Fuzzy multi-objective optimization model to design a sustainable closed-loop manufacturing system

Introduction

It is an eminent fact that major environmental concerns have arisen because of global megatrends such as the exponential growth of population and pollution due to the disposal of untreated and unavoidable manufacturing waste, carbon dioxide $C{O}_2$ emissions, energy consumption, and depletion of natural resources. Manufacturing companies represent the backbone of contemporary industrialized civilization and have long been regarded as the fulcrum of the global economy. It has been indicated by Karuppiah, Sankaranarayanan & Lo (2024) that advancement in industrialization is adversely impacting the environment and human life on earth but it is yet uncertain how to evaluate their long-term viability. Sheng et al. (2023) identified that the manufacturing sector is a major contributor to huge energy consumption, waste generation, and greenhouse emissions, therefore researchers are required to propose/develop such kind of industrial model that may ensure enhanced resource efficiency, minimize waste, and reduce carbon emission. The waste generation and carbon emission rate in Fig. 1 (Kaza et al. (2018)) and Fig. 2 (IEA (2023)) indicate the scale of environmental degradation caused by these factors. Although new terms like environmentally conscious manufacturing, green manufacturing, and reverse manufacturing are thought to be connected to sustainability, there is a wide range of views and interpretations, due to which it has become difficult to find a definitive definition for sustainability in manufacturing firms (Sartal et al., 2020).

It is incumbent that industrial communities develop such modern enterprises which may ensure that enacting a sustainable system is an essential requirement (Anwar et al., 2023). Most definitions of sustainability include terms like green, ‘clean, keep, retain, stability, ecological balance, natural resources, and the environment. It can be difficult to define the term “sustainability,” usually it is understood as a way of nonlethal manufacturing products, and natural resources preservation, and frugal. In the meantime, there is still a disagreement over what constitutes and defines sustainability; promoting sustainable development has emerged as a significant worldwide goal. The phrase sustainable development (SD) is widely used to describe contemporary ideas of sustainability. SD has been regarded as a form of development that can gratify the needs of human beings devoid of jeopardizing the capability of future generations to gratify their desires. It is crucial to consider SD principles in every activity or choice made to achieve responsible and sustainable development. One of the most crucial topics for advancing the overall goal of SD is sustainable manufacturing. Designing for sustainability and sustainable development becomes the design of sustainable manufacturing firms. To accomplish sustainability, it is significant to appreciate the models and interrelated needs sustainably.

Over the last century, sustainability has been attracting both researchers and the industrial sector from almost all over the world (Bello-Pintado, Machuca & Danese, 2023). To start along a more sustainable road, several objectives and targets have been established. To encourage the concept of sustainability by confronting the three primary global concerns, the United Nations has defined seventeen sustainable development targets (SDGs) (see Table 1). The goals for sustainable development are interconnected, which means that achieving one requires addressing problems that are frequently linked with another. Just as the unadorned stressful situation on earth is caused by numerous interconnected global megatrends, so are the goals for sustainable development.

One of the important goals of SDGs is SDG9 which focuses on industry, innovation, and infrastructure. Here, it is important to mention that an important element of SD is sustainable industrialization. In developing countries, industrialization has produced several environmental impacts and sustainable industrialization is being promoted at a rapid pace; as it has been indicated by Chen et al. (2024) that due to it there is an increase in carbon emission in many countries. Regarding this, action has been taken on the defined principles to encourage fuel (energy) saving and the reduction in the emissions of $C{O}_2$ in many manufacturing industries. The expanded efforts (cost, energy, and emissions) of $C{O}_2$ at all levels and elements, from the process, product, and the linked system through the complete product life cycle, must be made to make a manufacturing firm more sustainable. It is necessary to define a sustainable manufacturing system to minimize environmental effects. Utilizing mathematical optimization approaches, the issues with sustainable industrialization are resolved. When solving problems with sustainable manufacturing, many goal functions must be taken into account at once. For instance, in the sugar sector, boosting sugar output while decreasing waste is a problem. Because of this, no one solution can be referred to as the best and a variety of trade-off options must be taken into account. These issues serve as illustrations of the class of optimization problems known as multi-objective optimization problems.

To handle the range of optimization approaches used in real-world circumstances, multi-objective optimization is crucial. For addressing several objectives and offering solutions that are more in line with system requirements, there are numerous specialized strategies. The construction of the non-inferior (Pareto optimum) solution is a general objective in multi-objective optimization due to the multi-criteria of such issues; a non-inferior solution is one in which an improvement in one objective necessitates the deterioration of another. Typically, several trade-off solutions (referred to as the Pareto set) are discovered using the stated concept of optimality. Some data used for analysis and other purposes in the actual world are questionable.

However, decision-makers must take this unpredictability into account while designing their networks. The creation of a multi-objective optimization helps to address uncertainty resulting from the impreciseness of data. Fuzzy multi-objective optimization is a method that relies only on the formulation of fuzzy information in terms of membership functions to address the optimality of the fuzziness model utilizing currently available multi-optimization tools and methodologies (Paksoy, Pehlivan & Özceylan, 2012).

Supply chain operations now inevitably influence several company sectors and international trade. Customers play a fundamental part in these processes, and markets work to meet their needs by taking lead time minimization, product quality improvement, quick delivery, and their prospective expectations into account. The lack of precious resources, the state of the economy, social and environmental effects, and the rise in customer requests aimed at the finest products, lead policymakers towards deliberation and reverse logistics networks (Soon et al., 2022; Kazancoglu et al., 2022). Reverse logistics advantages in this area let organizations best the most of their businesses through material recycling, return, and reuse in multi-level systems that help improve the environment (Tavana et al., 2022).

The modern closed-loop manufacturing system will unavoidably have an impact on several industrial sectors as well as global trade. The primary goal of the research is to minimize the cost, total energy consumption, and overall CO₂ emissions of a suggested closed-loop manufacturing system (CLMS) by choosing the overall quantity of existing services and significantly reducing the overall cost within multiple levels. Each place in this design seems to have a limited scope, and other places for each place are defined. The supply rates to every type of customer are stated for each period. Each warehouse has a certain capacity, and after each period, the inventory cost is taken into consideration for the items that are still supplied. Since the multi-objective issue has been taken into account, the goal of the research is to use optimization approaches to validate the model and produce the best outcomes. The closed-loop supply chain (CLSC) is thought to be a tactic for developing a sustainable supply network, as Roudbari, Fatemi Ghomi & Eicker (2024) stated that it can play two major roles; satisfying customers’ demands and effectively recapturing the value of the returned product. The following are the primary goals of the research in the manufacturing system:

• 1)

  Constructing a closed-loop, sustainable supply network for the manufacturing phase in the industrial sector.

• 2)

  Adopting a linear programming methodology to reduce the overall expenses of the manufacturing logistics chain.

• 3)

  To supply materials and transportation of returned goods for subsequent use, take into consideration forward and reverse logistics.

The methodology initially evaluates the various providers under sustainable standards. The network of closed-loop supply chains is assessed to get the best possible amount from the potential providers chosen based on sustainable standards to satisfy consumer requests. Supplier (SU) then ships the quantity $q_{SU.PL}^{Raw}$ through plant (PL) to retailer (RE), and then transport it to customer (CU). The product as waste material is collected by collection center (CC) by CU and transported to disassembly center (DC) to refurbishing center (RC) back on the road to PL using various modes of transportation. The next section of this study presents a comprehensive literature review along with the current study’s motivation, aim, and objectives. “Closed-Loop Manufacturing System Model Formulation” develops the thorough aspects of dilemma formulation, taking into consideration objective function and problem constraint. “Solution Methodology” presents the solution methodology followed by a case study, results, comparison of techniques, and conclusion.

Literature review

There have been several studies in supply chain systems, closed-loop supply chains, logistics system planning, and remanufacturing (Lo et al., 2021; Nunes, Causer & Ciolkosz, 2020; De & Giri, 2020). However, there are relatively few chronological writings on manufacturing supply chains. Initially, Lim et al. (2021) offers industrial sector approaches and created supply chain topics to draw scholars by providing an example of a specific supply chain network. Marić & Opazo-Basáez (2019) supports sustainable industrialization and ensures that providing an industrial supply chain for recycling products, results in more quality in the manufacturing industry. The planned measure was then given environmental and economic investigation in the same study. A detailed analysis of industrial practices and approaches for a design intended for sustainable actions is presented by Shekarian et al. (2022). They provided a new and enhanced classification of the work indicating more routes so it may be used to analyze how different firms reach sustainability.

The implementation of systems for closed-loop supply chain processes and reverse logistic (RL) has been studied in depth by a significantly as well as an increasing number of studies. RL is a mechanism in which products and other products are moved from their ultimate place to further destinations to be recycled, disposed of, or used in other operations (Fathollahi Fard & Hajaghaei-Keshteli, 2018; Cheraghalipour, Paydar & Hajiaghaei-Keshteli, 2017). Since RL promotes better growth and worth including both company owners and consumers, it has been adopted by the majority of emerging nations in various industries and enterprises (Hsu, Tan & Mohamad Zailani, 2016).

Keeping in view these advantages, previous research utilized reverse or closed-loop supply chain systems in their networks (Ritchie et al., 2000; Govindan, Soleimani & Kannan, 2015), and discussed closed-loop supply chain systems reversal logistics, too. As in research, they researched different aspects which include ecological factors, economic and legal aspects, and their positives from a supply chain and CLSC perspective, Özceylan, Paksoy & Bektaş (2014) suggesting a nonlinear model for constructing a closed-loop supply system in the accurate economic climate for balance separating and supply lines. The levels in the forward process include supplier, plant, retailer, and customer also the same number of levels are in the design of the reverse process which are the refurbishing center, waste center, disassembly center, and collecting center. To fulfill the need for product returns, certain writers, including Keyvanshokooh, Ryan & Kabir (2016), examined and improved the CLSC and presented models. The model used Benders’ approach with hybrid probabilistic and resilient programming to get the best solution to the issue. An increasing tendency towards networks of sustainability practices may be seen throughout the development of supply chain management. Rahemi et al. (2020), Li, Manier & Manier (2019), Balcombe et al. (2018) and Soleimani et al. (2017) included sustainability, optimization, and new distinct networks. They used multi-objective programming to construct a trustworthy and fuzzy CLSC that takes into account uncertainty. Suppliers, manufacturers, distributors, warehouses, return centers, and disposal facilities are all part of this supply chain. The maximization of overall revenue, taking sustainability principles, and enhancing customer response are examples of objective functions. To enhance the outcomes, a genetic algorithm (GA) was utilized. Devika, Jafarian & Nourbakhsh (2014) suggesting a linear model creates a closed-loop supply chain that is sustainable; to concurrently research the impacts of the environment, society, and economy in a specific context. The design contains six stages of collecting, recycled, reconstruction, refinement, waste, and supplementary consumers in the reverse direction, and four stages of supplies, manufacturing, distributing, and consumers in a forward manner. To cut carbon emissions, Talaei et al. (2016) created a closed-loop, environmentally friendly model of the supply chain. The suggested model tackles the position, in addition to optimal allocation problems, and has two objectives. The authors combined fuzzy sets and robust programming to deal with the uncertainty.

To solve the model, the epsilon constraint approach is employed. By creating a CLSC network and taking into account both optimizing profitability and social capacities by recycling items Pishvaee & Torabi (2010) and Panda, Modak & Cárdenas-Barrón (2017) attempted to maximize their suggested model by offering a bi-objective design framework about a closed-loop supply chain in an ambiguous setting. Although the secondary purpose seeks in minimizing the time needed to receive goods and services, the first goal seeks to decrease expenses. The model contains four layers of collecting, rebuilding, recycled, and third consumers in both forward and reverse directions, three phases of manufacturing, distribution, and consumers are also included. Soleimani & Kannan (2015) constructed a CLSC system on large systems. The suggested system is comprehensive, multi-level, multi-period, and multi-product. The model is solved using the GA, particle swarm optimization (PSO) technique, hybrids genetic algorithm particle swarm optimization (GAPSO), and the exact approach (CPLEX). Kaya & Urek (2016) created a CLSC network to place stocks and set prices in the best possible places. To find services and inventory, a nonlinear mixed integer programming approach is suggested in this study. The goal of the problem was to increase the suggested supply chain network’s overall profit.

While exploring the design of reversible logistics networks, taking into account multiple layers including warehouse, transit, logistic, manufacturer, distributor, and client, it was seen how each stage interacted with the supply chain (Fulconis & Philipp, 2018). Sadeghi, Mina & Bahrami (2020) used mixed integer linear programming (MILP) to create a reverse chain for industrial logistics. The levels of production, distribution, separation, and destruction were among those that were taken into account. To further clarify the problem, they developed a multi-period, multi-product model. Reverse supply networks’ environmental properties and their supply chain were analyzed and categorized by Islam et al. (2021). Govindan & Soleimani (2017) presented a review of reverse logistics and other earlier research in this field. Kazemi, Modak & Govindan (2019) developed evaluations of the work on reverse logistics and CLSC as well as a detailed analysis of content. For armed hardware including replacement parts, a closed-loop system was developed (Wang & Shi, 2019). Since it is essential to get to the military supplies, the target function of the minimal durations is used in this research’s design. To handle the suggested system, the Lion meta-heuristic method is employed. Torabi et al. (2016) created a reverse supply chain while taking the disrupting risk and operating limits into account. The suggested model is then solved using based on fuzzy logic programming, and even the goal function of minimizing cost is then optimized. Gholizadeh et al. (2022) added the creation of a renewable system to the list of priorities while utilizing reverse logistics to address both environmental and economic concerns. In this study, reproduction and recycling levels are used to reduce the manufacturing cost’s objectives functions. Zhang et al. (2021) proposes a problem with logistics management and choice in the production chain with an emphasis on the criteria for choosing the best RL providers. The best RL logistics provider for remanufacturing is suggested in this scheme, and the strategy is resolved using a multi-criteria decision-making (MCDM) approach that includes an analytical hierarchy process (AHP) also a technique for order of preference by similarity to the ideal solution (TOPSIS). Pourjavad & Mayorga (2019), Pourmehdi, Paydar & Asadi-Gangraj (2020), Gholizadeh & Fazlollahtabar (2020), Pahlevan, Hosseini & Goli (2021), Akbari-Kasgari et al. (2022), Amirian, Khalili & Mehrabian (2022) are some investigations on CLSC or RL in industrial manufacturing systems. Even if some earlier research created industrial manufacturing systems, there have been relatively few investigations on CLSC or RL for such items. Table 2 contains some studies related to sustainable industrialization.

The previous studies on CLSC and RL have mostly focused on developing new systems for selling rebuilt goods, recycling product returns, and converting waste into renewable energy. Although these efforts have expanded global attention, more comprehensive approaches toward closed and reverse supply chains, beyond just economic objectives, are necessary. In addition to addressing climate change, developing a holistic CLSC system presents wide-ranging opportunities for investors to maximize the value of products across their entire life cycle. Over the past two decades, significant developments in CLSC systems have simplified policymaking for addressing complex issues such as re-manufacturing used goods, meeting customer demand, and identifying marketing strategies for re-manufactured products. However, much of the existing research has focused on the forward supply chain and its role in meeting client demands, resulting in higher revenues and lower expenses for suggested systems. While such insights are valuable, they overlook critical aspects of closed-loop and backward logistics systems that are key to achieving a circular economy.

Despite the growing research on supply chain management, there is a noticeable gap in research addressing the full spectrum of CLSC systems. Specifically, the integration of circular economy principles, the development of multi-objective optimization models that consider both economic and environmental sustainability, and the need for stakeholder collaboration remain under-explored. Additionally, while most research has concentrated on the manufacturing industry, other sectors such as electronics, textiles, and consumer goods lack detailed studies on implementing efficient backward logistics systems. Additionally, the advent of digital transformation, AI, and IoT technologies presents new challenges and opportunities for enhancing the efficiency of CLSC networks. However, the role of these technologies in building resilient CLSC systems remains insufficiently explored in the current literature. Addressing these gaps could provide a more comprehensive framework for the implementation of CLSC, promoting sustainability and long-term profitability in various industries. Therefore, the following represent the key goals and suggestions to address the study gaps throughout the research:

• 1)

  Literature indicated that there are plenty of research studies on aforementioned topic in industrial supply chains (ISC), therefore, this study begins by examining the existing research on supply chain system architecture for the industrial sector.

• 2)

  Here on basis of its creative system, the present investigation has created revolutionary modeling. Although the majority of earlier studies recommended forward supply chain systems.

• 3)

  The suggested framework not only includes a tire backward route to preparing the gathered produced goods again for the secondary market, but also minimizes the total costs, overall energy, and CO₂ emissions in both forward and backward reverse directions.

• 4)

  Because of its distinctive system, the research presents a new mixed integer linear programming (MILP) approach from a mathematical standpoint.

• 5)

  The objective will be achieved using a four-valued refined neutrosophic optimization approach.

Closed-loop manufacturing system model formulation

The imperative for returned products from the manufacturing process is one of the emerging concerns in CLMS design for the industrial system. These materials can not only be obtained as raw materials in a range of industries and enterprises but they can also be recycled and used again. In this study, a CLMS (see Fig. 3) is developed to minimize total costs, energy use, and $C{O}_2$ emissions in the manufacturing system’s supply chain while taking into account returned goods on a reverse flow. Figure 3 encapsulates a schematic design of a CLMS. It moves ahead with production from the facility directed through the retailer to customers before going backward with the collection, disassembly, and refurbishment centers. During these phases, the recycling process is carried out to change the commodities into new ones. Our suggested model incorporates tactical decisions, such as production, inventory planning, capacity management, and transportation system, as well as strategic considerations, such as the best choice of facility sites made up of disassembly centers, recycling facilities, and collection centers. The locations of the factory and customers are taken for granted.

The following are the fundamental presumptions of the model:

• 1)

  The prospective sites of each center (Plant, Retailer, Collection center, Disassembly center, waste center, and Refurbishing center) are known and each has a limited capacity.

• 2)

  The predicted customer demand rate for each time.

• 3)

  There cannot be a shortage.

• 4)

  The inventory cost for the remaining products is calculated at the end of each period.

• 5)

  All costs are taken into account after each session.

Solution methodology

The developed multi-objective linear programming model is solved by using a four-valued refined neutrosophic optimization technique (FVRNOT) proposed by Freen et al. (2020). A four-valued refined neutrosophic set is one of several types of neutrosophic sets in which indeterminacy is divided into $\delta =$ uncertainty and $\eta =$ contradiction, where $\eta =\mu \wedge \nu$. The values of $\mu ,\delta ,\eta and\nu$ fall into the range $[0,1],$ and $0\le \mu +\delta +\eta +\nu \le 4.$ Mathematical expression of the set in given in Eq. (33).

(33) $$Z^{\sim RN}=\{(s,{\mu }_{Z^{\sim RN}}(s),{\delta }_{Z^{\sim RN}}(s),{\eta }_{Z^{\sim RN}}(s),{\nu }_{Z^{\sim RN}}(s)):s\in S\}.$$

The objective functions are transformed into neutrosophic fuzzy constraints when optimization is executed using FVRNOT. The uncertainty is effectively handled by this four-valued refined neutrosophic model that has been established and enhanced. The multi-objective four-valued refined neutrosophic linear programming model that is being described can handle uncertain data, which avoids unsustainable modeling. The computational algorithm of the stated approach includes the following steps:

1. One objective function from a set of N objectives should be used to solve the first objective function. Under the stated constraints, the values of the decision variables and objective functions will be determined.

2. Now, through using decision variables from step 1, calculate the values of the undetermined objective, i.e., $\left(N-1\right).$

3. Steps 1 and 2 must be repeated for the rest $\left(N-1\right)$ objective functions to get the pay-off matrix 34.

(34) $$\left[\begin{array}{cccc}{f_1}^{\ast }(s^1)& f_2(s^1)& \dots & f_n(s^1)\\ f_1(s^2)& {f_2}^{\ast }(s^2)& \dots & f_n(s^2)\\ ⋮& ⋮& ⋮& ⋮\\ f_1(s^n)& f_2(s^n)& \dots & {f_n}^{\ast }(s^n)\end{array}\right]$$

4. For every objective function $f_p$ determine the lower bound $L_p$ and the upper bound $U_p$ such that

$$L_p=\underset{r=1}{\overset{n}{max}}\{f_p(s^r)\}\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{U}}_{\mathrm{p}}=\underset{\mathrm{r}=1}{\overset{\mathrm{n}}{min}}\{{\mathrm{f}}_{\mathrm{p}}({\mathrm{s}}^{\mathrm{r}})\}.$$

5. The lower and upper bounds for truth membership function $\mu$, falsity membership function $\nu$, uncertainty membership function $\delta$ and contradiction membership function $\eta$ of the objective function $f_p$ are computed in Eqs. (35)–(38).

(35) $$L_p^{\mu }=L_p;U_p^{\mu }=U_p$$

(36) $$L_p^{\nu }=L_p^{\mu }+t_1(U_p^{\mu }-L_p^{\mu });U_p^{\nu }=U_p^{\mu }$$

(37) $$L_p^{\delta }=L_p^{\mu }+t_2(U_p^{\mu }-L_p^{\mu });U_p^{\delta }=U_p^{\mu }$$

(38) $$L_p^{\eta }=L_p^{\mu }\wedge L_p^{\nu };U_p^{\eta }=U_p^{\mu }\wedge U_p^{\nu }+t_3(U_p^{\mu }\wedge U_p^{\nu }-L_p^{\mu }\wedge L_p^{\nu })$$where $t_1,t_2,t_3\in (0,1)$.

6. Define the truth, uncertain, false, and contradiction membership functions as in Eqs. (39)–(42)

(39) $${\mu }_p(f_p(s))=\{\begin{array}{cc}1\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\le L_p^{\mu },\hfill \\ \frac{U_p^{\mu }-f_p(s)}{U_p^{\mu }-L_p^{\mu }}\hfill & \mathrm{i}\mathrm{f}{L}_p^{\mu }\le f_p(s)\le U_p^{\mu },\hfill \\ 0\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\ge U_p^{\mu },\hfill \end{array}$$

(40) $${\delta }_p(f_p(s))=\{\begin{array}{cc}1\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\le L_p^{\delta },\hfill \\ \frac{U_p^{\delta }-f_p(s)}{U_p^{\delta }-L_p^{\delta }}\hfill & \mathrm{i}\mathrm{f}{L}_p^{\delta }\le f_p(s)\le U_p^{\delta },\hfill \\ 0\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\ge U_p^{\delta },\hfill \end{array}$$

(41) $${\nu }_p(f_p(s))=\{\begin{array}{cc}0\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\le L_p^{\nu },\hfill \\ \frac{f_p(s)-L_p^{\nu }}{U_p^{\nu }-L_p^{\nu }}\hfill & \mathrm{i}\mathrm{f}{L}_p^{\nu }\le f_p(s)\le U_p^{\nu },\hfill \\ 1\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\ge U_p^{\nu },\hfill \end{array}$$

(42) $${\eta }_p(f_p(s))=\{\begin{array}{cc}1\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\le L_p^{\eta },\hfill \\ \frac{U_p^{\eta }-f_p(s)}{U_p^{\eta }-L_p^{\eta }}\hfill & \mathrm{i}\mathrm{f}{L}_p^{\eta }\le f_p(s)\le U_p^{\eta },\hfill \\ 0\hfill & \mathrm{i}\mathrm{f}{f}_p(s)\ge U_p^{\eta },\hfill \end{array}$$

7. Therefore, after finding the membership and non-membership functions, an equivalent crisp optimization problem is formed in Equation (43).

(43) $$\mathrm{m}\mathrm{a}\mathrm{x}(\varphi -\sigma +\psi +\theta ),$$such that

(44) $${\mu }_p(f_p(s))\ge \varphi ,{\nu }_p(f_p(s))\le \sigma ,{\delta }_p(f_p(s))\ge \psi ,{\eta }_p(f_p(s))\ge \theta$$with

(45) $$0\le \varphi +\sigma +\psi +\theta \le 4\mathrm{a}\mathrm{n}\mathrm{d}\varphi \ge \sigma ,\psi ,\theta$$and subject to the original constraints.