1 Introduction

In the highly competitive environment of the industry and the rapid growth of technology, the use of Additive Manufacturing (AM) can provide many competitive advantages as one of the critical factors in Industry 4.0 (I4.0). Some of these advantages include enhanced production speed, ability to generate complicated structural models, reduced work in process (WIP), reduced waste, no need for mold, no need to store materials, shortened supply chain (make-to-order production), increased flexibility to adapt to customer needs (customized demand) and reduce uncertainty (Ford & Despeisse, 2016; Khorram Niaki & Nonino, 2017; Kleer & Piller, 2019; Majeed et al., 2021). Therefore, AM not only provides lean manufacturing frameworks to the companies but also significantly impacts the supply chain (Chan et al., 2018).

AM has received considerable critical attention. More than 200 companies in various eras applied 3D printing technologies, such as Lima Corporate (medical), Launcher, ESA’s ArianeGroup (space), BMW, Audi, Volkswagen (automotive), Norsk Titanium, Boeing, Rolls Royce, and Boom Supersonic (aerospace), Specialized and Fizik (sports), UK trains and Mobility goes Additive (railway), US Army, the US Navy, the US Air Force, the Russian Army, and the Spanish Navy (defense), Wilhelmsen, Thyssenkrupp and Navantia (maritime). Hence, AM holds significant promise for application across diverse industries, including both military and humanitarian missions (den Boer et al., 2020), foundry industries (Ngo et al., 2018), buildings (Li et al., 2020), electrochemical industry (Hashemi et al., 2020), healthcare (Bose et al., 2018; Ghomi et al., 2021), fashion (McCormick et al., 2020), aerospace (Fasel et al., 2020), education (Assante et al., 2020), jewelry (Martinelli, 2018).

According to the Senvol database, in 2020, 2246 different materials could be used in AM, and this number is growing. Among all types of materials, polymers have the largest share of use (cf. Table 1). According to the Additive Manufacturing Landscape 2020 report, over 80 percent of companies use polymers in 3D printing (Akinsowon & Nahirna, 2020). Therefore, the supply process and cost of raw materials must be investigated. For example, the US military uses plastic waste and PET bottles as raw materials to diminish cost and environmental impact (Fey, 2017).

Table 1 Different types of materials used for AM (Wohlers et al., 2020)
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Global production rates of plastic waste are rising sharply. In 2018, the total generated plastic waste in the US was 35.68 million tons. Figure 1 represents that in managing collected plastic waste in the US in 2018, among three common waste disposal methods, recycling, combusting with energy recovery, and landfilling, landfilled waste has the largest share, while we face the problem of limited land (U.S. Environmental Protection Agency, 2020).

Fig. 1
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Waste management in the US in 2018

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The global plastic waste crisis is dire, with 400 million metric tons produced annually. 10 million tons end up in the oceans. Figure 2 shows the magnitude of global plastic production, while Fig. 3 illustrates the extent of plastic waste generation on a global scale. China leads in plastic production, but the U.S. tops in annual plastic waste at 42 million tons. Shockingly, 50 billion plastic water bottles are sold in the U.S. each year, and only 18% of plastic is recycled on average. India has the highest plastic recycling rate at 60%. Our oceans currently hold 5.25 trillion pieces of plastic. A 2016 assessment ranked the U.S., India, and China as the top three global plastic waste producers. Per capita, the U.S. remains a significant contributor, although some studies rate China as the largest overall producer. Other countries in the top ten include Brazil, Indonesia, Russia, Germany, and the UK. Individual lifestyle changes worldwide can help combat this pollution crisis. In 2018, the U.S. generated 130 kg of plastic waste per person, showing a slowing increase in the past three years. The challenge is to shift from slowing waste growth to reducing it.

Fig. 2
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Plastic production (https://theroundup.org/)

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Fig. 3
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Plastic waste generation (https://theroundup.org/)

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Figure 4 represents the plastic recycling rate. India leads the world with an impressive 60% plastic recycling rate, demonstrating a significant commitment to sustainable waste management. Following closely behind are South Africa, the Netherlands, South Korea, Norway, and Spain, each making commendable efforts in plastic recycling. South Korea, in particular, is working towards raising its rate to 70% by 2030, further exemplifying dedication to the cause. In stark contrast, the global average plastic recycling rate lags significantly at a mere 18%, underscoring the importance of widespread improvement in recycling practices worldwide.

Fig. 4
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Plastic recycling rate (https://theroundup.org/)

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Some of these materials, such as polymers, can be recycled or reused if collected and delivered to reclaimers. As shown in Fig. 5, the collected PET bottles in the US waste is less than half of the total generated plastic waste, and the rest is released into the environment and ocean (Aslani et al., 2021). Furthermore, These material is made of fossil fuels that emit greenhouse gases when exposed to sunlight, so they have many destructive effects on ocean ecosystems, marine species, and the environment. By 2030, the emitted CO2 from these materials will be approximately over 296 five-hundred-megawatt coal plants (1.34 gigatons per year). In this regard, It is vital to improve ongoing waste management methods (Hamilton et al., 2019).

Fig. 5
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Generated and collected Pet bottles in the US (1995–2017)

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The key elements, emphasizing the urgency and global impact of the challenges are given as follows:

  1. I.

    Rising Global Adoption of AM

The global adoption of AM has witnessed an unprecedented surge, with more than 200 companies across diverse industries integrating 3D printing technologies into their operations. In 2020 alone, 1095 different polymers were utilized in AM processes, accounting for a substantial portion of the total materials used (Wohlers et al., 2020). This widespread acceptance reflects the transformative impact of AM on manufacturing processes.

  1. II.

    Environmental Impact of Plastic Waste

The plastic waste crisis has reached alarming proportions, with an annual production of 400 million metric tons globally. This crisis is vividly illustrated in Fig. 2, showcasing the magnitude of plastic production. Despite the staggering numbers, only 18% of plastic is recycled on average, contributing to environmental pollution and degradation. The severity of the issue is further highlighted by the fact that oceans currently hold 5.25 trillion pieces of plastic (see Fig. 3).

  1. III.

    Recycling Rates and Sustainability Efforts

Figure 4 presents a snapshot of global plastic recycling rates, emphasizing India’s remarkable leadership with a 60% recycling rate. Countries such as South Africa, the Netherlands, South Korea, Norway, and Spain are making commendable efforts to enhance plastic recycling. However, the global average recycling rate remains low at 18%, underscoring the need for substantial improvements in recycling practices worldwide.

  1. IV.

    PET Bottle Waste and Environmental Impact

The environmental impact of PET bottle waste is substantial, with Fig. 5 revealing that collected PET bottles in the US represent less than half of the total generated plastic waste. This incomplete collection leads to adverse effects on ocean ecosystems, marine species, and the environment. The projected CO2 emissions from these materials by 2030 further highlight the urgency to improve waste management practices.

There has been a growing concentration on configuring a Closed-Loop Supply Chain (CLSC) in recent years. Using reverse flows, recyclable, repairable, or reusable materials can return to the value cycle at the end of their life. In addition, designing a CLSC provides sustainable advantages and creates a circular environment (Ali et al., 2023; Sazvar et al., 2022). This network can be designed for plastic waste such as PET bottles or 3D printing waste. Consequently, it reduces the environmental impact and creates job opportunities and economic benefits. According to the American Chemistry Council report in 2019, 600 advanced recycling facilities provide 38,500 jobs (American Chemistry Council Economics and Statistics, 2019). Besides these social benefits, some polymeric materials, such as PET bottles, can be used instead of raw polymeric materials. Due to the high price of these polymers, recycled PET bottles could provide economic benefits (Mikula et al., 2021). In this regard, designing an appropriate CLSC positively impacts waste management. Ottosson and Oweini (2023) contributed to the circular economy discourse by proposing a CLSC for reusable plastic products. Aligned with EU directives on phasing out single-use plastics, their study provided a circular business model (CBM) involving collaboration, reusable product design, and implementation calculations. This research finally offered practical insights for businesses seeking sustainable alternatives to single-use plastics. Chowdary and Rayside (2024) explored circular economy strategies in beverage manufacturing, using discrete event simulation modeling. Their research pinpointed recycling PET bottles into 3D printing filament as the most economically efficient method, making a meaningful contribution to reducing plastic waste and fostering environmental sustainability.

The lack of an efficient supply chain for PET bottles and waste from 3D printing centers (waste of post-processing) and customers (prototyping, R and D processes, education, etc.) led us to develop a circular CLSC which is used to collect, sell and recycle these wastes as an important part of the circular economy. Accordingly, a novel optimization model is built up to formulate the problem which is then evaluated in terms of applicability and validity using metaheuristic solution algorithms.

The rest of this work is structured in the following way. The literature review of related works in this area is provided in Sect. 2. Moreover, Sect. 3, explains the problem statement and the proposed mathematical model. Section 4 is concerned with the optimization methods used for this study. Furthermore, to evaluate proposed solution approaches, computational results are analyzed in Sect. 5, and at the end, Sect. 6 summarizes the research and offers directions for further study.

2 Literature survey

In this section, the literature concerning CLSCs and sustainable supply chains is evaluated, especially those examining the possibility of converting PET bottles and plastic waste to 3D printing filament.

2.1 CLSC network design

There is a reverse follow in the CLSC for recycling, disassembling, reusing, and integrating environmental remarks into the conventional design of supply chains (Ghayebloo et al., 2015). Therefore, it can be applied to many products, such as gold, to collect and use again, which was investigated by Zohal and Soleimani (2016). They designed a CLSC and paid attention to CO2 emissions as an environmental impact. Shokouhyar and Aalirezaei (2017) considered social, environmental, and economic effects. They proposed a reverse logistic network for waste electrical material. Therefore, they offered a MILP to control the influence of hazardous materials on both human health and environment. In their work, Metaheuristics are employed as a solution approach. In the same way, Rentizelas et al. (2018) introduced a supply chain network that minimizes the total cost of recycling plastic waste. Their proposed model was validated with a confirmed case of agricultural plastic waste in Scotland.

As mentioned, a substantial and expanding body of literature has explored recycling networks. Arampantzi and Minis (2017) investigated a sustainable supply chain design problem to minimize the cost (investment, operational, and emissions costs) and the environmental effects (greenhouse gas emissions and waste generation) along with the public consequences (employment situations and social community progress). Overall, these studies highlight the need for designing an appropriate supply chain to add social and environmental aspects to traditional models. Likewise, Sahebjamnia et al. (2018) suggested a supply chain for plastic waste but with different aspects and cases. They consider the sustainable part and design a sustainable tire CLSC. To solve their offered model they applied metaheuristics and hybrid metaheuristics models. According to Paydar and Olfati (2018), a reverse logistic (RL) can be designed to decrease recycling PET bottles costs. The four-level RL is solved by two metaheuristics and validated considering a real case.

Similarly, Accorsi et al. (2020) configured a CLSC network paying attention to reusable containers in the food industry and proposed a MILP model. Rezaei and Maihami (2020) explored a CLSC involving a manufacturer, retailer/remanufacturer, and a government-run collection center operating in primary and secondary markets. They focused on sustainability and emissions reduction in manufacturing, remanufacturing, and delivery. Game theory was used to address low-carbon customer preferences. Numerical analysis showed that reducing emissions positively affects profitability, particularly in remanufacturing. There are also many other similar studies in the literature addressing the CLSC design using different types of models and solution methods such as Abdolazimi et al. (2022), Sajadiyan et al. (2022), and Rajabi-Kafshgar et al. (2023).

Akbari-Kasgari et al. (2022) addressed the growing copper demand due to industrialization, emphasizing the need for sustainable copper supply chains. They suggested a unique network design that considers resilience, especially in earthquake-prone areas. Their research presented two models, one with backup suppliers and one without, aiming to optimize profit, minimize environmental impacts, and enhance social aspects. The results revealed that the model with backup suppliers improves supply chain responsiveness and economic and social performance but lags in environmental sustainability due to the environmental impact of backup suppliers. Momenitabar et al. (2022) investigated sustainable supply chains and offered a novel approach considering lateral transshipment along with backup suppliers in configuring a sustainable CLSC network. Their model aimed to minimize shortages, optimize costs, and enhance environmental performance while raising job opportunities. They tackled demand uncertainty and employed fuzzy robust optimization for efficient decision-making.

Shahidzadeh and Shokouhyar (2023) examined the role of reverse logistics in promoting the circular economy within supply chains, particularly focusing on consumer behavior’s impact on sustainability. It expanded the concept of sustainability to include consumers, profitability, environmental concerns, and employee well-being. The research introduces an extended sustainability model and employs a unique linguistic interval-value hesitant fuzzy decision-making trial and evaluation laboratory (FDEMATEL) approach to examine the relationship between reverse logistics performance and sustainability.

Given all that has been mentioned so far, one gap is the lack of investigation at different levels after the treatment process. In these levels, industries with the potential of being reclaimers of these materials could be studied (see Tables 2 and 3).

Table 2 Abbreviations
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Table 3 Comparison between the most relevant SC and CLSC design studies and the current research
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2.2 Use of recycled PET bottles and plastic waste in 3D printing technology

In the last decades, 3D printing technology has attracted increasing attention, which is one of the critical factors in I4.0. As mentioned in the previous section, AM can benefit enterprises. Since most of the used materials in this technology are polymers, recycled plastic waste can be used, and consequently, costs and environmental impact will be reduced. Therefore, a number of research works have been performed to check the conditions of the required recycling process. These works are summarized in Table 4.

Table 4 Studies related to the feasibility of recycling PET plastic into filament
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Lehrer and Scanlon (2017) examined the viability of extruding recyclable plastic into filament. to provide a sustainable and cost-effective method to satisfy the demand for 3D printing filament. They determined viscosity and melting temperature as vital influential factors. Since Polyethylene Terephthalate (PET) plastics have a high melting temperature, they do some modifications to access higher temperature extrusion. Moreover, the drying process influences shredded PET bottles. Moreover, viscosity is the most influential factor in the possibility of extruding filament. This study used a mix of cracked PET bottles with PETG pellets to improve viscosity. Exconde et al. (2019) stated that reusing and recycling plastic waste to produce filament could be an alternative to conserve energy and sustain the environment. Hence, they utilized a Multi-Criteria Decision-Making (MCDM) approach for materials selection. Recycled post-consumer plastics and virgin polymer resins for consumption in 3D printer filaments as possible choices.

From a logistics viewpoint, Santander et al. (2020) evaluated the economic and ecological feasibility of the distributed plastic recycling method. A Mixed-Integer Linear Programming (MILP) model was applied to assess a local CLSC network. The proposed model was elucidated through a case study involving a university aiming to execute a distributed recycling demonstrator. This initiative focused on recovering 3D printing wastes from secondary schools in France. According to the reviewed articles, there is a lack of an integrated supply chain that combines reverse logistics for collecting PET bottles and a CLSC of a 3D printing network.

PET polyethylene terephthalate, PS polystyrene, P.P. Polypropylene, PLA polylactic acid, FPF fused particle fabrication, FDM fused deposition modeling, ABS acrylonitrile butadiene styrene, rPET recycled polyethylene terephthalate

2.3 Research gap

This research paper addresses the critical gap in the current body of literature by proposing a comprehensive and integrated MILP model to configure a circular CLSC network that optimally incorporates the recycling of plastic bottle waste and waste from 3D printing processes as 3D printing filament within an I4.0 context. This innovative model is designed to minimize overall costs while considering environmental, social, and economic objectives, reflecting the tenets of sustainable development and circular economy. Leveraging three prominent metaheuristics—SA, GA, and PSO—fine-tuned via the Taguchi design method, the study aims to provide practical solutions for configuring a CLSC that seamlessly integrates 3D printing technology with recycled materials. The gap addressed lies in the holistic approach to CLSC design within a circular economy, emphasizing both materials and industries while achieving optimal environmental, social, and economic outcomes.

3 Problem definition

Here, a 3D printing filaments CLSC network is configured in a circular environment. There has been an accumulative interest in employing 3D printing technologies in recent years that provide competitive advantages, such as time-saving, customizing, and satisfactory quality. Unfortunately, the high price of filament produced by raw material is an important issue and might affect this product’s consumption. However, applying recycled filament or using recycled material to make filament can be employed to solve this problem. One of these materials can be PET bottles. If we use recycled materials, we convert them into valuable substances and reduce environmental impacts. Therefore, ecological problems and total costs can be efficiently diminished by location-allocation of recycling and designing a CLSC network, collection, and treatment hubs. In the following, we will describe the proposed CLSC.

The suggested network is presented in Fig. 6. This network is a combination of two sub-network and includes eight echelons: 3D printing center (DPC), treatment center (TC), recycling center (RC), filament customer (FC), processing center (PC), collection center (CC), markets (recycling companies), and end-user. In the first network, filaments are transported from recycling centers to DPCs. In addition, the waste of the printing process can be transformed into TCs. Next, 3D-printed objects are the products of DPCs that are shipped to customers. Afterward, in a reverse flow, a fraction of received products by 3D-printed customers are sent to treatment centers. PET bottles collected from end-users in another network are shipped to CCs. Then, after separating extra material from PET bottles, they are sent to PCs. In PCs, Pet bottles are sorted and pressed into the bales. After that, bales are transformed into reclaimers (i.e., markets and TCs).

Fig. 6
figure 6

Proposed 3D printing CLSC network

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On the other hand, transformed bales to TCs connect two sub-networks. In TCs, all received material from PCs, DPCs, and FCs is crushed into flakes. After the treatment (crushing, washing, and drying), it can be sent to RCs. In these RCs, chips are combined with other raw materials to enhance filament quality. Then, they are converted to 3D printing filament.

Next, we introduce the assumptions underlying the offered model:

  1. I.

    Capacity of all centers is limited,

  2. II.

    Demand of customers must be satisfied,

  3. III.

    Potential location of each echelon is predefined,

  4. IV.

    Production capacity and customers’ needs are determined,

  5. V.

    Number of facilities and their potential sites are specified,

  6. VI.

    Amount of 3D printed products sent to the treatment centers is a fraction of the received products,

  7. VII.

    Color of all PET bottles is considered the same. In other words, color does not affect the demand for bales.

3.1 Model

The suggested mathematical model is developed based on Pishvaee et al. (2010)‏‏. Table 5 represents the notations building up the model.

Table 5 Notations used in the suggested mathematical model
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3.1.1 Objective functions

$${\text{minimize}} \,Z={f}_{1}+{f}_{2}+{f}_{3}$$
(1)
$${f}_{1}=\sum_{i}{Fu}_{i}{U}_{i}+\sum_{j}{Fy}_{j}{Y}_{j}+\sum_{k}{Fv}_{j}{V}_{i}+\sum_{t}{Fx}_{j}{X}_{i}+\sum_{l}{Fg}_{j}{G}_{i}$$
$${f}_{2}=\sum_{k}\sum_{i}{Cf}_{ki} {F}_{ki}+\sum_{i}\sum_{n}{Cp}_{in}{P}_{in}+\sum_{n}\sum_{j}{Cd}_{nj} {D}_{nj}+\sum_{i}\sum_{j}{Cw}_{ij} {W}_{ij}$$
$$+\sum_{j}\sum_{k}{Cs}_{jk} {S}_{jk}+\sum_{b}\sum_{t}\sum_{j}{Cr}_{btj} {R}_{btj}+\sum_{t}\sum_{m}{Cq}_{tm} {Q}_{tm}+\sum_{l}\sum_{t}{Cz}_{lt} {Z}_{lt}$$
$$+\sum_{e}\sum_{l}{Cpb}_{el} {Pb}_{el}$$
$${f}_{3}=\sum_{i}{Cu}_{i}\left(\sum_{n}{P}_{in}+\sum_{j}{W}_{ij}\right)+\sum_{k}\sum_{j}{Cy}_{j}{S}_{jk}+\sum_{i}{\sum_{k}{Cv}_{k}F}_{ki}+\sum_{t}\sum_{l}{Cg}_{l}{Z}_{lt}$$
$$+\sum_{t}{Cx}_{t}\left(\sum_{j}{R}_{tj}+\sum_{m}{Q}_{tm}\right)$$

The object function minimizes the total cost, including fixed opening costs (\({f}_{1}\)), production costs (\({f}_{2}\)), and transportation costs (\({f}_{3}\)).

3.1.2 Model constraints

The main constrainsts are the model are listed as follows:

$$\sum_{k}{F}_{ki}\le Cp{u}_{i}\times {U}_{i} \quad\forall i\in I,$$
(2)
$$\sum_{b}\sum_{t}{R}_{tj}\le {\beta }_{j} \times {Cpy}_{j}\times {Y}_{j}\quad \forall j\in J,$$
(3)
$$\sum_{n}{D}_{nj}+\sum_{i}{W}_{ij}\le \left(1-{\beta }_{j}\right) \times {Cpy}_{j}\times {Y}_{j}\quad \forall j\in J,$$
(4)
$$\sum_{j}{S}_{jk}\le {\frac{1}{{\rho }_{k}}\times Cpv}_{k}\times {V}_{k}\quad \forall k\in K,$$
(5)
$$\sum_{l}{Z}_{lt}\le {Cpx}_{t}\times {X}_{t}\quad \forall t\in T,$$
(6)
$$\sum_{e}{Pb}_{el}\le {Cpg}_{l}\times {G}_{l}\quad \forall l\in L,$$
(7)
$$\sum_{l}{Pb}_{el}\le {Tp}_{e} \quad\forall e\in E.$$
(8)

Constraints (2)–(7) consider the finite capacity of each center. Hence, these constraints guarantee that the delivered product quantities to \({DPC}_{i}\), \({TC}_{j}\), \({RC}_{k}\), \({PC}_{t}\) and \({CC}_{l}\) are limited by the capacity of each center, respectively. Furthermore, Constraint (8) ensures that the quantity of PET bottles shipped to \({CC}_{l}\) does not violate the capacity of end-user \(e\).

$$\sum_{i}{F}_{ki}\le {\rho }_{k}\times \sum_{j}{S}_{jk}\quad \forall k\in K,$$
(9)
$$\sum_{n}{P}_{in}\times \frac{1}{(1-{\lambda }_{i})}\le \sum_{i}{F}_{ki}\quad \forall i\in I,$$
(10)
$$\mathop \sum \limits_{k} S_{jk} \le \mathop \sum \limits_{n} D_{nj} + \mathop \sum \limits_{i} W_{ij} + \mathop \sum \limits_{t} R_{tj} \quad\forall j \in J,$$
(11)
$$\sum_{j}{R}_{tj}+\sum_{m}{Q}_{tm}\le \sum_{l}{Z}_{lt}\quad \forall t\in T,$$
(12)
$$\sum_{t}{Z}_{lt}\le {(1-\gamma }_{l})\times \sum_{e}{Pb}_{el}\quad \forall l\in L,$$
(13)
$$\sum_{j}{D}_{nj}\le {\alpha }_{n}\times \sum_{i}{P}_{in} \quad\forall n\in N,$$
(14)
$$\sum_{j}{W}_{ij}\le \frac{{\lambda }_{i}}{\left(1-{\lambda }_{i}\right)}\times \sum_{n}{P}_{in}\quad \forall i\in I.$$
(15)

The number of sent products in a center should be less or equal to the received product quantity. Thus, Constraint (9) ensures that the amount of filament transported from \({RC}_{k}\) should be fewer than or equal to the material quantity which was received from \({TC}_{j}\). Like Constraint (9), Constraints (10)–(13) control the flows in \({DPC}_{i}\), \({TC}_{j}\), \({RC}_{k}\), \({PC}_{t}\) and \({CC}_{l}\). Moreover, Constraints (14)–(15) show the amount of waste transported from \({DPC}_{i}\) and \({FC}_{n}\) to \({TC}_{j}\).

$$\sum_{t}{Q}_{tm}\ge {DM}_{m} \forall b\in B,\quad\forall m\in M,$$
(16)
$$\sum_{i}{P}_{in}\ge {DC}_{n} \quad\forall n\in N.$$
(17)

Constraints (16)–(17) satisfy the demand of markets and 3D printed customers, respectively.

$$\begin{gathered} U_{i} , Y_{j} , V_{k} , X_{t} , G_{l} \in \left\{ {0,1} \right\}\quad \forall i, j, k, t, l, \hfill \\ F_{ki} , P_{in} , D_{nj} , W_{ij} , S_{jk} ,R_{btj} , Q_{btm} , Z_{lt} , Pb_{el} \ge 0\quad \forall i, j, k, t, l,n,b,t,e. \hfill \\ \end{gathered}$$
(18)

Constraint (18) presents positive and binary variables.

4 Solution algorithms

This section presents the encoding and decoding procedure and explains the metaheuristics used to treat the problem. The number of potential places (binary variables), especially in large-scale location problems, implies the complexity of the problem. It is known that the supply chain network design problems’ complexity is NP-hard (Seyedi & Maleki-Daronkolaei, 2013). Despite the reliability of the exact methods, these methods are not very efficient in large-scale problems (Abdi et al., 2021). In that respect, three distinguished metaheuristics (i.e., PSO, GA, and SA) are employed to seek a satisfactory solution. In the following, the applied solution approaches and representation approaches are detailed.

4.1 Encoding and decoding

Various methods have been proposed for encoding and decoding the solutions. In the present research, the priority-based encoding approach (Gen et al., 2006; Michalewicz et al., 1991) is employed. A small-scale problem is used to explain that the constraints are met. The number of 3D printing centers, treatment centers, recycling centers, processing centers, filament customers, markets, and end-users are assumed as 3, 2, 1, 1, 1, 1, and 1, respectively. Figure 7 shows the proposed chromosome. It is divided into seven segments. First, random numbers between 0 and 1 are randomly generated to make the chromosome. Then, for each segment, the random numbers are sorted based on priority to achieve the allocation sequence. For instance, consider Segment 3. According to the priority in Fig. 8, Constraints (3), (12), and (16) can be satisfied in the procedures of allocation. The further constraints procedures are the same. The applied algorithms are explained in the following.

Fig. 7
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Schematic design of the proposed chromosome

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Fig. 8
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Random key and priority-based chromosome of Segment 1

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4.2 Particle swarm optimization

Kennedy and Eberhart (1995) introduced PSO algorithm, which is a population-based algorithm inspired by the social behavior of birds, bees, or fishes (Barzinpour et al., 2013). Several versions have been also developed to improve the efficiency of PSO. For instance, weighted PSO (WPSO) (Dhivya & Meenakshi, 2015), adaptive PSO (APSO) (Dashora & Awwal, 2016), Levy Flight PSO (LFPSO) (Gupta et al., 2016), multi-vector PSO (MVPSO) (Fakhouri et al., 2020), etc. The first population of particles is placed randomly, and the objective function is evaluated. Then, each particle moves to a new position on the basis of its own best position, global best-known position, and velocity. Afterward, the history of the best location and global location is updated, and new movements are conducted. This procedure continues till the swarm is likely to attain a satisfactory best-known solution (Poli et al., 2007). The pseudo-code of the suggested PSO is displayed in Fig. 9.

Fig. 9
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Proposed pseudo-code of the suggested PSO

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4.3 Genetic algorithm

GA was first introduced by (Holland, 1992). Various engineering problems applied this population-based technique to solve their problems (Gholizadeh & Fazlollahtabar, 2020; Midaoui et al., 2021; Seyedi et al., 2022a, 2022b).

In this method, the algorithm starts with an initial randomly generated population. Next, the fitness value is evaluated for each individual. In each iteration, new solutions are developed by biologically inspired operators (crossover and mutation). The selection, crossover, and mutation operator continue until a termination condition is met (Esmaeili & Barzinpour, 2014). The pseudo-code of the offered GA is displayed in Fig. 10.

Fig. 10
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Proposed pseudo-code of the suggested GA

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4.4 Simulated Annealing

For the first time, SA was presented by Kirkpatrick et al., (1983). This single-solution-based metaheuristic algorithm is a standard solution method. Furthermore, SA has been used for treating many supply chain and location-allocation problems, such as (Fakhrzad & Goodarzian, 2021; Jabalameli et al., 2012; Liu et al., 2020; Seyedi et al., 2022a, 2022b).

In this method, the algorithm begins with a primary feasible solution, and then, a specific cost function is calculated for each solution. A new solution is generated in each iteration by slightly modifying one or some variables. An unfavorable neighbor is acknowledged with a probability established by the Boltzmann probability \(P={e}^{-\Delta \theta / T}\), while an improving move is always accepted. In the Boltzmann probability, \(\Delta \theta\) expresses the new and the best solutions difference, and \(T\) is the temperature (Garza-Santisteban et al., 2019). The procedure is repeated for a constant number of iterations. The pseudo-code of the proposed SA is given in Fig. 11.

Fig. 11
figure 11

Pseudo-code of the suggested SA

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5 Computational results

This section assess the model and the performance of applied methodologies. First, twenty-four small, medium, and large problems are randomly generated. Next, to tune the parameters of metaheuristics, the Taguchi design method is utilized.

Afterward, based on the results, solution approaches are compared. Finally, various criteria are used to select the best algorithm and solution.

5.1 Data

The performance of the employed metaheuristics is assessed in this sub-section. In this regard, twenty-four random problems in three different levels are produced and compared. These test problems are classified based on 3D printing centers (\(I\)), treatment centers (\(J\)), recycling centers (\(K\)), processing centers (\(T\)), collection centers (\(L\)), filament customers (\(n\)), markets (\(M\)), and end-user (\(E\)). The various dimensions of generated problems are presented in Table 6 and Fig. 12. In each echelon, the problem complexity is shown by the number of binary variables (potential places). Furthermore, Table 7 provides the other parameters’ values.

Table 6 Problems with different sizes
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Fig. 12
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Dispersion of problem size

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Table 7 Parameter setting of the suggested model
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5.2 Parameter setting of the algorithms

The performance of metaheuristics is highly influenced by the identity of the algorithm’s parameters. Therefore, the parameters of metaheuristics are tuned to improve the reliability of solution approaches. Many works have employed the complete factorial design to set the parameters. However, the more factors there are, the less effective this method is. In this respect, the experiments are designed by Taguchi Methods to decrease the test numbers and the examination complexity (Taguchi, 1986). First, each test problem is run ten times. Then, the number of objective functions is converted to relative percentage deviation (RPD) to balance the results’ measure. Next, the mean of RPD is utilized to compute signal-to-noise (S/N) ratios. Eventually, the best level was chosen.

Table 8 presents preferred parameters (factors) and their levels. Moreover, the amount of initial temperature in simulated annealing is selected adaptively. First, some neighborhoods are generated. After evaluating \(\Delta {\text{f}}\), Eq. (19) is used to calculate \(T0\):

Table 8 Parameter levels of the algorithms
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$$T0=\left|{\Delta f}_{{\text{min}}}+0.1({\Delta f}_{{\text{max}}}-{\Delta f}_{{\text{min}}})\right|.$$
(19)

Moreover, the temperature reduction ratio is done by Eqs. (20)–(22) (Hosseini Baboli et al., 2023; Liao et al., 2020).

$${T}_{i}={T}_{0}-i\times \left(\frac{\left({T}_{0}-{T}_{f}\right)}{N}\right) (i=1,\dots ,N),$$
(20)
$${T}_{i}={T}_{0}-\left(\frac{\left({T}_{0}-{T}_{f}\right)\left(N+1\right)}{N}\right)\left(1-\frac{1}{\left(i+1\right)}\right) (i=1,\dots ,N),$$
(21)
$${T}_{i}=\frac{1-tgh\left(\left(\frac{10i}{N} \right)-5\right)}{2}\left({T}_{0}-{T}_{f}\right)+{T}_{f} \left(i=1,\dots ,N\right).$$
(22)

According to the parameters and their levels, L9, L18, and L18 orthogonal arrays in Taguchi methods are used for the SA, GA, and PSO, respectively. To assess the result of each experiment, the RPD with the below equation is utilized:

$$RPD=\frac{{{\text{alg}}}_{{\text{sol}}}-{{\text{min}}}_{{\text{sol}}}}{{{\text{min}}}_{{\text{sol}}}},$$
(23)

where \({{\text{min}}}_{{\text{sol}}}\) is the minimum value of the cost function and \({{\text{alg}}}_{{\text{sol}}}\) is the attained solution (Ruiz & Stutzle, 2007). Tables 9, 10 and 11 provide orthogonal arrays and the average RPD of 10 times run for all 24 problems and all test problems.

Table 9 Orthogonal array L9 for SA
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Table 10 Orthogonal array L18 for GA
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Table 11 Orthogonal array L18 for PSO
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Taguchi design method seeks to maximize the controllable factor’s effects and minimize the impact of the noise. The S/N ratio can be used for providing both targets. In addition, there are three classifications for the Taguchi method: the more significant, the better, the smaller is better, and the nominal is better. In this work, RPD will be used as a response. Hence, “the smaller is better” is applied to adjust parameters, and Eq. (24) is utilized to calculate the value of the S/N ratio:

$$S/N=-10{{\text{log}}}_{10}\left(\sum \left({Y}^{2}\right)/n\right),$$
(24)

where \(Y\) is the response of each test problem and \(n\) is the number of orthogonal arrays. Minitab was employed to analyze the response (RPD) of designed experiments. Figures 13, 14 and 15 illustrate the mean of S/N. As a result, the level of SA’s parameters is 3,2, and 1. Moreover, the best levels for PSO and GA are 3, 3, 1, 3, and 3 and 3, 2, 1, 3, 1, and 2, respectively. Table 12 provides The best value of each factor.

Fig. 13
figure 13

Mean SN plot for levels of the factors in SA

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Fig. 14
figure 14

GA factors mean SN plot

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Fig. 15
figure 15

PSO factors mean SN plot

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Table 12 Best level for the parameters of the suggested metaheuristic algorithms
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5.3 Experimental results

When the appropriate value for the parameters of the algorithms is chosen, all twenty-four test problems are executed thirty times. RPD and hitting time are two metrics that are employed to appraise the performance of the algorithms. Hitting time lasts until the algorithm finds the minimum solution for the first time. The averages of these criteria are summarized in Table 13. Moreover, to clarify differences, the results are illustrated in Figs. 16 and 17.

Table 13 Computation results obtained by the algorithms
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Fig. 16
figure 16

RPD values

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Fig. 17
figure 17

First time find the best cost (FTBC)

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In conclusion, for small and medium size problems, GA provides a better, best-known solution. However, SA has less than 10% deviations from GA’s solutions. On the other hand, GA needs more time than SA to find this solution. Thus, from the point of view of time or quality, SA or GA, i.e., selected respectively. For large-scale problems, SA and GA have almost the same function. Since SA significantly needs less time, this algorithm is chosen to treat large-scale problems.

6 Conclusion and further studies

In this study, we delved into the intricate design and configuration of a circular CLSC network for an AM process. This network encompasses two vital sub-networks. In the first sub-network, we focused on the collection and baling of PET bottles, which cater to market demand and the needs of the second sub-network. The second sub-network involves the collection of waste from 3D printing processes, consumer waste, and bales, which are then processed in treatment centers to produce flakes. These flakes are further transformed into 3D printing filaments in recycling centers. To optimize this complex network, we introduced a novel MILP model, with the primary objective of minimizing total costs, encompassing location costs, transportation costs, and processing costs, all while addressing specific environmental and economic targets. We harnessed the power of three well-known metaheuristics as solution methods in our quest for efficient optimization. Prior to comparing the algorithm results, we employed the Taguchi design method to fine-tune their parameters. Based on the outcomes, we tailored our selection of either SA or GA, depending on factors such as time and quality. For large-scale problems, SA and GA exhibited similar performance, but the former was notably faster, leading to our choice for large-scale scenarios. The results obtained through this rigorous approach offer valuable insights into the feasibility of utilizing plastic waste in filament production. Our proposed CLSC model not only aims to reduce network expenses but also implicitly addresses other crucial objectives, including mitigating environmental impacts and cutting the costs associated with sourcing filament from raw materials. The summary of findings can be classified into the following items.

  • Algorithm Performance: For small and medium-sized problems, GA consistently provided better-known solutions compared to SA and PSO. Moreover, SA exhibited less than a 10% deviation from GA’s solutions, making it a competitive alternative. On the other hand, GA achieved better solutions but required more time, leading to a trade-off between time and solution quality.

  • Large-Scale Problem Solution: SA and GA demonstrated similar performance for large-scale problems. Notably, SA exhibited significantly faster convergence, making it the preferred choice for large-scale scenarios.

  • Optimal Algorithm Selection: The choice between SA and GA depended on the specific requirements of the problem, with SA being more time-efficient and GA offering potentially superior solutions.

  • Implications for CLSC Networks: This study emphasized the importance of considering both time efficiency and solution quality in choosing metaheuristic algorithms for optimizing CLSC networks. The proposed model and optimization strategies contributed to reducing network expenses and address environmental concerns associated with filament production.

Of course, there are various gaps which could be considered for future work:

  1. (i)

    Environmental Footprint: Delving deeper into the environmental impact of CLSC networks by conducting comprehensive carbon emissions assessments, especially in collection and transportation phases,

  2. (ii)

    Socioeconomic Impacts: Exploring the social and economic implications, focusing on job creation, local community benefits, and cost-effectiveness of using recycled materials in filament production,

  3. (iii)

    Uncertainty Management: Applying uncertainty-handing techniques to enhance the network’s adaptability to real-world uncertainties and unexpected events (Baltas et al., 2022; Kara et al., 2019; Özcan et al., 2023; Özmen et al., 2017; Palancı et al., 2016; Savku & Weber, 2018, 2022; Üstünkar et al., 2012),

  4. (iv)

    Technology-Specific Networks: Investigating the enhancement of CLSC networks through establishing different 3D printing technologies and processes, acknowledging their unique requirements,

  5. (v)

    Bale Differentiation: Examining how differentiating bales, such as by color or other characteristics, impacts network efficiency and performance,

  6. (vi)

    Real-Case Implementation: Validating the proposed models and strategies through real-case studies, collaborating with industries to apply them in diverse contexts.

These research directions will contribute to the ongoing development and optimization of CLSC networks, aligning them with sustainability, efficiency, and environmental responsibility in supply chain management.