Optimal Pricing and Retailing Strategy for an Assembled Product Manufacturing–Remanufacturing Process under Carbon Emission Regulations and Autonomation

Abstract

Online-to-offline (O2O) retailing offers unique opportunities for customizable assembled products with spare parts. Customers can browse and configure their desired product online, selecting from various components. Imperfect production, where a certain percentage of products have defects, can be amplified in the manufacturing system. Stricter carbon emission regulations put pressure on manufacturers to minimize waste. This creates a tension between discarding imperfect products, generating emissions, and potentially offering them at a discount through the O2O channel, which could raise quality concerns for consumers. In this study, an imperfect single-stage production process is examined, incorporating manufacturing–remanufacturing within a single stage for assembled products containing various spare parts. The study explores an investment scenario aimed at enhancing the environmental sustainability of the product. Additionally, two carbon emissions regulation strategies, specifically carbon cap-and-trade regulation and carbon taxation, are evaluated for their effectiveness in mitigating carbon footprints. The identification of waste, particularly in the form of defective items, is achieved through automated inspection techniques. The demand for spare parts associated with the assembled products is intricately linked to the selling prices set across diverse channels. Finally, the total profit of the manufacturing system is maximized with the optimized value of the selling prices, order quantity, backorder quantity, and investments in autonomated inspection, setup cost, and green technology. Numerical illustrations show that system profit was optimized when the defective rate followed a triangular distribution under carbon cap-and-trade regulation and when green technology investment helped to enhance retailer profit by $18.12\%$, whereas autonomated inspection increased retailer profit by $10.27\%$.

1. Introduction

Selling customizable products with spare parts through O2O retailing is a great opportunity for companies. Customers can make their products on the Internet by choosing from many parts. Customers can use the ease of shopping and setting things up online. They create the product they want at home by choosing from many different parts. This level of customization allows for a wider variety of products to be offered without needing to stock a massive physical inventory [1]. Furthermore, O2O retailers can conveniently showcase various replacement parts on their website for customers to purchase alongside the product or for future needs. Nowadays, some customers love to get the product on their doorstep, whereas some customers still believe in choosing the product by physical touch in the offline store [2]. However, this system can have problems with making things perfect, so some of the finished products might not be good. Disposing of these faulty products can result in garbage and environmental damage, whereas marketing them at a lower cost online can cause skepticism about their effectiveness. This makes it hard for O2O retailers to handle making products that are not perfect while also trying to reduce waste and keep customers happy [3]. In O2O retailing, customers may place an order online and pick up the product from the store or choose the product by visiting a local store and ordering online. The product’s price is usually influenced by the seller, leading to fluctuations in consumer interest [4]. Therefore, proper management is required to obtain optimized profit for the industry.

The regulations on carbon emissions from manufacturing plants may have both positive and negative impacts. On the one hand, more stringent regulations such as carbon cap-and-trade (CCT) or carbon taxes (CTs) encourage manufacturers to reduce their waste [5]. This is nice because it corresponds to reducing the amount of defective products as a natural consequence of manufacturing in an imperfect environment. A CCT policy type is more effective for an assembly-type product with imperfect manufacturing processes than a stricter environmental regulation [6]. The cap on the emissions motivates companies to innovate and bring down their environmental footprint. This in turn might incentivize the adoption of greener technologies or make production processes more efficient, which reduces waste [7]. By combining pieces from various sources, the demand for materials and the energy consumed during production can be reduced. Of course, the performance also depends on how one designs the cap-and-trade system. In a hypothetical scenario where it is easy or economical to obtain permission for carbon emissions, there will be minimal motivation to explore alternative solutions for decreasing pollution. This could indirectly and significantly harm the environment [8]. In a similar way, to make the imperfect manufacturing process profitable and to optimize the carbon footprint, we need appropriate CT policy as well [9]. When policies like these are in place, they can often lead to making sure that all products will come out of the manufacturing plant, no matter how defective. This could potentially lead to discounted sales through online channels, raising concerns among consumers about product quality and potentially undermining the intended environmental benefits [4]. Therefore, for carbon regulations to be truly effective, industries need to be coupled with strategies for managing defective products responsibly, such as proper recycling or refurbishment programs. This would allow manufacturers to comply with regulations while maintaining consumer trust in the quality of assembled products.

During the long-run production process, the manufacturing system may shift to an out-of-control state and start to produce faulty items randomly, which increases waste generation and enhances system costs [10]. Those faulty items need to be identified perfectly and repaired or remanufactured to make them perfect and increase customer satisfaction levels [11]. However, errors cannot be entirely avoided in human inspection. Thus, in an imperfect manufacturing process for assembled products, employing an autonomated inspection strategy can significantly improve efficiency [4]. Using machines to find defects in products helps separate the good ones from the bad ones quickly. This prevents faulty products from being missed during manual checks, which reduces the amount of wasted parts. Therefore, components that are in good condition can be removed from damaged items and reused in the production of new items. This helps to reduce waste and make the most of resources. This translates to cost savings and potentially higher profits. It ensures that defects are discovered with greater accuracy and reliability compared to human inspection. This reduces the risk of faulty products slipping through quality control and reaching customers [12]. Once that is set up, autonomation makes the quality inspection easier, enabling identification and kick-out of defective products from an assembly line without compromising the inspection speed. It streamlines the process to avoid wasted resources on things that will be deleted. When looking at O2O retailing with spare parts, autonomation can help efficiently manage defective assembled products as well. However, it is important to consider the initial investment cost of implementing an autonomated inspection system and ensure its capabilities align with the complexity of the assembled products.

An assembled product is defined as an object created from individual components, spanning a wide range of items, from basic furniture necessitating assembly with a screwdriver to intricate electronics featuring elaborate circuitry. The process of assembly itself can be complex, encompassing manual labor in manufacturing facilities or automated assembly systems (Capponi et al. [13]). Typically, manufacturers furnish consumers with instructions to assist them in completing the assembly process at home. The concept of assembling products has historical roots, with early instances found in furniture and tools. Nevertheless, the advent of mass production and modular design in the 20th century significantly propelled the expansion of this product category. This advancement facilitated the efficient production of intricate products in separate parts, subsequently assembled for broader dissemination and affordability (Eswaran et al. [14]). Assembled products offer numerous benefits, such as enhanced cost-effectiveness in production and transportation owing to optimal space utilization. Moreover, modularity enables customization and facilitates simpler repairs or upgrades. Nonetheless, incorrect assembly by consumers can result in malfunctions, and some assembled products may be perceived as less robust or enduring compared to fully constructed items. The future landscape of assembled products is likely to witness increased automation and personalization. The utilization of 3D printing technology could potentially enable on-demand assembly of specific products, while progress in robotics could further streamline manufacturing assembly lines. Furthermore, modularity is anticipated to gain more prominence, empowering consumers to tailor products according to their individual requirements and preferences.

Investments in setup cost reduction can offer a double benefit for O2O retailers dealing with imperfect production of assembled products with various spare parts, while also facing carbon emission regulations [12]. Firstly, lower setup costs allow for more frequent production runs. This can be crucial in an O2O environment where customer demand for specific configurations fluctuates. Secondly, lower setup costs can incentivize smaller batch sizes. This becomes handy when there are imperfections in items, as it allows for quicker removal of defective items in smaller batches and reduces the volume of waste generated by defective parts in long production runs.

The following gaps in the literature have been identified based on the above explanations.

• Research on assembled products with multiple spare parts is common in the literature [11,12]. However, research on O2O retailing for assembled products under the channel’s selling price-dependent demand for different products with different carbon regulation policies is rare in the literature.
• The utilization of automated inspection for the detection of faulty products within an inefficient manufacturing process has been discussed in numerous prior research studies [4]. Nevertheless, a deficiency in the literature persists regarding a one-step, cleaner production framework designed for assembled goods characterized by varying defect rates in the presence of CCT laws as well as CT policies, a gap that is addressed in the present investigation.
• Green technology investments to reduce carbon emissions are prevalent in academic literature [7,9,15]. Nevertheless, to the best of our understanding, the literature has not yet addressed an imperfect manufacturing process for a final product subject to various carbon regulations, automated inspection, and cost reduction in the context of O2O retailing. Therefore, this study makes a pioneering effort to bridge this gap.

An O2O retailing system for assembled products with various spare parts is introduced in this research. Defective items are produced randomly when the manufacturing process is not controlled, and they may follow a specific distribution pattern that can be accurately detected through an advanced machine-based autonomous inspection method. This autonomous inspection guarantees a perfect inspection rate of $100\%$ and eliminates any waste in the production process. Two carbon management strategies, specifically carbon cap-and-trade regulations and CT policies, are examined to reduce the carbon footprint. Additionally, investments are made to enhance the environmental sustainability of the product through green technology initiatives. By investing in setup costs, the total system cost is reduced, leading to an increase in total system profitability. Ultimately, the total profit of the O2O retailing system is maximized by optimizing the decision variables, such as selling prices for two different channels, order quantity, backorder quantity, and investments to reduce setup cost and to reduce carbon emissions from the manufacturing process.

The arrangement of the remainder of the manuscript is outlined as follows. A comprehensive examination of the gaps in the current literature in this particular area and the rationale behind the research are presented as a literature review in the subsequent Section 2. The production model for assembled products within O2O retailing, incorporating a range of costs, assumptions, and notations, is expounded upon in Section 3. Instances demonstrating the O2O retailing for assembled products across different scenarios are delineated in Section 4. The impact of various parameters on determining the system’s profitability is discussed in Section 4.4. The practical application of this research and the advantages for management are elucidated in Section 5. Lastly, Section 6 presents the final remarks, including constraints and potential future expansions.

2. Literature Review

The motivation of the present study and existing literature gaps are comprehensively described in this section. Table 1 is structured to show the contributions of the current study over the existing literature.

2.1. Importance of O2O Retailing for Assembled Products

O2O retailing can offer a unique solution to imperfect production systems of assembled products with varying spare parts. By leveraging online platforms, retailers can provide customers with a wider selection of customizable products and real-time availability information. The O2O retailing system is a trending topic these days. The people of the Gen Z generation love to connect with the Internet most of the time. Industries and researchers have adapted to this behavior and proposed several new techniques for the O2O retail business. Customers can choose the product online and pick it up from the store or choose from the store and order online. However, for assembled products with different spare parts, customers can customize their desired products, which is cost-effective and profitable for the industries. Reward points are a good option for attracting customers to online channels, as discussed by Yan et al. [16]. They discussed the coordination between the players of the supply chain for an O2O retailing system under reward points and profit-sharing contracts. However, they avoided the concept of channel selection strategies for their study. Wan and Chen [17] put forward a proposal for an O2O retailing system focusing on platform selection and supplier efficiency within the realm of China’s e-commerce. Their research, however, omitted the consideration of variable demand. Omni-channel or dual channel retailing has become popular today [18]. An in-depth analysis of the literature regarding omni-channel retailing was performed by Cai and Lo [18]. They discussed the efficiency of the omni-channel model in business industries. An imperfect manufacturing and remanufacturing model with different spare parts was established by Sarkar et al. [19]. However, they developed their model under traditional retailing with constant demand. Similarly, Dey et al. [11] proposed a cost-effective production system for assembled products under intelligent autonomated inspection with budget and space constraints. An O2O supply chain system for newly launched products with the impact of customer reviews was developed by Li et al. [20]. They discussed the impact of online customer reviews in O2O on retention, with pricing strategies for the newly launched product in the market. However, they ignored the concept of assembled products and intelligent technologies for their model. Considering O2O retailing to be a flexible production system with defective generation was proposed by Sarkar et al. [21]. They considered two different quality products for their model. However, they discussed a single type of product for their model instead of assembled products with different spare parts. By considering assembled products with different spare parts, a defect generation system was illustrated by Dey et al. [12]. However, they considered traditional retailing for their production model. An assembled product production system considering defect prediction was developed by Verna et al. [22]. A systematic literature review of the assembled system under Industry 4.0 standards was conducted by Dolgui et al. [23]. They provided a detailed explanation of the assembly system under Industry 4.0 standards. From their study, it was clear that O2O retailing for assembled products still requires additional investigation.

Several studies have been published in the literature considering O2O retailing with different pricing decisions, and a huge number of publications regarding assembly systems or assembled products can be found in the literature. However, O2O retailing for an imperfect single-stage manufacturing–remanufacturing system for assembled products with different spare parts under specific pricing strategies is still a significant gap in the literature. In this study, we try to fill this literature gap.

Table 1. Comparison and contribution of the present study.

Author(s)	Model Type	Retailing	Carbon Emissions Regulation	Demand	Product Type	Investment
Mridha et al. [4]	EPQ	O2O	NC	CSP	Single	AI
Hasan et al. [7]	Inventory	Traditional	CCT, CT, LCE	Constant	Single	GT
Gupta & Khanna [9]	EPQ	Traditional	CCT & CT	SPD	Single	GT
Taleizadeh et al. [10]	CLSCM	O2O	NC	SPD	Single	GT, AD
Dey et al. [12]	EPQ	Traditional	NC	Constant	Assembled	SC, GT, & AI
Li et al. [20]	SCM	O2O	NC	Constant	Single	GT
Sarkar et al. [21]	EPQ	O2O	NC	CSP	Single	AI
Verna et al. [22]	Production	Traditional	NC	Constant	Assembled	NC
Konstantaras et al. [24]	CLSCM	Traditional	CT	Constant	Single	NC
Mahato et al. [25]	Inventory	Traditional	CT	Constant	Single	NC
Sana et al. [26]	EPQ	Traditional	CN	SPD	Single	GT
This study	EPQ	O2O	CCT & CT	CSP	AP	SC, GT, & AI

AD: Advertisement investment; AI: Autonomated inspection investments; AP: Assembled product with different spare parts, CT: Carbon taxation, CCT: Carbon cap-and-trade; CLSCM: Closed-loop supply chain system; CSP: Channel’s selling price; EPQ: economic production quantity, GT: Green technology; LCE: Limited carbon emission; NC: Not considered; SC: Setup cost reduction; SCM: Supply chain management; SPD: Selling price dependent demand.

Table 1. Comparison and contribution of the present study.

Author(s)	Model Type	Retailing	Carbon Emissions Regulation	Demand	Product Type	Investment
Mridha et al. [4]	EPQ	O2O	NC	CSP	Single	AI
Hasan et al. [7]	Inventory	Traditional	CCT, CT, LCE	Constant	Single	GT
Gupta & Khanna [9]	EPQ	Traditional	CCT & CT	SPD	Single	GT
Taleizadeh et al. [10]	CLSCM	O2O	NC	SPD	Single	GT, AD
Dey et al. [12]	EPQ	Traditional	NC	Constant	Assembled	SC, GT, & AI
Li et al. [20]	SCM	O2O	NC	Constant	Single	GT
Sarkar et al. [21]	EPQ	O2O	NC	CSP	Single	AI
Verna et al. [22]	Production	Traditional	NC	Constant	Assembled	NC
Konstantaras et al. [24]	CLSCM	Traditional	CT	Constant	Single	NC
Mahato et al. [25]	Inventory	Traditional	CT	Constant	Single	NC
Sana et al. [26]	EPQ	Traditional	CN	SPD	Single	GT
This study	EPQ	O2O	CCT & CT	CSP	AP	SC, GT, & AI

AD: Advertisement investment; AI: Autonomated inspection investments; AP: Assembled product with different spare parts, CT: Carbon taxation, CCT: Carbon cap-and-trade; CLSCM: Closed-loop supply chain system; CSP: Channel’s selling price; EPQ: economic production quantity, GT: Green technology; LCE: Limited carbon emission; NC: Not considered; SC: Setup cost reduction; SCM: Supply chain management; SPD: Selling price dependent demand.

2.2. Carbon Emissions Regulation Policies for Imperfect Production Systems

Carbon emission regulation policies can be a powerful tool to curb pollution in imperfect production systems, where defects and waste occur. These policies, like carbon taxes or emissions caps, help companies reduce their carbon footprint [27]. In an imperfect system, this could lead to investment in cleaner technologies or improved production processes to minimize waste and associated emissions [28]. In the logistic system, the impact of carbon emission was unavoidable when optimizing shipment size [29]. While some defective products and emissions might be unavoidable, regulations push manufacturers to innovate and operate more efficiently, ultimately leading to a decrease in overall carbon output [15]. Researchers have discussed the effect of government regulation policy on manufacturing industries. He et al. [30] put forth a rigorous carbon policy for managing supply chains in the context of emission regulations. They explored the impact of carbon cap policies on supply chain management, overlooking the concept of O2O retailing for assembled products within an imperfect production process. Meanwhile, Huang et al. [5] developed an inventory system considering various carbon emission regulation policies, encompassing production, transportation, and storage. Nevertheless, their focus was on single-product retailing through traditional channels rather than dual channels. Correspondingly, Rout et al. [31] examined the influence of carbon emission regulation policies on a sustainable supply chain system for a single deteriorating product, yet they neglected the concept of imperfect generation processes. Konstantaras et al. [24] introduced a closed-loop supply chain model under the CT policy to diminish the carbon footprint, optimizing inventory decisions accordingly. Hasan et al. [7] developed an inventory system that considered different carbon regulation policies for a single type of deteriorating item, omitting the notions of O2O retailing and imperfect generation. Dey et al. [12] proposed an imperfect production system for assembled products, integrating carbon emissions and green technology investments but overlooking the variations in carbon regulation policies. Lu et al. [6] modeled a global production–inventory system for deteriorating items under CT and CCT regulation policies, focusing on a single item with a flawless manufacturing process while excluding imperfect generations. Arora et al. [32] developed an imperfect inventory model incorporating remanufacturing amidst uncertain demand, demonstrating the impact of the CCT regulation policy on remanufacturing models, yet disregarding assembled products and the O2O retailing strategy. Gupta and Khanna [9] illustrated an imperfect production system under a flexible production process, considering various carbon regulation policies alongside reworking and green technological investments. Dey et al. [33] presented an imperfect production model that integrated green technology investments and partial outsourcing, optimizing product sustainability within a flexible manufacturing system.

Nowadays, researchers are focused on developing different strategies to reduce carbon emissions, and several models have already been published in the literature with different policies. Governments of different countries are also very careful and take several steps to reduce carbon emissions. However, an imperfect production system for assembled product production with different carbon emission regulation policies under O2O retailing is still a major gap in the literature.

2.3. Investment Management under Backorder

An imperfect single-stage manufacturing reworking production model is described in this study under O2O retailing. Imperfect items are generated randomly, which causes backorders [34]. Pal and Adhikari [34] described an imperfect production inventory system considering partial backordering. They considered a price-sensitive demand pattern for their study. A multi-item inventory system for deteriorating items considering partial backordering was proposed by Adak and Mahapatra [35]. They concluded that the effect of the reliability of the product was unavoidable in making the system cost-effective. However, they ignored dual-channel retailing for their study. Faulty items are harmful to industries, as they increase the waste and overall cost of the entire system [26]. Proper management is required for those faulty products to make the system cost-effective. By repairing or remanufacturing those faulty products, industries can reduce their entire system cost (Dey et al. [11]). Now, to identify those faulty products, a proper inspection is required. Human inspection cannot confirm a perfect error-free inspection [36]. Manna et al. [36] examined the impact of inspection errors on an imperfect production system while considering backorders. Nobil et al. [37] conducted an examination of a rigorous inspection strategy for such a system. However, these studies primarily focused on human inspection errors within the manufacturing process, whereas automated, machine-based inspection can offer error-free outcomes [11]. Dey et al. [11] explored the benefits of automated inspection in imperfect production processes and its role in enhancing cost-effectiveness. Recently, Mridha et al. [4] emphasized the significance of automated inspection in imperfect flexible production systems within the realm of O2O retailing. To implement this intelligent machine-based inspection, some investments are required.

Investment in environmentally friendly technology has the potential to enhance the biodegradability of a product and lessen its overall carbon footprint [12]. Dey et al. [12] devised an imperfect manufacturing system for assembled products, taking into account investments in green technology. Bian and Guo [38] also explored the incorporation of green technology investments in the manufacturing process to mitigate emissions, albeit without addressing dual-channel retailing. Jauhari et al. [39] outlined a supply chain system integrating carbon tax and green technology investment, while highlighting the importance of these factors. Establishing such a complex system with intelligent automated inspection necessitates a substantial initial investment, although such investments can ultimately lead to cost reductions [40]. Tiwari et al. [40] specifically examined the impact of inspection errors and investments in reducing setup costs for an imperfect production system. Investments in the setup cost helped to reduce setup cost and enhance total system profit, as proven by Dey et al. [12]. An inventory system by considering setup cost reduction was proposed by Khan and Dey [41]. They developed their study in an uncertain environment.

Several studies have discussed the effect of investments in production systems. Therefore, proper management in the investments is required in order to obtain the optimized profit. Several studies were developed in the literature by considering investments in different aspects, such as setup, green technology, inspection, and many more. However, as per the authors’ knowledge, proper investment management for setup, green technology, and autonomated inspection for imperfect single-stage manufacturing and remanufacturing for assembled products with different spare parts under backorder and O2O retailing has not yet been explored in the literature. Therefore, the current study is proposed to fill this literature gap.

3. Mathematical Model

In this section, the mathematical derivation of the imperfect production of assembled products with analytical optimum values of the decision variables is presented.

3.1. Hypothesis

The following crucial assumptions are considered in this study of O2O retailing for assembled products.

• An O2O retailing strategy is considered for assembled products wherein the generation process is imperfect. Different spare parts of the assembled item can be sold separately in the market.
• The demand for spare parts of an assembled product is influenced by the selling price of the product. There exists an inverse relationship between demand and selling price. This research explores how the demand for spare parts is impacted by varying selling prices across different distribution channels (Sarkar et al. [21]).
• Shortages may arise before a production that is fully backlogged starts. Linear and fixed backordering costs are considered in this study. The generation rate of defective products is random, and it follows a certain distribution. In this study, we show three different distributed defective rates (uniform, triangular, and double triangular) (Sarkar et al. [19]).
• Faulty items are identified through autonomated inspection and remanufactured or repaired in the same production cycle (Dey et al. [12]). Therefore, a single-stage production process with manufacturing and remanufacturing is considered in this study.
• A substantial quantity of carbon is released from the retailing and manufacturing system in the processes of production, raising, or storing the product. Consequently, in order to mitigate the impact of carbon emissions, a commitment is made to enhance the eco-friendly characteristics of the components. Furthermore, in an effort to control carbon emissions, two specific carbon regulatory measures known as CCT and CT policies are examined, yielding the findings presented in this research (Khan et al. [42]).
• For building a complex and smart retailing system, a massive amount of setup cost is required. In this study, we use continuous investments to lower the cost of setting up the process (Tiwari et al. [40]).

3.2. Different Costs and Solution Methodology

The inventory position for this imperfect production process is graphically presented in Figure 1. The process starts with a backorder and, up to time $z_2$, the production system produces perfect as well as imperfect items, where the rate of imperfection follows certain probability distributions. The time $z_1$ is for filling the backorder, and the reworking of the imperfect items is performed in time $z_3$. Finally, after time $z_3$, the processes reach the maximum inventory, and then up to time $z_4$, the downstream of the inventory occurs due to demand only. In the time interval $[z_4,z_5]$, the backorder arose again due to the demand.

Now, by following the calculation of the model in Dey et al. [11], the average inventory can be obtained as follows:

$$\begin{array}{ccc}\hfill I_{{avg}_{\tau }}& =& \frac{1}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2)+\frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2)\hfill \\ & +& 2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)].\hfill \end{array}$$

(1)

Several basic costs are associated with this smart O2O retailing system, which are discussed in the following.

3.2.1. Holding Cost of Spare Parts and Assembled Products

An O2O retailing strategy for assembled products is considered in this study, where products are sold through online and offline channels. Now, in order to keep the manufactured assembled products, the manufacturer spends a certain amount, which is known as the holding cost. If the cost of holding one item is $H_{C_{tau}}$, the total holding cost of multiple spare parts is:

$$\begin{array}{ccc}& =& \sum _{x=1}^n\frac{H_{C_{\tau }}}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2)+\frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2)\hfill \\ & +& 2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)].\hfill \end{array}$$

3.2.2. Setup Cost with Investment

In order to establish O2O retailing, advanced technology, home delivery of products, and intricate production processes necessitate a substantial amount of funding. Consequently, investments are integrated to mitigate the setup expenses involved in the entire operation. This study examines the application of a continuous logarithmic investment approach to minimize the setup costs associated with this intricate retailing system [40]. Hence, the costs linked to the establishment, along with the investment, can be represented mathematically.

$$\begin{array}{c}\hfill \sum _{x=1}^n(\frac{{\zeta }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+I_{sr}log\left(\frac{{\zeta }_{0_{\tau }}}{{\zeta }_{\tau }}\right))\end{array}$$

3.2.3. Production Cost

In the realm of manufacturing, the cost of production plays a vital role. This encompasses expenses related to raw materials, research and development, as well as tooling and die costs in order to manufacture both flawless and faulty spare parts of the final product. Thus, the calculation of production cost is an essential component in the manufacturing industry.

$$\begin{array}{c}\hfill \sum _{x=1}^n{C}_{P_{\tau }}{D}_{\tau }(1+{E\left[v\right]}_{\tau })\end{array}$$

3.2.4. Repairing Cost

In order to meet the needs of customers and uphold the industry’s brand reputation, it is essential to deliver flawless products. This research examines the scenario where the faulty products are either repaired or remanufactured within the same production cycle, thus incurring certain costs. Consequently, the cost of repairing defective items can be determined as outlined below:

$$\begin{array}{c}\hfill \sum _{x=1}^n{C}_{R_{\tau }}{R}_{P_{\tau }}E{\left[\nu \right]}_{\tau }\frac{D_{\tau }}{Q_m}.\end{array}$$

3.2.5. Cost Related to Carbon Emissions

• Setup of the process for this single-stage system produced a certain amount of carbon. Now, if ${ϵ}_{P_{cs}}$ is carbon transmitted due to setup, then the total emitted carbon for setting up the process is $\frac{D_{\tau }}{{\alpha }_{\tau }}\left({ϵ}_{P_{cs}}\right)$.
• Manufacturing and repairing of the perfect and imperfect items emitted carbon. If the amount of emitted carbon per unit in production and repair is ${ϵ}_{P_{\tau }}$ and ${ϵ}_{r_{\tau }}$, respectively, then the total emitted carbon related to manufacturing and repairing of the products is $D_{\tau }({ϵ}_{P_{\tau }}+{ϵ}_{r_{\tau }}{R}_{P_{\tau }}{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2)$.
• In order to keep the assembled product and spare parts of the assembled items, some carbon is emitted. Now, if ${ϵ}_{c{h}_{\tau }}$ is the unit of emitted carbon, then the total carbon emission to hold the spare parts of the assembled item is ${ϵ}_{c{h}_{\tau }}{I}_{av{g}_{\tau }}$.

To increase the green level of the products, some investments are incorporated in this study. The investments related to green technology are defined as ${\omega }_{\tau }(1-e^{-{\varphi }_{\tau }{\varpi }_{\tau }})+{\varpi }_{\tau }$ (Dey et al. [33]). Now, total carbon-related costs under different carbon regulation policies can be presented as follows.

3.2.6. Cost Associated with Carbon Taxation Policy

To reduce carbon emissions, the government may introduce some tax per unit of emitted carbon. If ${\xi }_{c_{\tau }}$ is the tax related to unit carbon emissions, then the total cost for this scenario with green technology investments is:

$$\begin{array}{ccc}& & \sum _{x=1}^n[{\xi }_{c_{\tau }}[\frac{D_{\tau }}{{\alpha }_{\tau }}{ϵ}_{P_{cs}}+{ϵ}_{P_{\tau }}{D}_{\tau }+{ϵ}_{r_{\tau }}{R}_{P_{\tau }}{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2{D}_{\tau }+\frac{{ϵ}_{c{h}_{\tau }}}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])\hfill \\ & +& \frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2)+\frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2)+2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)\left]\right]\left(1-{\omega }_{\tau }(1-e^{-{\varphi }_{\tau }{\varpi }_{\tau }})\right)+{\varpi }_{\tau }].\hfill \end{array}$$

3.2.7. Cost Associated with Cap-and-Trade Policy

In this regulation, the government sets a cap for the emitted carbon from the manufacturing industry. However, the industry can buy extra caps from other industries with some extra amount of payment. Now, if ${\lambda }_{\tau }$ is the per unit cost for emitted carbon and $U_{\tau }$ is the cap, then the total cost associated with this regulation is:

$$\begin{array}{ccc}& & \sum _{x=1}^n[{\lambda }_{\tau }[\frac{D_{\tau }}{{\alpha }_{\tau }}{ϵ}_{P_{cs}}+{ϵ}_{P_{\tau }}{D}_{\tau }+{ϵ}_{r_{\tau }}{R}_{P_{\tau }}{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2{D}_{\tau }+\frac{{ϵ}_{c{h}_{\tau }}}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}\left(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2\right)\hfill \\ & +& \frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2)+2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)\left]\right]\left(1-{\omega }_{\tau }(1-e^{-{\varphi }_{\tau }{\varpi }_{\tau }})\right)-{\lambda }_{\tau }{U}_{\tau }+{\varpi }_{\tau }].\hfill \end{array}$$

3.2.8. Autonomated Inspection Cost

To identify the defective product and to reduce the waste generation in terms of faulty products, a smart autonomated inspection is considered in this study. To implement this strategy, a manufacturer needs to invest some amount. The amount related to autonomated inspection can be calculated as follows (see Dey et al. [11]):

$$\begin{array}{c}\hfill \sum _{x=1}^n\frac{{\rho }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+\delta log\left(\frac{{\rho }_{0_{\tau }}}{{\rho }_{\tau }}\right).\end{array}$$

3.2.9. Cost of Backorder

Due to the high demand for the product, a backorder may arise. The customer can wait for a particular product or can visit a different store. In this study, an initial backorder will fill within time $[0,z_1]$, and $z_5-z_4=\frac{{\beta }_{\tau }}{D_{\tau }}$. Similar to Dey et al. [11], the average backorder is as follows:

$$\begin{array}{ccc}\hfill I_{bav{g}_{\tau }}& =& \frac{{\beta }_{\tau }^2(1-E\left[{\vartheta }_{\tau }\right])}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}.\hfill \end{array}$$

(2)

A fixed and a linear backorder cost is considered in this study. Therefore, the total backorder cost can be calculated by the following mathematical expression:

$$\begin{array}{c}\hfill \sum _{x=1}^n\left[\frac{F_{b{c}_{\tau }}{\beta }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+C_{l{b}_{\tau }}{I}_{bav{g}_{\tau }}\right].\end{array}$$

The product and its parts are sold online and in stores at different prices. Therefore, the combined earnings from both sources can be calculated in the following manner:

$$\begin{array}{c}\hfill \sum _{x=1}^n\left[s_{o{n}_{\tau }}({\psi }_{o{n}_{\tau }}-{\phi }_{o{n}_{\tau }}{s}_{o{n}_{\tau }})+s_{of{f}_{\tau }}({\psi }_{of{f}_{\tau }}-{\phi }_{of{f}_{\tau }}{s}_{of{f}_{\tau }})\right].\end{array}$$

3.2.10. Profit under Carbon Cap-and-Trade Regulations

Now, to obtain the profit expression under CCT regulations, we subtract all related costs under CCT regulations from the revenue, and then the total profit can be expressed as follows:

$$\begin{array}{ccc}& & T{P}_{ct}({\alpha }_{\tau },{\beta }_{\tau },{\rho }_{\tau },{\zeta }_{\tau },{\varpi }_{\tau })=\sum _{x=1}^n\left[s_{o{n}_{\tau }}{D}_{o{n}_{\tau }}+s_{of{f}_{\tau }}{D}_{of{f}_{\tau }}\right]-\sum _{x=1}^n[\frac{{\zeta }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+I_{sr}log\left(\frac{{\zeta }_{0_{\tau }}}{{\zeta }_{\tau }}\right)\hfill \\ & +& H_{C_{\tau }}\left[\frac{1}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}\right[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}\left(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2\right)+\frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}\left({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2\right)\hfill \\ & +& 2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)\left]\right]+\frac{F_{b{c}_{\tau }}{\beta }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+C_{l{b}_{\tau }}\left[\frac{{\beta }_{\tau }^2(1-E\left[{\vartheta }_{\tau }\right])}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}\right]+C_{P_{\tau }}{D}_{\tau }(1+E\left[{\vartheta }_{\tau }\right])+R_{P_{\tau }}E\left[{\vartheta }_{\tau }\right]C_{R_{\tau }}\frac{D_{\tau }}{{\alpha }_{\tau }}+\frac{{\rho }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}\hfill \\ & +& \delta log\left(\frac{{\rho }_{0_{\tau }}}{{\rho }_{\tau }}\right)+{\lambda }_{\tau }[\frac{D_{\tau }}{{\alpha }_{\tau }}{ϵ}_{P_{cs}}+{ϵ}_{P_{\tau }}{D}_{\tau }+{ϵ}_{r_{\tau }}{R}_{P_{\tau }}{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2{D}_{\tau }+\frac{{ϵ}_{ch}}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}(1+E\left[{\vartheta }_{\tau }\right]\hfill \\ & +& {\left(E\left[{\vartheta }_{\tau }\right]\right)}^2)+\frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2)+2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)\left]\right]\left(1-{\omega }_{\tau }(1-e^{-{\varphi }_{\tau }{\varpi }_{\tau }})\right)-{\lambda }_{\tau }{U}_{\tau }+{\varpi }_{\tau }].\hfill \end{array}$$

(3)

3.2.11. Profit under Carbon Taxation Policy

If $x{i}_{c_{tau}}$ is the tax for each unit of carbon emissions, we can find the profit by subtracting all the costs related to the carbon tax from the revenue. Then, we can express the total profit as:

$$\begin{array}{ccc}\hfill & & T{P}_t({\alpha }_{\tau },{\beta }_{\tau },{\rho }_{\tau },{\zeta }_{\tau },{\varpi }_{\tau })=\sum _{x=1}^n\left[s_{o{n}_{\tau }}{D}_{o{n}_{\tau }}+s_{of{f}_{\tau }}{D}_{of{f}_{\tau }}\right]-\sum _{x=1}^n[\frac{{\zeta }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+I_{sr}log\left(\frac{{\zeta }_{0_{\tau }}}{{\zeta }_{\tau }}\right)\hfill \\ & +& H_{C_{\tau }}\left[\frac{1}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}\right[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}\left(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2\right)+\frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}\left({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2\right)\hfill \\ & +& 2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)\left]\right]+\frac{F_{b{c}_{\tau }}{\beta }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+C_{l{b}_{\tau }}\left[\frac{{\beta }_{\tau }^2(1-E\left[{\vartheta }_{\tau }\right])}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}\right]+C_{P_{\tau }}{D}_{\tau }(1+E\left[{\vartheta }_{\tau }\right])\hfill \\ & +& R_{P_{\tau }}E\left[{\vartheta }_{\tau }\right]C_{R_{\tau }}\frac{D_{\tau }}{{\alpha }_{\tau }}+\frac{{\rho }_{\tau }{D}_{\tau }}{{\alpha }_{\tau }}+\delta log\left(\frac{{\rho }_{0_{\tau }}}{{\rho }_{\tau }}\right)+{\xi }_{c_{\tau }}[\frac{D_{\tau }}{{\alpha }_{\tau }}{ϵ}_{P_{cs}}+{ϵ}_{P_{\tau }}{D}_{\tau }+{ϵ}_{r_{\tau }}{R}_{P_{\tau }}{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2{D}_{\tau }\hfill \\ & +& \frac{{ϵ}_{c{h}_{\tau }}}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2)\hfill \\ & +& \frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2)+2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)\left]\right]\left(1-{\omega }_{\tau }(1-e^{-{\varphi }_{\tau }{\varpi }_{\tau }})\right)+{\varpi }_{\tau }].\hfill \end{array}$$

(4)

The above Equations (3) and (4) are highly non-linear equations. Classical optimization is only one method available in the literature through which a closed-form solution can be obtained with a global optimized value of the functions and decision variables. Therefore, we utilized this methodology to obtain global optimized results.

Necessary conditions: To obtain the optimum value of the profit function, it is required to obtain the critical points that the optimum value of the profit function reaches. The optimum points can be obtained by equating $\frac{\partial TP}{\partial s_{o{n}_{\tau }}}$, $\frac{\partial TP}{\partial s_{of{f}_{\tau }}}$, $\frac{\partial TP}{\partial {\beta }_{\tau }}$, $\frac{\partial TP}{\partial {\alpha }_{\tau }}$, $\frac{\partial TP}{\partial {\rho }_{\tau }}$, $\frac{\partial TP}{\partial {\zeta }_{\tau }}$, and $\frac{\partial TP}{\partial {\varpi }_{\tau }}$ to zero. Therefore, the optimum points or optimum value of the decision-making variables can be obtained as follows:

$$\begin{array}{ccc}\hfill s_{o{n}_{\tau }}& =& -\frac{{\mathsf{\Theta }}_{1_{o{n}_{\tau }}}}{3{\mathsf{\Theta }}_{2_{o{n}_{\tau }}}}-\frac{2^{\frac{1}{3}}\left(3{\mathsf{\Theta }}_{2_{o{n}_{\tau }}}{\mathsf{\Theta }}_{3_{o{n}_{\tau }}}\right)}{3{\mathsf{\Theta }}_{2_{o{n}_{\tau }}}{\mathsf{\Pi }}_{on}}+\frac{{\mathsf{\Pi }}_{on}}{2^{\frac{1}{3}}3{\mathsf{\Theta }}_{2_{o{n}_{\tau }}}}\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill s_{of{f}_{\tau }}& =& \frac{{\mathsf{\Theta }}_{1_{of{f}_{\tau }}}}{3{\mathsf{\Theta }}_{2_{of{f}_{\tau }}}}-\frac{2^{\frac{1}{3}}\left(3{\mathsf{\Theta }}_{2_{of{f}_{\tau }}}{\mathsf{\Theta }}_{3_{of{f}_{\tau }}}\right)}{3{\mathsf{\Theta }}_{2_{of{f}_{\tau }}}{\mathsf{\Pi }}_{off}}+\frac{{\mathsf{\Pi }}_{off}}{2^{\frac{1}{3}}3{\mathsf{\Theta }}_{2_{of{f}_{\tau }}}}\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\beta }_{\tau }& =& \frac{{\alpha }_{\tau }\left[{\mathsf{\Omega }}_1+H_{C_{\tau }}-\frac{F_{b{c}_{\tau }}{D}_{\tau }}{{\alpha }_{\tau }}\right]\left[1+E\left[{\vartheta }_{\tau }\right]-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}{\left(1-E\left[{\vartheta }_{\tau }\right]\right){\mathsf{\Omega }}_4},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\alpha }_{\tau }& =& {\left[\frac{{\mathsf{\Omega }}_4{\mathsf{\Omega }}_5+D_{\tau }{\mathsf{\Omega }}_3}{{\mathsf{\Omega }}_6}\right]}^{\frac{1}{2}}\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\rho }_{\tau }& =& \frac{\delta {\alpha }_{\tau }}{D_{\tau }}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }& =& \frac{I_{sr}{\alpha }_{\tau }}{D_{\tau }}\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\varpi }_{\tau }& =& \frac{1}{{\varphi }_{\tau }}log({\lambda }_{\tau }{\varphi }_{\tau }{\omega }_{\tau }[\frac{D_{\tau }}{{\alpha }_{\tau }}\left({ϵ}_{P_{cs}}\right)+{ϵ}_{P_{\tau }}{D}_{\tau }+{ϵ}_{r_{\tau }}{R}_{P_{\tau }}{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2{D}_{\tau }\hfill \\ & +& \frac{{ϵ}_{c{h}_{\tau }}}{2{\alpha }_{\tau }\left[(1-E\left[{\vartheta }_{\tau }\right])-\frac{D_{\tau }}{R_{P_{\tau }}}\right]}[({\alpha }_{\tau }^2+{\beta }_{\tau }^2)(1-E\left[{\vartheta }_{\tau }\right])+\frac{{\alpha }_{\tau }^2{D}_{\tau }^2}{P_{R_m}^2}(1+E\left[{\vartheta }_{\tau }\right]+{\left(E\left[{\vartheta }_{\tau }\right]\right)}^2)\hfill \\ & +& \frac{{\alpha }_{\tau }^2{D}_{\tau }}{R_{P_{\tau }}}({\left(E\left[{\vartheta }_{\tau }\right]\right)}^3-2)+2{\beta }_{\tau }{\alpha }_{\tau }\left(\frac{D_{\tau }}{R_{P_{\tau }}}+E\left[{\vartheta }_{\tau }\right]-1\right)\left]\right]).\hfill \end{array}$$

(11)

See Appendix A for detailed calculations.

Sufficient condition: This is a profit optimization problem, and thus, the sufficient condition for classical optimization states that a function with n variables will be concave at its optimal points when minors of an $n\times n$ Hessian matrix are alternately signed. The decision variables for this current study are $s_{o{n}_{\tau }}$, $s_{of{f}_{\tau }}$, ${\alpha }_{\tau }$, ${\zeta }_{\tau }$, ${\rho }_{\tau }$, ${\varpi }_{\tau }$, and ${\beta }_{\tau }$.

$$\begin{array}{ccc}\hfill \det \left(H_{77}\right)& =& \det \left(\begin{array}{ccccccc}\frac{{\partial }^{2}TP(.)}{\partial {\rho }_{\tau }^2}& \frac{{\partial }^{2}TP(.)}{\partial {\rho }_{\tau }\partial {\zeta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\rho }_{\tau }\partial G_m}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial {\rho }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }\partial {\rho }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\rho }_{\tau }\partial s_{o{n}_{\tau }}}& \frac{{\partial }^{2}TP(.)}{\partial {\rho }_{\tau }\partial s_{of{f}_{\tau }}}\\ \frac{{\partial }^{2}TP(.)}{\partial {\zeta }_{\tau }\partial {\rho }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial Y_m^2}& \frac{{\partial }^{2}TP(.)}{\partial {\zeta }_{\tau }\partial {\varpi }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial {\zeta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\zeta }_{\tau }\partial {\beta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\zeta }_{\tau }\partial s_{o{n}_{\tau }}}& \frac{{\partial }^{2}TP(.)}{\partial {\zeta }_{\tau }\partial s_{of{f}_{\tau }}}\\ \frac{{\partial }^{2}TP(.)}{\partial {\rho }_{\tau }\partial {\varpi }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\zeta }_{\tau }\partial {\varpi }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\varpi }_{\tau }^2}& \frac{{\partial }^{2}TP(.)}{\partial {\varpi }_{\tau }\partial {\alpha }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\varpi }_{\tau }\partial {\beta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\varpi }_{\tau }\partial s_{o{n}_{\tau }}}& \frac{{\partial }^{2}TP(.)}{\partial {\varpi }_{\tau }\partial s_{of{f}_{\tau }}}\\ \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial {\rho }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial {\zeta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial {\varpi }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }^2}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial {\beta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial s_{o{n}_{\tau }}}& \frac{{\partial }^{2}TP(.)}{\partial {\alpha }_{\tau }\partial s_{of{f}_{\tau }}}\\ \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }\partial {\rho }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }\partial {\zeta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }\partial {\varpi }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }\partial {\alpha }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }^2}& \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }\partial s_{o{n}_{\tau }}}& \frac{{\partial }^{2}TP(.)}{\partial {\beta }_{\tau }\partial s_{of{f}_{\tau }}}\\ \frac{{\partial }^{2}TP(.)}{\partial s_{o{n}_{\tau }}\partial {\rho }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{o{n}_{\tau }}\partial {\zeta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{o{n}_{\tau }}\partial {\varpi }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{o{n}_{\tau }}\partial {\alpha }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{o{n}_{\tau }}\partial {\beta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{p_{o{n}_{\tau }}}^2}& \frac{{\partial }^{2}TP(.)}{\partial s_{o{n}_{\tau }}\partial s_{of{f}_{\tau }}}\\ \frac{{\partial }^{2}TP(.)}{\partial s_{of{f}_{\tau }}\partial {\rho }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{of{f}_{\tau }}\partial {\zeta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{of{f}_{\tau }}\partial {\varpi }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{of{f}_{\tau }}\partial {\alpha }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{of{f}_{\tau }}\partial {\beta }_{\tau }}& \frac{{\partial }^{2}TP(.)}{\partial s_{o{n}_{\tau }}\partial s_{of{f}_{\tau }}}& \frac{{\partial }^{2}TP(.)}{\partial s_{p_{of{f}_{\tau }}}^2}\end{array}\right)\hfill \end{array}$$

Proof: The profit function in this paper is quite complex, as it is nonlinear and stochastic. This means that it is challenging to find a neat, straightforward solution. However, in our study, we have managed to come close to a solution that works well enough. The values of the principal minors are calculated numerically at the optimum point in the next section.

Due to the intricate nonlinearity of the profit function concerning the decision variables, manual computation becomes arduous. Therefore, employing computational tools like Mathematica 11.0 enables the determination of the optimal decision variables and profit function.

4. Results

This section presents the numerical outcomes derived from optimizing the decision variables to illustrate the O2O retailing under imperfect production.

4.1. Parameter Setup and Optimum Result

The data utilized here are sourced from Dey et al. [12], representing the most optimal form. The specific parameter values are detailed in the accompanying

Table 2. Parametric values for the assembled product with two spare parts.

Parameters	Value	Parameters	Value	Parameters	Value
${\psi }_{o{n}_{\tau }}$	470, 480 (unit/time)	${\phi }_{o{n}_{\tau }}$	4, 4	${\psi }_{of{f}_{\tau }}$	450, 450 (unit/time)
${\phi }_{of{f}_{\tau }}$	4.2, 4.2	$R_{P_{\tau }}$	650, 550 (unit/time)	$C_{l{b}_{\tau }}$	10, 11 (USD/unit/year)
$F_{b{c}_{\tau }}$	1, 1 (USD/unit)	$H_{C_{\tau }}$	50, 55 (USD/unit/year)	$C_{P_{\tau }}$	7, 8 (USD/unit)
$C_{R_{\tau }}$	5, 7 (USD/unit)	$\delta$	140	${\rho }_{0_{\tau }}$	250, 300 (USD/lot)
${\xi }_{c_{\tau }}$	1.4, 1.6 (USD/kg)	${ϵ}_{P_{cs}}$	20, 25 (kg/setup)	${ϵ}_{P_{\tau }}$	5, 6 (kg/cycle)
${ϵ}_{r_{\tau }}$	2, 2 (kg/lot)	${\omega }_{\tau }$	0.1, 0.1	${\varphi }_{\tau }$	0.03, 0.02
$I_{sr}$	230	${\zeta }_{0_{\tau }}$	250, 300 (USD/setup)

.

An IPS is considered in this study, where the generation rate of imperfect items during an out-of-control state is random. This random defective generation rate follows a certain distribution. To discuss different situations, in this study, we calculated the total profit under three different distributions. We considered that a faulty generation rate may follow any one of the uniform, triangular, or double triangular distributed defective rates. The parameters related to calculation of the mean for different distributions are $(0.03,0.07;0.03,0.07;0.04,0.08)$, $(0.03,0.04,0.07;0.03,0.04,0.07;0.04,0.04,0.07)$, and $(0.03,0.04,0.07;0.03,0.04,$$0.07;0.04,0.04,0.07)$ for uniform distribution, triangular distribution, and double triangular distribution, respectively.

Based on the above numerical values of the parameters, the optimum result for this study under different carbon regulation policies and different distributed defective rates is presented in

Table 3. Optimum result for O2O retailing for an assembled product with two spare parts (i.e., $\tau =2$).

Decision Variable	CCT			CT
	Uniform	Triangular	Double Triangular	Uniform	Triangular	Double Triangular
$s_{o{n}_{\tau }}$ (USD/unit)	75.05, 71.12	75.58, 70.41	76.89, 71.71	75.87, 71.94	76.36, 71.18	77.83, 72.65
$s_{of{f}_{\tau }}$ (USD/unit)	80.28, 76.54	82.81, 76.39	82.87, 76.44	80.31, 76.57	82.84, 76.42	82.91, 76.49
${\alpha }_{\tau }$ (Units)	80.86, 82.16	81.93, 84.83	86.61, 87.51	80.36, 81.23	81.53, 83.89	85.85, 86.36
${\zeta }_{\tau }$ (USD/setup)	59.79, 70.14	58.53, 70.21	64.00, 72.55	60.73, 69.41	59.42, 69.49	65.05, 71.68
${\beta }_{\tau }$ (Units)	31.24, 31.19	30.59, 31.10	33.29, 31.76	31.79, 30.85	31.16, 30.77	33.91, 31.37
${\rho }_{\tau }$ (USD/batch)	36.39, 42.69	35.63, 42.74	38.96, 44.16	36.97, 42.25	36.17, 42.30	33.91, 31.37
${\varpi }_{\tau }$ (USD/cycle)	106.71, 139.79	107.01, 140.23	107.96, 141.66	107.43, 140.87	107.74, 141.33	108.71, 142.80
TP (USD/cycle)	13,621.20	15,065.50	14,045.10	13,120.00	14,573.40	13,486.60

.

4.1.1. Discussion of Result for Carbon Cap-and-Trade Regulations

In carbon CCT, a cap is fixed by the regulation authority. The industry can buy extra caps with extra amounts from different companies. The maximum cap for this study for different spare parts is considered as $7.8$ kg per lot, where the cost of buying an extra cap is USD 1.4 per kg. In this case, the profit is USD 13,621.20, USD 15,065.50, and USD 14,045.10 when the defective generation rate follows a uniform, triangular, or double triangular distribution, respectively. Based on the results, we can say that the most profit can be made when the rate of faulty generation follows a triangle-shaped pattern. The online selling prices for different spare parts are USD $75.58$, USD $70.41$ per unit, whereas the offline selling price is USD $82.81$, USD $76.39$ per unit. For this case, the optimum order quantity is $81.93,84.83$ units, and the backordered quantity is $30.59,31.10$ units for two different spare parts. Investing in making the products more eco-friendly and reducing the initial cost will be around USD $58.53$ to USD $70.21$ per setup, and USD $107.01$ to USD $140.23$ per cycle. To use technology that checks for bad products, the industry needs to spend USD $36.39$ for one part and USD $42.69$ for another part for each group of products.

All the principal minors of the 7 × 7 Hessian matrix are alternate in sign—$|H_{11}|=-8.69<0$, $|H_{22}|=71.73>0$, $|H_{33}|=-37.20<0$, $|H_{44}|=31.79>0$, $|H_{55}|=-3.27<0$, $|H_{66}|=0.19>0$, $|H_{77}|=-0.0026<0$—which stipulates the concavity of the profit function at the optimal points. The concavity of the profit function for this scenario concerning different decision variables is graphically presented in Figure 2.

4.1.2. Results for an Assembled Item with Two Spare Parts under a Carbon Taxation Policy

In this case, it was considered that the industry would pay a certain tax per unit of emitted carbon fixed by the regulatory authorities. In this research, the study examined the carbon tax rate per unit of emitted carbon for a pair of spare parts set at $USD1$. By varying parametric values, the profit per cycle is determined to be USD 13,120, USD 14,573.40, and USD 13,486.60 for defective rates that are uniformly, triangularly, and doubly triangularly distributed.

The most advantageous profit outcome is realized when the defective rate adheres to a triangular distribution pattern. The selling prices online for the two distinct spare parts are USD $76.36$, USD $71.18$ per unit, while offline prices are USD $82.84$, USD $76.42$ per unit. The optimum order quantity and backorder quantity for two different spare parts are $81.53,83.89$ units and $31.16,30.77$ units, respectively. Investments for reducing setup cost and increasing the green level of the products are USD $59.42$, USD $69.49$ per setup and USD $107.74$, USD $141.33$ per cycle, respectively. For implementing advanced autonomated inspection technology, the industry needs to invest USD $36.17$, USD $42.30$ per batch for two different spare parts.

The magnitudes of the leading sub-determinants of the $7\times 7$ Hessian matrix are as follows: $|H_{11}|=-8.68<0$, $|H_{22}|=71.64>0$, $|H_{33}|=-37.76<0$, $|H_{44}|=31.50>0$, $|H_{55}|=-3.14<0$, $|H_{66}|=0.18>0$, $|H_{77}|=-0.0024<0$. These alternating signs in the principal minors suggest that the overall profit function exhibits global concavity with respect to the decision variables. The graphical representation of the profit function’s concavity with respect to different decision variables can be observed in Figure 3.

4.2. Special Cases

The best outcome happens when the rate of defects follows a specific kind of distribution called a triangular distribution. This section discusses scenarios with defective rates distributed in a triangular pattern.

4.2.1. Results for an Assembled Product with Three Spare Parts ($\tau =3$)

If we consider a production industry where an assembled product is produced with three different spare parts, then the optimal values of decision variables and maximized profit are provided as follows in

Table 4. Optimum for the assembled product with three spare parts.

Decision Variable	Value
$s_{on}$ (USD/unit)	86.78, 88.78, 85.43
$s_{off}$ (USD/unit)	106.78, 106.50, 100.52
${\alpha }_{\tau }$ (Units)	47.22, 45.51, 42.65
${\zeta }_{\tau }$ (USD/setup)	37.12, 53.09, 52.95
${\beta }_{\tau }$ (Units)	18.6, 22.84, 21.87
${\rho }_{\tau }$ (USD/batch)	34.14, 32.23, 37.12
${\varpi }_{\tau }$ (USD/cycle)	110.92, 146.11, 146.11
TP (USD/cycle)	11,544.50

. The values of the parameters are the same as in Table 2, along with the following extra values for the extra spare parts: ${\psi }_{o{n}_{\tau }}$ = 480 (unit/time) ${\phi }_{o{n}_{\tau }}$ = 4 ${\psi }_{of{f}_{\tau }}$ = 450 (unit/time); ${\phi }_{of{f}_{\tau }}$ = 4.2; $R_{P_{\tau }}$ = 500 (unit/time); $C_{l{b}_{\tau }}$ = 11 (USD/unit/year); $F_{b{c}_{\tau }}$ = 1 (USD/unit); $H_{C_{\tau }}$ = 55 (USD/unit/year); $C_{P_{\tau }}$ = 8 (USD/unit); $C_{R_{\tau }}$ =7 (USD/unit);${\rho }_{0_{\tau }}$ = 300 (USD/lot); ${\xi }_{c_{\tau }}$ = 1.6 (USD/kg); ${ϵ}_{P_{cs}}$ = 22 (kg/setup); ${ϵ}_{P_{\tau }}$ = 6 (kg/cycle); ${ϵ}_{r_{\tau }}$ = 2 (kg/lot); ${\omega }_{\tau }$ = 0.1; ${\varphi }_{\tau }$ = 0.02; ${\zeta }_{0_{\tau }}$ = 300 (USD/setup).

From the above table, we find that the optimum profit for assembled product with three spare parts is USD 11,544.50 under a carbon cap-and-trade regulation policy. The concavity of the profit function with respect to different decision variables is graphically presented in Figure 4.

4.2.2. Results without Investments in Green Technology

In this research, the study examines the impact of green technology investments on enhancing the biodegradability of products, thereby leading to a reduction in carbon emissions.

Table 5. Optimal result for O2O retailing under different scenarios with an assembled product with two spare parts.

Decision Variable	Result without Green Technology		Result without Autonomation
	CCT	CT	CCT	CT
$s_{on}$ (USD/unit)	71.74, 78.30	72.66, 79.48	69.88, 76.76	70.68, 76.82
$s_{off}$ (USD/unit)	75.67, 82.05	76.59, 83.22	75.06, 83.19	75.85, 83.25
${\alpha }_{\tau }$ (Units)	57.75, 71.97	57.75,72.90	109.85, 128.95	108.76, 127.53
${\zeta }_{\tau }$ (USD/setup)	76.84, 79.71	74.95,77.65	77.43, 107.92	78.24, 106.93
${\beta }_{\tau }$ (Units)	30.23, 32.14	30.23,32.52	41.22, 48.89	41.78, 48.43
${\rho }_{\tau }$ (USD/batch)	35.15, 43.81	35.15,44.38	−	−
${\varpi }_{\tau }$ (USD/cycle)	−	−	106.81, 139.94	107.53, 141.03
TP (USD/cycle)	12,754.70	11,719.30	13,661.70	13,165.70

CCT: Carbon cap-and-trade; CT: Carbon tax.

displays the optimal profits for two products assembled with spare parts, noting that, without green technology investment, profits amount to USD 12,754.70 and USD 11,719.30 per cycle under CCT regulation and CT policy, respectively. Investing in green technology can decrease carbon emissions by 1679 kg each cycle under the CT policy, while also increasing the system’s profit by $19.58\%$.

Investments in green technology under CCT regulations can lead to a reduction in carbon emissions by 1359 kg per cycle and an increase in system profit by $18.12\%$. Consequently, allocating resources to improve the environmental friendliness of spare parts can contribute to the economic and environmental sustainability of the manufacturing process. The graphical representation of the profit function’s concavity is illustrated in Figure 5.

4.2.3. Results without Autonomated Inspection

Imperfect or defective item generation is a crucial problem for the production process. In this study, we examined the frequency of errors and discovered that they occur randomly and adhere to a specific pattern. In this study, it was considered that, in order to satisfy the customer, all defective products need to be repaired or remanufactured within the same generation cycle.

Now, to identify defective items perfectly, a machine-based autonomated inspection strategy is considered for this retailing system. However, without autonomated inspection, the profit for the system under CCT regulation and CT policy is USD 13,661.70 and USD 13,165.70, respectively. Due to the implementation of an autonomated inspection strategy, the retail industry can earn $10.27\%$ and $10.69\%$ more profit under the CCT regulation and CT policy, respectively. Therefore, investment in autonomated inspection helps to reduce waste for the retail industry and helps to enhance economic sustainability. The shape of the profit function is shown in the graph in Figure 6.

4.3. Case Study

A case study based on real data collecting from an industry is presented in this section. The study findings were tested on real data from an industry located in the West Bengal state in India. This is a manufacturing industry with assembled items with two spare parts. Due to industry’s policy we cannot reveal the name or any other details of the industry; however, the industry managers happily shared the data with us and allowed us to use those data for our study. The results found are similar to those of this study. After obtaining the raw data from the industry, we normalized the data, otherwise they will not fit with this study. The basic normalization towards the mean method was used for normalization of the data. The normalized data provided by the industry are as follows: ${\psi }_{o{n}_{\tau }}$ = 470, 480 (unit/time); ${\phi }_{o{n}_{\tau }}$ = 4, 4; ${\psi }_{of{f}_{\tau }}$ = 450, 450 (unit/time); ${\phi }_{of{f}_{\tau }}$ = 4.2, 4.2; $R_{P_{\tau }}$ = 650, 550 (unit/time); $C_{l{b}_{\tau }}$ = 10, 11 (USD/unit/year); $F_{b{c}_{\tau }}$ = 1, 1 (USD/unit); $H_{C_{\tau }}$ = 50, 55 (USD/unit/year); $C_{P_{\tau }}$ = 7, 8 (USD/unit); $C_{R_{\tau }}$ = 5, 7 (USD/unit); $\delta$ = 140; ${\rho }_{0_{\tau }}$ = 250, 300 (USD/lot); ${\xi }_{c_{\tau }}$ = 1.4, 1.6 (USD/kg); ${ϵ}_{P_{cs}}$ = 20, 25 (kg/setup); ${ϵ}_{P_{\tau }}$ = 5, 6 (kg/cycle); ${ϵ}_{r_{\tau }}$ = 2, 2 (kg/lot); ${\omega }_{\tau }$ = 0.1, 0.1; ${\varphi }_{\tau }$ = 0.03, 0.02; $I_{sr}$ = 230; ${\zeta }_{0_{\tau }}$ = 250, 300 (USD/setup). The company pays a tax for carbon emissions. The carbon tax rate per unit of emitted carbon for each spare part is $USD1$. After normalization of the given data, it was found that the rate of defective generation follows a triangular distribution. Moreover, the industry avoids the concept of investment in the setup to reduce setup cost.

Based on the given data from the industry, the optimum results are shown in

Table 6. Optimum results for the case study with two spare parts.

${\mathit{s}}_{\mathit{o}\mathit{n}}$ (USD/Unit)	${\mathit{s}}_{\mathit{o}\mathit{f}\mathit{f}}$ (USD/Unit)	${\mathit{\alpha }}_{\mathit{\tau }}$ (Units)	${\mathit{\beta }}_{\mathit{\tau }}$ (Units)	${\mathit{\rho }}_{\mathit{\tau }}$ (USD/Batch)	${\mathit{\varpi }}_{\mathit{\tau }}$ (USD/Cycle)	TP (USD/cycle)
71.12, 76.34	76.30, 82.76	105.55, 105.75	41.00, 39.16	46.76, 53.19	107.66, 141.21	13,921.20

. The optimum profit is USD 13,921.20, which is $8.22\%$ less compared to the findings of the present study. Thus, the company happily adopted the concept of the present study. Recently, they started to invest in the setup process and, instead of carbon taxation, they used carbon cap-and-trade regulations for their industry.

4.4. Sensitivity Analysis

Sensitivity analysis is a crucial tool for evaluating manufacturing models and making informed decisions in production. Sensitivity analysis helps figure out how much the model’s predictions change when the underlying assumptions are changed. This helps us better understand what could happen and makes it easier to make decisions based on data in manufacturing. This section explains how changing different parametric values by $\pm 50\%$ affects the total best profit.

(i)

The demand for products is very important in O2O retailing. It can be a double-edged sword when there is a high level of interest in something. It can drive sales through increased buy online, pick up in store (BOPS) options, but it can also strain inventory management and fulfillment if not anticipated. Online retailers who also have physical stores need to be good at predicting how much of a product they will need to have in stock so that they can make sure they have enough to sell to customers online and in their stores. In this research, a demand that depends on the selling price is explored for both distribution channels. The decrease in the initial demand poses a disadvantage in the pursuit of maximizing the profit of the system, a consequence that is inherent in such circumstances. Conversely, a decrease in the scaling factor associated with the selling price aids in augmenting the demand while concurrently enhancing the profitability of the overall system. The effect of the original demand and scaling parameters related to the selling price is graphically presented in Figure 7.

(ii)

The impact of varying unit costs on a single-stage manufacturing and remanufacturing system is visually depicted in Figure 8. It is evident that the per unit holding cost plays a critical role in optimizing the overall profit of the system. A reduction of $50\%$ in the unit holding cost is shown to result in an $8.58\%$ increase in profit. In contrast, remanufacturing, setup, and inspection costs exhibit relatively lower sensitivity in influencing the demand dynamics of the entire system.

(iii)

The production rate and unit manufacturing cost are the two most crucial factors for production industries. A $50\%$ increment in production rate will reduce total system profit by $27.5\%$; similarly, a $50\%$ reduction in unit manufacturing cost will help to enhance the system profit by $29.3\%$. A graphical representation is provided in Figure 9 to show the effect of production rate and unit manufacturing cost on the total system’s profit.

(iv)

The environment suffers from the negative impact of carbon emissions. Therefore, two different strategies are considered in this study to regulate carbon emissions. However, a reduction in per unit emitted carbon tax will help increase the profit, but the industry should be careful about the emissions. Therefore, the industry will benefit if they can reduce the carbon emissions for manufacturing, remanufacturing, and holding the products. A $50\%$ reduction in carbon emissions due to holding the product will help to increase the profit by $54.54\%$. Similarly, the retail industry can enhance its profit by $31.42\%$ and $18.97\%$ if it reduces carbon emissions by $50\%$ for manufacturing and remanufacturing, respectively. Figure 10 is provided to show the effect of a carbon tax and the amount of carbon emitted during the production process.

(v)

A complex retailing system is proposed in the study. Thus, setup-related costs and parameters play a significant role in profit determination. Reduction in initial setup cost will help increase the profit of the system. Similarly, a reduction in the emissions due to setup helps increase the profit of the entire process. The effects of setup-related costs and parameters are graphically presented in Figure 11.

(vi)

To reduce carbon emissions and increase the green level of the products, some investments were incorporated into this retailing system. The green technology investment fraction is quite sensitive to the determination of the optimum profit of the system. A $50\%$ reduction in the investment fraction will reduce profit by $5.04\%$, whereas a reduction in green technology efficiency will reduce the profit by $1.43\%$. A graphical representation to show the effect of green technology investment fraction and green technology efficiency is presented in Figure 12.

5. Managerial Insights

The study provides valuable insights for O2O retailers to navigate the challenges of imperfect production in a sustainable manner. By optimizing pricing, inventory, automation, and green practices, O2O businesses can achieve profitability while meeting stricter environmental regulations. The following major managerial insights can be found in this study.

• This study highlights the potential of O2O retailing for products with customizable parts. By offering online configuration and selection, businesses can cater to diverse customer needs while managing inventory efficiently.
• Imperfect production can be a bigger issue in O2O due to the combined online and offline channels. Defective products become waste if discarded but can raise quality concerns if discounted online.
• Minimizing waste due to imperfect production aligns with stricter carbon emission regulations. However, O2O retailers must find ways to manage defective products without compromising customer trust through deep discounts.
• The study suggests that investing in green technology can significantly increase profits for O2O retailers. Additionally, automating defect inspection can further improve profitability.
• O2O retailers can optimize their profits by strategically pricing assembled products and spare parts across online and offline channels. This may involve considering demand fluctuations for spare parts based on the final assembled product’s price.
• The study suggests that CCT regulations might be more effective than CT for O2O businesses with customizable products facing imperfect production.

6. Conclusions

An imperfect production system for assembled products under O2O retailing and carbon emissions regulation is developed in this study. Defective items were generated randomly and followed a certain probability distribution. From numerical illustrations, one can conclude that a dual-channel retailing for assembled products with different spare parts can earn a maximum profit if the defective rate follows a triangular distribution under CCT regulation. In this case, the retailer can earn a $3.38\%$ higher profit compared to other cases. Green technology investments help reduce carbon emissions by 1359 kg per cycle, which helps to increase system profit by $18.12\%$. On the other hand, autonomated inspection helps to create a zero-waste system with a $10.27\%$ higher profit for the retailer. Simultaneously, investment in setup costs helps reduce system costs and enhance system profit. Therefore, proper investment management for a complex imperfect manufacturing system under dual-channel retailing provides optimized results under CCT carbon emissions regulations.

Consideration of single-stage production process for a single assembled product under O2O retailing is one of the limitations of this study. The constant rate of production is another limitation of this research. We considered that the faulty generation rate followed a certain distribution, and one may extend this study by considering a fuzzy environment. We considered only two types of carbon regulation polices, whereas future researchers may consider other emission regulation polices to develop their study. By exploring the possibility of multi-stage production involving numerous participants within the realm of multi-channel retailing, it is conceivable to broaden the scope of this research in subsequent studies. Given the intricate nature of this retailing system, a significant financial investment is necessary. Consequently, it may be beneficial to investigate constraints related to budgetary limitations and spatial considerations in order to expand on this study. Rather than adhering to a fixed production pace, an avenue for further exploration lies in examining the potential for adaptability within the production process.