A Sustainable Supply Chain Model with Variable Production Rate and Remanufacturing for Imperfect Production Inventory System under Learning in Fuzzy Environment

Abstract

In the present paper, a sustainable supply chain model is investigated with a variable production rate and remanufacturing for the production of defective items under the effect of learning fuzzy theory, where the lower and upper variations in fuzzy demand rate are affected by learning parameters and backorders are also allowed. Our proposed model reveals a springy manufacturing inventory organization that makes various types of items, and imperfect items can be created through the method of manufacturing things in a fuzzy environment. When the screening process is completed, defective items are remanufactured immediately, and a limited financial plan and space limitations are assumed concerning the product assembly. We minimized the total fuzzy inventory cost with different distributions (beta, triangular, double triangular, uniform, and χ² (chi−square)) concerning the production rate, lot size, and backorder under learning in a fuzzy environment where the costs of screening, manufacturing, carrying, carbon emissions, backorders, and remanufacturing are included. The Kuhn–Tucker optimization technique is applied to solve non-linear equations that are based on some distributions. Numerical examples, sensitivity analysis, managerial insights and observations, limitations, future work, and applications are provided for the validation of our proposed model, and the industrial scope of this proposed work is included.

1. Introduction

It can be seen and observed that when the demand for a product is constant, then industries, production companies, and any other business organizations run smoothly and receive more profit as per expectation, but on the other hand, when the demand for a product on occasion increases and decreases from the customer side, several industries face more and more problems due to the imprecise nature of the demand rate. If the quality of the products is good and a feature of the product is also innovative, like good quality, flexible, and appealing, then the demand for the product from the customer side increases, and this is not true in general. In any production company, not all produced items may be good in features and quality. To safeguard their viability, commercial entities must first reduce costs and raise standards. They need to take such steps to keep their competitive position. Industries focus on manufacturing activity to decrease costs using innovative ideas while improving the quality of items using various strategies. Nowadays, the emission of carbon units from various sources is very harmful to the environment, and the impact is very dangerous to human life. From this point of view, we studied a number of literature reviews for the development of the proposed model, in which many strategies for maximizing profit during the variable production of defective items and reduction in carbon units with the help of carbon policies are provided through the contributions of authors. In this order, we collected a number of literature reviews that are related to the present proposed work and which are presented in the forthcoming section of the literature review.

2. Literature Review

2.1. Literature Related to the Flexible Production of Inventory

In this section, we discuss the literature reviews with the production system under different inventory situations. We discuss the contribution of Taleizadeh et al. [1], which is based on a supply chain model with carbon policies under remanufacturing policies and a quality improvement strategy. For faulty products, hybrid remanufacturing is being looked into as a way of cutting costs based solely on carbon emissions. A single-stage production’s quality improvement has been enhanced by Taleizadeh et al. [1]. Based on environmental concerns, several papers about supply chains and production platforms as well as fuzzy environments and potential modifications of put-together goods are examined. To further enhance the feedback in the production and supply chain, small and medium-sized industries collaborated, a strategy that was expanded upon in the paper by Machado et al. [2]. They overlooked potential models and disregarded put-together goods. Mishra et al. [3] disregarded the recycling of production products and permitted backorders in the production–inventory model. An imperfect production system was developed by Cárdenas-Barron [4] and Sarkar et al. [5] with fixed setup costs and manual inspections to identify defective products. This approach resulted in higher overall system costs and the potential for inspection errors to deliver defective items to the market. The product return rate for remanufacturing is considered by the closed-loop supply chain, which was centered on manufacturer competition and coordination. Aydin et al. [6] also considered newly manufactured and remanufactured goods. Both Cárdenas-Barron [4] and Sarkar et al. [5] suggested a subpar manufacturing system with specific setup costs and manually conducted product inspections to identify defective products. This resulted in higher overall system costs and could cause defective products being released onto the market as a result of examination mistakes. The supply chain was responsible for the back rate for manufacturing again, which was concentrated on the concurrence and coordination of businesses. Similarly, Aydin et al. [6] generalized a supply chain model with the policy of remanufacturing of products under a closed-loop supply chain. In this way, Hariga et al. [7] designed a supply chain model for manufacturing–remanufacturing for good transportation under consignment stock. Qingdi et al. [8] proposed a production-based inventory model with a timing matching method for mechanical items under different realistic situations. In this direction, Taleizadeh et al. [1], Sivashankari and Panayappan [9] and Sanjai and Periyasamy [10] made a fruitful contribution in the field of production of items under different approaches. Using channels A and B, Taleizadeh et al. [1] looked into the remanufacturing process. The purpose of introducing remanufacturing processes was to lower carbon emissions. Sivashankari and Panayappan [9] suggested a production–inventory model that took reworking into account for a single item. They considered model shortages. Nevertheless, Sivashankari and Panayappan [9] completely ignored the idea of an assembled item, in which every component is produced and also reproduced defective items with a constant defective rate under a variable production rate, greatly raising the cost of the system. Similar to this, a production–inventory system with imperfect generation under various delivery policies was proposed by Sanjai and Periyasamy [10]. They disregarded the notions of assembled products and variable production rates. Some authors have improved an inventory model with the use of an eco-label policy under a green supply chain, like Gao et al. [11], who designed a double-channel-based green supply chain model with the help of eco-label policy for the green products. From this point of view, Alexopoulos et al. [12] generalized a supply chain model for the manufacturing process, and Lagoudakis [13] developed a supply chain model with the effect of online shopping under brand choice, price margin, and item exploration. Mukherjee et al. [14] considered a smart sustainable supply chain model with the effect of carbon emissions under the reverse-docking effect. Yang et al. [15] assumed a smart sustainable supply chain model with information and digital technologies for Industry 4.9 enterprises. Bazan et al. [16] considered a model with the effect of carbon emissions and energy on the manufacturing inventory system. Bhatia and Elsayed [17] generalized a smart sustainable supply chain model using a fuzzy environment and smart manufacturing technologies. Chai et al. [18] developed an inventory model with the effect of government subsidies and environmental regulations on the remanufacturing of the product. Assid et al. [19] proposed an integrated supply chain model with the use of return policies for the manufacturing system under the inspection process. Sarkar and Park [20] developed a flexible production model for the reduction in makespan under remanufacturing and balanced the multi-stage complex production system. Zheng et al. [21] improved a supply chain model with unreliable suppliers under variable time-dependent demand rates. Saxena and Sarkar [22] proposed a production model using misplacement and the reliability of the production process for the industrial approach. Taleizadeh et al. [23] presented a manufacturing technology in the field of supply chain modeling where supply chain members obtained more ideas for the development of supply chain management, and Kanishka and Acherjee [24] contributed a literature review report of the different works of authors. Dey et al. [25] and Amrouche et al. [26] proposed a model supply chain model for the retail industry schemes under different approaches. Within the product’s life cycle, various services were offered to customers to preserve the company’s reputation. However, remanufacturing of the faulty products was not taken into consideration there. A supply chain model under various carbon reduction policies was developed by Zhu et al. [27]. Using mathematical modeling, Saxena and Sarkar [28] were able to visualize a policy for production coordination and replenishment. They compared the traditional supply chain with misplacement and demand; remanufacturing faulty products can be very advantageous. The remanufacturing of defective items has been taken into consideration in the development of several production inventory models. Our strategy of proposed work with the impact of changeable production rate has not, however, yet been examined in the literature that is currently available. The present scenario examines a manufacturing system in which the products are assembled (single type) and prepared for the next cycle, and due to system imperfections, defective items are generated at random chains with radio frequency identification and a supply chain powered by block chain with misplacement.

2.2. Literature Related with Remanufacturing and Defective Items

In this section, we discuss only those literature reviews that are related to the remanufacturing of defective items, and where some authors have improved the production model using an inspection process with various strategies. From this point of view, a cleaner production approach was employed in the development of a model by Cañrdenas-Barrón [4]. Wee et al.’s [29] model was expanded upon using a different approach to solve Cardenas-Barrón’s [4] model. Polotski et al. [30] proposed a model for semi-finished items under the production system. Their model’s hypothesis was created using a hybrid production process. M-Haidar and Nasr [31] talked about the post-production screening and reworking procedures. Short delivery times and on-time delivery were accomplished by Silva et al. [32]. In direct relation to the make-to-order policy, the TKS system implemented a make-to-stock strategy. Sarkar et al.’s [5] model was expanded by Kugele and Sarkar [33] by taking multi-item and multi-stage production processes into account. For their investigation, they did take a fixed production rate into account. A multi-item production model’s defective rate was discussed by Mridha et al. [34].

According to Chen’s [35] model, the performance of the supply chain was impacted by defective items. In this way, the significance of decisions about pricing, replenishment, and rework was clarified. Subpar products, good-quality items, and first and second production rates are the two categories of non-defective products that Mtibaa and Erray [36] examined. Second-rate and non-conforming products were given a reworking, which enhanced the product quality and allowed the products to be sold at the greatest value.

In this proposed model, the simulation process is followed, and the GKS system outperformed the POLCA structure. Ruidas et al. [37] proposed an EOQ model with an interval-valued system for two kinds of imperfect items. To investigate a production inventory model, Ruidas et al. [38] presented an inventory model with a production strategy for imperfect items under a rough interval strategy. The dynamic intelligent manufacturing–remanufacturing system was created by Li et al. [39]. Subsequently, Polotski et al. [30] extended the feasibility conditions of degrading products by using hybrid manufacturing and remanufacturing systems. Their model produced a cooperative production process and the best maintenance policy.

The effects of achieving mix flexibility in assembly systems on product manufacturing were examined by Asadi et al. [40]. The low degree of product modularity in mixed-product assembly lines (MPALs) was their main discovery. A multi-stage production system for defective items was developed by Malik et al. [41]. The reworking procedure was created with a random production defect rate in mind, which minimizes the waste generated by defective products. Their goal was to reduce production costs while maintaining the necessary quality standards.

A model was created by Centobelli et al. [42] to investigate how a circular economy with green economic incentives can maximize a production system’s profitability. Costs and waste can be reduced by remanufacturing. To test the model, the study used structural equation modeling (SEM) and confirmatory factor analysis (CFA). The results demonstrate the model’s beneficial impact on the supply chain’s economy. Li et al. [43] developed a smart manufacturing model by considering partial outsourcing and remanufacturing into account [43]. A focus group design was used in a qualitative study by Cifone et al. [44] to investigate how supply chain management and manufacturing professionals perceive the potential of digital technology.

The findings demonstrate that digital technologies can significantly increase the production economy. The buyer–supplier relationship in a psychological contract breach was investigated by Aslam et al. [45]. This paper examines the effects of inter-organizational and interpersonal relationships to improve the economy of the production system and increase profits. The study’s conclusions demonstrated how the buyer’s response was impacted by the early interpersonal and inter-organizational relationships. These were among the most crucial elements in enhancing a production system’s or supply chain’s financial circumstances. The effect of carbon ejection on a smart production system was described by Chaudhari et al. [46].

Sarkar and Guchhait [47] discussed the impact of irregular evidence distribution on the fuzzy demand rate because production conditions are uncertain. Within a supply chain, Zhang et al. [48] created a dual channel using blockchain technology. Profits were maximized for a range of scenarios through a critical examination of distinct factors. The model included a discussion of cost-effectiveness and promotional activities. Defective products can lead to supply chain shortages and backorders. Professionals in management saw the potential for digital technologies to boost trade.

To avoid this backlog, Bachar et al. [49] proposed a scenario for outsourcing and reworking imperfect items. They demonstrated that a randomized portion of the faulty goods could be fixed and remanufactured to boost the production system’s effectiveness. The model profitability was increased through the application of various decision-making strategies, and the best solutions were found using classical optimization techniques. Several manufacturing models that take remanufacturing into account have been created. Some studies have considered the simultaneous production cycles of manufacturing and remanufacturing. Nevertheless, a production model with a multiple-level inventory system developed for a solitary accumulated item with a variable manufacturing rate was considered in previous studies. Consequently, this study makes a groundbreaking attempt to close these gaps.

2.3. Literature Related with Optimized Backorder

In this section, we present the contributions of renowned authors who developed an inspection model using a backorder strategy. The model proposed by Huang and Wu [50] considered wholesalers’ concerns about batch demand and back ordering into account. By determining the ordering and back-ordering quantities, in the paper, the overall cost was reduced. An unreliable supply chain model presented by Mittal et al. [51] suggested that ordering policies could be influenced by inflation. A stock inventory system with a partial backlog that used stochastic demand was proposed by Xu et al. [52]. Mukherjee et al. [53] proposed a cross-dock inventory system in a similar vein. An inventory model for a single product with numerous demands and backorders was examined by Bao et al. [54]. The unit backlog costs of the corresponding demand classes were used to divide them. The economic lot-sizing issue was tackled by Kilic and Tunc [55], who used fixed production, holding, and back-ordering costs to optimize the total expected cost. A method for restocking spare components by permitting dependent backorders was proposed by Guo et al. [56]. The model’s isolated system consisted of backorders for various component types. For repairable isolated systems, the Markov model with dependent backorder was used. To significantly increase the economy, Bertazzi et al. [57] generalized a two-level supply chain model with some recent policies where the total inventory cost was minimized.

In the worst situation as well as on average, air freight shipments resulted in significant cost savings. By utilizing mixed-integer linear programming, optimality was attained. Retailers’ ordering policies and trade credit strategies were examined by Bi et al. [58]. To maximize the retailer’s profit, this paper sought to determine the best choices for the credit period, order quantity, and replenishment cycle. The advantages and disadvantages of mobile procurement platforms were noted by Charpin et al. [59].

A decision support system (DSS) was presented by Bouzekri et al. [60] to help plan actions to maximize the supply chain economic benefits. To obtain the best results for the production and port models, they created a new storage model. Inventory decisions for stochastic problems in a production system for improving products were examined by Buisman and Rohmer [61]. The results demonstrated how profit margins affect both the producer’s economic growth and the best course of action.

In a study based on third-party logistics, Ashraf et al. [62] investigated how production systems could efficiently maximize the advantages for their economy. The results guaranteed a notable enhancement in the economy of production. To account for the erratic nature of demand, a crisp model was created and then expanded to a fuzzy model, which dramatically reduced backorders. Three recently created algorithms were used to find the model’s optimal solution. Zhou et al. [63] proposed a model with a game theory approach under stochastic demand rate.

Two models, each with three sub-cases, were created based on revenue-sharing, non-coordination, and coordination contracts. In a modern socioeconomic setting, the models demonstrated how demand is dependent on the green degree, average selling price, and product quality. The implications of a finite lifetime for optimal firm investments were examined by Balter et al. [64], considering that investment decisions depend on the size and timing of the investments. Additionally, they implemented the finite product-life assumption in a duopoly framework, taking strategic implications into account.

Astvansh and Jindal [65] suggested appropriate trade-credit financing policies. The policy of multiple shipping was applied to minimize backorders and deliver inventory. Shajalal et al. [66] developed an algorithm-based inventory model with the impact of backorders for defective products by using some unbalanced data. Reduced backorders have been the subject of numerous studies. Heydari et al [67] developed supply chan model with stochastic demand rate under some realistic situations. In this order, Misra et al. [68] derived an inventory with selling price dependent demand under manufacturing system.

2.4. Literature Related to Environmental Issues

In this section, we discuss some selected research studies that are related to carbon emissions with other realistic situations. In this area, very renowned author Sebatjane [69] suggested an inventory model with the effect of carbon emissions and various types of emissions strategies under a production-based three-echelon system for economic situations. In this view, Sebatjane and Adetunji [70] developed a growing item-based inventory model for imperfect quality items by using a four-echelon system under the screening process. Saini et al. [71] also used transformation in greening management with supply chain management systems in some investigations. Wiredu et al. [72] developed a model and discussed the effect of green supply chain management on environmental performance and its advantages. Das et al. [73] improved the previous supply chain by using a green environment for the greening items with some realistic performances and their impacts during COVID-19. Khan et al. [74] developed an inventory model with a green supply chain in manufacturing unit firms as the focus and taken as a resource point. Similarly, Dzikriansyah et al. [75] explored the role of green supply chain practices with the help of a case study for an Indonesian enterprise and provided very valuable results.

2.5. Literature Related to Fuzzy and Learning in Environment

In this section, we discuss fuzzy-based inventory literature reviews that made positive contributions in this field and have selected literature reviews from 2022 to 2024. Padiyar et al. [76] proposed an inventory model with the preservation of technology for the imperfect production system in a fuzzy environment. Shaikh and Gite [77] developed an inflationary-based production model with the effect of a variable production model for deteriorating items under a fuzzy environment where the demand is the function of the selling price. Widowati et al. [78] assumed a model with a fuzzy environment for imperfect quality items under different programming approaches. Chaudhary et al. [79] designed a model with the effect of a fuzzy environment under the impact of carbon emissions. Taheri et al. [80] assumed a fuzzy-based ordering-based inventory model by using fuzzy linear programming methods. Tyagi et al. [80] defined an inventory fuzzy model for pharmaceutical and cosmetic products under some realistic situations. Jayaswal et al. [81] extended the previous contributions using trade-credit policy and deterioration. Aslam et al. [82] improved the previous inventory model using a fuzzy environment for a non-linear supply chain under some realistic situations. Garg et al. [83] developed a model with the fuzzy environment for a returnable container inventory system where triangular fuzzy numbers are allowed. Roy et al. [84] generalized the previous inventory model with carbon emissions using the fuzzy-2 environment for deteriorating items under the supply chain model. De and Ojha [85] developed an EOQ model with the fuzzy backlogging situation under the volume of a fuzzy Hasse diagram. Jayaswal et al. [86] proposed an inventory model with the effect of learning for deteriorating items under a fuzzy environment. In the direction of learning in a fuzzy environment, Alsaedi [87] developed a supply chain model with the effect of setting up a cost reduction policy for defective items under learning in a fuzzy environment. Kalaichelvan et al. [88] considered an EOQ model and optimized lot size using a fuzzy environment and machine learning for the pharmaceutical framework.

2.6. Research Gap and Proposed Work

In this section, we present the research gap of our proposed model with the help of previous contributions of renowned authors and have shown already in the literature reviews. We are discussing only the author’s contribution who worked in the field of inventory for ordering policies and supply chain management under different approaches like manufacturing, remanufacturing, carbon emissions, fuzzy environment, imperfect items, fuzzy learning theory, etc. A number of authors have contributed to this field and suggested future work. We studied a number of literature reviews in this direction and found some future work from this literature. Several authors have addressed the problems with conventional production behaviors, which involve the manufacturing of single or multiple products, in the literature that is currently available. As of yet, no methods for raising the efficiency of production systems have been mentioned. While improving manufacturing systems for novel products is the focus of almost every industry, the production of the economy must also be taken into consideration as well. The significance of the total inventory cost in single-product manufacturing facilities under single-stage production has been the focus of earlier models. In highly competitive business markets, addressing customer expectations and generating assembled products through assembly processes are critical. There is a lack of research regarding the possibility that a single product produced by a conventional business plan may result in greater quantities of defective products being returned by consumers, which could lower the system’s sales and profit.

As a result, the production economy might struggle. Creating innovative, assembled products through adaptable production systems may be a concern for every industry looking to boost their business and financial circumstances. The suggested approach is an assemblage of products, a completed screening process, reprocessing, shortages, as well as the reworking of the products at different steps to avoid any effective damages and improve the financial circumstances of the manufacturing processes hampered structure. The primary research goal was to manage defective merchandise. On the other hand, industries are constantly focused on meeting customer demands and ensuring dispute-free product delivery. The goal of flexible production systems is to enhance machine production yet offer flexibility. The degree to which a system can be altered to produce the assembled product is referred to as its flexibility. The purpose of this research is to help industries use the suggested strategy to advance their businesses. One of the biggest problems facing industries is the creation of defective products during production. Remanufacturing faulty goods could be one way to address this. Therefore, the suggested model uses flexible production to rework the assembled product, thereby reducing the waste represented by defective items under the learning fuzzy approach.

Taleizadeh et al. [1] proposed a supply chain model with carbon emissions and return policy under quality improvement where remanufacturing policy is allowed but fuzzy and learning fuzzy theory is not considered in this research study, whereas Machado et al. [2] designed a supply chain model for the production of items under different environmental issues where carbon emissions, remanufacturing policy, fuzzy, and learning fuzzy theory are not assumed. Cárdenas-Barron [4] developed a model for the production and remanufacturing of items under carbon emissions where fuzzy and learning fuzzy environments were not permitted in the research study. Our contribution is to develop a supply chain model with the effect of carbon emissions for variable production rates and remanufacturing for imperfect production inventory systems under learning fuzzy theory. Similarly, we selected some authors’ contributions and show them in

Table 1. Theprevious and proposed contributions from the literature.

Author (s)	Production System	Remanufacturing	Backorders	Carbon Emissions	Fuzzy Environment	Learning Fuzzy Environment
Taleizadeh et al. [1]	✓	✓		✓
Machado et al. [2]	✓
Cárdenas-Barron [4]	✓	✓	✓
Aydin et al. [6]	✓
Alexopoulos et al. [12]	✓
Malik et al. [41]	✓	✓
Huang and Wu [50]		✓	✓
Mittal et al. [51]			✓
Xu et al. [52]			✓
Sebatjane [69]	✓			✓
Alsaedi et al. [76]				✓	✓	✓
Choudhry et al. [77]				✓	✓
Present work	✓	✓	✓	✓	✓	✓

, and our proposed work is presented at the bottom of Table 1. The novelty of this paper is that the total inventory cost is much less as compared to other models and also learning fuzzy theory is not considered yet during the supply chain where variable production rate and remanufacturing is allowed. This study deals with a supply chain model with a variable production rate and remanufacturing for defective manufacturing process under learning fuzzy theory where the demand rate is considered as a fuzzy demand rate and the backorder situation is also considered.

The proposed research problem is presented through a flowchart in Figure 1.

2.7. Novelty and Research Contribution of Our Proposed Work

Our proposed contribution is as follows:

• The lack of learning fuzzy theory in the previous research work is completed with the help of this proposed model and the total inventory fuzzy cost is minimized using learning fuzzy theory. This approach is more and more beneficial for the cost reduction policy in the field of the production of electronic products.
• A lot of authors have developed an inventory model with a fuzzy environment under different approaches and have not considered the concept of learning in a fuzzy environment while the proposed model considers the concept of learning in a fuzzy environment and calculates the total fuzzy cost under learning in a fuzzy environment where left and right deviation in fuzzy demand follow the effect of learning.
• The suggested approach considers a flexible production system to close the aforementioned research gap. A key tactic for effectively enhancing market responsibility in the face of future demand uncertainty despite any shortages is to increase flexibility in production. Different product types might be built simultaneously in the same production facility thanks to flexible manufacturing.
• This model is intended for product assembly from multiple parts to meet the various needs of customers. Consumer expectations are continually linked to a product variation based on their personal preferences (this paper examines a particular electronic product). This need can only be met and the system profit raised by a merchandise assembly method built on a flexible production system.
• The suggested model applies an inspection strategy to prevent a defective final product. The production system will obtain significant monetary profit from this approach. Once a product is sold, the model lowers end-user backorders of goods after production but before being sold to the final consumer, the inspection strategy for defective assembled products is carried out, and the defective products are sent to be recycled.
• The role of statistical distributions in the field of the business sector is more applicable to the reduction in cost function. In this paper, we selected five types of statistical distributions, namely, uniform, triangular, beta (β), double-triangular probability distributions, and χ² (chi−square). It is assumed that the percentage of defective items follows these distributions and the total fuzzy cost varies concerning each distribution with its expected values. We calculate the total fuzzy cost under each distribution with its expected values and after that, we compare the total fuzzy cost under these distributions. For the sensitivity analysis, we select that distribution in which the total fuzzy cost is minimum and such distribution is more applicable for the proposed model and other input parameters receive a positive response on the total inventory fuzzy cost.
• Remanufacturing policies have the potential to recover defective products prior to sale and decrease the amount of end-user backorders. There has not been any prior research on this flexible production system research gap in the literature. The total fuzzy cost is minimum in the χ² distribution as compared to uniform, triangular, beta (β), and double-triangular probability distributions. This tactic can increase the system’s financial gains. Increasing sales and satisfying end-user demands are critical to a production system’s financial success. An industry can respond to end users more quickly by implementing flexible production. This study’s contribution is in the form of a mathematical model with a changeable rate of production for one type of item.
• Concern for practically all products, including electronic products, as they suggest that the inventory of work-in-process is under control. This is so that even though the process of supplying a product is dependent upon market demand, the production of any product can be predicted. The just-in-time production policy is used in this study to examine an inspection policy for product assembly. This policy, which is more successful than the conventional method of inspection, delivers all imperfect quality items for remanufacturing of imperfect quality items before selling the final goods to the end operators. This can help reduce wasteful spending, boost sales, and improve the overall profit. If defective items need to be remanufactured after they are made ready for sale, this method benefits industries by lowering the costs of holding space. As a result, with flexible and economical production, less waste is generated in the form of defective products. The quality of the final assembled products can be raised by managing the remanufacturing of defective products under flexible production. Moreover, boosting production flexibility will stop shortages from occurring as a result of smart production. Remanufacturing products helps reduce returns, which is a significant financial benefit for industries. If this were not the case, they might have to pay high maintenance costs for the vast majority of products that end users return.

2.8. Organization of Proposed Plan

The following sections determine how the paper is structured. The literature is reviewed in Section 2. Section 3 presents the study’s fundamental assumptions, notations, and problem definition. The five distribution functions’ mathematical model is explained in Section 4. In Section 5, the process for deriving the solution is covered. Section 6 discusses a few numerical examples, and Section 7 presents the sensitivity analysis of the numerical examples. Section 8 contains the managerial insights.

3. Notations and Assumptions

3.1. Notations

The notations used in the proposed model are described here, as given below:

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$j$: The number of batches to production a single amassed items where, $j=1,2,3,\dots ,m$

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$P_j$: Production rate for j products (units per time) (decision variable);

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$Y_j$: The production lot for j products (units per cycle) (decision variable);

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$B_j$: The number of shortages lot size for j products (units) (decision variable);

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$D_j$: The rate of order of j products (units per time);

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$S_j$: The shipment expenses of j products (USD per shipment);

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$M_j$: The manufacturing costs of j products (USD per unit);

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$H_j$: The carrying cost for j product (USD per unit per time);

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$B_l$: The linear shortage costs for j products (USD per unit per time);

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$m$: The lot number (in integers);

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$B_l$: The backorder cost for fixed shortages j product/backorder (USD per backorder);

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${\xi }_j$: The average backorder of item j (units);

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$T$: Cycle length (time units);

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${\psi }_1$: The whole inventory cost (units);

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${\psi }_2$: The whole inventory cost per unit time (units);

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$D_r$: The remanufacturing costs for $j$ products (USD per item);

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$Vj$: The budget of j products (units);

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$f$: The volume for storage of products (units);

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$\mathrm{g}$: Whole amount of budget (in USD);

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$I_j$: The regular inventory for $j \mathrm{product}$ (units);

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$I_{maxj}$: Maximum inventory level where, $I_{maxj}=I_{1j}+I_{2j}$ (units);

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$\widetilde{D_j}$: The rate of fuzzy demand of item j (units per time);

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$\widetilde{D_{Lj}}$: The lower deviation of fuzzy demand of item j (units per time);

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$\widetilde{D_{Uj}}$: The upper deviation of fuzzy demand of item j (units per time);

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$\widetilde{{\psi }_4}$: The whole fuzzy inventory cost per unit time (units);

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$\widetilde{{\psi }_5}$: The whole fuzzy inventory cost per unit time (units) under learning in fuzzy environment;

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$l$: The slope of learning input parameter;

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${\beta }_j$: The random percentage imperfect quality products $j$ in each cycle;

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$E[{\beta }_j]$: Expected percentage of imperfect products $j$;

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$BD:$ Stands for $\beta$ distribution;

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$TD:$ Stands for $\mathrm{triangular} \mathrm{distribution}$;

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$DTD:$ Stands for double–triangular distribution;

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$UD:$ Stands for $\mathrm{uniform} \mathrm{distribution};$

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$CD:$ Stands for ${\chi }^2 \mathrm{distribution}$ which represents not feasible.

3.2. Assumptions

The following are the assumptions for the proposed model adopted:

➢

In the model, we are assuming a single assembled item is manufactured during the production of items. An electronic product is an example of a single assembled product and is prepared from different electronic parts j which depends on the customer’s demand. The various components are sold individually from different warehouses. The effects of product assembly in a market with competition are the main topic of this study. A fuzzy demand under flexible production can be promptly satisfied without causing any displeasure to the customer.

➢

The production rate in this model is constantly higher than the fuzzy demand and is variable. According to Alexopoulos et al. [12], the model’s entire inventory system is predicated on an unlimited scheduling horizon under changeable manufacture, and it is feasible to determine the deepest manufacture inventory price under an unlimited time distance.

➢

Following production, every assembled product is examined, and any issues are promptly reported for remanufacturing so that the same production cycle can continue.

➢

Each product has a fixed inspection cost, although since it is much less than the other costs, it is adjusted based on setup costs. According to Dey et al. [25], the imperfect rate of imperfect items under variable production was permitted to be accidental. As a result, five distinct distribution functions (uniform, triangular, double-triangular, beta, and 𝜒²) are used to evaluate the optimal fuzzy cost individually under learning in a fuzzy environment.

➢

After selling the completed assembled products, the model first experiences shortages, but all of these are fully backlogged. No sales are lost. There are two different kinds of backorder costs in this model: fixed backorder costs, which apply to the maximum backorder level, and linear backorder costs, which apply to average backorders during fuzzy environment and also the concept of average inventory, changeable production, limitations of budget, space for holding items, and other assumptions are taken from (Sarkar et al. [5]).

➢

The production of a single product assembled from various components is examined in this paper. The objective of this model in a flexible production system is to minimize inventory costs under learning in a fuzzy environment and damage to the assembled product while taking the constraints of budget and space capacity into account. The model incorporates random defective rates and multi-item clean production, building upon the benchmark model of Carreras-Barron [4] during a fuzzy environment.

➢

It is also considered that the effect of learning is involved in the lower and upper fuzzy range of customer’s demand.

➢

Carbon emissions come from the storage of holding units and carbon emission costs are included in this manuscript.

4. Mathematical Formulation

In this section, the formulation of the proposed model is divided in two parts, namely, problem definition and formulation of the proposed model.

4.1. Problem Definition

This research examines a single assembled product that is made with a flexible production process and various parts. Unpredictability in the production processes, equipment failures, and quality problems are common characteristics of an imperfect manufacturing system. The suggested system uses a variable production rate to allocate resources efficiently to address these flaws. The same production cycle is used to rework defective products. The remanufacturing of the defective items is performed within the same manufacture length to control waste and offer a cost-effective manufacturing stock scenario.

Items with imperfections are produced at random and adhere to specific distribution patterns. Backorders are the first step in the process, which simultaneously yields perfect and flawed goods. Within time T₃, defective items are found by human inspection and remanufactured. Production systems’ flaws lead to shortages that are completely backlogged. Backorders can impact a manufacturing system in several ways. In an imperfect manufacturing system, backorders can cause customer dissatisfaction, lower sales, raise expenses, and draw attention to hidden operational problems. Backorders may make inventory control more difficult. Manufacturers have to weigh the costs of carrying excess inventory against the costs of lost sales as a result of unhappy customers to prevent this.

Manufacturers have to weigh the costs of carrying excess inventory against the costs of lost sales as a result of unhappy customers to prevent backorders. To increase customer satisfaction and the manufacturing system’s overall efficiency, it is imperative to address the underlying causes of backorders and put strategies in place to reduce their frequency. The average back-ordering cost as well as the linear backordering cost are both considered in this study. When creating and executing a production system, financial and spatial limitations are frequent obstacles. It is crucial to concentrate on cost-effective solutions when operating on a tight budget.

This could entail comparing various equipment choices, considering reconditioned or used machinery, or looking into leasing or other financing options. Remanufacturing reduced the amount of material wasted in this study. Workflow, material flow, and accessibility should all be considered. Overcoming these obstacles and attaining an effective and efficient production setup is feasible by constructively considering financial and spatial constraints, planning carefully, and keeping an eye on and enhancing the production system all the while. Our model considers financial and spatial constraints. By maximizing the production rate, the batch size of the item j used for assembly, and the number of backorders, the total production–inventory cost is reduced under learning in lower and upper deviation of fuzzy environment.

4.2. Formulation of the Proposed Model

From Figure 2, when the inventory level will be zero, then the initial backorders in the interval $\left[0,T_1\right]$ are $B_j$ where flexible production is allowed and the rate of inventory level is needed $P_j\left(1-{\beta }_j\right)-D_j$. In the interval $\left[T_2, T_3\right]$, the imperfect items/defective items start to be remanufactured at the rate $\left( P_j-D_j\right)$ and after time $t=T_2$. Let us consider that ($I_{ma{x}_j})$ be the maximum capacity at $T_3$ which is the sum of ${I_1}_j$ and ${I_2}_j$ and it decreases due to the rate of demand $D_j$ as well as the inventory level becoming zero in the interval $[T_3,T_4]$. Now, it is also considered that in the interval $\left[T_3,T_4\right],$ the rate of demand is $D_j$ and the rate of backorders is $B_j$. The total cycle length is $T$ the sum of which is $TP \left(\mathrm{production} \mathrm{time}\right), TRM \left(\mathrm{remanufacturing} \mathrm{time}\right), \mathrm{and} TNP \left(\mathrm{non}-\mathrm{production} \mathrm{time}\right)$ where $TP=T_1+T_2$, $TRM=T_3$ and $TNP=T_4+T_5$ and mathematically, it can be written $T=TP+TRM+TNP$ or $T=T_1+T_2+T_3+T_4+T_5$ where $T_1 \mathrm{and} T_5$ are the time periods in which the system includes backorders, $T_2$ stands for the time period at which the system includes pure production of product, $T_3$ stands for the time period at which the system includes pure remanufacturing of the product, and $T_4$ is the pure consumption time.

The time period for the production can be calculated from Figure 2, which is:

$$T_1+T_2={\displaystyle \frac{\left(I_{1_j}+B_j\right)}{\left(P_j\left(1-E\left[{\beta }_j\right]\right)-D_j\right)}}={\displaystyle \frac{Y_j}{P_j}},$$

(1)

From Equation (1), we obtain the following:

$$I_{1_j}=Y_j\left(\left(\left(1-E\left[{\beta }_j\right]\right)\right)-{\displaystyle \frac{D_j}{P_j}}\right)-B_j$$

(2)

Similarly, we calculate $T_3$ from Figure 2,

$$T_3={\displaystyle \frac{{I_2}_j}{\left(P_j-D_j\right)\prime }}$$

As $T_3={\displaystyle \frac{E\left[{\beta }_j\right]Q_j}{P_j}}$.

Therefore,

$${\displaystyle \frac{E\left[{\beta }_j\right]Y_j}{P_j}}={\displaystyle \frac{{I_2}_j}{\left(P_j-D_j\right)}}$$

(3)

From Equation (3), we calculate the value of ${I_2}_j$,

$${I_2}_j=E\left[{\beta }_j\right]Y_j\left(1-{\displaystyle \frac{D_j}{P_j}}\right)$$

(4)

The optimal inventory level ($I_{ma{x}_j}$) is the sum of ${I_1}_j$ and ${I_2}_j$ then from Equations (2) and (4), we obtain the following:

$$I_{ma{x}_j}={I_1}_j+{I_2}_j=Y_j\left(1-{\displaystyle \frac{D_j}{P_j}}\left(1+E\left[{\beta }_j\right]\right)\right)-B_j$$

(5)

For the calculation of the average inventory level ($I_{AV{G}_j})$, then it is required to compute the whole stock for the total cycle length ($T$) and split it by total phase length ($T$); now,

$$I_{AV{G}_j}={\displaystyle \frac{1}{T}}\left[{\displaystyle \frac{1}{2}}{T}_2{I}_{1_j}+{\displaystyle \frac{1}{2}}{T}_3{I}_{1_j}+{\displaystyle \frac{1}{2}}{T}_3{I_{max}}_j+{\displaystyle \frac{1}{2}}{T}_4{I_{max}}_j\right]$$

(6)

The values of $I_{1_j}, I_{2_j}$ and $I_{ma{x}_j}$ from Equations (2), (4) and (5) and also put $\mathrm{values} \mathrm{of} T_1,T_2, T_3, T_4 \mathrm{and} T_5$ replaced in Equation (6)

$$I_{AV{G}_j}={\displaystyle \frac{1}{2Q_j\left(\left(1-E\left[{\beta }_j\right]\right)-\frac{D_j}{P_j}\right)}}\left[\left(Y_j^2+B_j^2\right)\left(1-E\left[{\beta }_j\right]\right)+{\displaystyle \frac{Y_j^2{D}_j^2}{P_j^2}}{\left(1+E\left[{\beta }_j\right]+E\left[{\beta }_j\right]\right)}^2+{\displaystyle \frac{Y_j^2{D}_j}{P_j}}\left(\left(E{\left[{\beta }_j\right]}^3-2\right)\right)+2B_j{Y}_j\left({\displaystyle \frac{D_j}{P_j}}+E\left[{\beta }_j\right]-1\right)\right]$$

(7)

Note: $\mathrm{the} \mathrm{calculation} \mathrm{of} T_1,T_2, T_3, T_4, \mathrm{and} T_5 \mathrm{have} \mathrm{been} \mathrm{shifted} \mathrm{into}$ Appendix A.

Now, we calculate the average backorder inventory $(I_{BAV{G}_j})$ which is as follows:

$$I_{BAV{G}_j}={\displaystyle \frac{1}{T}}\left[{\displaystyle \frac{1}{2}}{B}_j{T}_1+{\displaystyle \frac{1}{2}}{\displaystyle \frac{B_j^2}{D_j}}\right]={\displaystyle \frac{B_j^2\left(1-E\left[{\beta }_j\right]\right)}{2Y_j\left[\left(1-E\left[{\beta }_j\right]\right)-\frac{D_j}{P_j}\right]}}$$

(8)

Now, the total inventory costs are the sum of costs related to our proposed work which are (i) screening cost $\left(K={\displaystyle \frac{S_j{D}_j}{Y_j}}\right)$ which is incorporated in the set up cost; (ii) the cost per cycle time for manufacturing the product $\left(C_M={\displaystyle \frac{M_j{P}_j\left(1+E\left[{\beta }_j\right]\right)}{\frac{P_j}{D_j}}}\right)$; (iii) the cost per cycle time for holding of the product $\left(C_H={\displaystyle \frac{H_j}{T}}\left[{\displaystyle \frac{{\left\{\left[Y_j\left[\left(1-E\left[{\beta }_j\right]\right)\right]-\frac{D_j}{P_j}\right]-B_j\right\}}^2}{2\left[P_j\left(1-E\left[{\beta }_j\right]\right)\right]}}\right]+{\displaystyle \frac{E\left[{\beta }_j\right]Y_j\left\{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2\right\}}{P_J}}+{\displaystyle \frac{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2}{2D_j}}\right)$; (iv) the carbon emission cost per cycle time due to holding of the product in the warehouse $\left(C_E={\displaystyle \frac{e_{j{D}_j}}{Y_j}}\left[{\displaystyle \frac{{\left\{\left[Y_j\left[\left(1-E\left[{\beta }_j\right]\right)\right]-\frac{D_j}{P_j}\right]-B_j\right\}}^2}{2\left[P_j\left(1-E\left[{\beta }_j\right]\right)\right]}}\right]+{\displaystyle \frac{E\left[{\beta }_j\right]Y_j\left\{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2\right\}}{P_J}}+{\displaystyle \frac{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2}{2D_j}}\right)$; (v) the backordering cost per cycle time $\left(C_B={\displaystyle \frac{B_{c_j}{B}_j{D}_j}{Y_j}}+B_{ij}{I}_{BAV{G}_j}={\displaystyle \frac{B_{c_j}{B}_j{D}_j}{Y_j}}+{B_i}_j{\displaystyle \frac{B_j^2\left(1-E\left[{\beta }_j\right]\right)}{2Y_j\left[\left(1-E\left[{\beta }_j\right]-\frac{D_j}{P_j}\right)\right]}}\right)$; (vi) the average remanufacturing cost due to random defective items $\left(C_{RM}={D_r}_j{P}_{j}E\left[{\beta }_j\right]{\displaystyle \frac{D_j}{Y_j}}\right)$. Therefore, the total inventory cost per cycle can be represented by

$$\begin{array}{ll}TC={\displaystyle {\displaystyle \sum }_{j=1}^m}[{\displaystyle \frac{S_j{D}_j}{Y_j}}+& H_j{I}_{AV{G}_j}+{\displaystyle \frac{B_{c_j}{B}_j{D}_j}{Y_j}}+{B_l}_j{I}_{BAV{G}_j}+M_j{D}_j\left(1+E\left[{\beta }_j\right]\right)+P_{j}E\left[{\beta }_j\right]D_{r_j}{\displaystyle \frac{D_j}{Y_j}}\\ & +{\displaystyle \frac{e_{j{D}_j}}{Y_j}}\left[{\displaystyle \frac{{\left\{\left[Y_j\left[\left(1-E\left[{\beta }_j\right]\right)\right]-\frac{D_j}{P_j}\right]-B_j\right\}}^2}{2\left[P_j\left(1-E\left[{\beta }_j\right]\right)\right]}}\right]+ {\displaystyle \frac{E\left[{\beta }_j\right]Y_j\left\{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2\right\}}{P_J}}\\ & +{\displaystyle \frac{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2}{2D_j}}]\end{array}$$

(9)

The values of $I_{AV{G}_j}$ and $I_{BAV{G}_j}$ from Equations (7) and (8) replaced in Equation (9), we obtain the total inventory cost per cycle, as follows:

$$\begin{array}{ll}TC={\displaystyle {\displaystyle \sum }_{j=1}^m}[{\displaystyle \frac{S_j{D}_j}{Y_j}}+& H_j[{\displaystyle \frac{1}{2Y_j\left(\left(1-E\left[{\beta }_j\right]\right)-\frac{D_j}{P_j}\right)}}[\left(Y_j^2+B_j^2\right)\left(1-E\left[{\beta }_j\right]\right)+{\displaystyle \frac{Y_j^2{D}_j^2}{P_j^2}}{\left(1+E\left[{\beta }_j\right]+E\left[{\beta }_j\right]\right)}^2\\ & +{\displaystyle \frac{Y_j^2{D}_j}{P_j}}\left(\left(E{\left[{\beta }_j\right]}^3-2\right)\right)+2B_j{Y}_j\left({\displaystyle \frac{D_j}{P_j}}+E\left[{\beta }_j\right]-1\right)]]+{\displaystyle \frac{B_{c_j}{B}_j{D}_j}{Y_j}}\\ & +{B_l}_j\left[{\displaystyle \frac{D_j}{Y_j}}\left[{\displaystyle \frac{1}{2}}{B}_j{T}_1+{\displaystyle \frac{1}{2}}{\displaystyle \frac{B_j^2}{D_j}}\right]={\displaystyle \frac{B_j^2\left(1-E\left[{\beta }_j\right]\right)}{2Y_j\left[\left(1-E\left[{\beta }_j\right]\right)-\frac{D_j}{P_j}\right]}}\right]+M_j{D}_j\left(1+E\left[{\beta }_j\right]\right)+P_{j}E\left[{\beta }_j\right]D_{r_j}{\displaystyle \frac{D_j}{Y_j}}\\ & +{\displaystyle \frac{e_{j{D}_j}}{Y_j}}\left[{\displaystyle \frac{{\left\{\left[Y_j\left[\left(1-E\left[{\beta }_j\right]\right)\right]-\frac{D_j}{P_j}\right]-B_j\right\}}^2}{2\left[P_j\left(1-E\left[{\beta }_j\right]\right)\right]}}\right]+{\displaystyle \frac{E\left[{\beta }_j\right]Y_j\left\{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2\right\}}{P_J}}\\ & +{\displaystyle \frac{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2}{2D_j}]}\end{array}$$

(10)

4.3. Formulation of Crisp Model under Fuzzy Environment

As per consideration, the rate of demand is treated as a triangular fuzzy number $\left(D_j-\widetilde{d_{Lj}},D_j,D_j+\widetilde{d_{Uj}}\right)$ and the signed distance method has been taken for the defuzzification of fuzzy inventory total cost and also presented in the Figure 3. We take the total inventory cost under fuzzy environment from Equation (10), and obtain the following:

$$\begin{array}{ll}TC={\displaystyle {\displaystyle \sum }_{j=1}^m}[{\displaystyle \frac{S_j\widetilde{D_j}}{Y_j}}+& H_j[{\displaystyle \frac{1}{2Y_j\left(\left(1-E\left[{\beta }_j\right]\right)-\frac{\widetilde{D_j}}{P_j}\right)}}[\left(Y_j^2+B_j^2\right)\left(1-E\left[{\beta }_j\right]\right)+{\displaystyle \frac{Y_j^2{\widetilde{D_j}}^2}{P_j^2}}{\left(1+E\left[{\beta }_j\right]+E\left[{\beta }_j\right]\right)}^2\\ & +{\displaystyle \frac{Y_j^2\widetilde{D_j}}{P_j}}\left(\left(E{\left[{\beta }_j\right]}^3-2\right)\right)+2B_j{Y}_j\left({\displaystyle \frac{\widetilde{D_j}}{P_j}}+E\left[{\beta }_j\right]-1\right)]]+{\displaystyle \frac{B_{c_j}{B}_j\widetilde{D_j}}{Y_j}}\\ & +{B_l}_j\left[{\displaystyle \frac{\widetilde{D_j}}{Y_j}}\left[{\displaystyle \frac{1}{2}}{B}_j{T}_1+{\displaystyle \frac{1}{2}}{\displaystyle \frac{B_j^2}{\widetilde{D_j}}}\right]={\displaystyle \frac{B_j^2\left(1-E\left[{\beta }_j\right]\right)}{2Y_j\left[\left(1-E\left[{\beta }_j\right]\right)-\frac{\widetilde{D_j}}{P_j}\right]}}\right]+M_j\widetilde{D_j}\left(1+E\left[{\beta }_j\right]\right)+P_{j}E\left[{\beta }_j\right]D_{r_j}{\displaystyle \frac{\widetilde{D_j}}{Y_j}}\\ & +{\displaystyle \frac{e_{j\widetilde{D_j}}}{Y_j}}\left[{\displaystyle \frac{{\left\{\left[Y_j\left[\left(1-E\left[{\beta }_j\right]\right)\right]-\frac{\widetilde{D_j}}{P_j}\right]-B_j\right\}}^2}{2\left[P_j\left(1-E\left[{\beta }_j\right]\right)\right]}}\right]+{\displaystyle \frac{E\left[{\beta }_j\right]Y_j\left\{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{D_j}{P_j}\right]-B_j^2\right\}}{P_J}}\\ & +{\displaystyle \frac{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{\widetilde{D_j}}{P_j}\right]-B_j^2}{2\widetilde{D_j}}]}\end{array}$$

(11)

Now, the total fuzzy cost is defuzzified using the signed distance method from Equation (11). The signed distance between the total fuzzy cost ($\widetilde{TC}$) and $\tilde{0}$, we obtain as follows:

$$\begin{array}{ll}d\left(\widetilde{TC},\tilde{0}\right)={\psi }_F[P_j,Y_j& ,B_j]\\ & ={\displaystyle {\displaystyle \sum }_{j=1}^m}[{\displaystyle \frac{S_{j}d\left(\widetilde{D_j},\tilde{0}\right)}{Y_j}}\\ & +H_j[{\displaystyle \frac{1}{2Y_j\left(\left(1-E\left[{\beta }_j\right]\right)-\frac{d\left(\widetilde{D_j},\tilde{0}\right)}{P_j}\right)}}[\left(Y_j^2+B_j^2\right)\left(1-E\left[{\beta }_j\right]\right)+{\displaystyle \frac{Y_j^{2}d{\left(\widetilde{D_j},\tilde{0}\right)}^2}{P_j^2}}{\left(1+E\left[{\beta }_j\right]+E\left[{\beta }_j\right]\right)}^2\\ & +{\displaystyle \frac{Y_j^{2}d\left(\widetilde{D_j},\tilde{0}\right)}{P_j}}\left(\left(E{\left[{\beta }_j\right]}^3-2\right)\right)+2B_j{Y}_j\left({\displaystyle \frac{\widetilde{D_j}}{P_j}}+E\left[{\beta }_j\right]-1\right)]]+{\displaystyle \frac{B_{c_j}{B}_{j}d\left(\widetilde{D_j},\tilde{0}\right)}{Y_j}}\\ & +{B_l}_j\left[{\displaystyle \frac{d\left(\widetilde{D_j},\tilde{0}\right)}{Y_j}}\left[{\displaystyle \frac{1}{2}}{B}_j{T}_1+{\displaystyle \frac{1}{2}}{\displaystyle \frac{B_j^2}{d\left(\widetilde{D_j},\tilde{0}\right)}}\right]={\displaystyle \frac{B_j^2\left(1-E\left[{\beta }_j\right]\right)}{2Y_j\left[\left(1-E\left[{\beta }_j\right]\right)-\frac{d\left(\widetilde{D_j},\tilde{0}\right)}{P_j}\right]}}\right]+M_{j}d\left(\widetilde{D_j},\tilde{0}\right)\left(1+E\left[{\beta }_j\right]\right)+P_{j}E\left[{\beta }_j\right]D_{r_j}{\displaystyle \frac{d\left(\widetilde{D_j},\tilde{0}\right)}{Y_j}}\\ & +{\displaystyle \frac{d\left(\widetilde{D_j},\tilde{0}\right)e_j}{Y_j}}\left[{\displaystyle \frac{{\left\{\left[Y_j\left[\left(1-E\left[{\beta }_j\right]\right)\right]-\frac{d\left(\widetilde{D_j},\tilde{0}\right)}{P_j}\right]-B_j\right\}}^2}{2\left[P_j\left(1-E\left[{\beta }_j\right]\right)\right]}}\right]+{\displaystyle \frac{E\left[{\beta }_j\right]Y_j\left\{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{d\left(\widetilde{D_j},\tilde{0}\right)}{P_j}\right]-B_j^2\right\}}{P_J}}\\ & +{\displaystyle \frac{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{\widetilde{D_j}}{P_j}\right]-B_j^2}{2 d\left(\widetilde{D_j},\tilde{0}\right)}]}\end{array}$$

(12)

Now, the value of signed distance, $d\left(\widetilde{D_j} ,\tilde{0}\right)$ by using the definition from Equation (10) replaced in Equation (12), we obtain:

The total inventory cost per unit cycle time, ${\psi }_F\left(P_j,Y_j,B_j\right)$;

$$\begin{array}{ll}{\psi }_F\left(P_j,Y_j,B_j\right)={\displaystyle {\displaystyle \sum }_{j=1}^m}& [{\displaystyle \frac{S_j\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4Y_j}}\\ & +H_j[{\displaystyle \frac{1}{2Y_j\left(\left(1-E\left[{\beta }_j\right]\right)-\frac{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4P_j}\right)}}[\left(Y_j^2+B_j^2\right)\left(1-E\left[{\beta }_j\right]\right)\\ & +{\displaystyle \frac{Y_j^2{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}^2}{16P_j^2}}+{\left(1+E\left[{\beta }_j\right]+E\left[{\beta }_j\right]\right)}^2+{\displaystyle \frac{Y_j^2\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4P_j}}\left(\left(E{\left[{\beta }_j\right]}^3-2\right)\right)\\ & +2B_j{Y}_j\left({\displaystyle \frac{\widetilde{\left(4\tilde{D_j}+\tilde{d_{Uj}}-\tilde{d_{Lj}}\right)}}{4P_j}}+E\left[{\beta }_j\right]-1\right)]]{\displaystyle \frac{B_{c_j}{B}_j\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4Y_j}}\\ & +{B_l}_j\left[{\displaystyle \frac{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4Y_j}}\left[{\displaystyle \frac{1}{2}}{B}_j{T}_1+{\displaystyle \frac{2B_j^2}{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}}\right]={\displaystyle \frac{B_j^2\left(1-E\left[{\beta }_j\right]\right)}{2Y_j\left[\left(1-E\left[{\beta }_j\right]\right)-\frac{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4P_j}\right]}}\right]\\ & +M_j\left({\displaystyle \frac{4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}}{4}}\right)\left(1+E\left[{\beta }_j\right]\right)+P_{j}E\left[{\beta }_j\right]D_{r_j}{\displaystyle \frac{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4Y_j}}\\ & +{\displaystyle \frac{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)e_j}{4Y_j}}\left[{\displaystyle \frac{{\left\{\left[Y_j\left[\left(1-E\left[{\beta }_j\right]\right)\right]-\frac{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4P_j}\right]-B_j\right\}}^2}{2\left[P_j\left(1-E\left[{\beta }_j\right]\right)\right]}}\right]\\ & +{\displaystyle \frac{E\left[{\beta }_j\right]Y_j\left\{Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}{4P_j}\right]-B_j^2\right\}}{P_J}}+{\displaystyle \frac{4Y_j\left[1-\frac{E\left[{\beta }_j\right]}{2}-\left(1+\frac{E\left[{\beta }_j\right]}{2}\right)\frac{\widetilde{D_j}}{P_j}\right]-4B_j^2}{2\left(4\widetilde{D_j}+\widetilde{d_{Uj}}-\widetilde{d_{Lj}}\right)}]}\end{array}$$

(13)

4.4. Formulation of the Proposed Model under Learning Fuzzy Environment

In this section, we apply the theory of learning as suggested by Wright [77] and as per assumption, mathematically given below:

$$L_N=L_{01}{N}^{-l}$$

(14)

Here, $L_N$ denotes the time for $N$-th shipment; $L_0$ is the initial time; and $l$ is the slope of learning. The upper (${\rho }_{U,i}^{\widetilde{D_j}}$) and lower (${\sigma }_{L,i}^{\widetilde{D_j}}$) deviation of fuzzy demand follow the effect of learning and its mathematical representation is defined in Equations (15) and (16), as follows:

$${\rho }_{U,i}^{\widetilde{D_j}}=\left\{\begin{array}{c}{\rho }_{U,i}^{\widetilde{D_j}}, i=1\\ {\rho }_{U,i}^{\widetilde{D_j}}{\left(\left(i-1\right){\displaystyle \frac{365}{N}}\right)}^{-l},i>1\end{array}\right.$$

(15)

$${\sigma }_{L,i}^{\widetilde{D_j}}=\left\{\begin{array}{c}{\rho }_{L,i}^{\widetilde{D_j}}, i=1\\ {\sigma }_{L,i}^{\widetilde{D_j}}{\left(\left(i-1\right){\displaystyle \frac{365}{N}}\right)}^{-l},i>1\end{array}\right.$$

(16)

Now, the total fuzzy inventory cost per cycle time under learning effect using Equations (13), (15) and (16), we obtain the following:

(17)

4.5. Financial Plan and Space Limitations

Our proposed model has some limitations regarding budget and space limitations due to the variable production rate and imprecise nature of the demand rate. Boundary conditions relating to budget and space capacity are suggested in order to restrict the inventory and lower investments. These can be acquired in the manner described as follows:

$${\displaystyle \sum }_{j=1}^m{Y}_j{V}_j\le g$$

(18)

$${\displaystyle \sum }_{j=1}^m{Y}_j{S_c}_j\le f$$

(19)

5. Solution Method

The cost Equation (17) has three decision variables $P_j, Y_j \mathrm{and} B_j$ and these decision variables require optimization and cost Equation (17) including the financial plan and space constraints is as follows:

(20)

subjected to ${\displaystyle \sum }_{j=1}^m{Y}_j{V}_j-g\le 0$

$${\displaystyle \sum }_{j=1}^m{Y}_j{S_c}_j-f\le 0$$

and $P_j, Y_j \mathrm{and} B_j>0, j=1,2,3,\dots \dots m.$ Equation (20) represents a non-linear equation and we applied the Kuhn–Tucker condition for the solving of the total inventory costs used by calculating the Lagrange multiplier equation.

(21)

As per consideration, there are two constraints included in the expression of Equation (19) and we also considered two Lagrange multiplier coefficients, namely and, for the calculation of the decision variables. When we formulated the equation of Lagrange, then four cases are taken along with Equation (20) and all four cases are given for the calculation of the lot size, as follows:

(a) ${\gamma }_1=0, {\gamma }_2=0$; (b) ${\gamma }_1 \ne 0, {\gamma }_2=0$; (c) ${\gamma }_2\ne 0, {\gamma }_1=0$; (d) ${\gamma }_1 \ne 0, {\gamma }_2=0$.

Since $Y_j$ has different values in the first case (a), ${\gamma }_1=0, {\gamma }_2=0$ and this case is neglected. The value of $Yj$ differered for the remaining three cases (b) to (d), ${\gamma }_1 \ne 0, {\gamma }_2=0; {\gamma }_2\ne 0, {\gamma }_1=0;$ and ${\gamma }_1 \ne 0, {\gamma }_2=0$. Now, we differentiate the Lagrange multiplier ($L\left(P_j, Q_j, B_j, {\gamma }_1, {\gamma }_2\right)$) Equation (20) with respect to $Pj, Yj, Bj, {\gamma }_1$, and ${\gamma }_2$, we obtain the following:

$$\begin{array}{c}{\displaystyle \frac{\partial L\left(P_j,Y_j,B_j,{\gamma }_1,{\gamma }_2\right)}{\partial P_j}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=E\left[{\beta }_j\right]{\displaystyle \frac{4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right)\frac{365}{N}\right)}^{-l}\right)}{4Y_j}}{D}_{r_j}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \frac{H_j}{2Y_j}}[R_1{\displaystyle \frac{4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right)\frac{365}{N}\right)}^{-l}\right)}{4R_{*}^2{P}_j^2}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+R_2\left({\displaystyle \frac{2}{R_{*}^2{P}_j^4}}+{\displaystyle \frac{4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right)\frac{365}{N}\right)}^{-l}\right)}{4R_{*}^2{P}_j^4}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+R_3\left({\displaystyle \frac{1}{R_{*}^2{P}_j^4}}+{\displaystyle \frac{4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right)\frac{365}{N}\right)}^{-l}\right)}{4R_{*}^2{P}_j^3}}\right)]\hfill \end{array}$$

(22)

$$\begin{array}{c}{\displaystyle \frac{\partial L\left(P_j,Y_j, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial B_j}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{{B_c}_j}{4Y_j}}\left(4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right){\displaystyle \frac{365}{N}}\right)}^{-l}\right)\right)+{\displaystyle \frac{2B_j{H}_j\left(1-E\left[{\beta }_j\right]\right)}{2Y_j{R}_{*}}}-H_j+{\displaystyle \frac{{B_c}_j}{Y_j}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{B_j{B}_{l_j}\left(1-E\left[{\beta }_j\right]\right)}{2Y_j{R}_{*}}}\hfill \end{array}$$

(23)

Now, for solving $P_j$ and $B_j$ then we have to take, ${\displaystyle \frac{\partial L\left(P_j,Y_j, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial P_j}}=0$ and ${\displaystyle \frac{\partial L\left(P_j,Y_j, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial B_j}}=0,$ after solving, we obtain the following:

$$P_j={\displaystyle \frac{(R_{2}E\left[{\beta }_j\right]-R_2)+\sqrt{4{\left(R_{2}E\left[{\beta }_j\right]-R_2\right)}^2+R_2\left(R_3-E\left[\beta \right]R_3+R_1\right)\left(4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right)\frac{365}{N}\right)}^{-l}\right)\right)}}{2(R_3-E\left[{\beta }_j\right]R_3+R_1}}$$

(24)

$$B_j=\left[H_j-{\displaystyle \frac{{B_c}_j}{4Y_j}}\left(4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right){\displaystyle \frac{365}{N}}\right)}^{-l}\right)\right)\right]{\displaystyle \frac{Y_j\left(1-E\left[{\beta }_j\right]-\frac{1}{4P_j}\left(4\widetilde{D_j}+\left({\rho }_{U,i}^{\widetilde{D_j}}-{\sigma }_{L,i}^{\widetilde{D_j}}\right)\left({\left(\left(i-1\right)\frac{365}{N}\right)}^{-l}\right)\right)\right)}{\left(1-E\left[{\beta }_j\right]\right)\left[H_j+B_{i_j}\right]}}$$

(25)

We need to apply the conditions for the calculation of $Y_j$,${\gamma }_1 \mathrm{and} {\gamma }_2$, then

Condition (b): ${\gamma }_1\ne 0, {\gamma }_2=0:$

$${\displaystyle \frac{\partial L\left(P_j,Y_j, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial Y_j}}=-{\displaystyle \frac{R_4}{Y_j^2}}+H_j{R}_5+{\gamma }_1{V}_j$$

(26)

Now, for solving $Y_j$ then we have to take, ${\displaystyle \frac{\partial L\left(P_j,Y_j, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial Y_j}}=-{\displaystyle \frac{R_4}{Y_j^2}}+H_j{R}_5+{\gamma }_1{V}_j=0,$ after solving, we obtain the following:

$$Y_j=\sqrt{{\displaystyle \frac{R_4}{H_j{R}_5+{\gamma }_1{V}_j}}}$$

(27)

Now, for solving ${\lambda }_1$ then we have to take, ${\displaystyle \frac{\partial L\left(P_j,Y_j, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial {\gamma }_1}}=\left({\displaystyle \sum }_{j=1}^m{Y}_j{V}_j-g\right)=0,$ after solving, we obtain the following:

$${\gamma }_1={\displaystyle \frac{R_4{V}_j^2-H_j{R}_5{g}^2}{V_j{g}^2}}$$

(28)

Condition (c):${\gamma }_2\ne 0, {\gamma }_1=0:$

In this condition, the methodology is same for the calculation of $Y_j$ and ${\gamma }_2$.

Now, for solving $Y_j$ then we have to take, ${\displaystyle \frac{\partial L\left(P_j,Y, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial Y_j}}=-{\displaystyle \frac{R_4}{Y_j^2}}+H_j{R}_5+{\lambda }_2{S_c}_j=0,$ after solving, we obtain the following:

$$Y_j=\sqrt{{\displaystyle \frac{R_4}{H_j{R}_5+{\gamma }_2{S_c}_j}}}$$

(29)

Now, for solving ${\gamma }_2$ then we have to take, ${\displaystyle \frac{\partial L\left(P_j,Y_j, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial {\lambda }_2}}=\left({\displaystyle \sum }_{j=1}^m{Y}_j{S_c}_j-f\right)=0,$ after solving, obtain the following:

$${\lambda }_2={\displaystyle \frac{R_4{S_c}_j^2-H_j{R}_5{f}^2}{{S_c}_j{f}^2}}$$

(30)

Condition (c): ${\gamma }_2$:${\gamma }_1\ne 0, {\gamma }_2\ne 0:$

Now, for solving $Y_j$ then we have to take, ${\displaystyle \frac{\partial L\left(P_j,Y, B_j,{\gamma }_1, {\gamma }_2\right)}{\partial Y_j}}=-{\displaystyle \frac{R_4}{Y_j^2}}+H_j{R}_5+{\gamma }_2{S}_c{S_c}_j+{\gamma }_1{V_j}_j=0,$ after solving, obtain the following:

$$Y_j=\sqrt{{\displaystyle \frac{R_4}{H_j{R}_5+{\gamma }_1{V}_j+{\gamma }_2{S_c}_j}}}$$

(31)

Note: Here, the values of $R_1,R_2,R_3, \mathrm{and} R_4$ and the sufficient condition for the global solution using the Hessian matrix method are presented in Appendix B.

Numerical Example

This section covers all inventory parameters regarding our proposed study and presented in

Table 2. Model’s input parameters for assembled products when $j=2$.

Parameters ($\mathbf{}\mathit{j}=2$)	Values	Parameters	Values
$m$	2	$\left(\widetilde{D_j}-\widetilde{d_{Lj}},\widetilde{D_j},\widetilde{D_j}+\widetilde{d_{Uj}}\right)$	$\left(250, 300, 370\right)$,$\left(450, 500, 570\right)$
${\gamma }_2$	18	${B_l}_j$ (USD per unit)	10, 11
$\widetilde{d_{Uj}}$	70, 70	$\widetilde{d_{Lj}}$	50, 50
${\gamma }_2$	18	${B_i}_j$ (USD per unit)	10, 11
$c_e$	0.75, 0.75	$i$	2, 3
$H_j$ (USD per unit time)	50, 55	${B_c}_j$ (USD per unit)	1, 2
$S_j$ (USD per setup)	50.22, 55.22	$M_j$ (USD per unit)	7, 8
$V_j$ (USD per unit)	55, 65	${S_c}_j$ (USD per unit)	3, 4
$D_{r_j}$ (USD per item)	5, 7	$g$ (USD)	33,000
$f$	400	${\gamma }_1$	15
$l$	0.23, 0.23	$N^{*}$	5, 5

and

Table 3. Model’s input parameters for assembled products when $j=3$.

Parameters ($\mathbf{}\mathit{j}=3$)	Values	Parameters	Values
$m$	3	$\left(\widetilde{D_j}-\widetilde{d_{Lj}},\widetilde{D_j},\widetilde{D_j}+\widetilde{d_{Uj}}\right)$	$\left(300, 350, 420\right)$, $\left(270, 320, 370\right)$, $\left(250, 300, 370\right)$
${\gamma }_2$	0.008	${B_i}_j$ (USD per unit)	10, 10, 10
$\widetilde{d_{Uj}}$	70, 70, 70	$\widetilde{d_{Lj}}$	50, 50, 50
$c_e$	0.75, 0.75, 0.75	$i$	3, 3, 3
$H_j$ (USD per unit time)	50, 50, 50	${B_c}_j$ (USD per unit)	1, 1, 1
$H_j$ (USD per setup)	50.22, 50.22, 50.22	$M_j$ (USD per unit)	7, 7, 7
$V_j$ (USD per unit)	60, 60, 60	${S_c}_j$ (USD per unit)	2, 2, 2
$D_{r_j}$ (USD per item)	6, 6, 6	$g$ (USD)	58,000
$f$	400	${\gamma }_1$	0.009
$l$	0.23, 0.23, 0.23	$N^{*}$	5, 5, 5

for assembling items $j=2 \mathrm{and} j=3.$ We have shown the different distribution patterns, and their numerical values are given in

Table 4. Proposed distributions and its expected formula for assembled items $j=2$ and $j=3$.

Name’s Distribution	Expected Values of Probability Distribution E [${\mathit{\beta }}_{\mathit{j}}$]
Uniform probability distribution	${\displaystyle \frac{a_j+b_j}{2}}$, $a_j,b_j>0$
Triangular probability distribution	${\displaystyle \frac{a_j+b_j+C_j}{3}},a_j,b_j,c_j>0$
Double triangular probability distribution	${\displaystyle \frac{a_j+4b_j+c_j}{6}},a_j,b_j,c_j>0,$
Beta $\beta$ probability distribution	${\displaystyle \frac{a_j}{{\alpha }_j+{\beta }_j}},{\alpha }_j,{\beta }_j>0$
${\chi }^2$ chi-square probability distribution	$k_j,k_j>0$

and

Table 5. Proposed distributions and its expected values ($E\left[{\beta }_j\right]$).

Name’s Distribution	Uniform Probability Distribution	Triangular Probability Distribution	Double-Triangular	$\mathbf{Beta} \mathit{\beta }$ Probability Distribution	Chi Square Probability Distribution ${\mathit{\chi }}^2$
1	$\left(a_j,b_j\right)$	$\left(a_j,b_j,c_j\right)$	$\left(a_j,b_j,c_j\right)$	$\left({\alpha }_j,{\beta }_j\right)$	$\left(k_j\right)$
2	(0.03, 0.07)	(0.03, 0.04, 0.07)	(0.03, 0.04, 0.07)	(0.03, 0.07)	(0.03)
3	(0.03, 0.07)	(0.03, 0.04, 0.07)	(0.03, 0.04, 0.07)	(0.03, 0.07)	(0.03)
4	(0.04, 0.08)	(0.04, 0.04, 0.07)	(0.04, 0.04, 0.08)	(0.04, 0.08)	(0.04)
5	0.03, 0.04	0.047, 0.047, 0.05	0.043, 0.043, 0.047	0.03, 0.03, 0.33	(0.04)

. The values of coefficients are used to calculate the numerical solution along with these distributions.

As per consideration, $a_j, b_j, c_j, {\alpha }_j, {\beta }_j,$ and ${\kappa }_j$ are the scaling parameters and its numerical values are given in Table 5. These are more important and are required for the calculation of the decision variables and total fuzzy inventory cost under the effect of learning.

We calculate the optimal values of decision variables, Pj, Yj, and B_j for both two and three spare parts ($j=2$ and $j=3$) and also separately calculate the inventory total fuzzy cost concerning the optimal values of the decision variables along with each five proposed distributions. The optimal values of the decision variables and the minimum inventory fuzzy cost under five proposed distributions under learning in a fuzzy environment are presented in

Table 6. Optimal values of decision variables and corresponding total fuzzy cost under learning in fuzzy environment for assembled items $j = 2$.

Distribution	Production Rate ${\mathit{P}}_{\mathit{j}}$	Production Batch Size ${\mathit{Y}}_{\mathit{j}}$	Backorder Quantity ${\mathit{B}}_{\mathit{j}}$	Total Cost (USD per Year) under Learning in Fuzzy Environment
Uniform distribution	103, 27	789, 604	76, 57	221,504
Triangular distribution	109, 54	930, 187	91, 108	29,943
Double triangular distribution	112, 225	533, 309	69.45, 30.62	25,209
Beta distribution	99.31, 63.03	741, 508	98, 112	329,506
${\chi }^2$ distribution	32.27, 25.09	269.88, 220.12	61.05, 43.03	19,973

for two spare parts ($j=2$) and

Table 7. Optimal values of decision variables and corresponding total fuzzy cost under learning in fuzzy environment for assembled items $j = 3$.

Distribution	$\mathbf{Production} \mathbf{Rate} {\mathit{P}}_{\mathit{j}}$	$\mathbf{Production} \mathbf{Batch} \mathbf{Size} {\mathit{Y}}_{\mathit{j}}$	$\mathbf{Backorder} \mathbf{Quantity} {\mathit{B}}_{\mathit{j}}$	Total Cost (USD per Year) under Learning in Fuzzy Environment
Uniform distribution	619.09, 626.03, 563.04	69.08, 98.09, 105.41	7.06, 18.07, 22.72	24,262
Triangular distribution	785.05, 417.07, 429.05	85.61, 94.05, 208.07	19.05, 20.02, 19.04	23,142
Double triangular distribution	792.07, 716.07, 732.07	74.08, 97.08, 71.07	20.07, 27.07, 31.10	28,341
Beta distribution	648.08, 845.02, 842.02	49.01, 98.09, 90.07	58, 43, 41	39,675
${\chi }^2$ distribution	591.07, 483.07, 574.05	94.05, 47.04, 94.45	43.03, 40.03, 41.03	21,109

for three spare parts ($j=3$). Finally, the total fuzzy inventory cost is minimized under distribution (${\mathsf{\chi }}^2$ chi-square distribution) for both assembled items $j=2$ and $j=3$. Our proposed model’s financial plan and space are considered to be bounded, and it means that the set-up cost is raised. So, the manufactured bulk inclines to boost with the set-up cost and if the production rate increases then the batch size of the production increases. The objective of this proposed study is that the high production rate and high production batch size should be highest, but the total fuzzy inventory cost should be lowest.

Remark 1.  

If the proposed model only takes into account one type of product with constant production rates of defects, no inspection, no carbon emissions, no fuzzy environment, and no learning in the fuzzy environment, then it will converge to the Cardenas-Barrón model [4].

Remark 2.  

This model will converge to the Sarkar et al. [5] model if it considers a random imperfect rate, one type of assembled product, the rate of production is fixed, without carbon emissions, fuzzy environment, learning in fuzzy environment and inspection, an assembled product, and financial plan strategy and space limitations.

6. Discussion of Our Study with Previous Contributions

Our proposed model is more and more applicable for the production inventory system with an inspection process where the demand rate is imprecise (fuzzy environment) in nature which means that the rate of demand increases or decreases or is not constant from the customer side before production of items and the production system produces some defective items. The inspection is a good policy for the improvement of the quality of the produced lot before the delivery of the products and this policy sustains the reliability of the product among the buyer, seller/customers/clients, etc. who want to purchase this product from these production companies. If the quality of the product is reliable among the people then the selling of the products will increase and on the other hand, if the quality of products is not good then the buyers/sellers/customers will hesitate to purchase the products from these companies. For example, we are discussing some reliable companies like Sony, LG, Samsung, and Reliance that provide almost all electronic products and people are also reliable with these companies and other production companies (use and through). The production companies want more and more production of products based on selling products at low inventory costs with high production rates. The fuzzy environment nullifies the imprecise nature of the demand rate and the effect of learning designs the shape of the lot size and minimizes the total fuzzy cost. This study is different to others in using some concepts like (i) variable production rate, (ii) remanufacturing of defective items after inspection, (iii) single assembled items, (iv) random defective items, (v) carbon emissions, (vi) fuzzy environment, and (vii) learning in the fuzzy environment during the modelling of the proposed problems and also presented at bottom of

Table 8. Comparison of our proposed with previous contribution.

Existing Literature	Production Rate	Production Type	Defective Rate	Carbon Emissions	Fuzzy Environment	Learning in Fuzzy Environment	Total Cost
Cardenas Barron [4]	Fixed	Single	Fixed rate	N A	N A	N A	3430
Sarkar et al. [5]	Fixed	Single	Random	N A	N A	N A	2629
Sivashankari and Panayappan [9]	Fixed	Single	Shortages	N A	N A	N A	455,185
Sanjai and Periyasamy [10]	Fixed	Single	N A	N A	N A	N A	450,915
This study	Variable	Single assembled	Random	Considered	Considered	Considered	19,973

. The present study obtained a minimum inventory fuzzy cost from the previous fruitful contributions like Sivashankari and Panayappan [9], Sanjai and Periyasamy [10], Carreras-Barron [4], and Sarkar et al. [5]. Finally, our proposed model is more sensitive than the other models regarding the cost point of view and it can be seen in Table 8. These models have different setups. However, because variable production rates are used in a one-stage manufacturing-remanufacturing process with space and budget constraints, carbon emissions, fuzzy environment, and learning in a fuzzy environment, the results of this study are more realistic and reduce the total fuzzy cost.

7. Sensitivity Analysis

In this section, we show the behavior of all inventory known parameters and also include financial plan and budget limitations on the distribution functions as well as calculated separately the sensitivity values of the five different distributions along these input parameters. The variable effect of input parameters on the five different distributions is presented in

Table 9. Changeable effect of inventory parameters on the different distributions for sensitivity analysis.

Parameter	Variations	$\mathit{\beta }$-Distribution	Triangular Distribution	Double-Triangular Distribution	Uniform Distribution	${\mathit{\chi }}^2$-Distribution
$a_J$	−50	−34.08	−51.06	−43.03	−50.34	−54.01
	−25	−16.03	−29.07	−19.04	−24.06	−28.07
	+25	16.03	29.07	19.04	24.06	28.07
	+50	34.08	51.06	43.03	50.34	54.01
$b_J$	−50	−19.85	−2.26	−1.75	−4.12	−12.41
	−25	−5.10	−0.03	−0.65	−1.99	−5.88
	+25	3.07	1.03	0.61	2.12	5.08
	+50	0.009	2.09	1.12	4.03	9.07
$c_J$	−50	−9.05	−6.69	−11.09	−10.23	−9.06
	−25	−5.45	−4.98	−6.79	−4.98	−0.94
	+25	6.80	4.93	6.43	4.71	0.54
	+50	−0.007	6.17	9.26	8.68	−2.04
$k_J$	−50	−0.005	−0.44	−0.59	-	-
	−25	−0.002	0.039	−0.77	-	-
	+25	0.001	0.043	0.45	-	-
	+50	0.002	0.025	0.030	-	0.002
$f$	−50	+15.03	+31.01	+25.05	+21.06	+25.42
	−25	+8.08	+16.07	+12.03	+11.08	+13.05
	+25	−8.08	−16.07	−12.03	−11.07	−13.07
	+50	−15.03	+31.01	−25.05	−21.06	−25.42
$g$	−50	+0.23	+0.44	+0.41	+0.37	+0.45
	−25	+0.12	+0.25	+0.23	+0.19	+0.22.5
	+25	−0.12	−0.25	−0.23	−0.19	−0.45
	+50	−0.23	−0.44	−0.41	−0.37	−0.80
$m_J^{*}$	−50	−10.36	-	−22.33	-	-
	−25	-	−24.05	−31.41	−18.33	−20.01
	+25	13.53	88.55	60.02	43.06	21.07
	+50	16.96	28.45	68.33	58.08	70.08
$S_J$	−50	−12.16	−12.18	−15.27	−14.24	−15.98
	−25	−12.12	13.19	−15.23	−14.18	−13.18
	+25	12.03	12.05	14.31	14.08	12.65
	+50	22.63	14.07	15. 34	15.55	15.23
$H_J$	−50	−8.05	−3.23	−3.65	-	-
	−25	−3.04	−4.45	−4.45	−3.67	-
	+25	3.07	4.13	4.07	3.98	3.05
	+50	6.03	6.73	6.32	-	5.07
$B_{c_j}$	−50	−0.036	−0.32	−0.24	−1.07	−1.68
	−25	−0.19	−0.063	−0.64	−0.63	0.69
	+25	0.10	0.051	0.48	0.57	0.63
	+50	0.25	0.08	1.16	0.15	0.26
$B_{l_j}$	−50	−0.50	−0.60	−0.66	−0.61	−0.54
	−25	−0.25	−0.26	−0.32	−0.29	−0.35
	+25	0.20	0.25	0.29	0.26	0.23
	+50	0.40	0.50	0.53	0.50	0.45
$C_{ej}$	−50	−2.32	−2.95	−4.65	-	-
	−25	−1.65	−1.87	−2.81	−1.72	-
	+25	1.45	1.76	2.43	1.34	0.08
	+50	2.03	2.67	3.56	-	1.02
$l_j$	−50	−1.26	−1.45	−1.74	−1.97	−0.09
	−25	−1.06	−1.02	−1.10	−0.98	−0.04
	+25	1.05	0.98	1.03	0.85	0.05
	+50	1.03	1.29	1.32	1.01	0.08
$\widetilde{d_{Uj}}$	−50	−8.53	−9.42	−6.12	−4.06	−1.87
	−25	−4.46	−5.73	−4.21	−3.31	−0.98
	+25	4.46	5.73	4.21	3.31	0.97
	+50	8.53	9.28	6.11	4.06	1.67
$\widetilde{d_{Lj}}$	−50	−6.08	−5.33	−4.08	−5.33	−0.09
	−25	−3.21	−4.26	−3.01	−2.09	−0.05
	+25	3.21	4.26	3.01	2.08	0.05
	+50	6.07	5.32	4.08	5.28	0.09

. The impact of input parameters graphically have been presented in the Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19.

Observations and managerial insights:

➢

The variation in the value of $a_J$ with both negative and negative from −50% to +50% means then the total inventory fuzzy cost is directly affected by the change in the values of $a_J$. In the observations, the value of parameter $b_J$ and $c_J$ is more and more effective under the $\mathsf{\beta }$-distribution and χ²-distribution. From Table 9, the inventory input $k_J$ has a smaller effect on the value of total inventory fuzzy cost than the other parameters.

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In this model, the budget and space restrictions with corresponding modifications are more delicate. Table 9 shows that both constraints consistently affect the overall total inventory fuzzy cost.

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The manufacturing cost, carbon effect and holding cost are the two parameters that change significantly with both positive and negative changes. Additionally, each distribution has a negative sign, suggesting that the two parameters’ changes have little effect on the overall cost.

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At each stage of the distributions, there are small adjustments made to the setup cost, linear backorder cost, and fixed backorder cost parameters within the −50% to 50% range. Consequently, these small adjustments have an impact on the overall cost, showing that the total inventory fuzzy cost varies with different distributions.

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If we change with both negative and positive values of learning rates $l_J$ from −50% to +50%, then each distributions are varying according to the variation in learning rates with different numerical values but ${\chi }^2$ distribution has less value others distribution. If the rate of learning increases from −60% to +60%, then the value of ${\chi }^2$ distribution remains constant and the distribution of others varies with respect to the learning rate. Due to variation in the distribution of others with respect to the learning rate, the total fuzzy cost also varies. Finally, the total fuzzy cost remains cost under ${\chi }^2$ distribution if the rate of learning increases from −60% to +60%. It means that ${\chi }^2$ distribution is more beneficial for this proposed model. The total inventory fuzzy cost is minimized when the rate of learning achieves the position of saturation. If the decision-makers take the values of the learning rate in the form of a percentage from −60% to +60%, then the total fuzzy cost can be expected under such assumption. The concept of the learning theory gave positive effect in this model and is also applicable for the reduction in cost function.

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If we change the value of the upper and lower deviation of the fuzzy demand rate $\widetilde{d_{Uj}}$ and $\widetilde{d_{Lj}}$ then ${\chi }^2$ distribution has less value than the distribution of others and it means that the ${\chi }^2$ distribution is more beneficial for this proposed model. The total inventory fuzzy cost is minimized when ${\chi }^2$ distribution is allowed for this proposed model.

8. Conclusions and Future Work

Our proposed model deals with a sustainable supply chain model with variable production rates and remanufacturing for an imperfect production inventory system under learning in a fuzzy environment where financial and budget limitations are also allowed. In this proposed model, the total fuzzy cost is minimized by learning fuzzy theory for the two $\left(j=2\right)$ and three $\left(j=3\right)$ assembled items under five different distributions ($\mathsf{\beta }$-distribution, triangular distribution, double-triangular distribution, uniform distribution, and ${\chi }^2$ distribution) where carbon emissions and financial and budget limitations are also added during cost minimization. Finally, we observed that ${\chi }^2$ distribution is more applicable for the variable production system where the rate of demand is imprecise and the production system produces some defective items and remanufactures defective items after the inspection process of produced products. The financial plan and budget constraints (limitations) suggest there is always a good payoff for any business sector and these limitations are always needed for variable production where the rate of demand is not constant. If the owners of the production companies, seller, buyer customer, etc. ignore the financial plan and budget constraints during decision making then there may be some losses. For the controlling of financial economics and safety for future ‘losses’, we applied a financial plan and budget limitations and obtained a positive effect in our proposed study. Furthermore, in the five different distributions, the ${\chi }^2$ distribution has a smaller impact on other distributions when learning the rate of learning changes and its presentation is presented in Table 9. The total fuzzy cost also changes concerning the variation in the learning effect and this variation may be more harmful to the production companies but when the rate of learning varies, then the total fuzzy cost remains constant per cycle and production companies make more profit using the proposed strategies. It means that the learning in the lower and upper deviation of fuzzy demand rate has been given a positive response regarding the reduction in the total fuzzy cost and also in the optimal decisions variable. The theory of fuzzy is more beneficial for the nullification of the imprecise nature of the demand rate, especially the lower and upper deviation of demand rate. In this view, the lower and upper deviation of the fuzzy demand rate are also responsible for controlling the total inventory fuzzy cost during the uncertain production of a product when the demand comes suddenly. In our study, the assembled items can be taken to be electronic goods like mobile phones and also for complex production processes, like car production and assembled items that could be remanufactured without any shortages and include less maintenance costs in the total costs. This is more important for the organization to classify and report on the limitations in their business systems to progress the overall recital, efficiency, and quality of the product. The practical application of this model can be used in the production of electronic products like refrigerators, coolers, ACs, and mobiles, as well as the manufacturing of cars in companies like Tata Motors, Honda, Suzuki Motors, etc. For example, nowadays the demand for the Tata Nexon and the Tata Punch cars is very high and the demand for these products is also imprecise. In this case, the total inventory fuzzy cost of manufacturing cars may cause some economic problems for the companies due to the uncertainty of the demand rate for the assembled items. The payoff of this production’s company varies due to variations in Tata Nexon and punch car total fuzzy cost and our proposed model is more applicable in this case. The proposed study is more and more beneficial for such types of production of electronic items. In this scenario, during the production process, defective items/those which are not good in quality might be produced randomly and the defective items might be remanufactured. The production policies are changing day by day due to people’s new ideas and our proposed model can be extended using a two-level trade–credit strategy, warehouses, and a cloudy fuzzy environment. The variable effect of the learning and cost inputs is also presented and this scenario is more and more applicable for the production of electronic products like refrigerators, coolers, ACs, mobiles, etc.

8.1. Uniqueness of Our Proposed Model

The uniqueness of our proposed model is unique by applying a variable production rate, inspection process of defective items during production of products, remanufacturing of defective items, five different types of distributions ($\beta$-distribution, triangular distribution, double-triangular distribution, uniform distribution, and ${\chi }^2$ distribution), leaning in lower and upper deviation of fuzzy demand rate, assembled items (mobile phones, freeze, and car (complex production)), backorders, financial and budget limitations, and carbon emissions. The total fuzzy cost of manufacturing of assembled products and complex assembled products is minimized using of learning fuzzy theory under these assumptions. Our proposed model is more and more beneficial to others’ contributions regarding total inventory cost and our proposed work has been presented in Table 8.

8.2. Application of Our Proposed Model

Our proposed model is more and more applicable for production inventory systems where production companies produce only electronic items (assembled items during production of electronic items). As we know, some electronic items are used only in a seasonal period and the demand for the products also increases and decreases during the seasonal period. In the year 2019, the demand for electronic items like mobile phones, laptops, high-quality cameras, and electronic boards for teaching increased due to the effect of COVID-19. Due to variable fuzzy demand, the production companies were not providing the customer’s demand for electronic items and had to face some losses due to lower production of demanded lots and budget limitations. But almost all production companies generated a lot of pay off during the duration of COVID-19 due to increased sales of electronic items, especially mobile phones, electronic teaching boards, and laptops as per the institutional demand for electronic items. It means that our scenario is beneficial in this situation and if some situations become like this in the future, then the present study will work well for the variable production rate of electronic items where the demand rate of electronic products increases/decreases or is not constant. The decision-makers can take more and more advantage during the production of electronic products and earn a lot of profit. The present study is more applicable especially for single and complex assembled products. Our proposed work can be more applicable for the extension of Jayaswal et al. [89], Alsaedi et al. [90], Kalaichelvan et at. [91] and Pandey et al. [92].

8.3. Social Implications of Our Proposed Model

In our study, the decision maker, seller, buyer, and others who want to produce other products like greening items, deteriorating items, and fast food items can obtain an understanding of methodology, fuzzy demand rate, and also the cost functions. The production of bread, biscuits, and raw materials for the Maggi products, are recent opportunities for the business and can earn more profit using an idea of our research study. These are positive social implications of our proposed model in the recent global trade market.

8.4. The Implications of Our Proposed Model in the Waste Management Policy

Our proposed model has more and more future work when a production company produces electronic products and after the inspection process, almost all defective items are remanufactured but the remaining defective items whose qualities are not able to be recycled, reworked, and remanufactured, these types of items are called waste items which gives a negative impact on the total revenue of the system. Our model suggests that if the waste management costs are included in this situation, then the decision maker can save the profit from this waste product in the future.

8.5. Limitations of Our Proposed Model

Our proposed model has some limitations for obtaining more payoff, which are given below:

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The present proposed model is beneficial for the production of electronic items not for ordering policies.

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The variation in fuzzy input parameters should be according to the proposed study.

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The values of learning inputs should be according to the proposed model.

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The deviations of fuzzy demand, namely lower and upper, should be according to the proposed model.

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The financial plan and budget constraints should be allowed.