An Investigation of Subsidy Policies on Recycling and Remanufacturing System in Two-Echelon Supply Chain for Negative Binomial Distribution

Abstract

This study investigates a two-stage production–inventory model with subsidy policies for paper cup recycling. The model includes remanufacturers, recyclers, and consumers, taking into account their preferences for different recycling channels. The negative binomial distribution of investment fund w is introduced and briefly studied. The influence of various subsidy strategies on the optimal pricing, profit, and recycling volume of the reverse supply chain is discussed. Numerical simulations show that increased consumer recycling preferences positively impact the recycling volume and profit. When subsidies are limited, subsidizing remanufacturers leads to higher recycling volumes, while subsidizing consumers results in higher profits at lower-to-middle subsidy levels. The findings suggest that policymakers can leverage different subsidy strategies to effectively manage the paper cup recycling supply chain and promote sustainability by incentivizing key stakeholders to participate in the recycling process. For example, subsidizing remanufacturers can increase the overall recycling volume by making it more financially viable for them to collect and process used cups, while subsidizing consumers can boost their participation and willingness to properly dispose of cups for recycling, leading to higher profits for the reverse supply chain.

1. Introduction

Over the past seven decades, the use of paper materials in various structural and non-structural applications has seen considerable expansion, garnering attention from numerous industrial sectors (Khalid et al. [1]). The production and disposal of paper materials significantly influence a company’s carbon footprint through various stages of their life cycle. Understanding these impacts is crucial for companies aiming to reduce their environmental footprint. The carbon footprint of paper production varies widely, with specialty paper products showing emissions between 0.95 to 1.83 tCO₂e/t paper, influenced by raw material and energy consumption (Lai et al. [2]). Nonfood residues, while promoting circularity, can have a higher carbon footprint than wood-based products due to inefficient recycling and production processes (Suarez et al. [3]). Managing the recovery of paper containers presents unique challenges for organizations. The disassembly process can be complex, with a need to consider the various components and materials that must be handled differently (White et al. [4]). Researchers have explored various perspectives on product recovery, including identifying unique characteristics of disassembly processes (Priyono [5]) and developing taxonomies for resource recovery from end-of-life products. One study examined the specific challenges managers face in computer and electronics product recovery, as well as the environmental considerations involved in these decisions. Researchers have also investigated the concept of a “sustainable supply chain”, proposing a re-classification of the 6R methodology to optimize product utilization and reduce waste at the post-use stage (Kuik et al. [6]). Effective information sharing between the end-of-life stage and the product design stage is crucial for enabling more sustainable product development (Lee et al. [7]). To establish an effective unit recycling subsidy per return, a multifaceted approach is necessary, integrating economic incentives, regulatory frameworks, and operational strategies. The following key aspects are essential for optimizing recycling subsidies. Implement a deposit–refund policy that charges disposal fees for new products while providing subsidies for recycling used products. This dual regulation can help balance the financial aspects of recycling operations (Liu et al. [8]). Utilize a methodological framework to determine optimal subsidies based on the Relative Cost Decrease (RCD) from recycling, which aids in decision-making for subsidizing recycling investments (Batzias [9]). Differentiate subsidy rates based on the recycling process’s environmental impact and efficiency, encouraging industries to adopt advanced technologies for better recycling outcomes (Chang et al. [10]; Shih et al. [11]; Yuan et al. [12]; Yu et al. [13]). While these strategies can enhance recycling rates, challenges remain, such as ensuring sufficient funding and managing the complexities of subsidy distribution effectively. Recycling paper faces significant challenges due to insufficient infrastructure and the mixed quality of recovered materials. A substantial gap remains between paper production and recycling, despite progress in this area. Enhancing paper recovery and recycling is an ongoing effort, with the ultimate goal of a closed-loop system where paper and packaging waste re-enters the production cycle, minimizing the need for raw materials. There has been little literature published on subsidy distribution. The purpose of the research presented in this study is to examine the relationship between recycling rates and subsidy distribution.

2. Literature Review

The economic production quantity model proposed by Taft [14] in 1918 examined a single-product, single-stage production system with a perfect process and constant demand. The objective is to determine the optimal production quantity that minimizes total costs, including setup and holding costs. Researchers have expanded this model to address real-world manufacturing scenarios, such as multi-stage production systems, multiple products, imperfect processes, inspection and rework, preventive maintenance, resource constraints, and complex scheduling. Price has the most significant impact on market demand and profits. When demand is sensitive to price, the quantities ordered or produced and the prices are closely linked. Many researchers have studied inventory systems assuming that demand is dependent on price in a linear manner. Su et al. [15] examined the impact of two common corporate social responsibility initiatives on a two-stage assembly production system characterized by multiple components and imperfect processes. The analysis is conducted under the assumption that consumer demand is influenced by both selling price and the firm’s CSR initiatives. Taleizadeh et al. [16] introduced an EPQ inventory model integrating pricing, backorders, and rework. It optimizes selling price, lot size, and backorders to maximize total profit, enhancing inventory management realism. Pando et al. [17] presented an inventory model with price- and stock-dependent demand to maximize profitability, determining optimal pricing strategies integrated with inventory management decisions. Saraswat and Sharma [18] considered price-dependent demand, mortality, and deterioration, aiming to optimize total cost by determining optimal ordered quantity and cycle length, with partial backlogging of shortages. Tsoularis and Wallace [19] addressed optimal inventory pricing and ordering strategies considering stock- and price-dependent demand, deriving optimal policies for inventory, order rate, price, and maximum price bounds. San-José et al. [20] addressed an EOQ model with price-dependent demand, aiming to determine optimal pricing, lot size, and inventory cycle for maximizing profit under full backordering and time-dependent demand. Pando et al. [17] explore maximizing returns on inventory management expenses in a system with price- and stock-dependent demand rates, indicating a potential link between optimal pricing and inventory management strategies. Singer and Khmelnitsky [21] addressed a production–inventory problem with price-sensitive demand, focusing on optimal pricing strategies within an integrated inventory model to meet price-dependent demand effectively. Widyadana et al. [22] considered pricing for items with price-dependent demand, aiming to determine the best price and replenishment time to maximize profit efficiently. Price is the factor with the greatest influence over market demand and profits. In cases where demand is price sensitive, order/production quantities and prices tend to be strongly interdependent. Many researchers have considered inventory systems under the assumption of linear price-dependent demand. Reverse logistics, which handles product returns, is another key component of a circular economy. Achieving a circular economy requires the implementation of effective product recovery systems that can maximize the value from returned products through various strategies such as reuse, remanufacturing, and recycling. Yu et al. [13] addressed the optimal pricing model for recycling products that are returned, recovered, and shipped to a secondary market for resale. Lin and Lin [23] examined the problem of a single-vendor, single-buyer integrated supply chain inventory system with price-dependent demand and product recovery. Karim and Nakade [24] employed systematic methods to undertake a survey that evaluated studies across two scenarios: a sustainable economic production quantity model accounting for carbon emissions from inventory storage and production, and a sustainable EPQ model incorporating product recycling, as well as a reverse logistics model that considers emissions and product recycling. Hallak et al. [25] focused on developing environmentally responsible inventory policies and models that could help green supply chains. Their research also presents a numerical study to compare the proposed models and quantify the trade-off between in-house and outsourced product recovery processes. Parsa et al. [26] examined a closed-loop supply chain system where a recycling facility and a supplier, respectively, provide recycled and virgin raw materials to a manufacturer. Karmakar and Das [27] examined an integrated supply chain model with concurrent production and refurbishment operations, where the manufacturer aggregates all returned items in a dedicated inventory, then dispatches them for refurbishment. Liao et al. [28] applied an optimizing mathematical analysis to model automobile engine remanufacturing in a joint manufacturing system, where the quantity and quality of procured materials, as well as market demand, are uncertain. Their goal was to maximize resource utilization and profitability. Rabta [29] introduced an EOQ inventory model in the context of a circular economy. We assume the manufactured product can be produced with varying degrees of circularity, quantified by an index. Some researchers have investigated the impact of carbon pricing and emissions policies (Huang et al. [30]; Tseng et al. [31]). A higher circularity index corresponds to a greater proportion of recycled materials incorporated into the product. This research investigates the management of a recycling fund through the use of differentiated subsidy rates, with the aim of incentivizing greater effort towards environmental sustainability. The Recycling Fund Management Board in Taiwan targets the environmental goal of maximizing the overall recycling rate by providing subsidies to the paper industry. For the same reason,

Table 1. Comparison between present model and related previous research.

Authors (Year)	Model Type	Product Reuse	Recycling Fund	Recovery Rate	Price- and Stock- Dependent Demand
Taleizadeh et al. [16]	EPQ				√
Saraswat and Sharma [18]	EPQ				√
Tsoularis and Wallace [19]	EPQ				√
San-José et al. [20]	EOQ				√
Pando et al. [17]	EOQ				√
Singer and Khmelnitsky [21]	EPQ				√
Widyadana et al. [22]	EPQ				√
Yu et al. [32]	EOQ	√		√	√
Lin and Lin [23]	EOQ			√	√
Present model	EOQ	√	√	√	√

summarizes previous related research works.

This gap in the research motivated us to define an alternative demand model capable of accounting for other types of common recycling activity.

3. Problem Description

The carbon emissions generated by producing a paper cup are actually lower than those of a reusable cup. However, a reusable cup can be used more than 300 times under normal wear and tear. If used without a lid, a reusable cup only needs to be used 17 times to produce less carbon emissions than a paper cup. However, a reusable cup with a lid needs to be used more than 549 times. Both disposable paper cups and reusable cups incur delivery costs to the store and post-use transport costs, generating carbon emissions from logistics. However, transportation is less significant, and the key lies in the environmental impact of cleaning. Reusable cups sent back to the factory for cleaning use industrial dishwashers, allowing for large-scale cleaning that consumes significantly less water compared to individual hand washing. After using a reusable cup, the general public can simply rinse it with cold water without needing detergent or excessive wiping, minimizing the waste of repeated cleaning. Three issues that need to be resolved in this paper are:

(1)

What is the optimal production run time?

(2)

What is the optimal recycling subsidy per return?

(3)

How to reduce the production cost and optimize recovery rate?

This paper examines a two-stage production–inventory system with imperfect processes, focusing on subsidy policies, recycling, and remanufacturing using the system. Section 4 presents the notations and assumptions, which serve as the basis for the mathematical formulations and theoretical results. Section 5 introduces a real-world reverse supply chain scenario from the pulp and paper industry to enhance the relevance of the model, including numerical examples and a sensitivity analysis to provide managerial insights. Finally, Section 6 offers concluding remarks and suggestions for future research.

4. Model Formulation

Before developing the mathematical model, this section first lists the notation used and the assumptions required for the proposed model. It is hereby stated as follows:

4.1. Notation

$k$	set-up cost
$s_p$	selling price per unit item
$\mathrm{n}$	number of required components for a green finished product
$p_i$	production rate of semi-finished products $\mathrm{i}$ in units per unit time, where $\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}$ and ${\mathrm{p}}_1>{\mathrm{p}}_2>\dots >{\mathrm{p}}_{\mathrm{n}}$
${\lambda }_{gf}$	assembly rate of the green finished product in units per unit time
$p_{re}$	production rate of the returned product
$\sigma$	return rate of reusable product
$\phi$	coefficient of social cost of work stress
$c_p$	purchasing cost of FSC material per unit
$h_i$	holding cost of semi-finished products $\mathrm{i}$ per unit time, where $\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}$
$h_{gf}$	holding cost of green finished product per unit time
${\theta }_i$	defect rate of semi-finished products
${\theta }_{gf}$	defect rate of green finished product
$t_{id}$	time period prior to depletion of inventory of semi-finished products $\mathrm{i}$
$t_{gfd}$	time period prior to depletion of inventory of green finished product
$t_{gf}$	production run time of green finished product
$\mathrm{T}$	length of cycle
$H_i$	maximum inventory level of semi-finished products $\mathrm{i}$
$H_{gf}$	maximum inventory level of green finished product
$H_{re}$	maximum inventory level of reusable product
$t_i$	the production run time of semi-finished products $\mathrm{i}$ (decision variables)
$\mathrm{s}$	unit recycling subsidy per return (decision variables)
$\mathrm{r}$	recovery rate of reusable product (decision variables)

4.2. Assumptions

• To avoid the situation where one stage starves due to a lack of input from the previous stage, it is necessary that the minimum production rate in Stage 1, the assembly rate in Stage 2, and demand rate satisfy the following condition: ${\mathrm{p}}_{\mathrm{n}}>{\mathsf{\lambda }}_{\mathrm{g}\mathrm{f}}>\mathrm{D}$.
• Process quality is assumed to be independent in the two stages, and the inspection time is so short that it can be disregarded. The rework time for defective items is also disregarded.
• To reduce the usage of disposable cups, provide a reusable cup rental service by improving one’s corporate image. This inevitably requires investment, including the fixed costs for equipment per cycle ${\mathrm{F}}_{\mathrm{g}}$ (e.g., reusable cup vending machine service, APPs) and variable costs per unit time of operations ${\mathrm{V}}_{\mathrm{g}}$ (e.g., material, water, and labor).
• To extend the demand pattern proposed by Modak et al. [33], demand is affected by selling price, recycling subsidy, and investment in borrow-and-return cup programs. Figure 1 presents the relationship between the impact of two sustainable product activities and customer demand. Model formulation was facilitated by assuming that demand is a simple linear function, $\mathrm{D}=\mathrm{a}-\mathrm{b}{\mathrm{s}}_{\mathrm{p}}+\mathsf{\rho }\mathrm{s}+\mathrm{w}\mathrm{v}$, where $\mathrm{a}>0$ is the market potential, $\mathrm{b}$ is the elasticity factor of selling price, $\mathsf{\rho }$ is the elasticity factor of manufacturer recycling subsidy, and $\mathrm{w}~\mathrm{B}\mathrm{i}\mathrm{n}(\mathrm{n},\mathrm{p})$ is investing in a fund for borrow-and-return cup programs, where $\mathrm{n}$ is the frequency of investment decisions; $\mathrm{p}$ is the probability of success in a borrow-and-return cup program investment. In the current paper, we set $\mathrm{v}$ as Return on Investment (ROI) to determine whether the investment in borrow-and-return cup program is indeed advantageous. Further, it is common that ODMs company decides the optimal recovery rate instead of adjusting the selling price due to carbon price to maximize profit. Therefore, the selling price is assumed to be a given parameter in this model.
• When estimating the carbon footprint of a mechanical product at the conceptual design stage, a carbon footprint calculation model is firstly needed. According to the definition of carbon footprint (PAS-2050 [34]) and product life cycle, the contribution of carbon footprint could be classified into five stages for the entire life cycle of a product: acquisition of raw materials stage (design stage ${\mathsf{\delta }}_1$), manufacturing stage (finished product ${\mathsf{\delta }}_{21}$; components ${\mathsf{\delta }}_{22}$; scrap returns ${\mathsf{\delta }}_{23}$), transportation stage, usage stage, and recycle and disposal stage (He et al. [35]).

Figure 1 graphs the inventory levels for the proposed two-stage assembly production system throughout one cycle. This figure reveals the relationships between green finished goods, semi-finished goods, and scrap returns. The researchers developed production models to investigate the effects of various subsidy policies, including recycling subsidies, on the remanufacturing system in the context of Taiwan disposable paper and plastic packaging manufacturers. They conducted a detailed analysis to assess the individual and comparative impacts of four distinct subsidy policies as well as mixed-subsidy approaches on both recycling and remanufacturing activities.

• R1. The volume of required components, end product yield, and total demand do not change within a given cycle; i.e.,

  $$t_i={\displaystyle \frac{p_n{t}_n}{p_i}}$$

  (1)

  and

  $$T={\displaystyle \frac{p_n{t}_n}{D}},$$

  (2)

  where $i=\mathrm{1,2},\dots ,n.$
• R2. The maximum inventory level of component $i$ can be written as

  $$Q_i=\left(p_i-{\lambda }_{gf}\right)t_i={\lambda }_{gf}{t}_{id}.$$

  (3)

After rearranging Equation (3), we obtain the following:

$$\begin{array}{ll}{t}_{id}& ={\displaystyle \frac{\left(p_i-{\lambda }_{gf}\right)t_i}{{\lambda }_{gf}}}\\ & ={\displaystyle \frac{(p_i-{\lambda }_{gf})p_n{t}_n}{{\lambda }_{gf}{p}_i}}.\end{array}$$

(4)

where $\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}.$

• R3. The maximum inventory level of the finished product can be described as follows:

  $$\begin{array}{ll}{H}_e& =\left({\lambda }_{gf}-D\right)t_e\\ & =\left({\lambda }_{gf}-D\right){\displaystyle \frac{p_n}{{\lambda }_{gf}}}{t}_n.\end{array}$$

  (5)
• R4. The maximum inventory level of the returned product can be described as follows:

  $$H_{re}=\sigma D{t}_{red}=p_{re}{t}_{red}.$$

  (6)

Based on the above results, the elements of the total profit per cycle are established as follows:

(a)

Sales revenue (denoted by SR): The sales revenue is equal to the actual operating revenue, ${\mathrm{s}}_{\mathrm{p}}-\mathrm{s}$, multiplied by the total demand, $\mathrm{D}\mathrm{T}$, as follows:

$$\begin{array}{cc}SR& =\left(s_p-s\right)DT\\ & =\left(s_p-s\right){p_{n}t}_n.\end{array}$$

(7)

(b)

Design cost (denoted by DC): In the design stage, the total carbon footprint, ${\mathsf{\delta }}_1$, which includes the costs associated with changing tools or molds, moving materials or components, and checking the initial output.

$$DC={\delta }_1\sum _{i=1}^n{Q}_i={\delta }_1\left(p_n-{\lambda }_{gf}\right)t_n.$$

(8)

(c)

 

(c-1)

Holding cost of finished product (denoted by HCf): Figure 1 presents the per cycle holding cost of the end product, which is calculated as follows:

$$\begin{array}{ll}{HC}_f& ={\displaystyle \frac{{\delta }_{21}{h}_{gf}{H}_{e}T}{2}}\\ & ={\displaystyle \frac{{\delta }_{21}{h}_{gf}({\lambda }_{gf}{\theta }_{i}r-D)}{2{\lambda }_{gf}D}}{p}_n^2{t}_{n }^2(\mathrm{from}\mathrm{Equations}(2)\mathrm{and}(5)).\end{array}$$

(9)

(c-2)

Holding cost of all components (denoted by HCs): Similarly, the total holding cost for n components per cycle is calculated as follows:

$$\begin{array}{ll}{\mathrm{H}\mathrm{C}}_s& =\sum _{i=1}^n{\displaystyle \frac{{\delta }_{22}{h}_i{Q}_i\left(t_i+t_{id}\right)}{2}} \\ & ={\delta }_{22}{\displaystyle \frac{p_n^2{t}_n^2}{2}}\left[\sum _{i=1}^n{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]\end{array}$$

(10)

(c-3)

Holding cost of scrap returns (denoted by HCr):

$${HC}_r={\delta }_{23}{h}_r\left[{\displaystyle \frac{1}{2}}\left({\lambda }_{gf}{\theta }_{i}r-D\right)t_e+{\displaystyle \frac{1}{2}}D{t}_{red}\right].$$

(11)

(d)

Return cost (denoted by RC): To extend the return cost proposed by Soleymanfar et al. [36], the total quantity of the return products in a cycle is

$$\mathrm{R}\mathrm{C}={\delta }_3[\left(w-\phi \right)\sigma D+{\lambda }_{gf}{\theta }_{i}r(t_e+t_{ed})].$$

(12)

(e)

Investment to reduce the amount of waste in landfill (denoted by IC): The investment cost is the sum of fixed cost, ${\mathrm{F}}_{\mathrm{g}}$ (annual investment amount for disposable cups reduction per cycle, such as machines equipment), and variable cost, ${\mathrm{V}}_{\mathrm{g}}{\mathsf{\lambda }}_{\mathrm{g}\mathrm{f}}{\mathrm{t}}_{\mathrm{e}}$ (reduce the use of consumable, such as cleaning and logistics costs), $\mathrm{v}$ (ROI of investment in borrow-and-return cup program), that is

$$\begin{array}{ll}\mathrm{I}\mathrm{C}& =v{\delta }_4\left(F_g+V_g{\lambda }_{gf}{t}_e\right)\\ & =v{\delta }_4\left(F_g+V_g{p}_n{t}_n\right)\end{array}$$

(13)

(f)

Purchasing cost of SFP material (denoted by PC): This cost is the unit purchasing cost multiplied by the ordering quantity of SFP material, which is:

$$\begin{array}{ll}\mathrm{P}\mathrm{C}& =c_p{{\delta }_{5}Q}_i\\ & =c_p{\delta }_5\left(p_i-{\lambda }_{gf}\right).\end{array}$$

(14)

To summarize the above results, the total cost per unit time (denoted by $\mathrm{A}\mathrm{P}(\mathrm{s},\mathrm{r}{,\mathrm{t}}_{\mathrm{n}}$)) can be obtained as follows:

$$\begin{array}{ll}\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)& =\left\{\left(s_p-s\right)D-\right.{\delta }_1\left(p_n-{\lambda }_{gf}\right){\displaystyle \frac{D}{p_n}}-{\displaystyle \frac{{\delta }_{21}{h}_{gf}({\lambda }_{gf}{\theta }_{i}r-D)}{2{\lambda }_{gf}}}{p}_n{t}_n\\ & -{\delta }_{22}D{\displaystyle \frac{p_n{t}_n}{2}}\left[{\displaystyle \sum _{i=1}^n}{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]-{\delta }_{23}{h}_r{H}_{sr}D\left[\left({\displaystyle \frac{1}{2}}{\lambda }_{gf}{\theta }_{i}r{\displaystyle \frac{1}{p_n{t}_n}}-{\displaystyle \frac{D}{{\lambda }_{gf}}}\right)+{\displaystyle \frac{1}{2}}\right]\\ & \left.-{\delta }_3\left[\left(w-\phi \right)\sigma {\displaystyle \frac{{2\rho D}^2}{p_n{t}_n}}+{\lambda }_{gf}{\theta }_{i}r\right]-v{\delta }_{4}D\left({\displaystyle \frac{F_g}{p_n{t}_n}}+V_g\right)-c_p{\delta }_5\left(p_n-{\lambda }_{gf}\right){\displaystyle \frac{D}{p_n}}\right\}.\end{array}$$

(15)

The purpose of this paper is to determine the optimal unit recycling subsidy per return and reusable product, and the production run time of component such that the total profit per unit time is maximized. The purpose is to determine ${\mathrm{s}}^{\mathrm{*}}$, ${\mathrm{r}}^{\mathrm{*}}$, and ${\mathrm{t}}_{\mathrm{n}}^{\mathrm{*}}$, which maximize the total profit $\mathrm{A}\mathrm{P}(\mathrm{s},\mathrm{r}{,\mathrm{t}}_{\mathrm{n}}$) shown in Equation (14). First, for given $\mathrm{r}$ and ${\mathrm{t}}_{\mathrm{n}}$, the necessary condition for the total profit in Equation (16) is $\partial \mathrm{A}\mathrm{P}(\mathrm{s},\mathrm{r}{,\mathrm{t}}_{\mathrm{n}}$)/ $\partial \mathrm{s}$, which give:

$$\begin{array}{ll}{\displaystyle \frac{\partial \mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial s}}& =\left\{-D+\left(s_p-s\right)\rho -\right.{\delta }_1\left(p_n-{\lambda }_{gf}\right){\displaystyle \frac{\rho }{p_n}}+{\displaystyle \frac{{\delta }_{21}{h}_{gf}\rho }{2{\lambda }_{gf}}}{p}_n{t}_n\\ & -{\delta }_{22}\rho {\displaystyle \frac{p_n{t}_n}{2}}\left[\sum _{i=1}^n{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]-{\delta }_{23}{h}_r{H}_{sr}{\displaystyle \frac{\rho }{2}}\left[{\displaystyle \frac{{\lambda }_{gf}{\theta }_{i}r}{p_n{t}_n}}-{\displaystyle \frac{4D}{{\lambda }_{gf}}}+1\right]\\ & \left.-{\delta }_3\left[\left(w-\phi \right)\sigma {\displaystyle \frac{2\rho D}{p_n{t}_n}}\right]-v{\delta }_{4}D\left({\displaystyle \frac{F_g}{p_n{t}_n}}+V_g\right)-c_p{\delta }_5\left(p_n-{\lambda }_{gf}\right){\displaystyle \frac{D}{p_n}}\right\}.\end{array}$$

(16)

Theorem 1. 

For given $r$ and ${ t}_n$, the total profits $AP(s,r{,t}_n$) have unique global maximum values at the points $s=s_1$, where ${ s}_1$ and ${ s}_2$ can be solved by Equation (16).

Proof. 

 Next, for given $\mathrm{s}$, taking the first-order and second-order partial derivatives of $\mathrm{A}\mathrm{P}(\mathrm{s},\mathrm{r}{,\mathrm{t}}_{\mathrm{n}}$) with respect to $\mathrm{r}$ and ${\mathrm{t}}_{\mathrm{n}}$, we have

$$\begin{array}{ll} {\displaystyle \frac{\partial \mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial r}}& =-\left\{{\displaystyle \frac{{\delta }_{21}{h}_{gf}{\theta }_i}{2}}{p}_n{t}_n+{\delta }_3{\lambda }_{gf}{\theta }_i\right.\\ & +\left.{\displaystyle \frac{{\delta }_{23}{h}_r{H}_{sr}D{\lambda }_{gf}{\theta }_i}{2p_n{t}_n}}\right\},\end{array}$$

(17)

$$\begin{array}{ll}{\displaystyle \frac{\partial \mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial t_n}}& =\left\{-{\displaystyle \frac{{\delta }_{21}{h}_{gf}\left({\lambda }_{gf}{\theta }_{i}r-D\right)}{2{\lambda }_{gf}}}{p}_n\right.\\ & -{\delta }_{22}D{\displaystyle \frac{p_n}{2}}\left[{\displaystyle \sum _{i=1}^n}{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]+{\displaystyle \frac{{\delta }_{23}{h}_r{H}_{sr}D{\lambda }_{gf}{\theta }_{i}r}{2p_n{t}_n^2}}\\ & \left.+{\delta }_3\left(w-\phi \right)\sigma {\displaystyle \frac{D^2}{p_n{t}_n^2}}+v{\delta }_{4}D{\displaystyle \frac{F_g}{p_n{t}_n^2}}\right\}\end{array}$$

(18)

and

$$\begin{array}{ll}{\displaystyle \frac{{\partial }^2\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial t_n^2}}=& -\left\{{\displaystyle \frac{{\delta }_{23}{h}_r{H}_{sr}D{\lambda }_{gf}{\theta }_{i}r}{p_n{t}_n^3}}\right.\\ & \left.+{2\delta }_3\left(w-\phi \right)\sigma {\displaystyle \frac{D^2}{p_n{t}_n^2}}+2v{\delta }_{4}D{\displaystyle \frac{F_g}{p_n{t}_n^3}}\right\}<0. \end{array}$$

(19)

Furthermore, for given $s$, the following can be obtained:

$${\displaystyle \frac{{\partial }^2\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial t_n\partial \mathrm{r}}}={\displaystyle \frac{{\partial }^2\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial \mathrm{r}\partial t_n}}=0.$$

Therefore, for given $s$, the determinant of the Hessian matrix at the point $(\mathrm{r}{,t}_n$) is:

$$\left|\begin{array}{cc}{\displaystyle \frac{{\partial }^2\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{{\partial r}^2}}& {\displaystyle \frac{{\partial }^2\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial r\partial t_n}}\\ {\displaystyle \frac{{\partial }^2\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{\partial t_n\partial r}}& {\displaystyle \frac{{\partial }^2\mathrm{A}\mathrm{P}\left(s,\mathrm{r}{,t}_n\right)}{{\partial t_n}^2}}\end{array}\right|={\displaystyle \frac{{2\delta }_3{\lambda }_{gf}{\theta }_i}{p_n{t}_n^3}}\left\{{\displaystyle \frac{{{\delta }_{23}h}_r{H}_{sr}D{\lambda }_{gf}{\theta }_{i}r}{2}}+{\delta }_3\left(w-\phi \right)\sigma D^2+v{\delta }_{4}D{F}_g\right\}>0.$$

(20)

Hence, the Hessian matrix is a negative definite at the point $(\mathrm{r}{,t}_n$). Consequently, the optimal solution occurs at the point $(\mathrm{r}{,t}_n$), which satisfies, $\partial \mathrm{A}\mathrm{P}(s,\mathrm{r}{,t}_n$)/ $\partial \mathrm{r}$ = 0 and $\partial \mathrm{A}\mathrm{P}(s,\mathrm{r}{,t}_n$)/ $\partial t_n$ = 0, simultaneously.

Summarizing the above results, we establish the following theorem to help the manufacturer to obtain the optimal recycling policy. It is not easy to find the closed-form solution of ${\mathrm{t}}_n$ and $r$ form Equations (17) and (18). But, we can prove that the value of ${\mathrm{t}}_n$ which satisfies Equations (17) and (18) not only exists but also is unique, following Theorem 2. Let $(t_{n1},{\mathrm{r}}_1$) and $(t_{n2},{\mathrm{r}}_2$) be the solutions of Case (a) and Case (b), respectively. □

Theorem 2. 

For given $s$ and $w\ge \phi$, we have

(a) 

If $w\ge \phi$, the optimal solution is $r=r^{*}$ and ${ t}_n=t_n^{*}$ given in Equations (17) and (18).

(b) 

If $w<\phi$, the optimal solution is $r^{*}\to 0$ and $t_n^{*}\to 0$ given in Equations (17) and (18).

This paper determines the optimal value of $t_n$ to maximize its own profit for the value of ($s,r$), determined by the vendor. The necessary condition for $AP(s,r{,t}_n$) to be maximum is $\partial AP(s,r{,t}_n$)/$\partial t_n$ = 0, which gives

$$\begin{array}{c}\left\{-{\displaystyle \frac{{\delta }_{21}{h}_{gf}\left({{\lambda }_{gf}\theta }_{i}r-D\right)}{2{\lambda }_{gf}}}{p}_n-{\displaystyle \frac{{{\delta }_{22}Dp}_n}{2}}\right.\left[{\displaystyle \sum _{i=1}^n}{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]+{\displaystyle \frac{{\lambda }_{gf}{\delta }_{23}D{h}_r{H}_{sr}{\theta }_{i}r}{2p_n{t}_n^2}}\\ \left.+{\displaystyle \frac{{\delta }_3\left(w-\phi \right)\sigma D^2}{p_n{t}_n^2}}+v{\delta }_4{\displaystyle \frac{F_g}{p_n{t}_n^2}}\right\}=0\end{array}$$

Proof. 

Let

$$\begin{array}{c}F\left(t_n\right)=\left\{-{\displaystyle \frac{{\delta }_{21}{h}_{gf}\left({{\lambda }_{gf}\theta }_{i}r-D\right)}{2{\lambda }_{gf}}}{p}_n-{\displaystyle \frac{{{\delta }_{22}Dp}_n}{2}}\right.\left[{\displaystyle \sum _{i=1}^n}{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]+{\displaystyle \frac{{\lambda }_{gf}{\delta }_{23}D{h}_r{H}_{sr}{\theta }_{i}r}{2p_n{t}_n^2}}\\ \left.+{\displaystyle \frac{{\delta }_3\left(w-\phi \right)\sigma D^2}{p_n{t}_n^2}}+v{\delta }_4{\displaystyle \frac{F_g}{p_n{t}_n^2}}\right\}=0\end{array}$$

for $t_n\in \left(0,\infty \right)$. Taking the first derivative of $F\left(t_n\right)$ with respect to $t_n\in \left(0,\infty \right)$, it gives

$${\displaystyle \frac{dF\left(t_n\right)}{{dt}_n}}={\displaystyle \frac{2}{p_n{t}_n^3}}\left\{{\displaystyle \frac{{{{\delta }_{23}\theta }_{i}r\lambda }_{gf}{h}_r{H}_{sr}D}{2}}+{\delta }_3\left(w-\phi \right)\sigma D^2+{v\delta }_{4}D{F}_g\right\}.$$

(a)

$w\ge \phi$

$${\displaystyle \frac{dF\left(t_n\right)}{{dt}_n}}={\displaystyle \frac{2}{p_n{t}_n^3}}\left\{{\displaystyle \frac{{{{\delta }_{23}\theta }_{i}r\lambda }_{gf}{h}_r{H}_{sr}D}{2}}+{\delta }_3\left(w-\phi \right)\sigma D^2+{v\delta }_{4}D{F}_g\right\}<0.$$

$F\left(t_n\right)$ is a strictly decreasing function of $t_n\in \left(0,\infty \right)$. Further, we have

$$\begin{array}{c}\underset{t_n\to 0}{\lim }F\left(t_n\right)=\left\{-{\displaystyle \frac{{\delta }_{21}{h}_{gf}\left({{\lambda }_{gf}\theta }_{i}r-D\right)}{2{\lambda }_{gf}}}{p}_n-{\displaystyle \frac{{{\delta }_{22}Dp}_n}{2}}\right.\left[{\displaystyle \sum _{i=1}^n}{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]+{\displaystyle \frac{{\lambda }_{gf}{\delta }_{23}D{h}_r{H}_{sr}{\theta }_{i}r}{2p_n{t}_n^2}}\\ \left.+{\displaystyle \frac{{\delta }_3\left(w-\phi \right)\sigma D^2}{p_n{t}_n^2}}+v{\delta }_{4}D{\displaystyle \frac{F_g}{p_n{t}_n^2}}\right\}=\infty >0.\end{array}$$

and $F\left(t_n\right)$ is a strictly decreasing function of $t_n\in \left(0,\infty \right)$. Further, we have

$$\underset{t_n\to \infty }{\lim }F\left(t_n\right)=-\left\{{\displaystyle \frac{{\delta }_{21}{h}_{gf}\left({{\lambda }_{gf}\theta }_{i}r-D\right)}{2{\lambda }_{gf}}}{p}_n+{\displaystyle \frac{{{\delta }_{22}Dp}_n}{2}}\left[{\sum }_{i=1}^n{h}_i\left({\displaystyle \frac{1}{{\lambda }_{gf}}}-{\displaystyle \frac{1}{p_i}}\right)\right]\right\}<0.$$

By applying the Intermediate Value Theorem, there exists a unique $t_n\in \left(0,\infty \right)$, such that $F\left(t_n\right)=0$. This completes the proof.

(b)

$w<\phi$

The optimal value of $t_n$ is obtained as ${\mathrm{t}}_n^{*}\to 0$. The recycling system should not be implemented. Summarizing the above results, the following Figure 2 and Algorithm 1 were used to obtain the optimal solution of our problem. □

Algorithm 1: the optimal solution of ${\mathrm{s}}^{\mathrm{*}}$ $,{\mathrm{r}}^{\mathrm{*}}$ $, \mathrm{and}{\mathrm{t}}_{\mathrm{n}}^{\mathrm{*}}$

Step 1. $\mathrm{Start} \mathrm{with} \mathrm{j}=0$ $\mathrm{and} \mathrm{the} \mathrm{initial} \mathrm{value} \mathrm{of} s=s_j, s=s_{j+1}$. Step 2. $\mathrm{Check} \mathrm{the} \mathrm{values} \mathrm{of} w\ge \phi$. Step 2-1. $\mathrm{If} w\ge \phi$ $, \mathrm{calculate} \mathrm{the} \mathrm{values} \mathrm{of} t_j=t_{n1,j}$, $t_j=t_{n2,j}$ and $r_j=r_{1,j}$, $r_j=r_{2,j}$ put $(t_{nj},{\mathrm{r}}_j$) into Equation (17) and Equation (18) to solve the value of $s=s_j$, and go to Step 3. Step 2-2. $\mathrm{If} w<\phi$ $, \mathrm{calculate} \mathrm{the} \mathrm{values} \mathrm{of} {\mathrm{t}}_{n1,j}\to 0,$ ${\mathrm{t}}_{n2,j}\to 0$ $\mathrm{and} {\mathrm{s}}_{1,j}\to 0,{\mathrm{s}}_{2,j}\to 0.$ Step 3. $\mathrm{If} \mathrm{the} \mathrm{difference} \mathrm{between} s_{j+1}$ $\mathrm{and} s_j$ $\mathrm{is} \mathrm{tiny}, \mathrm{set} s^{*}=s_{j+1}$ $, \mathrm{and}{r}^{*}=r_{j+1}$ $, (s^{*},{t_n}^{*},{\mathrm{r}}^{*}$) $\mathrm{is} \mathrm{the} \mathrm{optimal} \mathrm{solution}. \mathrm{Otherwise}, \mathrm{set} j=j+1$ and go back to Step 2. Step 4. Stop.

5. Application Example

ABC, founded in 2002, is a leading Taiwanese manufacturer of disposable paper and plastic packaging products. The company offers a wide range of items, including hot and cold beverage cups, accompanying lids, food containers, boxes, bowls, paper bags, sleeves, carriers, and cutlery. ABC’s headquarters and two production facilities occupy a total area of 67,100 square meters in the Taichung industrial district, providing convenient logistics for import and export operations with easy access to the Taichung port. To reduce the use of disposable paper and plastic cups, ABC has implemented a reusable cup borrowing and return system for drink containers.

5.1. Lend System

• When purchasing a drink, consumers can pick up a reusable cup from the borrowing area, which is recommended to be located next to the counter.
• Consumers scan to borrow a cup at the scanning station. When borrowing, log the borrowing time using a credit card tap or electronic payment. The borrowing period for reusable cups is limited to 5 days. If exceeded, a deposit of TWD 50 will be charged to the account.
• Scan the QR code at the bottom of the cup when borrowing.

5.2. Return System

• Consumers scan to return the cup at the scanning station. If returned late, the electronic payment registered when borrowing will incur a TWD 49 deposit payment.
• For every 40 cups collected, FamilyMart logistics will transport them to the logistics center. ABC will arrange freight to transport the reusable cups back to the logistics center for cleaning. After cleaning, disinfection, and inspection, they will be returned to the logistics center.
• Cross-platform system, allowing cup borrowing and return between different brand stores. When borrowing a cup from Brand A, you can return it to Brand B. Lidian will charge Brand A as a reusable cup service fee (the fee is based on the brand from which the cup was borrowed; for example, if borrowed from FamilyMart, the fee is charged to FamilyMart). Brand B will assist Lidian in reverse logistics back to the Distribution Center (DC), and Lidian will subsidize Brand B for the reverse logistics costs (0.3 yuan per cup).

In Taiwan, most businesses and brands pay container recycling and disposal fees. However, the actual amount recycled by these companies, what is remanufactured from the recycled materials, or how much is regenerated remains unknown. The high-quality wood fibers in paper cups come from sustainably managed forests. According to research, the carbon footprint of paper cups significantly decreases by 46% after recycling. It can achieve 100% waste reduction and environmental recycling by recycling paper cups into other usable or marketable products. Superior quality product lines and competitive value are obtained by utilizing state of the art manufacturing equipment’s with the newest technology, and our strict, unparalleled quality controls ensure the highest quality product line is produced. We respect and maintain Earth-friendly environmental policies, delivering exceptional quality while having the smallest environmental impact. Su et al. [37] considered an imperfect multiple-stage production system that manufactures paired products made from mixed materials containing scrap returns, in which the scrap returns are converted from defective products. The appearance (Figure 3) and production systems (Figure 4) of paper cup are shown below.

Figure 4 indicates that ABC may take FSC material (paper cup or borrowed cup) to a paper cup recycler in the recovery process. The material is also recyclable, easy to process and is made in Taiwan for a lower carbon footprint. In particular, the main purpose of green manufacturing is reducing material wastages and selecting innovative and more sustainable materials. In this circular perspective, the ecosystem materials should be environmentally harmless, and require a low energy consumption.

5.3. Numerical Examples

Base settings were established for the model by conducting interviews and surveys with relevant staff in the firm. In the current COVID-19 pandemic situation, the supplier offers advance payments to the firm so that they will not cancel the order. Due to the shortages in demand, the supplier offers a discount rate that is dependent on the number of installments. The firm also offers delays in payments for customers who do not have transportation and goods available. The values presented here were altered to preserve the confidentiality of the commercial information.

Example 1. 

Let us consider an inventory system with the following data:

$\mathrm{k}=50$	${\mathrm{h}}_1=1$	${\mathrm{h}}_2=2$	${\mathrm{h}}_3=3$
${\mathrm{h}}_{gf}=4$	${\mathrm{h}}_r=5$	$\mathrm{w}=0.4$	${\mathrm{c}}_p=0.2$
${\mathrm{F}}_g=40$	${\mathrm{V}}_g=0.01$	$\mathrm{n}=2$	$\mathrm{p}=0.6$
$\mathrm{a}=50$	$\mathrm{b}=0.1$	$\mathsf{\phi }=0.1$	$v=0.5$
${\mathsf{\theta }}_1=0.4$	${\mathsf{\theta }}_2=0.3$	${\mathsf{\theta }}_3=0.2$	${\mathsf{\theta }}_{gf}=0.1$
${\mathrm{p}}_1=100$	${\mathrm{p}}_2=200$	${\mathrm{p}}_3=300$	${\lambda }_{gf}=500$
${\mathsf{\delta }}_1=0.2$	${\mathsf{\delta }}_2=0.3$	${\mathsf{\delta }}_{21}=0.1$	${\mathsf{\delta }}_{22}=0.4$
${\mathsf{\delta }}_{23}=0.5$	${\mathsf{\delta }}_3=0.3$	${\mathsf{\delta }}_4=0.5$	${\mathsf{\delta }}_5=0.3$
$\mathsf{\sigma }=0.2$	$\mathsf{\rho }=0.1$

Example 2. 

Let us consider an inventory system with the following data:

$\mathrm{k}=50$	${\mathrm{h}}_1=1$	${\mathrm{h}}_2=2$	${\mathrm{h}}_3=3$
${\mathrm{h}}_{gf}=4$	${\mathrm{h}}_r=5$	$\mathrm{w}=0.1$	${\mathrm{c}}_p=0.2$
${\mathrm{F}}_g=40$	${\mathrm{V}}_g=0.01$	$\mathrm{n}=2$	$\mathrm{p}=0.6$
$\mathrm{a}=50$	$\mathrm{b}=0.1$	$\mathsf{\phi }=0.4$	$v=0.5$
${\mathsf{\theta }}_1=0.4$	${\mathsf{\theta }}_2=0.3$	${\mathsf{\theta }}_3=0.2$	${\mathsf{\theta }}_{gf}=0.1$
${\mathrm{p}}_1=100$	${\mathrm{p}}_2=200$	${\mathrm{p}}_3=300$	${\lambda }_{gf}=500$
${\mathsf{\delta }}_1=0.2$	${\mathsf{\delta }}_2=0.3$	${\mathsf{\delta }}_{21}=0.1$	${\mathsf{\delta }}_{22}=0.4$
${\mathsf{\delta }}_{23}=0.5$	${\mathsf{\delta }}_3=0.3$	${\mathsf{\delta }}_4=0.5$	${\mathsf{\delta }}_5=0.3$
$\mathsf{\sigma }=0.2$	$\mathsf{\rho }=0.1$

5.4. Sensitivity Analysis

The optimal solutions for Examples 1 and 2 are listed in

Table 2. Results of Examples 1 and 2 for three trade credit policies.

		$\mathbf{Case} \left(\mathbf{a}\right):\mathit{w}\ge \mathit{\phi }$			$\mathbf{Case} \left(\mathbf{b}\right):\mathit{w}<\mathit{\phi }$
Parameter		$\mathit{s}$	$\mathbf{r}$	$\mathbf{A}\mathbf{P}$	$\mathit{s}$	$\mathbf{r}$	$\mathbf{A}\mathbf{P}$
$k$	+50%	101.6558	0.7824	1836.04	99.9612	0.6122	1619.08
	+25%	114.6177	0.9823	1607.02	108.1411	0.6958	1116.26
	−25%	124.6938	1.1824	1235.06	113.4821	0.7067	1092.94
	−50%	124.7319	1.2822	1004.08	113.4911	0.7505	995.59
ACT		5.0191	6.9280	7.1254	6.8913	7.8341	9.1245
$h_1$	+50%	124.1880	1.2556	1590.13	113.9012	0.7442	1351.11
	+25%	124.2240	1.1646	1589.25	113.8112	0.6247	1140.12
	−25%	124.2940	0.9828	1587.49	113.9621	0.4991	1133.55
	−50%	124.3300	0.8920	1586.61	113.0211	0.4770	1131.08
ACT		4.2345	5.1236	6.3412	7.3451	7.5412	8.8712
$h_2$	+50%	124.1110	1.4535	1592.05	113.8711	1.3430	1132.90
	+25%	124.1850	1.2635	1590.21	113.8901	1.1474	1129.47
	−25%	124.3330	0.8841	1586.53	113.9711	0.8661	1118.84
	−50%	124.4070	0.6947	1584.69	113.9921	0.3937	1109.78
ACT		5.7811	6.0712	7.8901	5.7810	5.9102	6.9102
$h_3$	+50%	124.0341	1.6516	1593.97	113.1701	1.3263	1167.22
	+25%	124.1471	1.3624	1591.17	113.1811	1.1393	1163.31
	−25%	124.3710	0.7854	1585.57	113.2701	0.8709	1124.68
	−50%	124.4840	0.4977	1582.78	113.3711	0.7901	1117.00
ACT		6.7189	6.8901	7.5671	8.1901	8.5671	9.1014
$h_{gf}$	+50%	124.4771	0.1135	1583.00	113.2711	0.1101	1119.12
	+25%	124.3970	0.5175	1584.95	113.1631	0.3541	1122.57
	−25%	124.0020	1.9357	1594.74	113.0011	1.3541	1159.57
	−50%	123.4511	3.5689	1608.52	111.1711	2.3541	1219.57
ACT		7.1892	5.1631	7.8914	7.2561	7.8951	8.9182
$\sigma$	+50%	127.2591	1.0731	1618.37	119.0121	1.0141	1311.21
	+25%	125.1490	1.0734	1608.37	117.1103	1.0142	1301.91
	−25%	122.1290	1.0740	1538.36	114.0112	1.0145	1109.31
	−50%	120.1561	1.0743	1518.36	112.0011	1.0149	1102.21
ACT		5.6718	5.8910	6.7241	6.7451	7.8192	8.2791
$w$	+50%	126.7841	1.1932	1690.86	118.8711	1.0541	1240.08
	+25%	125.7341	1.0926	1599.59	116.5401	1.0312	1194.83
	−25%	121.6345	1.0313	1581.04	111.4011	1.0241	1104.28
	−50%	120.5846	1.0206	1572.77	109.5812	1.0141	1058.98
ACT		6.7810	6.9123	7.1235	8.9901	9.1682	9.6781
$\phi$	+50%	161.6844	1.0910	1537.32	151.6221	1.0525	1238.18
	+25%	141.6843	1.0819	1564.14	131.5512	1.0233	1293.89
	−25%	101.6843	1.0619	1618.38	100.3112	1.0149	1405.22
	−50%	91.6842	1.0418	1643.39	81.1512	0.9566	1460.84
ACT		6.0012	6.5532	6.5611	8.5191	7.2451	7.4511
$c_p$	+50%	124.7128	1.0614	804.591	112.5801	1.0565	792.261
	+25%	124.6986	1.0717	1204.95	112.0412	1.0753	1120.95
	−25%	124.1711	1.0822	1670.68	112.0121	1.0829	1778.12
	−50%	124.1558	1.0854	1804.04	112.0008	1.0917	1806.61
ACT		6.9011	6.9801	7.0011	7.2912	7.6712	8.8911
$F_g$	+50%	124.0545	1.0792	1598.07	111.7011	1.0579	1594.21
	+25%	124.1551	1.0808	1591.06	111.8112	1.0633	1481.72
	−25%	124.6564	1.0840	1576.02	111.8261	1.0794	1385.32
	−50%	124.6571	1.0856	1536.01	111.8311	1.0811	1191.01
ACT		6.9151	6.8751	7.2351	7.5619	7.8765	8.1023
$V_g$	+50%	101.6558	1.0524	1806.04	99.0311	0.9932	1614.90
	+25%	111.1558	1.0624	1706.04	100.9051	1.0486	1368.03
	−25%	142.6557	1.0924	1306.04	121.5121	1.3599	1329.61
	−50%	161.1557	1.1124	1206.04	141.1211	1.3661	1308.21
ACT		7.6711	7.8901	8.0123	8.9112	9.1231	10.2121

Note: ACT: average CPU time (second).

. The solution procedures are implemented using Mathematica Version 11 on a personal computer with Intel Core i7 processor under Microsoft Windows 7 Pro. In order to verify the performance of the algorithm in our problem, Examples 1 and 2 were repeated with 100 cycles of the algorithm. Results show that the algorithm is a good compromise between the quality of obtained solutions and the used average CPU time. The numerical example presented in Section 4.1 was used to assess the effects of changes to system parameters ( $k$, $h_1$, ${ h}_2$, ${ h}_3$, $h_{gf}$, $\sigma$, $w$, $\phi$, $c_p$, $F_g$, and $V_g$) on the values of $s^{*}$, $r^{*}$ and ${\mathrm{A}\mathrm{P}}^{*}$. Each parameter was adjusted separately (i.e., the others parameters were left unchanged) by +50%, +25%, −25%, or −50%. Our analytical results in Table 2 permit the following interesting observations and managerial insights which could be used to guide decision-making. The following two cases are examined.

• Case(a) ($\mathbf{w}\ge \mathsf{\phi }$)

a1. An increase in the defect parameter ( ${\mathrm{h}}_1,{\mathrm{h}}_2,{\mathrm{h}}_3,{\mathrm{h}}_{gf}$) led to a corresponding increase in the total profit per unit time. The implies that holding excessive inventory levels can increase associated carrying costs, thereby diminishing the company’s profit margins. Moreover, an oversupply of stock may also lead to issues such as product obsolescence or spoilage, particularly with perishable or fashion-oriented goods, further eroding profitability. Green materials and processes can help mitigate these issues by enhancing resource utilization, minimizing waste generation, and optimizing energy consumption, ultimately improving operational efficiency and profitability.

a2. An increase in the defect parameter $\mathrm{k}$ led to a corresponding increase in the total profit per unit time. The implies that minimizing initial costs and maximizing per-unit profits for borrow-and-return cup programs is crucial for improving financial performance and operational efficiency. However, considering the social costs of worker stress adds a more comprehensive perspective. Both the sustainability program and employee well-being impact the bottom line, although in different ways. Let us examine the connection between these two objectives.

a3. An increase in the defect parameter $\sigma$ led to a corresponding increase in the total profit per unit time. The implies that the total profit per unit time refers to the net earnings generated by the reusable product system over a defined period, such as a month, quarter, or year. This metric is calculated by subtracting all associated costs, including manufacturing, operational, and overhead expenses, from the total revenue generated by the reusable product system during the specified timeframe.

a4. An increase in the defect parameter $\mathrm{w}$ led to a corresponding increase in the total profit per unit time. The implies that investing in a borrow-and-return cup program can be a strategic move that supports sustainability goals while generating long-term profits. Understanding the revenues and costs associated with such a system is crucial for calculating and maximizing the total profit over time.

a5. An increase in the defect parameter $\phi$ led to a corresponding increase in the total profit per unit time. The implies that analyzing the relationship between a recycling system, work-related stress, and profitability requires considering the interplay between the costs of stress, the efficiency of the recycling system, and overall profitability. Work-related stress can impact a company’s productivity, while a recycling system introduces new cost and revenue factors affecting total profit.

a6. An increase in the defect parameter ${\mathrm{c}}_p$ led to a corresponding increase in the total profit per unit time. The implies that rising costs of FSC-certified materials per unit can impact a company’s overall profitability. FSC materials are often preferred for their sustainability, but they can be more expensive than non-certified alternatives, which adds complexity to the company’s cost structure. An increase in the cost of FSC materials may affect the company’s total profit over time, and businesses can employ strategies to mitigate this impact.

a7. An increase in the defect parameter ${\mathrm{F}}_g,{\mathrm{V}}_g$ led to a corresponding increase in the total profit per unit time. The implies that the fixed costs for equipment per cycle, can significantly affect the company’s total profit per unit of time. These fixed costs remain constant across each operational cycle and do not depend on the number of units produced or sold, making them crucial in calculating profitability.

• Case(b) ($\mathit{w}<\mathit{\phi }$)

b1. An increase in the defect parameter ( ${\mathrm{h}}_1,{\mathrm{h}}_2,{\mathrm{h}}_3,{\mathrm{h}}_{gf}$) led to a corresponding increase in the total profit per unit time. The implies that work-related stress is expensive. Tackling stress and psychosocial risks can be viewed as too costly, but the reality is that it costs more to ignore them. When a returnable cup system is utilized, it can have a lower environmental impact than single-use cups, as the production and disposal of single-use cups can generate more carbon emissions than the repeated use of reusable cups.

b2. An increase in the defect parameter $\mathrm{k}$ led to a corresponding increase in the total profit per unit time. The implies that reducing initial costs and increasing per-unit profits is crucial for improving the financial and operational performance of borrow-and-return cup programs. However, the social costs of work stress from implementing and managing these programs must also be considered. Implementing employee support and fostering a healthy work environment can help ensure a more balanced and sustainable approach to improving the overall performance and viability of these programs.

b3. An increase in the defect parameter $\sigma$ led to a corresponding increase in the total profit per unit time. The implies that a reusable product system and the social costs of work stress are two interconnected aspects that can significantly impact a company’s financial health and overall social responsibility. The success of a reusable product system can enhance sustainability and reduce operational costs, while addressing work stress can boost employee productivity and well-being. These two factors intersect in ways that influence a company’s long-term performance.

b4. An increase in the defect parameter $\mathrm{w}$ led to a corresponding increase in the total profit per unit time. The implies that the social costs associated with work-related stress can have a significant impact on a company’s financial performance and overall workplace culture. It is crucial to understand and manage these factors in order to maximize profits and ensure employee well-being. How the social costs of work stress can affect a business’s profitability and how to calculate the total profit per unit time in this context.

b5. An increase in the defect parameter $\phi$ led to a corresponding increase in the total profit per unit time. The implies that mitigating the costs associated with work-related stress, and maximizing the total profit generated per unit of time while considering environmental, social, and governance performance to create long-term value for the firm and its stakeholders.

b6. An increase in the defect parameter ${\mathrm{c}}_p$ led to a corresponding increase in the total profit per unit time. The implies that managing work-related stress and the higher costs of FSC certified materials requires a balanced approach. These factors can significantly impact a firm’s operational expenses and profitability, so addressing them together can help maintain or improve overall business performance.

b7. An increase in the defect parameter ${\mathrm{F}}_g,{\mathrm{V}}_g$ led to a corresponding increase in the total profit per unit time. The implies that the costs that fluctuate with production volume refer to the expenses that vary with the number of units produced or processed in a given period. In the context of a borrow-and-return cup program, these costs are incurred based on the number of cups used, returned, and processed. Effectively managing these variable costs is crucial for the profitability of such a program.

We then examine how investing versus not investing in borrow-and-return cup program impacts the optimal strategies for both Case (a) and Case (b). This study examines the borrowed cup system, a mug library that eliminates the need for single-use cups. To simplify the explanation, we use the numerical values in Example 1 and 2, adopt a value for $\mathsf{\sigma }\in \left\{0.1,0.2,\cdots ,1\right\}$, and draw a line chart according to the analysis results, as shown in Figure 5. Companies that have implemented energy-saving and carbon-reduction technologies achieve higher profitability compared to those that have not. Governments worldwide have introduced subsidy policies and paper cup recycling systems to promote environmentally responsible practices. Businesses must manage production costs efficiently and account for environmental protection and carbon tax expenses.

6. Conclusions

This paper examines a two-stage production–inventory system that incorporates subsidy policies and a borrow-and-return cup program, using the economic production quantity model. The study establishes the necessary and sufficient conditions for the existence and uniqueness of the optimal solution. A key finding is that subsidizing remanufacturers can increase recycling volume when fewer subsidies are provided. Conversely, subsidizing consumers can also generate recycling volume. Next, we presented a straightforward algorithm to identify the optimal solution $\left(s^{*},{{r^{*},t}_n}^{*}\right)$ that maximizes total profit per unit of time. The effects of model parameters on the optimal solution and total profit were analyzed through two numerical examples in the pulp and paper industry, providing valuable insights for ABC. The key findings of this study are as follows:

• The study examined the impact of increased consumer preference for a borrow-and-return cup program on the profitability of the reverse supply chain for end-of-life vehicles. Two scenarios were analyzed: government subsidies for remanufacturers and consumer subsidies. The findings suggest that when designing subsidy policies, the government should prioritize supporting and incentivizing productivity enhancements driven by technological advancements. Furthermore, the borrow-and-return cup program provides a convenient recycling service for consumers handling end-of-life vehicles, and consumer preference has facilitated its integration into recycling practices.
• The effect of government subsidies on the reverse supply chain depends on the level of subsidies and consumer preferences. Lower subsidy levels favor subsidizing remanufacturers to increase recycling, while higher levels favor subsidizing consumers. The goal of subsidies is to boost recycling rates and promote remanufacturing. For reverse supply chain profitability, subsidizing remanufacturers is better at lower-to-middle subsidy levels, and subsidizing consumers is better at higher levels.
• This study suggests some areas for future research. Researchers could look at how government subsidies affect online recyclers, how subsidies and regulations impact remanufacturers, and how consumer preferences influence recycling of used vehicles. The study also looked at how government financial incentives and penalties affect remanufacturing firms. Additionally, it examined the pros and cons of recycled paper and plastic containers under policies limiting plastic use.