takes maximum value at $\psi^{ms1} = \frac{h(C_s-x C_{mc})(D^{\prime}_1+D^{\prime}_2)}{h(D^{\prime}_1+D^{\prime}_2)-2l}$,

$p^{ms1}_b = \frac{(C_s-x C_{mc})(hD^{\prime}_1+hD^{\prime}_2-l)}{h(D^{\prime}_1+D^{\prime}_2)-2l}$ if the condition $2l > h(D^{\prime}_1+D^{\prime}_2)$ holds.

Proof. See the Appendix B

Knowing the responses of the retailers and recycler, the manufacturer takes decisions on $g$ and $s_{mi}$. Substituting the values of $\psi^{ms1}$ and $p_b^{ms1}$ and then replacing $s^{ms1}_{r1}, s^{ms1}_{r2}, \rho^{ms1}_1$, and $\rho^{ms1}_2$ in equation (1), we get the profit of the manufacturer as

Now, our target is to Maximize $\Pi^{ms1}_M(g, s_{m1}, s_{m2})$ subject to $g > 0$, $s_{m1} > s_{r1}$, and $s_{m2} > s_{r2}$

Equations $\frac{\partial \Pi^{ms1}_M}{\partial g} = 0$, $\frac{\partial \Pi^{ms1}_M}{\partial s_{m1}} = 0$, and $\frac{\partial \Pi^{ms1}_M}{\partial s_{m2}} = 0$ yield values of $g = g^{ms1}$, $s_{m1} = s^{ms1}_{m1}$, and $s_{m2} = s^{ms1}_{m2}$; this will be the optimal solution if the $j^{th}$ order leading principal minor, $\Delta_j$ of the $HN_m$ corresponding to the profit function $\Pi^{ms1}_M(g^{ms1}, s^{ms1}_{m1}, s^{ms1}_{m2})$ take the sign $(-1)^j$, $j = 1, 2, 3$, i.e., $\Delta_1 < 0$, $\Delta_2 > 0$, and $\Delta_3 < 0$, where

Due to the longer expression of the manufacturer's profit function, we verify the condition numerically.

Replacing $g$, $s_{m1}$, and $s_{m2}$ by $g^{ms1}$, $s^{ms1}_{m1}$, and $s^{ms1}_{m2}$ respectively, in equation (10), we get the manufacturer's optimum profit.

4.5.2.   Manufacturer-Stackelberg (MS2) model 2

This is another case of the manufacturer Stackelberg model. In this game, the manufacturer plays the role of leader and other members are followers. Here, the two retailers play as a single member and jointly make strategies on selling prices and PEs to maximize their integrated profit.

The decision making power structures are:

Level 1: Manufacturer, max $\Pi^{ms2}_M(g, s_{m1}, s_{m2})$ subject to $g > 0$, $s_{m1} > s_{r1}$, and $s_{m2} > s_{r2}$

Level 2: Recycler, max $\Pi^{ms2}_{RC}(\psi, p_b)$ subject to $0 < (\psi+p_b) < \frac{1}{h}$, $p_b > 0$

Level 3: Two retailers (play jointly), max $\Pi^{ms2}_{JR}(s_{ri}, \rho_i)$ subject to $s_{ri} > 0$, $\rho_i > 0$ According to the Stackelberg game principle, the retailers jointly find out decisions on selling prices and PEs to maximize the integrated profit. Seeing the retailers' strategies, the recycler derives optimal $\psi$ and $p_b$ to optimize its own profit. Knowing the other members' strategies on $\psi$, $p_b$, $s_{ri}$, and $\rho_i$, the manufacturer optimizes its own profit for the decisions on $g$, $s_{m1}$, and $s_{m2}$.

Now, joint profit of the retailers,

Our objective is to Maximize $\Pi^{ms2}_{JR}(s_{ri}, \rho_i)$ subject to $s_{ri} > 0$, $\rho_i > 0$.

Proposition 4.3. $\Pi^{ms2}_{JR}(s_{ri}, \rho_i)$ is concave function of $s_{ri}$ and $\rho_i$ if the conditions in (13), (14), and (15) are satisfied.

Proof. See the Appendix C

Solving the equations $\frac{\partial \Pi^{ms2}_{JR}}{\partial s_{r1}} = 0$, $\frac{\partial \Pi^{ms2}_{JR}}{\partial s_{r2}} = 0$, $\frac{\partial \Pi^{ms2}_{JR}}{\partial \rho_1} = 0$, and $\frac{\partial \Pi^{ms2}_{JR}}{\partial \rho_2} = 0$, we get the optimal solution $(s^{ms2}_{r1}, s^{ms2}_{r2}, \rho^{ms2}_1, \rho^{ms2}_2)$.

The target of the recycler is to Maximize $\Pi^{ms2}_{RC}(\psi, p_b)$ subject to $0 < (\psi+p_b) < \frac{1}{h}$, $p_b > 0$.

Using proposition 2, it can be shown that $\Pi^{ms2}_{RC}(\psi, p_b)$ takes maximum value at

$\psi^{ms2} = \frac{h(C_s-x C_{mc})(D^{\prime \prime}_1+D^{\prime \prime}_2)}{h(D^{\prime \prime}_1+D^{\prime \prime}_2)-2l}$, $p^{ms2}_b = \frac{(C_s-x C_{mc})(hD^{\prime \prime}_1+hD^{\prime \prime}_2-l)}{h(D^{\prime \prime}_1+D^{\prime \prime}_2)-2l}$ if the condition $2l > h(D^{\prime \prime}_1+D^{\prime \prime}_2)$ holds, where $D^{\prime \prime}_1$ and $D^{\prime \prime}_2$ are obtained by substituting the values of $s^{ms2}_{r1}, s^{ms2}_{r2}, \rho^{ms2}_1$, and $\rho^{ms2}_2$ in $D_1$ and $D_2$.

The manufacturer makes own strategies on $g$ and $s_{mi}$ knowing the responses of the rest of the members.

In equation (1), after substituting the values of $\psi^{ms2}$, $p^{ms2}_b$, we replace $s^{ms2}_{r1}$, $s^{ms2}_{r2}$, $\rho^{ms2}_1$, $\rho^{ms2}_2$ and get the manufacturer's profit

Now, our target is to Maximize $\Pi^{ms2}_M(g, s_{m1}, s_{m2})$ subject to $g > 0$, $s_{m1} > s_{r1}$ and $s_{m2} > s_{r2}$.

Solving the simultaneous equations $\frac{\partial \Pi^{ms2}_M}{\partial g} = 0$, $\frac{\partial \Pi^{ms2}_M}{\partial s_{m1}} = 0$, and $\frac{\partial \Pi^{ms2}_M}{\partial s_{m2}} = 0$, we get a solution $g = g^{ms2}$, $s_{m1} = s^{ms2}_{m1}$, and $s_{m2} = s^{ms2}_{m2}$; it will be the optimal solution if all the eigenvalues of the $HN_m$ corresponding to the profit function $\Pi^{ms2}_M(g^{ms2}, s^{ms2}_{m1}, s^{ms2}_{m2})$ are negative. Substituting $g^{ms2}$, $s^{ms2}_{m1}$, and $s^{ms2}_{m2}$ in equation (7), manufacturer's optimum profit is obtained.

4.5.3.   Retailer Recycler-Stackelberg (RCS) model

In this game, the two retailers and the recycler unitedly play as a leader, whereas the manufacturer performs the follower's role. We consider fixed wholesale prices of the manufacturer equal to the obtained wholesale prices in the CS.

Here, the decision making power structures are:

Level 1: Retailers and recycler, max $\Pi^{rcs}_J(s_{ri}, \rho_i, \psi, p_b)$ subject to $s_{ri} > 0$, $\rho_i > 0$, $p_b > 0$, $0 < (\psi+p_b) < \frac{1}{h}$

Level 2: Manufacturer, max $\Pi^{rcs}_M(g)$ subject to $g > 0$.

The manufacturer optimizes its own profit for the decision on $g$. Knowing the manufacturer's response, the retailers and recycler unitedly find out optimal strategies on $s_{r1}, s_{r2}, \rho_1, \rho_2, \psi$, and $p_b$ to maximize their joint profit.

Now, our target is to Maximize $\Pi^{rcs}_M(g)$ subject to $g > 0$.

Proposition 4.4. $\Pi^{rcs}_M(g)$ is concave function of $g$ if the condition $g < \frac{1}{6\beta \delta}\Big{(}4bC_e\delta+\beta(-\gamma_1-\gamma_2+s_{r1}\zeta+s_{r2}\zeta-s_{r1}\eta-s_{r2}\eta-\lambda\rho_1+\mu\rho_1-\lambda \rho_2+\mu \rho_2)\Big{)}$ holds.

Proof. See the Appendix D

Solving $\frac{\partial \Pi^{rcs}_M}{\partial g} = 0$, we have the optimal value of $g$ = $g^{rcs}$.

The retailers and the recycler jointly decide their optimal strategies knowing the decision of the manufacturer.

Joint profit of the retailers and the recycler is

The problem is to

Maximize $\Pi^{rcs}_J(s_{ri}, \rho_i, \psi, p_b)$ subject to $s_{ri} > 0$, $\rho_i > 0$, $p_b > 0$, $0 < (\psi+p_b) < \frac{1}{h}$.

To obtain the optimum value of $\Pi^{rcs}_J(s_{ri}, \rho_i, \psi, p_b)$, we derive all the first and second order derivatives of the profit function with respect to $s_{r1}, s_{r2}, \rho_1$, and $\rho_2$. Solving the first order equations $\frac{\partial \Pi^{rcs}_J}{\partial s_{r1}} = 0$, $\frac{\partial \Pi^{rcs}_J}{\partial s_{r2}} = 0$, $\frac{\partial \Pi^{rcs}_J}{\partial \rho_1} = 0$, $\frac{\partial \Pi^{rcs}_J}{\partial \rho_2} = 0$, $\frac{\partial \Pi^{rcs}_J}{\partial \psi} = 0$, and $\frac{\partial \Pi^{rcs}_J}{\partial p_b} = 0$, we obtain the values $s_{r1} = s^{rcs}_{r1}$, $s_{r2} = s^{rcs}_{r2}$, $\rho_1 = \rho^{rcs}_{1}$, $\rho_2 = \rho^{rcs}_{2}$, $\psi = \psi^{rcs}$, and $p_b = p_b^{rcs}$. These values will be the optimal values if the $HN_m$ of $\Pi^{rcs}_J$ is negative definite at ($s^{rcs}_{r1}$, $s^{rcs}_{r2}$, $\rho^{rcs}_{1}$, $\rho^{rcs}_{2}$, $\psi^{rcs}$, $p_b^{rcs}$). Due to the complicated form of the profit function, we verify the above condition numerically by using Mathematica 11.1.1.

4.5.4.   Vertical Nash (VN1) model 1

In the Nash game, the players have the same decision power and have set their respective decisions independently and simultaneously. The manufacturer's target is to acquire optimal profit for the decision on the green level, whatever others may make. Irrespective of others, each retailer finds its strategies for selling prices and PEs to optimize its profit. In contrast, the recycler plan of action includes $EA_e$ and buy-back price to achieve maximum profit, ignoring others' plans.

To validate the Nash game in the proposed model, we assume that the manufacturer takes the decision on green level (g) only and wholesales the products to the retailers at a fixed price.

Here, the decision making power structures are:

Level 1: Manufacturer, max $\Pi^{vn1}_M(g)$ subject to $g > 0$

Level 1: Two retailers (play individually), max $\Pi^{vn1}_{Ri}(s_{ri}, \rho_i)$ subject to $s_{ri} > 0$, $\rho_i > 0$

Level 1: Recycler, max $\Pi^{vn1}_{RC}(\psi, p_b)$ subject to $0 < (\psi+p_b) < \frac{1}{h}$, $p_b > 0$.

Recalling propositions 4.4, 4.1, and 4.2, it can be verified that $\Pi^{vn1}_M(g)$ is concave on $g$, $\Pi^{vn1}_{R1}(s_{r1}, \rho_1)$ is a concave function of $s_{r1}$ and $\rho_1$, and $\Pi^{vn1}_{R2}(s_{r2}, \rho_2)$ is concave on $s_{r2}$ and $\rho_2$, $\Pi^{vn1}_{RC}(\psi, p_b)$ is a concave function of $\psi$ and $p_b$.

Therefore, solving the simultaneous equations $\frac{\partial \Pi^{vn1}_M}{\partial g} = 0$, $\frac{\partial \Pi^{vn1}_{R1}}{\partial s_{r1}} = 0$, $\frac{\partial \Pi^{vn1}_{R1}}{\partial \rho_1} = 0$, $\frac{\partial \Pi^{vn1}_{R2}}{\partial s_{r2}} = 0$, $\frac{\partial \Pi^{vn1}_{R2}}{\partial \rho_2} = 0$, $\frac{\partial \Pi^{vn1}_{RC}}{\partial \psi} = 0$, and $\frac{\partial \Pi^{vn1}_{RC}}{\partial p_b} = 0$, we get optimal solution $g^{vn1}, s^{nv}_{r1}, \rho^{vn1}_1, s^{vn1}_{r2}, \rho^{vn1}_2, \psi^{vn1}$, and $p^{vn1}_b$. Using the optimal values, individual profit of each member is obtained.

4.5.5.   Vertical Nash (VN2) model 2

In this model structure, all the chain members establish their own decisions independently with the condition that the two retailers play as a single member. In this model structure, manufacturer takes the decision on green level (g) only, and wholesales the products to the retailers at a fixed price.

The decision making power structures are:

Level 1: Manufacturer, max $\Pi^{vn2}_M(g)$ subject to $g > 0$

Level 1: Two retailers (play jointly), max $\Pi^{vn2}_{JR}(s_{ri}, \rho_i)$ subject to $s_{ri} > 0$, $\rho_i > 0$

Level 1: Recycler, max $\Pi^{vn2}_{RC}(\psi, p_b)$ subject to $0 < (\psi+p_b) < \frac{1}{h}$, $p_b > 0$.

The manufacturer aims to derive optimal profit for the decision on the green level irrespective of others' decisions. Without concerning others, both retailers jointly make decisions on RPs and PEs to optimize their profit. The recycler plans to find out $EA_e$ and buy-back price to achieve maximum profit regardless of others' decisions.

Recalling the propositions 4.4, 4.3, and 4.2, and solving the equations $\frac{\partial \Pi^{vn2}_M}{\partial g} = 0$, $\frac{\partial \Pi^{vn2}_{R1}}{\partial s_{r1}} = 0$, $\frac{\partial \Pi^{vn2}_{R1}}{\partial \rho_1} = 0$, $\frac{\partial \Pi^{vn2}_{R2}}{\partial s_{r2}} = 0$, $\frac{\partial \Pi^{vn2}_{R2}}{\partial \rho_2} = 0$, $\frac{\partial \Pi^{vn2}_{RC}}{\partial \psi} = 0$, and $\frac{\partial \Pi^{vn2}_{RC}}{\partial p_b} = 0$, we get the optimal solution $g^{vn2}, s^{nvc}_{r1}, \rho^{vn2}_1, s^{vn2}_{r2}, \rho^{vn2}_2, \psi^{vn2}$, and $p^{vn2}_b$. Using the optimal values, individual profit of the members is achieved.

5.   Discussion of results

In this section, with the help of a numerical illustration, we examine the sensitivity of the essential parameters as well as the behavior of the present model.

5.1.   Numerical example

We analyze the proposed model numerically under different model structures. Due to the difficulty of accessing accurate industry data, we considered some hypothetical data from previous related research that was compatible with our model assumption. We adopt input parameter values of earlier studies (Bai et al. (2019) and Mondal and Giri (2022)) as far as possible. As our model is somehow dissimilar to the previous literature, some additions and modifications of data are made without violating model assumptions. We use input parameters data of Table 3 to perform the numerical experiments. Tables 4 and 5 present the various optimal outcomes of different systems for the input data.

5.1.1.   Numerical observation

The following observations are drawn from Tables 4 and 5. The chain profit meets with the highest value in the CS among all other Stackelberg and Nash models. All the decision variables take the highest value in the CS compared with other systems, and the resultant effect lifts chain profit to the peak. The manufacturer collects maximum individual profit in the MS1 model. It is evident since the manufacturer makes extreme GT investments and charges a higher wholesale price to retailers. Each retailer acquires the highest personal profit in the VN2 model, where they jointly play to optimize their profit. It is entirely rational as the retailers' RPs are reasonably high with moderate PEs compared with other models. The recycler achieves maximum individual profit in the CS, where PR meets the desired level due to the highest value $EA_e$ and buy-back price. Among DS, maximum PR occurs in the RCS model as the retailers and recyclers jointly play the role of leader. The green level of the product takes the highest value in the manufacturer's Stackelberg model, among other decentralized structures, as the manufacturer performs as a leader. The green investment works significantly in the proposed model; whenever $g$ increases, the emission amount decreases correspondingly. The used PR is also effective in the model; increasing $EA_e$ and the buy-back price increases the recovery rate. From a profit perspective, a thorough inspection reveals that the CS is the most acceptable and desirable model for all DS for the chain.

5.2.   Discussion on parameters' sensitivity

We examine the sensitivity of the decision variables along with individual member profit, chain profit, CE, and PR rate for all scenarios with the changes of the critical parameters $k_1, k_2, \delta, \beta, \eta, \mu$, and $\lambda$ by fixing the remaining parameters' value as mentioned in the Subsection 5.1. The sensitivity analysis with respect to the parameters shows the stability and reliability of the work. The analysis shows that the model is not only appropriate for fixed data, but it is also applicable within a range of the given data. Table 5 (see Appendix E) presents the variation of decision variables, CE, and PR rate. Moreover, percentage changes of the individual profit and chain profit corresponding to the changes of the parameters for all game approaches are depicted in Figures 3 to 6.

5.2.1.   Effects of promotional effort cost coefficients $k_1$ and $k_2$

From Table 6, the impact of $k_1$ and $k_2$ are quite significant on effort level $\rho_1$, and $\rho_2$ respectively, for all scenarios, whereas trivial changes are noticed in all other decision variables. Both parameters have a marginal effect on CE in all model structures. Since increasing $k_1$ results in higher promotional costs, the first retailer makes a substantial decrease in $\rho_1$, corresponding to lower customer demand. Consequently, individual and chain profits take downward movement in all game approaches except the second retailer's profit, which increases interestingly due to the competing behavior among the retailers (see Figure 3). Figure 3b reveals that the first retailer's profit percentage change is lower in the VN1 model, as chain members make independent decisions. Again, higher values of $k_2$ make promotional costs more significant for the second retailer, which is why they have to reduce effort level to balance expenditure. Therefore, individual member and chain profits decrease except the first retailer's profit, which catches upward movement (depicted in Figure 4).

The above result indicates that each retailer's promotion effort positively influences individual profit. If one retailer's PE cost coefficient increases, then the PE level automatically decreases; for that, the retailer has to decrease the retail price to achieve higher profit. Due to the retailers' rivalry, one retailer's increasing PE cost coefficient provides an opportunity for the other retailer to achieve higher profit.

5.2.2.   Effect of parameters $\beta$ and $l$

The green level and CE are susceptible to the parameter $\beta$ compared with others for every system, presented in Table 6. We notice that CE boosts up exceptionally due to a splendid reduction of $g$ with higher $\beta$. For the higher green cost, the manufacturer has to reduce the product's green level, which decreases customer demand. As a consequence, the individual profit of players and chain profit move downwards in every game structure with increasing $\beta$ (Figure 5). Again, Table 6 illustrates that $l$ significantly impacts $EA_e$ and moderately impacts buy-back price in all game models. Increasing $l$ leads $EA_e$ level downwards, which leads to a lower PR rate. In Figure 6, recycler profit and chain profit fall in all game approaches due to the reduction of PR. Each retailer's profit is inversely proportional to $l$ for all scenarios except RCS, where corresponding profits are lifted due to retailers' dominating powers.

The above result shows that the higher green investment cost coefficient lowers the green level, and for that, all the members have to decrease their corresponding selling prices to avoid a non-profitable situation. Again, when the investment cost coefficient is lower, the manufacturer can reduce the emissions amount by spending more on green investments. A lower EA cost coefficient increases PR, and all members achieve higher profits.

5.2.3.   Effect of $\delta$

Table 6 reflects that increasing $\delta$ generates a higher green level, leading to greater customer demand. From Figure 7, we observe that each member's and chain profits hike up in all model structures due to the collective positive impact of demand increment.

Here, the higher values of $\delta$ make CE lower, and the reduction in CE cost compels all the chain members into a better profitable situation.

5.2.4.   Effect of price-sensitive parameters $\zeta$ and $\eta$

The $\zeta$ is the most hypersensitive parameter among all. Table 6 and Figure 8 reveal that all the decision variables, CE, along with individual and chain profit, are severely affected by $\zeta$ for all model structures. With higher $\zeta$, RP and PE catch downward movement, resulting in unprecedented customer demand, which leads to acute decrement in both retailers' profit. Surprisingly, the green level increases in CS, MS1, and MS2 but decreases in RCS, VN1, and VN2 models, reflecting its impact on CE. Rigorous changes in manufacturer profit in CS, MS1, and MS2, retailers' profit in RCS, VN2, and VN2, and recycler's profit in VN1 and VN2 are noticed with switching $\zeta$ because each player achieves a higher profit in the mentioned scenarios than in other scenarios. Again, $\eta$ is the second hypersensitive parameter. In Table 5, remarkable RP, PE, and $EA_e$ increments are observed with uprising $\eta$ for CS and all DS. Increasing RP and PE corresponds to higher demand for that individual and chain profits rise in all game approaches (see Figure 9).

The above result shows that both retailers must decrease their corresponding retail prices to maintain a profitable situation with the increasing price-sensitive parameter.

5.2.5.   Effect of PE-sensitive parameters $\lambda$ and $\mu$

The $\lambda$ causes sharp changes in PE levels for all game models, whereas the remaining decision variables are minor sensitive. Table 6 reveals that, with ascending $\lambda$, the effort level of both retailers $\rho_1$ and $\rho_2$ take upward values, which corresponds to higher demand. Due to the impact of demand hiking, the members' profit and chain profit lift for all scenarios (depicted in Figure 10). Again, we observe from Table 6 that $\mu$ creates minor changes to all the decision variables. As $\mu$ increases, effort levels of both retailers decrease, which impacts negatively the customer demand and that individual and chain profits fall in all game structures (illustrated in Figure 11).

The above result reveals that increasing promotional influence parameter makes higher PE and, for that reason, both retailers' acquire lofty profit.

5.2.6.   Effect of carbon cap parameter $C$

Figure 12 illustrates that the manufacturer's profit is influenced by the carbon emission quota. With the rising carbon emission limit, profit of the manufacturer increases in all the model structures.

Here, the manufacturer has to pay for lower additional emissions units than before if the offered carbon quota is higher. As a result, the CE costs of the manufacturer have been reduced and the manufacturer achieves greater profit.

5.2.7.   Effect of the parameter $\alpha$

Figure 13 illustrates that the manufacturer's profit is severely influenced by $\alpha$. With the increasing $\alpha$, profit of the manufacturer decreases in all the model structures. Table 5 reveals that $\alpha$ causes sharp changes in CE amount for all game models, whereas the changes of remaining decision variables can be neglected. Due to the increasing production rate of imperfect items, the manufacturer has to increase overall production quantity to satisfy both retailers' demand. Consequently, more production generates higher CE. As a result, the production costs and carbon emissions costs of the manufacturer increase, which causes lower profit for the manufacturer.

5.3.   Managerial implications

In the proposed study, some significant findings with managerial implications are derived, which can be utilized by the chain members.

● The model explores that GT investment is effective in reducing CE. When the green investment cost coefficient is higher (the green level is lower), the manufacturer and the retailers must decrease their selling prices to maintain the market demand; otherwise, they face profit loss. Therefore, business organizations could abate and restrict emissions using green investment and gain higher economic and environmental growth.

● The sensitivity results highlight that PE positively impacts product selling. If the PE cost coefficient of one retailer is more significant (PE level is lower), then the retailer has to decrease their retail price to achieve more substantial profit, whereas another retailer may hike up their retail price due to the competitive behavior between them. Therefore, chain members who know the PE strategy could increase market demand and their profits to a satisfactory level.

● The selling price plays an essential role in enhancing market sales. When the price sensitivity parameter ($\zeta$) increases, both players have to lower their corresponding selling prices and may increase GT investment to adjust demand and reputation in the market. Therefore, chain members with the proper pricing strategy could enhance chain operations and attain profit goals.

● To reduce production costs, curtail emissions, manage waste, balance natural resources, and move towards sustainable development, chain members prefer PR. The model presents that the $EN_e$ and buy-back price-sensitive PR rate effectively recover used products. When the $EN_e$ cost coefficient increases, the recycler has to offer a lower buy-back price to adjust the PR rate. Therefore, a chain member who is aware of product recycling strategies could promote chain performance by fulfilling environmental goals for more significant economic benefit.

6.   Concluding remarks

In the present situation, due to shortages of natural resources and rapid increment in environmentally conscious customers, product recycling and low-carbon products are getting intense attention not only from the manager of the supply chain but also from researchers in supply chain management. This article explores a green environment SCnM with imperfect production and recycling of used products under the governmental initiative CT policy. The recycler invests in EA and offers the best buy-back price to the end customers to enhance PR. This study considers the rivalry between the retailers in the RP and PE-based market. A CS and five different DS are presented to analyze the proposed model. In environmental and economic aspects, the following results are examined:

1. Green level and recovery rate attain the highest value in the CS, and their effectiveness is satisfactory in the present study.

2. In a competitive market, one retailer's demand is more severely sensitive to RP than PE.

3. The CS yields a more significant overall profit by enhancing chain performance compared with DS.

4. Among the DS, integrated chain profit is highest in the Vertical Nash 1 model, close to the CS.

From the above insights, the model's implications are as follows: The model demonstrates that GT investments effectively reduce CE. In this way, business organizations could reduce their emissions and achieve higher economic and environmental growth using green investment. PE has a positive impact on product sales. As a result, chain members who know the PE strategy could increase their profits to a satisfactory level. A chain member can increase profits and enhance chain operations with the appropriate pricing strategy. Chain members prefer PR to reduce production costs and manage waste. Therefore, a chain member who is aware of product recycling strategies can contribute to chain performance by meeting environmental goals while gaining significant economic benefits. By exercising this model, the chain managers with detailed operational information on proper pricing strategy, green investment, knowledge of the PE, $EA_e$, and proper buy-back price technique could enhance the chain performance from both environmental and economic perspectives.

According to the present study, considering a single green manufacturer without a separate remanufacturing unit under deterministic demand is the main limitation of our research. The proposed model should have considered product and product quality shortages due to uncertainty phenomena in the supply and production system. The proposed model may be extended immediately, incorporating the above issues. For future research, one can extend the model under trade credit policy with partial payment and inflation. Another extension may be possible by including multiple manufacturers, retailers, etc. One is to introduce manufacturers and retailers instead of recyclers for used PR in future studies. To ensure a win-win situation, an agreement between the players to coordinate the chain members will be worth investigating in a future study. The present model can be explored by analyzing other emission reduction incentives implemented by the government in further research.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflicts of interest.

A.   Proof of the Proposition 4.1

Proof. Solving $\frac{\partial \Pi^{ms1}_{R1}}{\partial s_{r1}} = 0$, $\frac{\partial \Pi^{ms1}_{R1}}{\partial \rho_1} = 0$, $\frac{\partial \Pi^{ms1}_{R2}}{\partial s_{r2}} = 0$, and $\frac{\partial \Pi^{ms1}_{R2}}{\partial \rho_2} = 0$, we obtain the solution

where, $E_1 = k_1 k_2(4 \zeta ^2-\eta ^2)+(k_1+k_2)(\eta \lambda \mu -2 \zeta \lambda ^2) + \lambda ^2(\lambda ^2 - \mu ^2)$,

$E_2 = k_1 k_2(\gamma_2 \eta +2 \gamma_1 \zeta +\zeta \eta s_{m2}+2 \zeta ^2 s_{m1})+k_2(\eta \lambda \mu s_{m1}-2 \zeta \lambda ^2 s_{m1})$+$k_1(\zeta \lambda \mu s_{m2}-\gamma_2 \lambda \mu -\gamma_1 \lambda ^2 -\eta \lambda ^2 s_{m2}-\zeta \lambda ^2 s_{m1})+\lambda ^2 s_{m1}(\lambda ^2-\mu ^2)$,

$E_3 = k_1 k_2(\gamma_1 \eta +2 \gamma_2 \zeta +\zeta \eta s_{m1}+2 \zeta ^2 s_{m2})+k_1(\eta \lambda \mu s_{m2}-2 \zeta \lambda ^2 s_{m2})$+$k_2(\zeta \lambda \mu s_{m1}-\gamma_1 \lambda \mu -\gamma_2 \lambda ^2 -\eta \lambda ^2 s_{m1}-\zeta \lambda ^2 s_{m2})+\lambda ^2 s_{m2}(\lambda ^2-\mu ^2)$,

$E_4 = k_2(\gamma_2 \eta \lambda +2 \gamma_1 \zeta \lambda +\zeta \eta \lambda s_{m2}-2 \zeta ^2 \lambda s_{m1}+\eta ^2 \lambda s_{m1})$+$(\zeta \lambda ^2 \mu s_{m2}-\gamma_2 \lambda ^2 \mu -\gamma_2 \lambda ^3 -\eta \lambda ^3 s_{m2}+\zeta \lambda ^3 s_{m1}-\eta \lambda ^2 \mu s_{m1})$,

$E_5 = k_1(\gamma_1 \eta \lambda +2 \gamma_2 \zeta \lambda +\zeta \eta \lambda s_{m1}-2 \zeta ^2 \lambda s_{m2}+\eta ^2 \lambda s_{m2})$+$(\zeta \lambda ^2 \mu s_{m1}-\gamma_1 \lambda ^2 \mu -\gamma_2 \lambda ^3 -\eta \lambda ^3 s_{m1}+\zeta \lambda ^3 s_{m2}-\eta \lambda ^2 \mu s_{m2})$

The $HN_m$ of the profit function $\Pi^{ms1}_{Ri}$ at $(s^{ms1}_{ri}, \rho^{ms1}_i)$ is

Therefore, $H_{Ri}$ is negative definite if $2\zeta k_i-\lambda^2 > 0$.

Hence, the profit function $\Pi^{ms1}_{Ri}(s_{ri}, \rho_i)$ is maximum at $(s^{ms1}_{ri}, \rho^{ms1}_i)$ if the condition $2\zeta k_i-\lambda^2 > 0$ holds.

B.   Proof of the Proposition 4.2

Proof. Solving the equations $\frac{\partial \Pi^{ms1}_{RC}}{\partial \psi} = 0$, $\frac{\partial \Pi^{ms1}_{RC}}{\partial p_b} = 0$, we get the solution $\psi^{ms1} = \frac{h(C_s-x C_{mc})(D^{\prime}_1+D^{\prime}_2)}{h(D^{\prime}_1+D^{\prime}_2)-2l}$, $p^{ms1}_b = \frac{(C_s-x C_{mc})(hD^{\prime}_1+hD^{\prime}_2-l)}{h(D^{\prime}_1+D^{\prime}_2)-2l}$.

This solution is optimal if the $HN_m$ of the profit function is negative definite at $(\psi^{ms1}, p^{ms1}_b)$.

The $HN_m$ of the profit function $\Pi^{ms1}_{RC}$ at $(\psi^{ms1}, p^{ms1}_b)$ is

Hence, the $HN_m$ is negative definite if the condition $2l > h (D^{\prime}_1+D^{\prime}_2)$ is satisfied.

Therefore, $\Pi^{ms1}_{RC}$ is maximum at $\psi^{ms1} = \frac{h(C_s-x C_{mc})(D^{\prime}_1+D^{\prime}_2)}{h(D^{\prime}_1+D^{\prime}_2)-2l}$, $p^{ms1}_b = \frac{(C_s-x C_{mc})(hD^{\prime}_1+hD^{\prime}_2-l)}{h(D^{\prime}_1+D^{\prime}_2)-2l}$, if the condition $2l > h(D^{\prime}_1+D^{\prime}_2)$ holds.

C.   Proof of the Proposition 4.3

Proof. The $HN_m$ of the profit function $\Pi^{ms2}_{JR}(s_{ri}, \rho_i)$ is

The matrix $H^{ms2}_{JR}$, is negative definite if the $p^{th}$ order leading principal minor, $\Delta_p$ take the sign $(-1)^p$ for $p=1, 2, 3, 4$, i.e., $\Delta_1 < 0$, $\Delta_2 > 0$, $\Delta_3 < 0$, $\Delta_4 > 0$.

Here,

Hence, the profit function $\Pi^{ms2}_{JR}$ is concave on $s_{r1}$, $s_{r2}$, $\rho_1$, and $\rho_2$ if the conditions of (13), (14), and (15) hold.

D.   Proof of the Proposition 4.4

Proof. Substituting the values of $D_1$ and $D_2$ in equation (11), we obtain the first and second order derivative of $\Pi^{rcs}_M(g)$ with respect to $g$

Now, $\Pi^{rcs}_M(g)$ will be concave if the second order condition $\frac{\partial^2 \Pi^{rcs}_M}{\partial g^2} < 0$ is satisfied.

From the equation (16),

Hence, $\Pi^{rcs}_M(g)$ is concave in $g$ if the condition $g < \frac{1}{6\beta \delta}\Big{(}4bC_e\delta+\beta(-\gamma_1-\gamma_2+s_{r1}\zeta+s_{r2}\zeta-s_{r1}\eta-s_{r2}\eta-\lambda\rho_1+\mu\rho_1-\lambda \rho_2+\mu \rho_2)\Big{)}$ holds.

E.   Sensitivity table