Optimal Rebate Policy in a Green Product Design and Development System with Price-Sensitivity Demand

1. Introduction

Understanding the consumption of limited resources in the Green Product Design and Development (GPDD) system leads to better determination of rebate policy and demand characteristics. The GPDD system has received increasing research interest over the past decade, with a particular focus on Corporate Social Responsibility. However, there are few studies that shed light on the interplay between corporate operations and environmental improvement. Numerous studies[1-2] emphasise that the selection process of lightweight materials has minimal environmental impact throughout its life cycle. Interest in the environment has increased due to the damage caused by the development of industry[3-4]. Early theories on the economic order quantity (EOQ) model and economic production quantity (EPQ) model can be traced back to Taft[5] and Harris[6]. Karim and Nakade[7] offered a comprehensive evaluation of the sustainable EPQ model in terms of carbon emissions and product recycling. In the literature, Jiaqi and Meizhang[8] proposed a remanufacturing product structure information model with remanufacturing design parameters and features. Digiesi et al.[9] studied sustainability aspects in the field of spare parts logistics, considering the environmental costs associated with both the production of new parts and the disposal of used ones. Modern complex manufacturing systems involve multiple stages to produce complex products, such as aviation inertial navigation product manufacturing and automobile assembling. As a result, multistage manufacturing systems (MMSs) are becoming increasingly common in the production process. The MMSs consist of multiple stages, and the products are manufactured in the order of the stages. Wang et al.[10] proposed a production quality prediction framework for multi-task joint deep learning, which evaluates the multi-task quality of all stages. In a multi-stage manufacturing environment, all these different actors will be interacting with each other. The manufacturer is responsible for selecting the green materials with which the product can be manufactured. As recommended by Han et al.[11], when consumers have the right to return second-hand products relying on manufacturers' rebates, it affects the prices of new products. Rebate determination is a common price elasticity of demand problem in a recycling program. To correct this, the return activity is one of the most important points in the production planning system. The rebates collected through this program are transferred to the government by the manufacturers and importers. The government uses these funds to support recycling initiatives, including education campaigns and the establishment of new recycling programs. In 2010, the recycling rate in Taiwan was 94.1 tonnes but this had risen to over 100 tonnes in some areas by 2016. The local government provides funding to support bike recycling program. Figure 1 shows the recovery amount by year. Approximately 148 tonnes were accounted in 2020 (148) and 2019 (141.6), 2018 (140.2), 2017 (128.1), 2016 (112.9), 2015 (98.5), 2014 (101.3), 2013 (92.2), 2012 (91), 2011 (94.4) (EPA (2023)).

Fig. 1 Recover amount statistics and analytics. Source: Environmental Protection Police (EPA (2023))

Reusing/remanufacturing recovers the value of end-of-life products by reusing their durable components to manufacture a product with the original functionality. Green investments are important investments such as remanufacturing policies and mutual funds that target sustainable strategies. Here is an example from a real manufacturing company in Taiwan. At the end of a bike’s life, it can either end up in a landfill or its components can be recycled, saving large amounts of energy. The company has an advanced recycling system to reduce the waste of components and raw materials. Figure 2 summarizes the flow chart of the recycling system. The system comprises one manufacturer and one retailer. Rebate agreements play a crucial role in the supply chain as they provide companies with an effective tool to incentivize their customers. The program encourages retailers to engage in manufacturer rebates within a green supply chain and examines its impact on the green product sales.

Fig. 2 Bike recycling system

However, there is little research that has empirically documented the relationship between inventory-production systems and rebate policies. Chen[12] studied the news-vendor problem in the case of a two-level supply chain consisting of one manufacturer and one retailer and investigated the combined effects of the cooperative advertising mechanism, the return policy, and the channel coordination. Hu et al.[13] studied how retailers under pressure to move inventory should use these two pricing tools to maximize revenue by determining the optimal use of shipping rebates together with dynamic pricing for limited inventory. The contribution of this paper is a better understanding of rebate and return policy under the green product design principle in two areas. First, new insights into the literature on demand patterns are provided, as previous studies on return policy strategy have mainly focused on a constant. In addition, this paper introduces a two-stage production system that considers the factors influencing demand, introduces ecological principles for material selection in product design, and provides guidance for designers to achieve sustainable development in their designs. The system is divided into two stages: the manufacturing of components (Stage 1) and the assembly of the end product (Stage 2). These two stages are usually studied independently of each other which does not lead to ideal results. The elements of total profit per cycle are sales revenue, set-up costs, holding costs of the end product, holding costs of all components, rework costs for all defective items, production costs and investment costs. Next, a product-inventory system was designed to investigate the identification and application of rebate policy. This is done with the motivation that it can provide an alternative solution to the rebate policy problem. Rebate programs can increase sales and demand, reduce inventory levels, and stimulate product return. Manufacturers often focus on the following three questions as key points of interest:

To answer these questions, this paper presents a new model that examines the effects of the return policy on the production system. Overall, this paper determines the optimal production time, rebate, and return rate to maximize the total profit per unit time. This paper is organized into six key sections. Section 1 provides background information on the return policy, demand patterns, and life cycle assessment considerations. Section 2 outlines the description of green product design. Section 3 presents the industrial context and a list of notations and assumptions. Section 4 discusses the problem-solving procedure of the algorithm. Section 5 presents application examples, numerical illustrations, and sensitivity analysis. Finally, Section 6 summarizes the key findings, discussions, and recommendations for future research.

2. Literature review

2.1 Green design

As populations and economies continue to expand, the depletion of natural resources is accelerating, and environmental pollution is becoming increasingly severe. The concept of a green supply chain aims to reduce environmental degradation and pollution by incorporating eco-friendly practices into business operations. A manufacturer rebate, in which the manufacturer refunds the consumers after the purchase, becomes an effective promotional tool. Lin[14] examined the impact of the manufacturer rebate strategy on the outcomes of channel members within a two-stage green supply chain model characterized by information asymmetry. Taleizadeh and Heydarian[15] applied a rebate strategy in a two-tier supply chain consisting of a single supplier and a manufacturer producing green and non-green products.

2.2 Green product design and development

Sustainable development represents a new approach to human survival. It not only includes the responsible use of resources and environmental protection for a sustainable ecological existence, but also serves as a basis for the development of economic and social life. Hong and Guo[16] examined various cooperation contracts within a green product supply chain and evaluated their environmental performance. Fadavi et al.[17] investigated the extent of environmental sustainability and price competition in price-sensitive markets, involving both a manufacturer and a retailer.

2.3 Optimal rebate policy for green design

A durable consumer good can consist of numerous components, with each part designed by a separate team that selects type of plastic or other material best suited to its purpose. Adopting green design (GD) combined with a rebate policy (RP) can be an important solution. Manufacturers must also befully informed about demand, price, and rebate, but may face uncertain demand and selling through a competitive retail market. Giri and Chakraborty[18] investigated an imperfect production system with uncertain demand and rebate warranty. Arcelus et al.[19] studied a price-dependent stochastic demand that depends directly on the selling price of the rebate value, followed by discussions on final customer demand. Mishra et al.[20] presented a deteriorating inventory model with four-stage production rates and derived demand based on the rebate value, considering the selling price of the product at shortage. Ganguly et al.[21] presented a versatile production-inventory system designed for a single-stage assembled item where dfferent parts are both manufactured and remanufactured within the same generation cycle. Defective items occur randomly due to the imperfections inherent in the system. Different approaches for a sustainable green economy with remanufacturing are discussed in Karmakar et al.[22] Shah et al.[23] Roy and Sana[24]. Zwolinski and Brissaud[25] illustrated how the methodology is applied in the two primary design activities: redesigning products from a remanufacturing perspective and developing new products. Haziri and Sundin[26] presented a framework that supports design for remanufacturing by implementing structured feedback from remanufacturing to design. On the other hand, green technologies and green products startups offer numerous benefits beyond just being environmentally friendly. The government provides subsidies to encourage the development of green products. To deepen our understanding of subsidy and tax rebate policies for the research and development (R&D) of green enterprises in China, Chang et al.[27] called for further research on the global Malmquist–Luenberger (GML) index method. In the papers by Fadavi et al.[28], the applications of new green product structures that can help companies reduce their environmental impacts were discussed in detai. For example, marketers of environmentally friendly products can boost future purchases by promoting the company's green image. A growing number of research studies provide insights into solutions for sustainable Economic Order Quantity (EOQ) and Economic Production Quantity (EPQ) problems by taking into account various sustainability aspects such as economic[27-28], environmental[29-34], and social issues[35-37]. There is a fairly extensive literature on the traditional EOQ/EPQ model. However, within that literature, there is a surprising lack of information on rebate policies or green supply chains. Using price or rebate to increase demand for new or existing products or services can be a good way to attract customers or boost lagging sales. Mishra et al.[20] have done extensive work with four-stage production rates and demand based on rebate value, considering the product selling price. Meng et al.[38] studied a cooperative collaborative pricing policies for products in a dual-channel green supply chain and compared the optimal solutions in two cases. However, in many real-world supply chain systems, demand patterns can be price-dependent[38-42], stock-dependent[43-44], freshness-dependent[45], age-dependent[46-47], quality-dependent[48], time-dependent[49-51], CSR impact-dependent[52-57]. Life cycle assessment (LCA) is a widely used method for measuring the environmental costs that can be attributed to a product design or service. In the following years, numerous studies have been conducted on the effectiveness of LCA for green product design[58-60]. This paper may lead to a better understanding of rebate and return policy under the principle of green product design. This paper is divided into six main sections. Section 1 provides some background information about the rebate policy, demand patterns and LCA issues. Section 2 outlines the description of green product design. Section 3 presents the industrial background and a list of notations and assumptions. Section 4 discusses the development of mathematical formulations describing some of the theoretical results encountered. Section 5 presents application examples, some numerical examples, and sensitivity analysis. Finally, Section 6 outlines some results, discussions, and conclusions for further research. Table 1 provides a brief comparison of the above demand patterns mentioned. It should be noted that demand affected by the rebate impact has been the focus of many researchers in recent years. However, it seems that various other types of product return activities can influence demand in the real world. There has been far less research on the impact of rebates and green product design development in a sustainable EPQ model. By conducting these demand classifications in Table 1, several research gaps in sustainable EOQ and EPQ models were identified. The intention is to look at the economic and social sustainability of manufacturer rebates, where the manufacturer refunds the consumers after the purchase behaviors, which is an effective promotional tool.

Table 1 Comparison tables for different demand patterns

Note: price-dependent demand (PD), stock-dependent demand (SD), freshness-dependent demand (FD), age-dependent demand (AD), quality-dependent demand (QD), time-dependent demand (TD), and CSR impact-dependent demand (CD).

3. Notations and assumptions

In the following, an EPQ model that considers a single product, two-stage production systems, and the component replacement problem is developed. For convenience, the notations and assumptions required to develop this integrated model are given first.

3.1 Notations

The following system parameters are used to develop the model:

System parameters

3.2 Assumptions

The following assumptions are made in our production model:

4. Mathematical model formulation

This paper is primarily concerned with sustainable return on investment (S-ROI) issues through data analysis of an inventory model. Figure 3 summarizes the results of the production system. The bottom three sketches represent the inventory level of the raw material, the inventory level of the end product and the inventory level of the return product, respectively.

Assumption 1, as stated, implies that total demand remains constant within a given cycle, i.e., [TeX:] p_{1}t_{1} = p_{2}t_{2} = \text{...} = p_{i}t_{i} = \text{...} = p_{n}t_{n} = \chi_{e}t_{e} = DT$p_1{t}_1=p_2{t}_2=\mathrm{\text{...}}=p_i{t}_i=\mathrm{\text{...}}=p_n{t}_n={\chi }_e{t}_e=DT$, leading to:

[TeX:] t_{i} = \frac{p_{n}t_{n}}{p_{i}},$t_i=\frac{p_n{t}_n}{p_i},$ (1)

and

[TeX:] T = \frac{p_{n}t_{n}}{D},$T=\frac{p_n{t}_n}{D},$ (2)

where [TeX:] i = 1,2,\ldots,n$i=1,2,\dots ,n$.

Fig. 3 Graph illustrating inventory levels for components, end products, and returned items

Deriving the maximum inventory level of the component i based on Eqs. (1) and (2) can be formulated as:

[TeX:] H_{i} = \left( p_{i} - \chi_{e} \right)t_{i} = \chi_{e}t_{\text{id}}.$H_i=(p_i-{\chi }_e)t_i={\chi }_e{t}_{\mathrm{\text{id}}}.$ (3)

At the period t_id, inventory depletion of the component i is:

[TeX:] t_{\text{id}} = \frac{H_{e}}{\chi_{e}} = \frac{\left( p_{i} - \chi_{e} \right)t_{i}}{\chi_{e}},$t_{\mathrm{\text{id}}}=\frac{H_e}{{\chi }_e}=\frac{(p_i-{\chi }_e)t_i}{{\chi }_e},$ (4)

where [TeX:] i = 1,2,\ldots,n$i=1,2,\dots ,n$.

The maximum inventory level of the end product can be expressed as follows:

[TeX:] \begin{matrix} H_{e} = \left( \chi_{e} - D \right)t_{e} = \left( \chi_{e} - D \right)\left( t_{n} + t_{\text{nd}} \right) \\ \ \ \ \ \ \ = \left( \chi_{e} - D \right)\frac{p_{n}}{\chi_{e}}t_{n}.\ \ \ \ \ \ (from\ Eq.\ (4)) \\ \end{matrix}$\begin{array}{c}\hfill H_e=({\chi }_e-D)t_e=({\chi }_e-D)(t_n+t_{\mathrm{\text{nd}}})\hfill \\ \hfill =({\chi }_e-D)\frac{p_n}{{\chi }_e}{t}_n.(fromEq.(4))\hfill \end{array}$ (5)

At the period t_rd, inventory depletion of the return product is:

[TeX:] H_{r} = p_{i}\left( 1 - \theta_{r} \right)t_{\text{rd}} = rt_{r}.$H_r=p_i(1-{\theta }_r)t_{\mathrm{\text{rd}}}=r{t}_r.$ (6)

This model is composed of the following nine elements:

[TeX:] SR = \left( s_{p} - s \right)DT = \left( s_{p} - s \right)p_{n}t_{n}$SR=(s_p-s)DT=(s_p-s)p_n{t}_n$.

To summarize the above results, total profit per unit time (denoted by[TeX:] AP\left( s,t_{n},r \right)$AP(s,t_n,r)$) can be obtained as follows:

[TeX:] \begin{matrix} AP\left( s,t_{n},r \right) = \frac{1}{T}\left( \text{SR} - SC - HC_{e} - HC_{s} - HC_{r} - RC - RC_{r} - PC - IC \right) \\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } = \frac{1}{T}\left\{ \left( s_{p} - s \right)p_{n}t_{n} - k - \frac{h_{e}\left( \chi_{e} - D \right)}{2\chi_{e}D}p_{n}^{2}t_{n}^{2} - \frac{p_{n}^{2}t_{n}^{2}}{2}\left\lbrack \sum_{i = 1}^{n}h_{i}\left( \frac{1}{\chi_{e}} - \frac{1}{p_{i}} \right) \right\rbrack \right.\ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - r\left\lbrack c_{r}\theta_{r} + \theta_{e}DT + \sum_{i - 1}^{n}{\theta_{i}\text{DT}} \right\rbrack - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right)p_{n}t_{n} \\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. \ - \frac{h_{r}r{t_{r}}^{2}}{2} - c_{w}r(1 - \theta_{r}) - \gamma\left\lbrack F_{r} + V_{r}(rt_{r})t_{\text{rd}} \right\rbrack \right\}. \\ \end{matrix}$\begin{array}{c}\hfill AP(s,t_n,r)=\frac{1}{T}(\mathrm{\text{SR}}-SC-H{C}_e-H{C}_s-H{C}_r-RC-R{C}_r-PC-IC)\hfill \\ \hfill =\frac{1}{T}\{(s_p-s)p_n{t}_n-k-\frac{h_e({\chi }_e-D)}{2{\chi }_{e}D}{p}_n^2{t}_n^2-\frac{p_n^2{t}_n^2}{2}\left[{\sum }_{i=1}^n{h}_i(\frac{1}{{\chi }_e}-\frac{1}{p_i})\right]\hfill \\ \hfill -r[c_r{\theta }_r+{\theta }_{e}DT+{\sum }_{i-1}^n{\theta }_i\mathrm{\text{DT}}]-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)p_n{t}_n\hfill \\ \hfill -\frac{h_{r}r{t_r}^2}{2}-c_{w}r(1-{\theta }_r)-\gamma [F_r+V_r(r{t}_r)t_{\mathrm{\text{rd}}}]\}.\hfill \end{array}$ (7)

Now, the first-order partial derivatives of the total profit should be calculated per unit time with respect to s, t_n, and r. Then the following is obtained:

[TeX:] \frac{\partial AP\left( s,t_{n},r \right)}{\partial s} = \frac{1}{T}\left\{ - p_{n}t_{n} + \left\lbrack \frac{h_{e}}{2D^{2}}\delta + \frac{h_{e}}{2\chi_{e}} \right\rbrack p_{n}^{2}t_{n}^{2} - r\left\lbrack \theta_{e}\delta T + \delta\sum_{i - 1}^{n}{\theta_{i}T} \right\rbrack \right\},$\frac{\partial AP(s,t_n,r)}{\partial s}=\frac{1}{T}\{-p_n{t}_n+[\frac{h_e}{2D^2}\delta +\frac{h_e}{2{\chi }_e}]p_n^2{t}_n^2-r[{\theta }_e\delta T+\delta {\sum }_{i-1}^n{\theta }_{i}T]\},$ (8)

[TeX:] \frac{\partial AP\left( s,t_{n},r \right)}{\partial t_{n}} = \frac{1}{T}\left\{ \left( s_{p} - s \right)p_{n} - \frac{h_{e}\left( \chi_{e} - D \right)}{\chi_{e}D}p_{n}^{2}t_{n}^{} \right.\ - p_{n}^{2}t_{n}^{}\left\lbrack \sum_{i = 1}^{n}h_{i}\left( \frac{1}{\chi_{e}} - \frac{1}{p_{i}} \right) \right\rbrack\left. \ - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right)p_{n} \right\},$\frac{\partial AP(s,t_n,r)}{\partial t_n}=\frac{1}{T}\{(s_p-s)p_n-\frac{h_e({\chi }_e-D)}{{\chi }_{e}D}{p}_n^2{t}_n^{}-p_n^2{t}_n^{}\left[{\sum }_{i=1}^n{h}_i(\frac{1}{{\chi }_e}-\frac{1}{p_i})\right]-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)p_n\},$ (9)

and

[TeX:] \begin{matrix} \frac{\partial AP\left( s,t_{n},r \right)}{\partial s} = \frac{1}{T}\left\{ - p_{n}t_{n} + \left\lbrack \frac{h_{e}}{2D^{2}}\delta + \frac{h_{e}}{2\chi_{e}} \right\rbrack p_{n}^{2}t_{n}^{2} - r\left\lbrack \delta\theta_{e}T + \delta\sum_{i - 1}^{n}{\theta_{i}T} \right\rbrack \right\}, \\ \frac{\partial AP\left( s,t_{n},r \right)}{\partial r} = \frac{1}{T}\left\{ \left\lbrack \frac{h_{e}}{2D^{2}}\delta + \frac{h_{e}}{2\chi_{e}} \right\rbrack p_{n}^{2}t_{n}^{2} \right.\ + \left\lbrack \frac{h_{e}}{2D^{2}}\left( 1 - \delta - \theta_{r} \right) - \frac{h_{e}}{2\chi_{e}} \right\rbrack p_{n}^{2}t_{n}^{2} - \left\lbrack c_{r}\theta_{r} + D\theta_{e}T + D\sum_{i - 1}^{n}{\theta_{i}T} \right\rbrack \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - r\left\lbrack \theta_{e}\delta T + \delta T\sum_{i - 1}^{n}\theta_{i} \right\rbrack - r\left( 1 - \delta - \theta_{r} \right)\left\lbrack \theta_{e}T + T\sum_{i - 1}^{n}\theta_{i} \right\rbrack\left. \ - \frac{h_{r}{t_{r}}^{2}}{2} - c_{w}(1 - \theta_{r}) - \gamma V_{r}t_{r}t_{\text{rd}} \right\}. \\ \end{matrix}$\begin{array}{c}\hfill \frac{\partial AP(s,t_n,r)}{\partial s}=\frac{1}{T}\{-p_n{t}_n+[\frac{h_e}{2D^2}\delta +\frac{h_e}{2{\chi }_e}]p_n^2{t}_n^2-r[\delta {\theta }_{e}T+\delta {\sum }_{i-1}^n{\theta }_{i}T]\},\hfill \\ \hfill \frac{\partial AP(s,t_n,r)}{\partial r}=\frac{1}{T}\{[\frac{h_e}{2D^2}\delta +\frac{h_e}{2{\chi }_e}]p_n^2{t}_n^2+[\frac{h_e}{2D^2}(1-\delta -{\theta }_r)-\frac{h_e}{2{\chi }_e}]p_n^2{t}_n^2-[c_r{\theta }_r+D{\theta }_{e}T+D{\sum }_{i-1}^n{\theta }_{i}T]\hfill \\ \hfill -r[{\theta }_e\delta T+\delta T{\sum }_{i-1}^n{\theta }_i]-r(1-\delta -{\theta }_r)[{\theta }_{e}T+T{\sum }_{i-1}^n{\theta }_i]-\frac{h_r{t_r}^2}{2}-c_w(1-{\theta }_r)-\gamma V_r{t}_r{t}_{\mathrm{\text{rd}}}\}.\hfill \end{array}$ (10)

The optimal solutions for [TeX:] (s,\ \ t_{n},\ \ r)$(s,t_n,r)$ satisfy the equations [TeX:] \partial AP(s,\ \ t_{n},\ \ r)/\partial s = 0$\partial AP(s,t_n,r)/\partial s=0$, [TeX:] \partial AP(s,\ \ t_{n},\ \ r)/\partial t_{n} = 0$\partial AP(s,t_n,r)/\partial t_n=0$ and [TeX:] \partial AP(s,\ \ t_{n},\ \ r)/\partial r = 0$\partial AP(s,t_n,r)/\partial r=0$, simultaneously. The necessary condition for [TeX:] \text{AP}(s,\text{\ \ }t_{n},\text{\ \ r})$\mathrm{\text{AP}}(s,t_n,\mathrm{\text{r}})$ to be maximum is [TeX:] \partial AP(s,\ \ t_{n},\ \ r)/\partial t_{n} = 0$\partial AP(s,t_n,r)/\partial t_n=0$, which gives:

[TeX:] \left( s_{p} - s \right)p_{n} - \frac{h_{e}\left( \chi_{e} - D \right)}{\chi_{e}D}p_{n}^{2}t_{n}^{} - p_{n}^{2}t_{n}^{}\left\lbrack \sum_{i = 1}^{n}h_{i}\left( \frac{1}{\chi_{e}} - \frac{1}{p_{i}} \right) \right\rbrack - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right)p_{n} = 0$(s_p-s)p_n-\frac{h_e({\chi }_e-D)}{{\chi }_{e}D}{p}_n^2{t}_n^{}-p_n^2{t}_n^{}\left[{\sum }_{i=1}^n{h}_i(\frac{1}{{\chi }_e}-\frac{1}{p_i})\right]-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)p_n=0$ (11)

It is not easy to find the closed-form solution of t_n from Eq. (11). But the value of t_n that satisfies Eq. (11) not only exists, but is also unique, as stated by the following lemma:

Lemma 1.

The solution of [TeX:] {t_{n}}^{}${t_n}^{}$ (say [TeX:] {t_{n}}^{*}${t_n}^{*}$) profit per profit per unit time [TeX:] \text{AP}(s,\ \ t_{n},\ \ r)$\mathrm{\text{AP}}(s,t_n,r)$ has the global maximum value at the point [TeX:] {t_{n}}^{} = {t_{n}}^{*}${t_n}^{}={t_n}^{*}$, where [TeX:] {t_{n}}^{*} \in \left( 0,\infty \right)${t_n}^{*}\in (0,\infty )$ and satisfies Eq. (11).

Proof.

Let:

[TeX:] F\left( t_{n}^{} \right) = \left( s_{p} - s \right)p_{n} - \frac{h_{e}\left( \chi_{e} - D \right)}{\chi_{e}D}p_{n}^{2}t_{n}^{} - p_{n}^{2}t_{n}^{}\left\lbrack \sum_{i = 1}^{n}h_{i}\left( \frac{1}{\chi_{e}} - \frac{1}{p_{i}} \right) \right\rbrack - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right)p_{n}$F\left(t_n^{}\right)=(s_p-s)p_n-\frac{h_e({\chi }_e-D)}{{\chi }_{e}D}{p}_n^2{t}_n^{}-p_n^2{t}_n^{}\left[{\sum }_{i=1}^n{h}_i(\frac{1}{{\chi }_e}-\frac{1}{p_i})\right]-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)p_n$ (12)

for [TeX:] {t_{n}}^{} \in \left( 0,\infty \right)${t_n}^{}\in (0,\infty )$. Taking the first derivative of [TeX:] F(t_{n}^{})$F(t_n^{})$ with respect to t_n, it gives:

i.e.,[TeX:] F(t_{n}^{})$F(t_n^{})$ is a strictly decreasing function of t_n. Then, it is obtained [TeX:] \lim_{t_{n} \rightarrow \infty}F\left( t_{n}^{} \right) = - \infty${lim}_{t_n\to \infty }F\left(t_n^{}\right)=-\infty$ and [TeX:] \lim_{t_{n} \rightarrow 0}F\left( t_{n}^{} \right) = \left\lbrack \left( s_{p} - s \right) - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right) \right\rbrack p_{n}${lim}_{t_n\to 0}F\left(t_n^{}\right)=[(s_p-s)-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)]p_n$.

Following two cases are possible:

CASE 1 If [TeX:] \left( s_{p} - s \right) - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right) > 0$(s_p-s)-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)>0$ , then the solution [TeX:] {t_{n}}^{*}${t_n}^{*}$ which maximizes [TeX:] \text{AP}(s,\text{\ \ }t_{n},\text{\ \ r})$\mathrm{\text{AP}}(s,t_n,\mathrm{\text{r}})$ not only exists, but is also unique when applying the Intermediate Value Theorem, where [TeX:] {t_{n}}^{*} \in \left( 0,\infty \right)${t_n}^{*}\in (0,\infty )$ .

In this case, t_n is also greater than or equal to zero. The most important finding in this case is that the return profits for defective products are greater than the production costs. If the return is made within a limited time frame of 30 days, the manufacturer will provide a courtesy credit to the account for a cash refund for all defective products. Since the sufficient condition is met, taking the second derivative of [TeX:] \text{AP}(s,\ \ t_{n},\ \ r)$\mathrm{\text{AP}}(s,t_n,r)$with respect to t_n, and then substituting [TeX:] {t_{n}}^{} = {t_{n}}^{*}${t_n}^{}={t_n}^{*}$into it, the following is obtained:

Therefore, [TeX:] {t_{n}}^{*}${t_n}^{*}$ is the global maximum point of [TeX:] \text{AP}(s,\ \ t_{n},\ \ r)$\mathrm{\text{AP}}(s,t_n,r)$ . The proof is now complete.

CASE 2If[TeX:] \left( s_{p} - s \right) - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right) < 0$(s_p-s)-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)<0$, then the optimal solution is[TeX:] {t_{n}}^{*} = 0${t_n}^{*}=0$.

In this case, t_n is negative and against [TeX:] {t_{n}}^{} \geq 0${t_n}^{}\ge 0$ because return profits for defective products are less than production costs. At this point, the manufacturer stops offering a refund policy. Therefore, the optimal length of the production run time of the component n should approach zero. Then substituting [TeX:] {t_{n}}^{} = {t_{n}}^{*}${t_n}^{}={t_n}^{*}$into Eq. (7), the total profit per unit time [TeX:] AP\left( s,t_{n},r \right)$AP(s,t_n,r)$can bemodified to a new function of (s, r), given by:

[TeX:] \begin{matrix} AP\left( s,r\left| t_{n} = {t_{n}}^{*} \right.\ \right) = \frac{1}{T}\left\{ \left( s_{p} - s \right)p_{n}{t_{n}}^{*} - k - \frac{h_{e}\left( \chi_{e} - D \right)}{2\chi_{e}D}p_{n}^{2}{t_{n}}^{2*} - \frac{p_{n}^{2}{t_{n}}^{2*}}{2}\left\lbrack \sum_{i = 1}^{n}h_{i}\left( \frac{1}{\chi_{e}} - \frac{1}{p_{i}} \right) \right\rbrack \right.\ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - r\left\lbrack c_{r}\theta_{r} + \theta_{e}DT + \sum_{i - 1}^{n}{\theta_{i}\text{DT}} \right\rbrack - \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right)p_{n}{t_{n}}^{*} \\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. \ - \frac{h_{r}r{t_{r}}^{2}}{2} - c_{w}r(1 - \theta_{r}) - \gamma\left\lbrack F_{r} + V_{r}(rt_{r})t_{\text{rd}} \right\rbrack \right\}. \\ \end{matrix}$\begin{array}{c}\hfill AP(s,r|t_n={t_n}^{*})=\frac{1}{T}\{(s_p-s)p_n{t_n}^{*}-k-\frac{h_e({\chi }_e-D)}{2{\chi }_{e}D}{p}_n^2{t_n}^{2*}-\frac{p_n^2{t_n}^{2*}}{2}\left[{\sum }_{i=1}^n{h}_i(\frac{1}{{\chi }_e}-\frac{1}{p_i})\right]\hfill \\ \hfill -r[c_r{\theta }_r+{\theta }_{e}DT+{\sum }_{i-1}^n{\theta }_i\mathrm{\text{DT}}]-({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)p_n{t_n}^{*}\hfill \\ \hfill -\frac{h_{r}r{t_r}^2}{2}-c_{w}r(1-{\theta }_r)-\gamma [F_r+V_r(r{t}_r)t_{\mathrm{\text{rd}}}]\}.\hfill \end{array}$ (13)

Due to the complexity of the model, it is difficult to find the close form of (s, r) and check the concavity of the manufacturer’s profit function directly. Alternatively, in the following section, the concavity by numerical analysis will be verified and then a simple algorithm developed to obtain the solutions for the manufacturer.

4.1 Algorithm

The model proposed in Section 3 is solved using a solution procedure. First, the problem-solving procedure of the algorithm is introduced. The aim of this algorithm is to determine the optimal solution[TeX:] (s^{*},\ \ t_{n}^{*},\ \ r_{}^{*})$(s^{*},t_n^{*},r_{*}^{})$for maximizing the total annual profit per unit time.

Step 1. Start with [TeX:] \tau = 0$\tau =0$and[TeX:] s_{j,\text{\ \ τ}} = 0$s_{j,\mathrm{\text{τ}}}=0$

Step 2. Substitute [TeX:] s = s_{j,\text{\ \ τ}}$s=s_{j,\mathrm{\text{τ}}}$ into Eq. (11) and solve for [TeX:] t_{n}$t_n$.

Step 3. Solve Eq. (12) to find the optimal value of [TeX:] t_{n}$t_n$ (say [TeX:] t_{n}(s)$t_n(s)$ which is a function of s) and then substitute [TeX:] t_{n}(s)$t_n(s)$ into Eq. (13) to obtain[TeX:] AP\left( s,r\left| t_{n} = {t_{n}}^{*}(s) \right.\ \right)$AP(s,r|t_n={t_n}^{*}(s))$.

Step 4. Find the value of [TeX:] s$s$ and [TeX:] r$r$by setting [TeX:] \partial AP\left( s,r\left| t_{n} = {t_{n}}^{*}(s) \right.\ \right)/\partial r = 0$\partial AP(s,r|t_n={t_n}^{*}(s))/\partial r=0$and [TeX:] \partial AP\left( s,r\left| t_{n} = {t_{n}}^{*}(s) \right.\ \right)/\partial s = 0$\partial AP(s,r|t_n={t_n}^{*}(s))/\partial s=0$.

Step 5. If the difference between [TeX:] s_{j,\ \ \tau}$s_{j,\tau }$ and [TeX:] s_{j,\ \ \tau + 1}$s_{j,\tau +1}$ is sufficiently small, set [TeX:] {\widetilde{s}}_{j} = s_{j,\ \ \tau + 1}${\tilde{s}}_j=s_{j,\tau +1}$. Otherwise, set [TeX:] s_{j,\ \ \tau + 1} = s_{j,\ \ \tau} + \varepsilon$s_{j,\tau +1}=s_{j,\tau }+\epsilon$, where [TeX:] \varepsilon$\epsilon$ is any small positive number, and set [TeX:] \tau = \tau + 1$\tau =\tau +1$, then, go back to Step 2.

Step 6. Substitute [TeX:] t_{n} = {\widetilde{t}}_{\text{n\ j}}$t_n={\tilde{t}}_{\mathrm{\text{n j}}}$ and [TeX:] s = {\widetilde{s}}_{j}$s={\tilde{s}}_j$ into Eq. (7) to calculate the value of[TeX:] \text{AP}(\ \ {\widetilde{t}}_{\text{nj}},\ \ {\widetilde{s}}_{j},\ \ {\widetilde{\gamma}}_{j})$\mathrm{\text{AP}}({\tilde{t}}_{\mathrm{\text{nj}}},{\tilde{s}}_j,{\tilde{\gamma }}_j)$.

Step 7. If [TeX:] \text{AP}({\widetilde{t}}_{\text{nj}},{\widetilde{s}}_{j},{\widetilde{r}}_{j}) < \text{AP}({\widetilde{t}}_{nj + 1},{\widetilde{s}}_{j + 1},{\widetilde{r}}_{j + 1})$\mathrm{\text{AP}}({\tilde{t}}_{\mathrm{\text{nj}}},{\tilde{s}}_j,{\tilde{r}}_j)<\mathrm{\text{AP}}({\tilde{t}}_{nj+1},{\tilde{s}}_{j+1},{\tilde{r}}_{j+1})$, then [TeX:] (t_{n}^{*},\text{\ \ }s^{*},\text{\ \ }r^{*})$(t_n^{*},s^{*},r^{*})$=[TeX:] ({\widetilde{t}}_{nj + 1},{\widetilde{s}}_{j + 1},{\widetilde{r}}_{j + 1})$({\tilde{t}}_{nj+1},{\tilde{s}}_{j+1},{\tilde{r}}_{j+1})$ is the optimal solution. Otherwise, [TeX:] (t_{n}^{*},\text{\ \ }s^{*},\text{\ \ }r^{*})$(t_n^{*},s^{*},r^{*})$=[TeX:] ({\widetilde{t}}_{\text{nj}},{\widetilde{s}}_{j},{\widetilde{r}}_{j})$({\tilde{t}}_{\mathrm{\text{nj}}},{\tilde{s}}_j,{\tilde{r}}_j)$.

Step 8. Substitute [TeX:] t_{n}^{*}$t_n^{*}$, [TeX:] s^{*}$s^{*}$ and [TeX:] r^{*}$r^{*}$ into Eqs. (1), (2), and (13) to calculate the values of [TeX:] {t_{1}}^{*},{t_{2}}^{*},\ldots,{t_{n - 1}}^{*}${t_1}^{*},{t_2}^{*},\dots ,{t_{n-1}}^{*}$, and [TeX:] \text{AP}(t_{n}^{*},\ \ s^{*},\ \ r^{*})$\mathrm{\text{AP}}(t_n^{*},s^{*},r^{*})$.

5. Application of the proposed algorithm

Rebate can be used for many recycling purposes, such as establishing new programs or collection points and identifying markets for recovered materials. The practicality of the proposed model was evaluated through a case study with the own-brand manufacturers (OBMs) of bike companies in Taiwan. Numerical examples were solved to test the robustness and reliability of the proposed model and examine trends in the optimal policies, and managerial insights for the OBMs.

5.1 Industrial background

Bike manufacturers have been trying to rethink the way bicycles are made and delivered in terms of a circular economy. The aim is to design bicycles so that they last much longer and all raw materials can be separated and reused. Ensuring a prompt return to small and medium-sized enterprise (SME) manufacturers following the consumption phase is crucial for maximizing the recovery value of durable products and, in general, minimizing negative environmental impacts[63-64]. SME manufacturers often emphasize the benefits of their products, leveraging this positive information to influence consumers in the target market during their purchasing decisions and increase their awareness of green environmental protection.

5.2 Numerical examples

Manufacturers and importers of newly regulated recyclable waste (RRW) products, as well as their packaging, containers, and specific raw materials, are obliged to take responsibility for collecting these used bicycles and old bike parts when they reach the end of their life cycle. Numerical examples were employed to validate our analytical results in the scenario and a sensitivity analysis conducted to identify trends in the optimal policies. This approach aims to provide managerial insights to the bicycle manufacturer.

Example 1. Let us consider an inventory system with the following data. A two-stage assembly system is observed. It consists of 3 component processes (n=3) in stage 1, and 3 components are required to assemble an end product in Stage 2.

Example 2. Let us consider an inventory system with the following data. A two-stage assembly system is observed. It consists of 3 component processes (n=3) in Stage 1 and 3 components are required to assemble an end product in stage 2.

The algorithm presented was utilized to calculate the following optimal solution: [TeX:] s_{}^{*} = 25.4488$s_{*}^{}=25.4488$, [TeX:] t_{n}^{*} = 1.0176$t_n^{*}=1.0176$, [TeX:] r_{}^{*} = 6.7411$r_{*}^{}=6.7411$, [TeX:] \gamma^{*} = 1${\gamma }^{*}=1$,[TeX:] ACT = 9.6519.$ACT=9.6519.$ Note: ACT refers to the average CPU time (in seconds).

5.3 Sensitivity analysis

The numerical example provided in Section 5.2 was used to evaluate the impact of modifications to the system parameters ([TeX:] \beta_{0}${\beta }_0$, [TeX:] \beta_{1}${\beta }_1$, [TeX:] \beta_{2}${\beta }_2$, a, b, [TeX:] \delta$\delta$, w, [TeX:] F_{r}$F_r$, [TeX:] V_{r}$V_r$, [TeX:] h_{1}$h_1$, [TeX:] h_{2}$h_2$, [TeX:] h_{3}$h_3$, [TeX:] h_{e}$h_e$, [TeX:] h_{r}$h_r$, [TeX:] \theta_{1}${\theta }_1$, [TeX:] \theta_{2}${\theta }_2$, [TeX:] \theta_{e}${\theta }_e$, [TeX:] \theta_{r}${\theta }_r$) on the values [TeX:] s^{*}$s^{*}$, [TeX:] t_{n}^{*}$t_n^{*}$, [TeX:] r^{*}$r^{*}$, and [TeX:] AP(s^{*},t_{n}^{*},r^{*})$AP(s^{*},t_n^{*},r^{*})$. Each parameter was individually adjusted (keeping other parameters constant) by +50%, +25%, −25%, or −50%. Analytical results analytical results for examples 1 and 2 are shown in Tables 2 and 3, respectively. These results provider noteworthy observations and managerial insights that can guide decision-making. Specifically, rebate programs are found to boost revenue, incentivize customers to choose one manufacturer over the competition, and foster long-term buyer-seller relationships due to their higher return profit i.e., [TeX:] \left( s_{p} - s \right) > \left( \beta_{0} + \frac{\beta_{1}}{\chi_{e}} + \beta_{2}\chi_{e} \right)$(s_p-s)>({\beta }_0+\frac{{\beta }_1}{{\chi }_e}+{\beta }_2{\chi }_e)$. This would help quickly set up rebate programs based on product lines, customer segments, or other criteria. Additional detailed results are tabulated in Table 4 to provide some managerial insights for bike firms.

Table 2 Effect of changes in various parameters of the model for Example 1