An Expected Value-Based Symmetric–Asymmetric Polygonal Fuzzy Z-MCDM Framework for Sustainable–Smart Supplier Evaluation

Abstract

Background: Nowadays, traditional supply chain management (SCM) processes are undergoing a profound transformation enabled by advanced technologies derived from Industry 4.0. The rapid adoption of these technologies has led to the emergence of smart SCM, which integrates modern technologies in sourcing, production, distribution, and sales. Supplier evaluation and selection (SES) in smart SCM is a strategic decision impacting the entire supply chain. Organizations must also incorporate sustainability principles into their strategic decisions alongside smart production and efficiency. Methods: The main objective of this study is to develop a multi-criteria decision-making (MCDM) approach under uncertainty to address sustainable–smart supplier evaluation and selection problems. The approach integrates polygonal fuzzy numbers (POFNs), Z-numbers, expected interval (EI), and expected value (EV) to develop methods such as the logarithmic methodology of additive weights (LMAW) and the weighted aggregated sum product assessment (WASPAS), which are used to prioritize criteria and rank suppliers. Furthermore, novel approaches are introduced for calculating membership functions, $\mathcal{a}$-cut formulations, and the crispification process in POFNs. Results: A real case study in the home appliance industry revealed that cost reduction through smart technologies, green and smart logistics and manufacturing, and smart working environments are the most critical evaluation criteria. Suppliers three and four, excelling in these areas, were identified as top suppliers. Conclusions: The proposed approaches effectively addressed hybrid uncertainty in SES problems within smart SCM. Finally, sensitivity and comparative analysis confirmed their robustness and reliability.

1. Introduction

Organizations are compelled to implement fundamental changes aligned with new and emerging technologies influenced by Industry 4.0 across supply chain management (SCM) to maintain or enhance their competitive position. Core processes in SCM, including sourcing, production, distribution, and sales, are all impacted by Industry 4.0 innovations such as artificial intelligence (AI), big data, the Internet of Things (IoT), and more, experiencing transformative changes [1]. These rapid advancements in technology have revolutionized markets and had a direct impact on people’s lives [2]. From a broader perspective, Industry 4.0 seeks to fully automate production processes, minimizing or eliminating human intervention—a concept commonly known as intelligent factories [3]. In this context, organizations leverage the concept of smart SCM to empower their infrastructure. A smart supply chain (SSC) is defined as the utilization of intelligent technologies as fundamental enablers to enhance the performance of flexibility, transparency, responsiveness, integration, and collaboration within the supply chain [4]. Therefore, SSCs offer insights and structures distinct from traditional supply chains. Under these circumstances, new and enhanced requirements have emerged to address the issue of supplier evaluation and selection (SES), which is considered a key factor in the success of supply chains [5].

The SES is one of the most critical decisions in the sourcing and procurement functions of organizations, significantly influencing product costs, organizational expenses, competitiveness, and overall performance [6]. Choosing the right supplier can substantially enhance an organization’s flexibility, stability, and productivity [7]. Organizations must ensure that the evaluation criteria are efficient, decision-making tools are effective, and the evaluation results are reliable, as the outcome of this process leads to the inclusion of new members as the initial link in the supply chain [8]. It is evident that the SES in smart SCM requires the identification of appropriate evaluation criteria, the application of suitable decision-making tools, and the assurance of reliable evaluation outcomes that clearly reflect the adherence of potential suppliers to the principles of SSCs. On the other hand, given recent concerns about climate change, greenhouse gas emissions, environmental pollution, and related issues, corporate social responsibility, alongside economic factors, demands particular attention and alignment with the current business environment. In this context, the importance of adhering to sustainability principles—encompassing economic, environmental, and social dimensions—has become more critical than ever before [9]. On the other hand, smart technologies are regarded as key enablers for enhancing the productivity of sustainable SCM [10].

Smart SCM, along with the issue of smart SES, is still considered a contemporary and emerging concept [11]. As such, numerous research gaps remain unaddressed in the existing literature. From a practical perspective, studies focusing on smart SES in diverse organizations and industries with varying characteristics are scarce, making the contribution of each study significant when specific industries and products are considered. Furthermore, research integrating various dimensions of supply chains, such as sustainability, greenness, circularity, etc., alongside smart evaluation criteria, remains limited. Addressing these gaps could pave the way for more comprehensive evaluations in the future. From a methodological perspective, there is a clear need for further development of robust and capable decision support approaches and systems that can effectively address high levels of combined uncertainty in the process of smart SES. This necessity is strongly emphasized in the literature [12,13].

The primary objective of this study is to address the smart SES in the home appliance supply chain using a proposed multi-criteria decision-making (MCDM) approach under uncertainty. The SES problem is inherently an MCDM problem, comprising two main components: assigning importance to evaluation criteria and prioritizing suppliers [14].

In this study, an integrated approach under uncertainty is proposed for the sustainable smart SES by employing concepts such as symmetric and asymmetric polygonal fuzzy numbers (POFNs), Z-numbers, expected intervals (EIs), and expected values (EVs). The proposed approach develops the logarithm methodology of additive weights (LMAW) and the weighted aggregated sum product assessment (WASPAS) methods to address uncertainties in the decision-making space. These methods, named pentagonal fuzzy Z-LMAW (PZ-LMAW) and pentagonal fuzzy Z-WASPAS (PZ-WASPAS), are developed for assigning importance to criteria and prioritizing suppliers, respectively. By utilizing symmetric and asymmetric pentagonal fuzzy numbers (PFNs), Z-number fuzzy constraint functions, and Z-number reliability functions, the proposed approach effectively addresses high levels of inherent uncertainty in the SES process.

Moreover, innovative and efficient approaches for calculating membership functions, $\mathcal{a}$-cut formulations, and crispification for symmetric and asymmetric POFNs have been proposed. To evaluate the applicability and demonstrate the capabilities of the proposed approach, a real-world case study was conducted to address the problem of evaluating and selecting sustainable smart suppliers in the supply chain of a home appliance manufacturer. Additionally, sensitivity analysis was performed on the research findings.

Overall, this study offers significant contributions both in terms of literature and practical applications, which can be summarized as follows:

• Conducting a literature review and identifying the factors and criteria influencing the process of sustainable smart SES, complemented by a real-world case study in the home appliance supply chain.
• Developing an integrated MCDM approach incorporating the PZ-LMAW and PZ-WASPAS methods to address combined uncertainty, utilizing the characteristics of symmetric and asymmetric POFNs, fuzzy constraint functions of Z-numbers, fuzzy reliability functions of Z-numbers, and the EI and EV concepts in symmetric and asymmetric POFNs.
• Proposing efficient approaches for calculating membership functions, a-cut formulations, and crispification processes for symmetric and asymmetric POFNs.
• Developing a robust and efficient decision support system capable of handling high levels of combined uncertainty in decision-making environments to address various MCDM and managerial issues.
• Conducting sensitivity analysis based on diverse approaches in the literature to validate and ensure the reliability of the results obtained through the proposed approach.

The structure of this paper is organized as follows. Section 2 provides a literature review on smart SCM and sustainable smart SES. Section 3 discusses the materials and methods, including preliminaries, the crispification process for symmetric and asymmetric POFNs, and the proposed PZ-LMAW and PZ-WASPAS approaches. Section 4 presents the case study and analysis of the results. Section 5 covers the sensitivity comparative analysis of this study’s primary findings. Section 6 includes a discussion of the results and the contributions of this study. Finally, the conclusions, limitations, and future research directions are presented in Section 7.

2. Literature Review

In recent years, SCM has undergone changes to adapt to advancements, and the importance of sustainability and smart technologies has increased due to the depletion of available resources, the emergence of novel technologies, and the impacts of climate change [15]. Supplier selection is one of the critical processes influenced by the smart transformation of SCM [10,16]. Evaluation criteria for suppliers in various supply chains, such as sustainable and green ones, focus not only on overall organizational efficiency but also on the economic, social, and environmental responsibilities of manufacturing companies [17,18]. Furthermore, incorporating smartness aspects into the criteria for evaluating and selecting suppliers emphasizes not only structural factors like cost, quality, and responsiveness but also the smart capabilities of organizations, leveraging emerging and advanced technologies [19,20,21,22,23]. Frank et al. [24] proposed a conceptual framework to understand the adoption patterns of Industry 4.0 technologies in manufacturing companies, categorizing these technologies into Front-end and base technologies. Front-end technologies consider four dimensions: smart production, smart products, SSCs, and smart work. Base technologies include four elements: the IoT, cloud services, big data, and analytics. Therefore, SSCs fall under the category of Front-end technologies, driven by the integration of innovative and emerging technologies into the traditional supply chain structure.

Decision support tools and methodologies can assist organizations and supply chain managers in making more effective decisions [25]. Moreover, advanced decision-making tools that address the inherent combined uncertainties in the decision-making process lead to more realistic and reliable outcomes [26]. Therefore, the proposed approach in this study effectively addresses the inherent hybrid uncertainties in the process of evaluating and selecting sustainable smart suppliers by employing concepts such as symmetric and asymmetric POFNs, fuzzy constraint functions of Z-numbers, and fuzzy reliability functions of Z-numbers in developing the PZ-LMAW and PZ-WASPAS approaches.

In the following, several studies that have evaluated and selected suppliers based on smart principles alongside other perspectives such as sustainability, greenness, circularity, and so on are reviewed. This review aims to identify the structural characteristics, criteria, and prevalent methodologies in the literature on sustainable smart SES.

Demiralay et al. [15] examined the impact of different fuzzy environments and decision-making methods on the selection of smart and sustainable suppliers. Through a case study, they demonstrated that although changes in environments and methods alter the weight of criteria, they do not affect the final evaluation of suppliers. Additionally, scenario analysis revealed that shifting priorities between smart and sustainable criteria influence costs, carbon emissions, and delivery performance. Çalık [27] acknowledged that advancements in information and communication technologies have led to the emergence of innovative technologies such as cloud computing, the IoT, big data analytics, and AI, which have made production systems smarter. To select the best green supplier, they proposed a novel group decision-making approach based on Industry 4.0 components, utilizing the analytical hierarchy process (AHP) and TOPSIS methods in a Pythagorean fuzzy environment. Finally, the effectiveness of the proposed approach was evaluated through a case study in the field of agricultural tools and machinery. In another study, a novel framework for selecting sustainable suppliers in an SSC was proposed using a hybrid DEMATEL-TOPSIS approach in a rough–fuzzy environment. The proposed approach simultaneously managed internal and external uncertainties, and its effectiveness was demonstrated in selecting suppliers for sustainable vehicle transmission systems. The results were analyzed and validated [11].

Kaur et al. [28] believed that the need for structural changes in supply chains, influenced by factors such as Industry 4.0 and disruptions, has become increasingly critical. They proposed a multi-stage hybrid model for segmentation, selection, and order allocation to suppliers, considering risks and disruptions. Suppliers were evaluated and prioritized using data envelopment analysis (DEA) and FAHP-TOPSIS methods. Additionally, a Mixed-Integer Programming (MIP) model was employed to optimize order allocation with the goal of reducing costs and risks. The applicability of the proposed approach was demonstrated through a case study in the automotive industry. In another study, the SWARA-WASPAS methods were employed to evaluate the influencing factors for selecting digital suppliers and identifying top alternatives. The results indicated that supplier competence, including responsiveness, sustainability, and digital innovation, was the most critical factor for enhancing the quality of products and services. The research recommended that companies consider these key factors in supply chain decision-making to develop sustainable and transparent supply chains [29]. Torğul et al. [30] proposed a novel model using the interval type-2 fuzzy AHP method for selecting smart and sustainable suppliers. The evaluation criteria were identified and defined based on smart and sustainable aspects. The proposed model was applied in the automotive industry, where smart and sustainable suppliers were prioritized. The results demonstrated that the proposed decision-making method provided more reliable outcomes for supplier selection. This study was particularly innovative due to its integration of smart and sustainable criteria within a new framework. Ali et al. [31] proposed an integrated approach for supplier selection in the beverage sector, incorporating the concepts of sustainability, circular economy, and Industry 4.0. Through a thorough literature review and expert feedback, fourteen sub-criteria were identified and categorized into three main dimensions: economic, social, and circular. The full consistency method (FUCOM) was employed to calculate the relative weights of the criteria and sub-criteria, while the MULTIMOORA method was used to evaluate the suppliers. The results highlighted the significance of sub-criteria such as cost, GIS/GPS-assisted logistics, employee training in Industry 4.0, cyber–physical production, and environmentally friendly packaging. Sensitivity and comparative analyses confirmed that the proposed hybrid approach provided reliable and robust results.

Xu et al. [32] emphasized that in Industry 4.0, the development of SSCs is crucial for improving organizational efficiency and customer satisfaction. They highlighted that the use of blockchain ensures supply chain optimization, data exchange security, and effective interaction among various suppliers. In this context, a reputation-aware supplier assessment system was proposed. This system utilized the canopy and k-medoids methods for the initial categorization of suppliers and then employed a backpropagation neural network for supplier evaluation. The results demonstrated that the reputation-aware supplier assessment system is effective in evaluating suppliers through extensive testing. Bonab et al. [33] proposed an integrated approach based on an extended version of MCDM methods in the spherical fuzzy sets environment for selecting resilient and sustainable IoT suppliers. In the proposed approach, the main criteria of resilience and sustainability were utilized in the supplier selection process. These criteria were then weighted using the spherical fuzzy set (SFS) and best–worst method (BWM), which reduced uncertainty in pairwise comparisons. Subsequently, 14 selected IoT suppliers were evaluated and ranked using the SFS-multi-normalization multi-distance assessment (TRUST) method. The results of this study showed that the sub-criteria of pollution control and risk-taking ranked first and second in priority, respectively. Comparisons of SFS-TRUST with other MCDM methods and sensitivity analyses demonstrated the effectiveness and robustness of the proposed approach across various scenarios. Another study concluded that in recent years, companies have faced international changes driven by technological advancements, market globalization, or natural disasters, leading organizations to strive for improved performance to enhance their competitiveness. It was further noted that the competitiveness of organizations is highly dependent on their suppliers. Accordingly, an adaptive fuzzy-neuro approach was proposed for SES based on the criteria of resilience, sustainability, and intelligence [34].

Kusi-Sarpong et al. [35] proposed a decision-making framework based on Industry 4.0 components for implementing the circular economy to evaluate and select sustainable suppliers. An MCDM approach, consisting of the BWM and VIKOR, was used to assist in the sustainable SES in a textile manufacturing company. The findings of the study revealed that technology and infrastructure, along with a positive organizational culture, are the most important criteria for evaluating suppliers in the context of implementing Industry 4.0 initiatives and the circular economy. Ghadimi et al. [36] proposed a multi-agent systems approach for processing the sustainable SES to establish an appropriate communication channel, structured information exchange, and shared perspectives between suppliers and manufacturers. The results demonstrated that the proposed approach can assist decision-makers within manufacturing companies in making faster decisions with fewer human interactions. The advantages of the developed multi-agent systems were illustrated through implementation in a real-world case study involving a medical device manufacturer. Gai et al. [37] proposed a decentralized feedback mechanism designed to evaluate suppliers within a smart logistics network, aiming to assist large-scale decision-makers (DMs) in achieving consensus while considering the bounded compromise behavior of subgroups. In their proposed approach, the compromise behavior of subgroups was analyzed during the feedback process, and the concept of compromise thresholds was introduced to assess the bounded compromise behavior. The results showed that the proposed approach significantly improves the efficiency of decision-makers’ preferences in decision-making processes. Tavana et al. [38] proposed an integrated method for evaluating sustainable suppliers in smart circular supply chains using a fuzzy inference system (FIS) and MCDM approaches. In the first phase of the proposed method, SES sub-criteria from economic, social, circular, and Industry 4.0 perspectives were identified and weighted using the fuzzy BWM method. Then, suppliers were scored based on each criterion. In the second phase, suppliers were ranked and selected based on the overall score determined by the FIS. Finally, the applicability of the proposed method was demonstrated using data from a public–private partnership project in an offshore wind farm. In another study, with the aim of evaluating and selecting suppliers in the Industry 4.0 era, an integrated decision-making approach based on a hybrid intuitionistic fuzzy entropy weight-based MCDM model with TOPSIS was proposed. In the proposed approach, the importance of evaluation criteria was determined by using the entropy approach in the intuitionistic fuzzy space, and supplier prioritization was performed using the TOPSIS method. The results showed that considering the importance of decision-makers in the evaluation process of criteria and supplier ranking is effective in improving efficiency and reducing ambiguity in the supplier selection process [39].

Hosseini Dolatabad et al. [40] argued that in today’s world, with technological advancements and the emergence of industrial revolutions, organizations must seek to gain competitive advantages to maintain their position in the competitive market and adapt to new trends. They utilized fuzzy cognitive maps and the hesitant fuzzy linguistic VIKOR method to evaluate and prioritize suppliers in the Industry 4.0 era. In another study, a multi-criteria decision-making model based on the BWM and the Additive Ratio Assessment (ARAS) method in a hesitant fuzzy linguistic environment was proposed for selecting the appropriate supplier in the digital supply chain financing for small and medium-sized enterprises. A case study was then conducted to demonstrate the applicability of the proposed approach in the supplier selection process. Sensitivity analysis and comparative analysis showed the effectiveness and superiority of this method [41]. Hasan et al. [7] developed a decision support system that processes heterogeneous and ambiguous data related to Logistics 4.0 and ranks suppliers. The proposed method utilized triangular fuzzy numbers (TFNs) and fuzzy linguistic variables to process both qualitative and quantitative data. Through the use of fuzzy TOPSIS and the multi-criteria goal programming model, the decision-making problem was solved. Sensitivity analysis also examined the variations in the key performance indicators of suppliers across different priorities. Kayapinar Kaya et al. [42] stated that with the advent of Industry 4.0 in supply chains, extensive digitalization has begun at all stages of the supply chain, transforming the supplier selection process. They identified key criteria related to Industry 4.0 technologies and evaluated them for selecting appropriate suppliers. Their proposed approach included type-2 fuzzy AHP and the Grey COPRAS method for weighting the criteria and prioritizing suppliers. The results of this study provided valuable insights for practitioners and researchers in understanding the impact of Industry 4.0 strategies on the supplier selection process.

Fallahpour et al. [43] proposed a new integrated model that considers sustainability and Industry 4.0 criteria in supplier selection problems. This approach included the fuzzy BWM (FBWM) and a two-stage FIS to evaluate supplier performance. A case study in a textile company demonstrated the applicability of the proposed model, and sensitivity analysis confirmed the effectiveness of the proposed approach. Fallahpour et al. [44] believed that with the advancement of Industry 4.0 and digitalization, traditional supplier selection models have become inefficient. Therefore, they proposed a new framework using fuzzy preference programming (FPP) to determine the importance of one supplier attribute over another, and multi-objective optimization based on ratio analysis (MOORA) to prioritize suppliers based on fuzzy performance ratings. The proposed approach, evaluated in the supply chain of a food company, demonstrated how sustainable and digital supplier selection could enhance the quality of the supply process. In another study, the concept of Fermatean fuzzy sets (FFSs) was employed, and the AHP method was developed for more accurate modeling of expert preferences in addressing a real-world supplier selection issue related to the transition to Industry 4.0. The results of the study indicated that supplier selection, which holds strategic importance in the current environment, faces challenges such as the need for greater integration between suppliers and customers. Ultimately, criteria for selecting suppliers in alignment with Industry 4.0 requirements were proposed [45].

Matthess et al. [6], in a study with a qualitative and interview-based approach, examined the sustainability evaluation processes of suppliers in the electronics industry and the role of digital technologies. The findings revealed that companies do not consistently use digital tools for sustainability information exchange, and data collection is typically carried out manually. They ultimately recommended standardizing sustainability requirements with a focus on implementing digital technologies to improve data accessibility and credibility. Wang et al. [46] proposed a novel model for SES by employing two MCDM approaches—Ordinal Priority Approach (OPA) and measurement alternatives and ranking based on the compromise solution (MARCOS)—in a fuzzy environment. The model integrated Industry 4.0 technologies and sustainability considerations. The results of the study conducted in the leather and footwear industry revealed that criteria such as green image, innovation in green products, and cloud computing significantly influence the sustainability evaluation of the supply chain. In another study, the issue of SES in the medical equipment industry was examined by considering agility, sustainability, and Industry 4.0 indicators. The importance of the criteria was determined using the rough BWM (RBWM) method, followed by the ranking of suppliers with the interval rough multi-attributive border approximation area comparison (IR-MABAC) method. The results indicated that criteria such as agility, sustainability, production flexibility, cost, and quality are highly significant in supplier selection. This research was the first to utilize the combined RBWM-IR-MABAC approach to address the supplier selection problem and provided practical guidance for supply chain managers to better understand sustainability, agility, and Industry 4.0 criteria [47].

A summary of the reviewed studies along with their features, information forms, applied approaches, and application domains, is presented in

Table 1. A summary of the literature review on studies conducted in the field of smart SES.

No.	Studies	Methodology/Approach	Information Type	Application Area
1	Demiralay et al. [15]	BWM, AHP, TOPSIS	Pythagorean fuzzy sets	Automobile spare parts
2	Çalık [27]	AHP, TOPSIS	Pythagorean fuzzy sets	Agricultural tools
3	Chen et al. [11]	DEMATEL, TOPSIS	Rough–fuzzy	Vehicle transmission system
4	Kaur et al. [28]	DEA, AHP, TOPSIS, MIP	Triangular fuzzy numbers	Automobile
5	Sharma et al. [29]	SWARA, WASPAS	Crisp values	Manufacturing firms
6	Torğul et al. [30]	AHP	Interval type-2 fuzzy sets	Automobile
7	Ali et al. [31]	FUCOM, MULTIMOORA	Triangular fuzzy numbers	Beverage
8	Xu et al. [32]	Classification, neural networks	Crisp values	Blockchain
9	Bonab et al. [33]	BWM, TRUST	Spherical fuzzy sets	IoT
10	Zekhnini et al. [34]	Adaptive fuzzy-neuro system	Fuzzy sets	Digital supply chain
11	Kusi-Sarpong et al. [35]	BWM, VIKOR	Crisp values	Textile
12	Ghadimi et al. [36]	Fuzzy inference system	Fuzzy sets	Medical devices
13	Gai et al. [37]	Large-scale group decision-making	2-tuple linguistic	Smart logistics
14	Tavana et al. [38]	BWM, fuzzy inference system	Triangular fuzzy numbers	Offshore wind farm
15	Sachdeva et al. [39]	Entropy, TOPSIS	Intuitionistic fuzzy sets	Automobile
16	Hosseini Dolatabad et al. [40]	Fuzzy cognitive map, VIKOR	Hesitant fuzzy sets	Electronics
17	Liao et al. [41]	BWM, ARAS	Hesitant fuzzy sets	Finance
18	Hasan et al. [7]	Goal programming, TOPSIS	Triangular fuzzy numbers	Hypothetical case study
19	Kayapinar Kaya et al. [42]	AHP, COPRAS	Interval type-2 fuzzy sets, grey numbers	Textile
20	Fallahpour et al. [43]	BWM, fuzzy inference system	Triangular fuzzy numbers	Textile
21	Fallahpour et al. [44]	FPP, MOORA	Triangular fuzzy numbers	Food
22	Camci et al. [45]	AHP	Fermatean fuzzy sets	Industry 4.0
23	Matthess et al. [6]	Interview-based approach	Qualitative	Electronics
24	Wang et al. [46]	OPA, MARCOS	Triangular fuzzy numbers	Leather and footwear
25	ForouzeshNejad [47]	BWM, MABAC	Rough sets	Medical devices
26	Current study	LMAW, WASPAS	Polygonal fuzzy Z-numbers	Home appliances

.

3. Materials and Methods

In this section, the proposed methodology for evaluating and selecting sustainable smart suppliers is explained. First, an introduction to fuzzy sets, POFNs, Z-numbers, and the proposed approaches for calculating membership functions and formulating $\mathcal{a}$-cuts in symmetric and asymmetric POFNs is provided in Section 3.1. Then, the proposed crispification approach, based on the expected interval (EI) and expected value (EV), is introduced in Section 3.2 for crispification of symmetric and asymmetric POFNs. Finally, the steps required for implementing the proposed PZ-LMAW and PZ-WASPAS approaches are detailed in Section 3.3 and Section 3.4, respectively. It is worth noting that the proposed approaches in this section are developed using PFNs, which are a specific case of POFNs. Since these approaches can be generalized to other types of POFNs, the term POFNs has been used.

3.1. Preliminaries

This section summarizes some basic definitions and fundamental concepts of fuzzy sets, POFNs, and Z-numbers. Also, the proposed approaches for calculating membership functions and formulating $\mathcal{a}$-cuts in symmetric and asymmetric POFNs are described in this section.

Definition 1.

Fuzzy set. A fuzzy set is defined by its membership function, which maps elements from a universal set $\mathcal{K}$ or domain of discourse to the unit interval $[0,1]$. Formally, a fuzzy set $\tilde{\mathcal{A}}$ in $\mathcal{K}$ can be represented as $\tilde{\mathcal{A}}=\left\{〈\mathcal{k},{\mu }_{\tilde{\mathcal{A}}}\left(\mathcal{k}\right)〉|\mathcal{k}\in \mathcal{K},{\mu }_{\tilde{\mathcal{A}}}(\mathcal{k})\in [0,1]\right\}$, where ${\mu }_{\tilde{\mathcal{A}}}\left(\mathcal{k}\right)$ is the membership grade of $\mathcal{k}$ in $\tilde{\mathcal{A}}$. The membership function ${\mu }_{\tilde{\mathcal{A}}}$ indicates the degree to which each element in $\mathcal{K}$ belongs to the fuzzy set $\tilde{\mathcal{A}}$ [48].

Definition 2.

Fuzzy number. A fuzzy set $\tilde{\mathcal{A}}$, defined over the universal set of real numbers $\mathbb{R}$, is called a fuzzy number if it satisfies the following properties [49].

• $\tilde{\mathcal{A}}$  is a convex fuzzy set.
• $\tilde{\mathcal{A}}$ is normal, meaning there exists at least one ${\mathcal{k}}_0\in \mathbb{R}$ such that the membership function ${\mu }_{\tilde{\mathcal{A}}}\left({\mathcal{k}}_0\right)=1$.
• The membership function  ${\mu }_{\tilde{\mathcal{A}}}\left(\mathcal{k}\right)$  is piecewise continuous.
• For every  $\mathcal{a}\in [0,1]$, the  $\mathcal{a}\text{-}cut$ of  $\tilde{\mathcal{A}}$, denoted as  ${\mathcal{A}}_{\mathcal{a}}$, forms a closed interval.
• The support of  $\tilde{\mathcal{A}}$,  $sup\left(\tilde{\mathcal{A}}\right)$, is bounded.

Definition 3.

PFN. A fuzzy number $\tilde{\mathcal{A}}=\left({\mathcal{g}}_1,{\mathcal{g}}_2,{\mathcal{g}}_3,{\mathcal{g}}_4,{\mathcal{g}}_5\right)$ is called a PFN if it satisfies the following conditions [50].

• The membership function ${\mu }_{\tilde{\mathcal{A}}}\left(\mathcal{k}\right)$ is continuous over the interval $[0,1]$.
• The membership function ${\mu }_{\tilde{\mathcal{A}}}\left(\mathcal{k}\right)$ is strictly a non-decreasing continuous function on the intervals $[{\mathcal{g}}_1,{\mathcal{g}}_2]$ and $\left[{\mathcal{g}}_2,{\mathcal{g}}_3\right]$.
• The membership function ${\mu }_{\tilde{\mathcal{A}}}\left(\mathcal{k}\right)$ is strictly a non-increasing continuous function on the intervals $[{\mathcal{g}}_3,{\mathcal{g}}_4]$ and $\left[{\mathcal{g}}_4,{\mathcal{g}}_5\right]$.

Definition 4.

PFNs are divided into symmetric and asymmetric types, and each type can be linear or nonlinear. In this study, linear PFNs in both symmetric and asymmetric forms are utilized. Each PFN has two weights, denoted as ${\Psi }_1$ and ${\Psi }_2$. For simplicity and to avoid confusion, in PFN $\tilde{\mathcal{A}}$ these weights are represented as ${\Psi }_i^{\tilde{\mathcal{A}}}$ for $i=\mathrm{1,2}$. If, in an asymmetric PFN, ${\Psi }_1^{\tilde{\mathcal{A}}}={\Psi }_2^{\tilde{\mathcal{A}}}=1$, the PFN becomes symmetric. Additionally, in an asymmetric PFN, ${\Psi }_1^{\tilde{\mathcal{A}}}$ can be smaller or larger than ${\Psi }_2^{\tilde{\mathcal{A}}}$ [51]. Figure 1 compares symmetric and asymmetric linear PFNs.

Definition 5.

Membership functions of POFNs. A symmetric linear PFN is represented as ${\tilde{\mathcal{A}}}_{SL}=\left({\mathcal{g}}_1,{\mathcal{g}}_2,{\mathcal{g}}_3,{\mathcal{g}}_4,{\mathcal{g}}_5:\Psi \right)$, while an asymmetric linear PFN is represented as ${\tilde{\mathcal{A}}}_{AL}=\left({\mathcal{g}}_1,{\mathcal{g}}_2,{\mathcal{g}}_3,{\mathcal{g}}_4,{\mathcal{g}}_5:{\Psi }_1,{\Psi }_2\right)$.

In the literature, various formulations are used to represent the membership functions of POFNs. Although most of the proposed formulations lead to similar results and are essentially equivalent, they may cause confusion. Therefore, we propose a simple and efficient approach to derive the formulations for different types of POFN membership functions, called the schematic calculation of POFN membership functions. Using this approach, the membership function of POFNs under various conditions of ${\mathcal{k}}_i$ can be obtained with a straightforward formula. To derive each formulation, the starting point of the linear curve is determined, followed by the calculation of the slope of the desired linear curve, which is then multiplied by the desired ${\mathcal{k}}_i$ value. Figure 2 separately illustrates the linear curves for symmetric and asymmetric linear PFNs. Furthermore, Equation (1) shows how each formulation can be derived based on its corresponding linear curve:

$${\mu }_{{\tilde{\mathcal{A}}}_{Polygonal}}\left(\mathcal{k};{\Psi }_i\right)=\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t} \mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}+\left(\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r} \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e} \mathrm{s}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\right).\left(\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{d} \mathcal{k} value\right).$$

(1)

For example, if in Figure 2A, which represents a symmetric linear PFN, the membership degree for the condition ${\mathcal{g}}_2\le \mathcal{k}\le {\mathcal{g}}_3$ is to be determined, the corresponding linear curve, specifically Curve 2, is considered. Then, the starting point of the linear curve is identified, which in this case is $\mathsf{\Psi }$. Next, the slope of the linear curve (considering whether it is increasing or decreasing) is determined. Here, the slope is increasing, so it is calculated as $\left(1-\Psi \right)/\left({\mathcal{g}}_3-{\mathcal{g}}_2\right)$. Subsequently, depending on whether the function is increasing or decreasing, the desired value $\mathcal{k}$ is determined, which in this case is $(\mathcal{k}-{\mathcal{g}}_2)$. Therefore, Equation (2) demonstrates how to calculate the formulation for ${\mathcal{g}}_2\le \mathcal{k}\le {\mathcal{g}}_3$ in a symmetric linear PFN:

$${\mu }_{{\tilde{\mathcal{A}}}_{SL}}\left({\mathcal{g}}_2\le \mathcal{k}\le {\mathcal{g}}_3;\Psi \right)=\mathsf{\Psi }+\left({\displaystyle \frac{1-\Psi }{{\mathcal{g}}_3-{\mathcal{g}}_2}}\right)\left(\mathcal{k}-{\mathcal{g}}_2\right).$$

(2)

Similarly, to calculate the membership degree for ${\mathcal{g}}_4\le \mathcal{k}\le {\mathcal{g}}_5$ in an asymmetric linear PFN, as shown in Figure 2B, Curve 4 is considered. The starting point ${\Psi }_2$ is added to the product of the slope of the linear curve (which in this case is decreasing), $\left(0-{\Psi }_2\right)/\left({\mathcal{g}}_5-{\mathcal{g}}_4\right)$, and the desired value $\mathcal{k}$. In the case of a decreasing function, this becomes $({\mathcal{g}}_5-\mathcal{k})$. Equation (3) demonstrates how to calculate the formulation for ${\mathcal{g}}_4\le \mathcal{k}\le {\mathcal{g}}_5$ in an asymmetric linear PFN:

$${\mu }_{{\tilde{\mathcal{A}}}_{AL}}\left({\mathcal{g}}_4\le \mathcal{k}\le {\mathcal{g}}_5;{\Psi }_2\right)={\Psi }_2+\left({\displaystyle \frac{0-{\Psi }_2}{{\mathcal{g}}_5-{\mathcal{g}}_4}}\right)\left({\mathcal{g}}_5-\mathcal{k}\right).$$

(3)

Equations (4) and (5) represent the membership functions of symmetric and asymmetric linear PFNs (see Figure 2), respectively, derived using the proposed approach. The membership functions of other POFNs, such as hexagonal fuzzy numbers (HFNs), can also be calculated in a similar manner.

$${\mu }_{{\tilde{\mathcal{A}}}_{SL}}(\mathcal{k};\mathsf{\Psi })=\left\{\begin{array}{c}\begin{array}{cc}0,if& \mathcal{k}\le {\mathcal{g}}_1\end{array}\\ \begin{array}{cc}0+\left({\displaystyle \frac{\Psi -0}{{\mathcal{g}}_2-{\mathcal{g}}_1}}\right)(\mathcal{k}-{\mathcal{g}}_1),if& {\mathcal{g}}_1<\mathcal{k}\le {\mathcal{g}}_2\end{array}\\ \begin{array}{cc}\Psi +\left({\displaystyle \frac{1-\Psi }{{\mathcal{g}}_3-{\mathcal{g}}_2}}\right)(\mathcal{k}-{\mathcal{g}}_2),if& {\mathcal{g}}_2<\mathcal{k}\le {\mathcal{g}}_3\end{array}\\ \begin{array}{cc}1+\left({\displaystyle \frac{\Psi -1}{{\mathcal{g}}_4-{\mathcal{g}}_3}}\right)({\mathcal{g}}_4-\mathcal{k}),if& {\mathcal{g}}_3<\mathcal{k}\le {\mathcal{g}}_4\end{array}\\ \begin{array}{cc}\Psi +\left({\displaystyle \frac{0-\Psi }{{\mathcal{g}}_5-{\mathcal{g}}_4}}\right)({\mathcal{g}}_5-\mathcal{k}),if& {\mathcal{g}}_4<\mathcal{k}\le {\mathcal{g}}_5\end{array}\\ \begin{array}{cc}0,if& \mathcal{k}>{\mathcal{g}}_5\end{array}\end{array}\right.$$

(4)

$${\mu }_{{\tilde{\mathcal{A}}}_{AL}}(\mathcal{k};{\Psi }_1,{\Psi }_2)=\left\{\begin{array}{c}\begin{array}{cc}0,if& \mathcal{k}\le {\mathcal{g}}_1\end{array}\\ \begin{array}{cc}0+\left({\displaystyle \frac{{\Psi }_1-0}{{\mathcal{g}}_2-{\mathcal{g}}_1}}\right)(\mathcal{k}-{\mathcal{g}}_1),if& {\mathcal{g}}_1<\mathcal{k}\le {\mathcal{g}}_2\end{array}\\ \begin{array}{cc}{\Psi }_1+\left({\displaystyle \frac{1-{\Psi }_1}{{\mathcal{g}}_3-{\mathcal{g}}_2}}\right)(\mathcal{k}-{\mathcal{g}}_2),if& {\mathcal{g}}_2<\mathcal{k}\le {\mathcal{g}}_3\end{array}\\ \begin{array}{cc}1+\left({\displaystyle \frac{{\Psi }_2-1}{{\mathcal{g}}_4-{\mathcal{g}}_3}}\right)({\mathcal{g}}_4-\mathcal{k}),if& {\mathcal{g}}_3<\mathcal{k}\le {\mathcal{g}}_4\end{array}\\ \begin{array}{cc}{\Psi }_2+\left({\displaystyle \frac{0-{\Psi }_2}{{\mathcal{g}}_5-{\mathcal{g}}_4}}\right)({\mathcal{g}}_5-\mathcal{k}),if& {\mathcal{g}}_4<\mathcal{k}\le {\mathcal{g}}_5\end{array}\\ \begin{array}{cc}0,if& \mathcal{k}>{\mathcal{g}}_5\end{array}\end{array}\right.$$

(5)

Definition 6.

$\mathcal{a}$-cut operations in POFNs (symmetric and asymmetric PFNs). The $\mathcal{a}$-cut in symmetric and asymmetric linear PFNs is defined by Equations (6) and (7), respectively:

$${\tilde{\mathcal{A}}}_{SL}^{\mathcal{a}}=\left\{〈\mathcal{k}\in \mathcal{K}〉\left|{\mu }_{{\tilde{\mathcal{A}}}_{SL}^{\mathcal{a}}}\right(\mathcal{k})\ge \mathcal{a}\right\},$$

(6)

$${\tilde{\mathcal{A}}}_{AL}^{\mathcal{a}}=\left\{〈\mathcal{k}\in \mathcal{K}〉\left|{\mu }_{{\tilde{\mathcal{A}}}_{AL}^{\mathcal{a}}}\right(\mathcal{k})\ge \mathcal{a}\right\}.$$

(7)

By applying $\mathcal{a}$-cut operations to PFNs, these numbers are transformed into closed intervals. $\mathcal{a}$-cut operations have numerous applications in fuzzy computations and fuzzy relations of PFNs; one example of their applications will be discussed in Section 3.2. The $\mathcal{a}$ level lies within the closed interval [0, 1], and the closer this level is to 1, the higher the reliability and the degree of membership of ${\mathcal{k}}_i$ in the given conditions. One simple approach for calculating $\mathcal{a}$-cut formulations is to set each of the equations in Equations (4) and (5) equal to $\mathcal{a}$. However, to enhance clarity, avoid confusion, and provide a precise explanation of the nature of $\mathcal{a}$-cut formulations, this section proposes an approach for calculating these formulations that can be applied to all POFNs.

$\mathcal{a}$-cut operations and their formulations differ in symmetric and asymmetric POFNs, and based on the authors’ best knowledge, ambiguities and shortcomings exist in formulating $\mathcal{a}$-cut rules for POFNs, particularly PFNs, in the literature. Therefore, we aim to propose an effective approach for formulating $\mathcal{a}$-cut rules for POFNs (here, PFNs). In the proposed approach applied to PFNs, the $\mathcal{a}$-cut rules for each of the symmetric and asymmetric linear PFNs will be calculated schematically based on the formula presented in Equation (8):

$${\tilde{\mathcal{A}}}_{polygonal}^{\mathcal{a}}=\left\{\begin{array}{c}linearcurveslope\oplus ,\\ Startpoint+{\displaystyle \frac{d_1}{d_2}}\times \left(distanceunderthelinearcurve\right).\\ linearcurveslope\ominus ,\\ Endpoint-{\displaystyle \frac{d_1}{d_2}}\times \left(distanceunderthelinearcurve\right).\end{array}\right.$$

(8)

Regarding Equation (8), when the slopes of the relevant curves are increasing, they approach complete certainty, meaning a membership degree of 1. Therefore, the starting point is added to the product of the ratio of distances ($d_1$ and $d_2$) and the distance under the curve. According to Figure 3, distance under the curve refers to the distance beneath the target curve on the horizontal axis. For each of the linear curves 1, 2, 3, and 4, this corresponds to ${\mathcal{g}}_2-{\mathcal{g}}_1$, ${\mathcal{g}}_3-{\mathcal{g}}_2$, ${\mathcal{g}}_4-{\mathcal{g}}_3$, and ${\mathcal{g}}_5-{\mathcal{g}}_4$, respectively. However, when the slopes of the relevant curves are non-increasing, they move away from certainty, or a membership degree of 1. In this case, the ending point is subtracted by the product of the ratio of distances $d_1$ and $d_2$ and the distance under the curve.

In symmetric linear PFNs, there are two intervals for applying the $\mathcal{a}$-cut: $\left[0,\Psi \right]$ and $\left[\Psi ,0\right]$. Additionally, each symmetric linear PFN has four linear curves: two on the left side of the PFN, named $A_{1L}^{SL}\left(\mathcal{a}\right)$ and $A_{2L}^{SL}\left(\mathcal{a}\right)$, with increasing slopes, and two on the right side of the PFN, named $A_{1R}^{SL}\left(\mathcal{a}\right)$ and $A_{2R}^{SL}\left(\mathcal{a}\right)$, with non-increasing slopes. According to Equation (8), the $\mathcal{a}$-cut in each interval and for each curve of symmetric linear PFNs has its own specific condition and formula.

To compute the $\mathcal{a}$-cut formula on the left side of symmetric linear PFNs (with increasing slopes), the following steps are performed:

• The starting point is first determined.
• The first distance ($d_1$) is divided by the second distance ($d_2$) and multiplied by the distance under the linear curve (according to Figure 3, for each of the linear curves 1, 2, 3, and 4, the distance under the curve corresponds to ${\mathcal{g}}_2-{\mathcal{g}}_1$, ${\mathcal{g}}_3-{\mathcal{g}}_2$, ${\mathcal{g}}_4-{\mathcal{g}}_3$, and ${\mathcal{g}}_5-{\mathcal{g}}_4$, respectively).
• Adding the results from steps 1 and 2 yields the $\mathcal{a}$-cut formula for the specified interval and linear curve.

To compute the $\mathcal{a}$-cut formula on the right side of symmetric linear PFNs (with non-increasing slopes), a similar process is followed. However, the endpoint of the curve is considered, and the results of steps 1 and 2 are subtracted instead of added. Figure 3 illustrates various scenarios of applying $\mathcal{a}$-cut operations to a symmetric linear PFN. Equation (9) provides the $\mathcal{a}$-cut formulas obtained using the proposed approach for symmetric linear PFNs.

$${\tilde{\mathcal{A}}}_{PFN}^{SL}(\mathcal{a})=\left\{\begin{array}{c}{A}_{1L}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_1+\left({\displaystyle \frac{d_1:\left(\mathcal{a}-0\right)}{d_2:\left(\Psi -0\right)}}\right)\left({\mathcal{g}}_2-{\mathcal{g}}_1\right),fora\in \left[0,\Psi \right]\\ A_{2L}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_2+\left({\displaystyle \frac{d_1:\left(\mathcal{a}-\Psi \right)}{d_2:\left(1-\Psi \right)}}\right)\left({\mathcal{g}}_3-{\mathcal{g}}_2\right),fora\in \left[\Psi ,1\right]\\ A_{1R}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_4-\left({\displaystyle \frac{d_1:\left(\mathcal{a}-\Psi \right)}{d_2:\left(1-\Psi \right)}}\right)\left({\mathcal{g}}_4-{\mathcal{g}}_3\right),fora\in \left[\Psi ,1\right]\\ A_{2R}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_5-\left({\displaystyle \frac{d_1:\left(\mathcal{a}-0\right)}{d_2:\left(\Psi -0\right)}}\right)\left({\mathcal{g}}_5-{\mathcal{g}}_4\right),fora\in \left[0,\Psi \right]\end{array}\right.$$

(9)

In asymmetric linear PFNs, there are four intervals for applying the $\mathcal{a}\text{-}cut$: $\left[0,{\Psi }_1\right]$, $\left[{\Psi }_1,1\right]$, $\left[{\Psi }_2,1\right]$, and $\left[0,{\Psi }_2\right]$. Similarly to symmetric linear PFNs, each asymmetric linear PFN contains four linear curves: two on the left side of the PFN, named $A_{1L}^{AL}\left(\mathcal{a}\right)$ and $A_{2L}^{AL}\left(\mathcal{a}\right)$, with increasing slopes, and two on the right side, named $A_{1R}^{AL}\left(\mathcal{a}\right)$ and $A_{2R}^{AL}\left(\mathcal{a}\right)$, with non-increasing slopes. Figure 4 illustrates various scenarios of applying $\mathcal{a}$-cut operations to an asymmetric linear PFN.

To compute the $\mathcal{a}$-cut formula for the left side of asymmetric linear PFNs (with increasing slopes), the following steps are performed:

• The starting point is first determined.
• The first distance ($d_1$) is divided by the second distance ($d_2$) and multiplied by the distance under the linear curve (according to Figure 4, for each of the linear curves 1, 2, 3, and 4, the distance under the curve corresponds to ${\mathcal{g}}_2-{\mathcal{g}}_1$, ${\mathcal{g}}_3-{\mathcal{g}}_2$, ${\mathcal{g}}_4-{\mathcal{g}}_3$, and ${\mathcal{g}}_5-{\mathcal{g}}_4$, respectively).
• Adding the results from steps 1 and 2 gives the $\mathcal{a}$-cut formula for the specified interval and linear curve.

A similar process is applied to the right side of asymmetric linear PFNs (with non-increasing slopes). However, the endpoint of the curve is considered, and the results of steps 1 and 2 are subtracted rather than added. Equation (10) presents the $\mathcal{a}$-cut formulas derived using the proposed approach for asymmetric linear PFNs.

$${\tilde{\mathcal{A}}}_{PFN}^{SL}(\mathcal{a})=\left\{\begin{array}{c}{A}_{1L}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_1+\left({\displaystyle \frac{d_1:\left(\mathcal{a}-0\right)}{d_2:\left({\Psi }_1-0\right)}}\right)\left({\mathcal{g}}_2-{\mathcal{g}}_1\right),fora\in \left[0,{\Psi }_1\right]\\ A_{2L}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_2+\left({\displaystyle \frac{d_1:\left(\mathcal{a}-{\Psi }_1\right)}{d_2:\left(1-{\Psi }_1\right)}}\right)\left({\mathcal{g}}_3-{\mathcal{g}}_2\right),fora\in \left[{\Psi }_1,1\right]\\ A_{1R}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_4-\left({\displaystyle \frac{d_1:\left(\mathcal{a}-{\Psi }_2\right)}{d_2:\left(1-{\Psi }_2\right)}}\right)\left({\mathcal{g}}_4-{\mathcal{g}}_3\right),fora\in \left[{\Psi }_2,1\right]\\ A_{2R}^{SL}\left(\mathcal{a}\right)={\mathcal{g}}_5-\left({\displaystyle \frac{d_1:\left(\mathcal{a}-0\right)}{d_2:\left({\Psi }_2-0\right)}}\right)\left({\mathcal{g}}_5-{\mathcal{g}}_4\right),fora\in \left[0,{\Psi }_2\right]\end{array}\right.$$

(10)

Similarly, by schematically illustrating the membership functions of fuzzy numbers and applying Equation (8), the $\mathcal{a}$-cut formulations for other linear POFNs, such as TFNs, trapezoidal fuzzy numbers (TRFNs), HFNs, and so on, can be determined.

Definition 7.

Algebraic operations of symmetric linear PFNs. The algebraic operations in symmetric and asymmetric linear PFNs differ, and these differences have often been overlooked in some studies. To ensure the reliability of results derived from approaches utilizing POFNs, it is essential that algebraic operations are performed correctly. If ${\tilde{\mathcal{A}}}_P^{SL}=\left({\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4,{\mathfrak{a}}_5;\Psi \right)$ under the conditions ${\mathfrak{a}}_1\ge 0,{\mathfrak{a}}_1\ge {\mathfrak{a}}_2\ge {\mathfrak{a}}_3\ge {\mathfrak{a}}_4\ge {\mathfrak{a}}_5$ and ${\tilde{\mathcal{B}}}_P^{SL}=\left({\mathfrak{b}}_1,{\mathfrak{b}}_2,{\mathfrak{b}}_3,{\mathfrak{b}}_4,{\mathfrak{b}}_5;{\Psi }^{\prime }\right)$ under the conditions ${\mathfrak{b}}_1\ge 0,{\mathfrak{b}}_1\ge {\mathfrak{b}}_2\ge {\mathfrak{b}}_3\ge {\mathfrak{b}}_4\ge {\mathfrak{b}}_5$ are two symmetric linear PFNs, the algebraic operations are represented by Equations (11) to (13).

$${\tilde{\mathcal{A}}}_P^{SL}+{\tilde{\mathcal{B}}}_P^{SL}=\left({\mathfrak{a}}_1+{\mathfrak{b}}_1,{\mathfrak{a}}_2+{\mathfrak{b}}_2,{\mathfrak{a}}_3+{\mathfrak{b}}_3,{\mathfrak{a}}_4+{\mathfrak{b}}_4,{\mathfrak{a}}_5+{\mathfrak{b}}_5;min(\Psi ,{\Psi }^{\prime })\right)$$

(11)

$${\tilde{\mathcal{A}}}_P^{SL}-{\tilde{\mathcal{B}}}_P^{SL}=\left({\mathfrak{a}}_1-{\mathfrak{b}}_5,{\mathfrak{a}}_2-{\mathfrak{b}}_4,{\mathfrak{a}}_3-{\mathfrak{b}}_3,{\mathfrak{a}}_4-{\mathfrak{b}}_2,{\mathfrak{a}}_5-{\mathfrak{b}}_1;min(\Psi ,{\Psi }^{\prime })\right)$$

(12)

$$-\left\{\begin{array}{c}{\tilde{\mathcal{A}}}_P^{SL}=\left({-\mathfrak{a}}_5,{-\mathfrak{a}}_4,-{\mathfrak{a}}_3,{-\mathfrak{a}}_2,{-\mathfrak{a}}_1;\Psi \right)\\ {\tilde{\mathcal{B}}}_P^{SL}=\left(-{\mathfrak{b}}_5,{-\mathfrak{b}}_4,{-\mathfrak{b}}_3,{-\mathfrak{b}}_2,{-\mathfrak{b}}_1;{\Psi }^{\prime }\right)\end{array}\right.$$

(13)

Similarly, the algebraic operations for multiplication, inverse, and division of these numbers are defined by Equations (14) to (16).

$${\tilde{\mathcal{A}}}_P^{SL}\times {\tilde{\mathcal{B}}}_P^{SL}\approx \left({\mathfrak{a}}_1\times {\mathfrak{b}}_1,{\mathfrak{a}}_2\times {\mathfrak{b}}_2,{\mathfrak{a}}_3\times {\mathfrak{b}}_3,{\mathfrak{a}}_4\times {\mathfrak{b}}_4,{\mathfrak{a}}_5\times {\mathfrak{b}}_5;min(\Psi ,{\Psi }^{\prime })\right)$$

(14)

$${\displaystyle \frac{1}{{\tilde{\mathcal{A}}}_P^{SL}}},{\displaystyle \frac{1}{{\tilde{\mathcal{B}}}_P^{SL}}}\left\{\begin{array}{c}{\tilde{\mathcal{A}}}_P^{SL}\approx ({\displaystyle \frac{1}{{\mathfrak{a}}_5}},{\displaystyle \frac{1}{{\mathfrak{a}}_4}},{\displaystyle \frac{1}{{\mathfrak{a}}_3}},{\displaystyle \frac{1}{{\mathfrak{a}}_2}},{\displaystyle \frac{1}{{\mathfrak{a}}_1}};\Psi )\\ {\tilde{\mathcal{B}}}_P^{SL}\approx ({\displaystyle \frac{1}{{\mathfrak{b}}_5}},{\displaystyle \frac{1}{{\mathfrak{b}}_4}},{\displaystyle \frac{1}{{\mathfrak{b}}_3}},{\displaystyle \frac{1}{{\mathfrak{b}}_2}},{\displaystyle \frac{1}{{\mathfrak{b}}_1}};{\Psi }^{\prime })\end{array}\right.,{\mathfrak{a}}_i,{\mathfrak{b}}_i\ne 0$$

(15)

$${\tilde{\mathcal{A}}}_P^{SL}÷{\tilde{\mathcal{B}}}_P^{SL}\approx \left({\displaystyle \frac{{\mathfrak{a}}_1}{{\mathfrak{b}}_5}},{\displaystyle \frac{{\mathfrak{a}}_2}{{\mathfrak{b}}_4}},{\displaystyle \frac{{\mathfrak{a}}_3}{{\mathfrak{b}}_3}},{\displaystyle \frac{{\mathfrak{a}}_4}{{\mathfrak{b}}_2}},{\displaystyle \frac{{\mathfrak{a}}_5}{{\mathfrak{b}}_1}};\mathit{min}\left(\Psi ,{\Psi }^{\prime }\right)\right),{\mathfrak{b}}_i\ne 0$$

(16)

To normalize symmetric linear PFNs, Equation (17) can be used.

$$\begin{gathered} {\tilde{\mathcal{A}}}_{\omega }^{SL}=\left({\displaystyle \frac{{\mathfrak{b}}_1^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_5^j}},{\displaystyle \frac{{\mathfrak{b}}_2^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_4^j}},{\displaystyle \frac{{\mathfrak{b}}_3^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_3^j}},{\displaystyle \frac{{\mathfrak{b}}_4^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_2^j}},{\displaystyle \frac{{\mathfrak{b}}_5^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_1^j}};\mathit{min}\left({\Psi }^{\omega },{\Psi }^j\right)\right), \\ where{\sum }_{j=1}^n{\mathfrak{b}}_i^j\ne 0,{\mathfrak{b}}_i^{\omega }>0,andj=\mathrm{1,2}\dots ,n. \end{gathered}$$

(17)

Definition 8.

Algebraic operations of asymmetric linear PFNs. If ${\tilde{\mathcal{A}}}_P^{AL}=\left({\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4,{\mathfrak{a}}_5;{\Psi }_1,{\Psi }_2\right)$ under the conditions ${\mathfrak{a}}_1\ge 0,{\mathfrak{a}}_1\ge {\mathfrak{a}}_2\ge {\mathfrak{a}}_3\ge {\mathfrak{a}}_4\ge {\mathfrak{a}}_5$ and ${\tilde{\mathcal{B}}}_P^{AL}=\left({\mathfrak{b}}_1,{\mathfrak{b}}_2,{\mathfrak{b}}_3,{\mathfrak{b}}_4,{\mathfrak{b}}_5;{\Psi }_1^{\prime },{\Psi }_2^{\prime }\right)$ under the conditions ${\mathfrak{b}}_1\ge 0,{\mathfrak{b}}_1\ge {\mathfrak{b}}_2\ge {\mathfrak{b}}_3\ge {\mathfrak{b}}_4\ge {\mathfrak{b}}_5$ are two asymmetric linear PFNs, the algebraic operations are represented by Equations (18) to (20).

$$\left\{\begin{array}{c}{\tilde{\mathcal{A}}}_P^{AL}+{\tilde{\mathcal{B}}}_P^{AL}=\left({\mathfrak{a}}_1+{\mathfrak{b}}_1,{\mathfrak{a}}_2+{\mathfrak{b}}_2,{\mathfrak{a}}_3+{\mathfrak{b}}_3,{\mathfrak{a}}_4+{\mathfrak{b}}_4,{\mathfrak{a}}_5+{\mathfrak{b}}_5;\mathit{min}\left({\Psi }_1,{\Psi }_1^{\prime }\right),\mathit{min}\left({\Psi }_2,{\Psi }_2^{\prime }\right)\right)\\ {{\mathsf{\Psi }}_1^{″}}_{\left({\mathfrak{a}}_2+{\mathfrak{b}}_2\right)}=\min \left({\Psi }_1,{\Psi }_1^{\prime }\right),{{\mathsf{\Psi }}_2^{″}}_{\left({\mathfrak{a}}_4+{\mathfrak{b}}_4\right)}=\mathit{min}\left({\Psi }_2,{\Psi }_2^{\prime }\right)\end{array}\right.$$

(18)

$$\left\{\begin{array}{c}{\tilde{\mathcal{A}}}_P^{AL}-{\tilde{\mathcal{B}}}_P^{AL}=\left({\mathfrak{a}}_1-{\mathfrak{b}}_5,{\mathfrak{a}}_2-{\mathfrak{b}}_4,{\mathfrak{a}}_3-{\mathfrak{b}}_3,{\mathfrak{a}}_4-{\mathfrak{b}}_2,{\mathfrak{a}}_5-{\mathfrak{b}}_1;\mathit{min}\left({\Psi }_1,{\Psi }_2^{\prime }\right),min({\Psi }_2,{\Psi }_1^{\prime })\right)\\ {{\mathsf{\Psi }}_1^{″}}_{\left({\mathfrak{a}}_2-{\mathfrak{b}}_2\right)}=\min \left({\Psi }_1,{\Psi }_2^{\prime }\right),{{\mathsf{\Psi }}_2^{″}}_{\left({\mathfrak{a}}_4-{\mathfrak{b}}_2\right)}=\min ({\mathsf{\Psi }}_2,{\mathsf{\Psi }}_1^{\prime })\end{array}\right.$$

(19)

$$-\left\{\begin{array}{c}{\tilde{\mathcal{A}}}_P^{AL}=\left({-\mathfrak{a}}_5,{-\mathfrak{a}}_4,-{\mathfrak{a}}_3,{-\mathfrak{a}}_2,{-\mathfrak{a}}_1;{\Psi }_2,{\Psi }_1\right)\\ {\tilde{\mathcal{B}}}_P^{AL}=\left(-{\mathfrak{b}}_5,{-\mathfrak{b}}_4,{-\mathfrak{b}}_3,{-\mathfrak{b}}_2,{-\mathfrak{b}}_1;{\Psi }_2^{\prime },{\Psi }_1^{\prime }\right)\end{array}\right.$$

(20)

Algebraic operations, including multiplication, inverse, and division for asymmetric linear PFNs, are presented in Equations (21) to (23).

$$\begin{gathered} {\tilde{\mathcal{A}}}_P^{AL}\times {\tilde{\mathcal{B}}}_P^{AL}\approx \left({\mathfrak{a}}_1\times {\mathfrak{b}}_1,{\mathfrak{a}}_2\times {\mathfrak{b}}_2,{\mathfrak{a}}_3\times {\mathfrak{b}}_3,{\mathfrak{a}}_4\times {\mathfrak{b}}_4,{\mathfrak{a}}_5 \\ \times {\mathfrak{b}}_5;\mathit{min}\left({\Psi }_1,{\Psi }_1^{\prime }\right),\mathit{min}\left({\Psi }_2,{\Psi }_2^{\prime }\right)\right) \end{gathered}$$

(21)

$${\displaystyle \frac{1}{{\tilde{\mathcal{A}}}_P^{AL}}},{\displaystyle \frac{1}{{\tilde{\mathcal{B}}}_P^{AL}}}\left\{\begin{array}{c}{\tilde{\mathcal{A}}}_P^{AL}\approx ({\displaystyle \frac{1}{{\mathfrak{a}}_5}},{\displaystyle \frac{1}{{\mathfrak{a}}_4}},{\displaystyle \frac{1}{{\mathfrak{a}}_3}},{\displaystyle \frac{1}{{\mathfrak{a}}_2}},{\displaystyle \frac{1}{{\mathfrak{a}}_1}};{\Psi }_2,{\Psi }_1)\\ {\tilde{\mathcal{B}}}_P^{AL}\approx ({\displaystyle \frac{1}{{\mathfrak{b}}_5}},{\displaystyle \frac{1}{{\mathfrak{b}}_4}},{\displaystyle \frac{1}{{\mathfrak{b}}_3}},{\displaystyle \frac{1}{{\mathfrak{b}}_2}},{\displaystyle \frac{1}{{\mathfrak{b}}_1}};{\Psi }_2^{\prime },{\Psi }_1^{\prime })\end{array}\right.,{\mathfrak{a}}_i,{\mathfrak{b}}_i\ne 0$$

(22)

$${\tilde{\mathcal{A}}}_P^{AL}÷{\tilde{\mathcal{B}}}_P^{AL}\approx \left({\displaystyle \frac{{\mathfrak{a}}_1}{{\mathfrak{b}}_5}},{\displaystyle \frac{{\mathfrak{a}}_2}{{\mathfrak{b}}_4}},{\displaystyle \frac{{\mathfrak{a}}_3}{{\mathfrak{b}}_3}},{\displaystyle \frac{{\mathfrak{a}}_4}{{\mathfrak{b}}_2}},{\displaystyle \frac{{\mathfrak{a}}_5}{{\mathfrak{b}}_1}};\mathit{min}\left({\Psi }_1,{\Psi }_2^{\prime }\right),min({\Psi }_2,{\Psi }_1^{\prime })\right),{\mathfrak{b}}_i\ne 0$$

(23)

To normalize asymmetric linear PFNs, Equation (24) can be used.

$$\begin{gathered} {\tilde{\mathcal{A}}}_{\omega }^{AL}=\left({\displaystyle \frac{{\mathfrak{b}}_1^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_5^j}},{\displaystyle \frac{{\mathfrak{b}}_2^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_4^j}},{\displaystyle \frac{{\mathfrak{b}}_3^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_3^j}},{\displaystyle \frac{{\mathfrak{b}}_4^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_2^j}},{\displaystyle \frac{{\mathfrak{b}}_5^{\omega }}{{\sum }_{j=1}^n{\mathfrak{b}}_1^j}};\mathit{min}\left({\Psi }_1^{\omega },{\Psi }_2^j\right),min({\Psi }_2^{\omega },{\Psi }_1^j)\right), \\ where{\sum }_{j=1}^n{\mathfrak{b}}_i^j\ne 0,{\mathfrak{b}}_i^{\omega }>0,andj=\mathrm{1,2}\dots ,n. \end{gathered}$$

(24)

Definition 9.

([52]). A Z-number is represented as an ordered pair $Z=(\tilde{A},\widetilde{{\rm B}})$, where $\left(\tilde{A}\right)$ is a fuzzy number describing the boundary or limit of a variable ${\rm X}$, and $\left(\widetilde{{\rm B}}\right)$ is a fuzzy number that signifies the reliability or certainty of $\left(\tilde{A}\right)$.

Z-numbers extend the concept of classical fuzzy numbers by incorporating additional uncertainty, enhancing their capability to handle imprecise information in decision-making processes. This structure allows Z-numbers to address uncertainties more effectively than classical fuzzy numbers, making them particularly suitable for MCDM. Furthermore, the components $\left(\tilde{A}\right)$ and $\left(\widetilde{{\rm B}}\right)$ in Z-numbers can be expressed as numerical values or natural language terms, providing flexibility in representing uncertain information [53].

Z-numbers have two fundamental components: a fuzzy constraint function $\left(\tilde{A}\right)$ and a fuzzy reliability function $\left(\widetilde{{\rm B}}\right)$. Let ${\mu }_{\tilde{A}}\left(x\right)$ represent the fuzzy constraint membership function and ${\mu }_{\widetilde{{\rm B}}}\left(x\right)$ denote the fuzzy reliability membership function. Accordingly, the components $\left(\tilde{A}\right)$ and $\left(\widetilde{{\rm B}}\right)$ are defined as per Equations (25) and (26) [54]:

$$\tilde{A}=\left\{〈x,{\mu }_{\tilde{A}}(x)〉|{\mu }_{\tilde{A}}(x)\in [0,1]\right\},$$

(25)

$$\widetilde{{\rm B}}=\left\{〈x,{\mu }_{\widetilde{{\rm B}}}(x)〉|{\mu }_{\widetilde{{\rm B}}}(x)\in [0,1]\right\}.$$

(26)

The weighted constraint function of a Z-number, expressed as $\left({\tilde{Z}}_{\Omega }\right)$, is obtained by integrating the reliability weight into the constraint, as outlined in Equation (27). Here, $\mathsf{\Omega }$ signifies the defuzzified value or score function of the fuzzy reliability [55]:

$${\tilde{Z}}_{\Omega }=\left\{〈x,{\mu }_{{\tilde{A}}^{\Omega }}(x)〉|{\mu }_{{\tilde{A}}^{\Omega }}\left(x\right)=\Omega {\mu }_{\tilde{A}}\left(x\right),{\mu }_{\tilde{A}}(x)\in [0,1]\right\}.$$

(27)

To convert ${\tilde{Z}}_{\Omega }$ into a standard fuzzy number $\tilde{Z}$, Equation (28) is applied.

$$\tilde{Z}=\left\{〈x,{\mu }_{\tilde{Z}}(x)〉|{\mu }_{\tilde{Z}}(x)={\mu }_{\tilde{A}}\left({\displaystyle \frac{x}{\sqrt{\Omega }}}\right),{\mu }_{\tilde{A}}\in [0,1]\right\}.$$

(28)

Z-numbers can be defined based on membership functions of various types of POFNs, such as TFNs, TRFNs, HFNs, and others. Figure 5 illustrates a symmetric linear pentagonal Z-number, where $\tilde{Z}=\left\{\tilde{A}:\left(a_1,a_2,a_3,a_4,a_5,\Psi \right),\widetilde{{\rm B}}:\left(b_1,b_2,b_3,b_4,b_5,\Psi \right)\right\}$ is defined based on pentagonal membership functions.

The topics discussed in this section, such as deriving membership functions, the formulation of $\mathcal{a}$-cuts, and arithmetic operations in symmetric and asymmetric linear PFNs, can be generalized to other symmetric and asymmetric linear POFNs, such as HFNs. However, to avoid redundancy and excessive length, their detailed explanations are omitted. It is also worth mentioning that after performing certain algebraic operations on PFNs within the proposed decision-making approaches, the results may not strictly remain PFNs. However, we approximately consider these numbers as PFNs.

3.2. The Crispification of POFNs

The crispification process in various types of linear POFNs is of significant importance. To accurately and reliably reflect the results of employing POFNs in different decision-making approaches, an appropriate crispification process that considers the characteristics of linear POFNs is required. Linear POFNs can be either symmetric (${\mathsf{\Psi }}_1={\mathsf{\Psi }}_2$) or asymmetric (${\mathsf{\Psi }}_1\ne {\mathsf{\Psi }}_2$), and obtaining their defuzzified values differs fundamentally in each case. These differences must be considered to ensure the results are accurate and realistic. In this section, the crispification process for linear POFNs is proposed for use in Z-transformation operations, where the crisp value of the fuzzy reliability function is required. Additionally, the proposed approach for the defuzzification of linear POFNs can be utilized in various stages of decision-making approaches as needed.

Various approaches for performing the crispification process on POFNs exist in the literature, such as the center of gravity approach [56], which is one of the commonly used methods. However, most crispification approaches for POFNs either involve computational complexities or are effective only in the symmetric case. Furthermore, crispification approaches for POFNs should be capable of accounting for the symmetry or asymmetry of the numbers in their computations. Therefore, by utilizing the concepts of EI and EV, a precise approach can be proposed for the crispification process of POFNs (specifically pentagonal and hexagonal). This approach is capable of defuzzifying both symmetric and asymmetric linear PFNs as well as symmetric and asymmetric linear HFNs. The proposed method considers the structural characteristics of these numbers during the crispification process, providing accurate results while reducing computational complexity compared to other approaches.

Definition 10.

The $\mathcal{a}$-cut of the fuzzy number $\tilde{\mathcal{A}}$ is represented as ${\tilde{\mathcal{A}}}_{\mathcal{a}}=\left\{〈\mathcal{k}\in \mathcal{K}〉\left|{\mu }_{\tilde{\mathcal{A}}}\right(\mathcal{k})\ge \mathcal{a}\right\}$. The $\mathcal{a}$-cuts are bounded and closed intervals, which are denoted as ${\tilde{\mathcal{A}}}_{\mathcal{a}}=\left[f_{\mathcal{A}}^{-1}\left(\mathcal{a}\right),g_{\mathcal{A}}^{-1}\left(\mathcal{a}\right)\right]$ [57]. The EI of the fuzzy number $\tilde{\mathcal{A}}$, denoted by $EI\left(\tilde{\mathcal{A}}\right)$, is expressed by Equation (29) [58]:

$$EI\left(\tilde{\mathcal{A}}\right)=\left[E_{\tilde{\mathcal{A}}}^L,E_{\tilde{\mathcal{A}}}^R\right]=\left[\underset{0}{\overset{1}{\int }}{f}_{\tilde{\mathcal{A}}}^{-1}\left(\mathcal{a}\right)d\mathcal{a},\underset{0}{\overset{1}{\int }}{g}_{\tilde{\mathcal{A}}}^{-1}\left(\mathcal{a}\right)d\mathcal{a}\right].$$

(29)

The expected value of the fuzzy number $\tilde{\mathcal{A}}$, denoted by $EV\left(\tilde{\mathcal{A}}\right)$, is calculated using Equation (30) [58].

$$EV\left(\tilde{\mathcal{A}}\right)={\displaystyle \frac{E_{\tilde{\mathcal{A}}}^L+E_{\tilde{\mathcal{A}}}^R}{2}}.$$

(30)

Definition 11.

Assume that  ${\tilde{\mathcal{A}}}_{SP}=\left({\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4,{\mathfrak{a}}_5;\Psi \right)$  is a symmetric PFN. According to Definition 6 and Equation (9),  $EI\left({\tilde{\mathcal{A}}}_{SP}\right)=\left[E_{{\tilde{\mathcal{A}}}_{SP}}^L,E_{{\tilde{\mathcal{A}}}_{SP}}^R\right]$  is defined as expressed in Equations (31) and (32):

$$\begin{gathered} E_{{\tilde{\mathcal{A}}}_{SP}}^L={\int }_0^{\Psi }\left({\mathfrak{a}}_1+{\displaystyle \frac{\mathcal{a}}{\Psi }}\left({\mathfrak{a}}_2-{\mathfrak{a}}_1\right)\right)d\mathcal{a}+{\int }_{\Psi }^1\left({\mathfrak{a}}_2+{\displaystyle \frac{\mathcal{a}-\Psi }{1-\Psi }}\left({\mathfrak{a}}_3-{\mathfrak{a}}_2\right)\right)d\mathcal{a}, \\ ={\displaystyle \frac{({\mathfrak{a}}_2+{\mathfrak{a}}_1)\Psi }{2}}+{\displaystyle \frac{({\mathfrak{a}}_3+{\mathfrak{a}}_2)(1-\Psi )}{2}}, \end{gathered}$$

(31)

$$\begin{gathered} E_{{\tilde{\mathcal{A}}}_{SP}}^R={\int }_0^{\Psi }\left({\mathfrak{a}}_5-{\displaystyle \frac{\mathcal{a}}{\Psi }}\left({\mathfrak{a}}_5-{\mathfrak{a}}_4\right)\right)d\mathcal{a}+{\int }_{\Psi }^1\left({\mathfrak{a}}_4-{\displaystyle \frac{\mathcal{a}-\Psi }{1-\Psi }}\left({\mathfrak{a}}_4-{\mathfrak{a}}_3\right)\right)d\mathcal{a}, \\ ={\displaystyle \frac{({\mathfrak{a}}_4+{\mathfrak{a}}_5)\Psi }{2}}+{\displaystyle \frac{({\mathfrak{a}}_4+{\mathfrak{a}}_3)(1-\Psi )}{2}}. \end{gathered}$$

(32)

Based on Equation (30) and after calculating and simplifying Equations (31) and (32), $EV\left({\tilde{\mathcal{A}}}_{SP}\right)$, which represents the crisp value of the symmetric linear PFN, is obtained as shown in Equation (33):

$$EV\left({\tilde{\mathcal{A}}}_{SP}\right)={\displaystyle \frac{\left({\mathfrak{a}}_2+{\mathfrak{a}}_1\right)\Psi +\left({\mathfrak{a}}_3+{\mathfrak{a}}_2\right)\left(1-\Psi \right)+\left({\mathfrak{a}}_4+{\mathfrak{a}}_5\right)\Psi +({\mathfrak{a}}_4+{\mathfrak{a}}_3)(1-\Psi )}{4}}.$$

(33)

Definition 12.

Consider that ${\tilde{\mathcal{A}}}_{AP}=\left({\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4,{\mathfrak{a}}_5;{\Psi }_1,{\Psi }_2\right)$ represents an asymmetric PFN. Based on Definition 6 and Equation (10), the $EI\left({\tilde{\mathcal{A}}}_{AP}\right)=\left[E_{{\tilde{\mathcal{A}}}_{AP}}^L,E_{{\tilde{\mathcal{A}}}_{AP}}^R\right]$ is defined as shown in Equations (34) and (35):

$$\begin{gathered} E_{{\tilde{\mathcal{A}}}_{AP}}^L={\int }_0^{{\Psi }_1}\left({\mathfrak{a}}_1+{\displaystyle \frac{\mathcal{a}}{{\Psi }_1}}\left({\mathfrak{a}}_2-{\mathfrak{a}}_1\right)\right)d\mathcal{a}+{\int }_{{\Psi }_1}^1\left({\mathfrak{a}}_2+{\displaystyle \frac{\mathcal{a}-{\Psi }_1}{1-{\Psi }_1}}\left({\mathfrak{a}}_3-{\mathfrak{a}}_2\right)\right)d\mathcal{a}, \\ ={\displaystyle \frac{({\mathfrak{a}}_2+{\mathfrak{a}}_1){\Psi }_1}{2}}+{\displaystyle \frac{({\mathfrak{a}}_3+{\mathfrak{a}}_2)(1-{\Psi }_1)}{2}}, \end{gathered}$$

(34)

$$\begin{gathered} E_{{\tilde{\mathcal{A}}}_{AP}}^R={\int }_0^{{\Psi }_2}\left({\mathfrak{a}}_5-{\displaystyle \frac{\mathcal{a}}{{\Psi }_2}}\left({\mathfrak{a}}_5-{\mathfrak{a}}_4\right)\right)d\mathcal{a}+{\int }_{{\Psi }_2}^1\left({\mathfrak{a}}_4-{\displaystyle \frac{\mathcal{a}-{\Psi }_2}{1-{\Psi }_2}}\left({\mathfrak{a}}_4-{\mathfrak{a}}_3\right)\right)d\mathcal{a}, \\ ={\displaystyle \frac{({\mathfrak{a}}_4+{\mathfrak{a}}_5){\Psi }_2}{2}}+{\displaystyle \frac{({\mathfrak{a}}_4+{\mathfrak{a}}_3)(1-{\Psi }_2)}{2}}. \end{gathered}$$

(35)

Using Equation (30), and after performing the calculations and simplifications outlined in Equations (34) and (35), the crisp value of the asymmetric linear PFN, denoted as $EV\left({\tilde{\mathcal{A}}}_{AP}\right)$, is determined as presented in Equation (36):

$$EV\left({\tilde{\mathcal{A}}}_{AP}\right)={\displaystyle \frac{\left({\mathfrak{a}}_2+{\mathfrak{a}}_1\right){\Psi }_1+\left({\mathfrak{a}}_3+{\mathfrak{a}}_2\right)\left(1-{\Psi }_1\right)+\left({\mathfrak{a}}_4+{\mathfrak{a}}_5\right){\Psi }_2+({\mathfrak{a}}_4+{\mathfrak{a}}_3)(1-{\Psi }_2)}{4}}.$$

(36)

Considering the formulation of $\mathcal{a}$-cut for various types of POFNs and the concepts discussed earlier, the $EV$ value for any POFN, whether symmetric or asymmetric, can be calculated using the proposed approach. For instance, if ${\tilde{\mathcal{A}}}_{SH}=\left({\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4,{\mathfrak{a}}_5,{\mathfrak{a}}_6;\Psi \right)$ and ${\tilde{\mathcal{A}}}_{AH}=\left({\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4,{\mathfrak{a}}_5,{\mathfrak{a}}_6;{\Psi }_1,{\Psi }_2\right)$ are considered symmetric and asymmetric linear HFNs, respectively, their $EV\left({\tilde{\mathcal{A}}}_{SH}\right)$ and $EV\left({\tilde{\mathcal{A}}}_{AH}\right)$ values can be obtained through similar operations, as shown in Equations (37) and (38):

$$EV\left({\tilde{\mathcal{A}}}_{SH}\right)={\displaystyle \frac{\left({\mathfrak{a}}_2+{\mathfrak{a}}_1\right)\Psi +\left({\mathfrak{a}}_3+{\mathfrak{a}}_2\right)\left(1-\Psi \right)+\left({\mathfrak{a}}_4+{\mathfrak{a}}_5\right)(1-\Psi )+({\mathfrak{a}}_5+{\mathfrak{a}}_6)\Psi }{4}},$$

(37)

$$EV\left({\tilde{\mathcal{A}}}_{AH}\right)={\displaystyle \frac{\left({\mathfrak{a}}_2+{\mathfrak{a}}_1\right){\Psi }_1+\left({\mathfrak{a}}_3+{\mathfrak{a}}_2\right)\left(1-{\Psi }_1\right)+\left({\mathfrak{a}}_4+{\mathfrak{a}}_5\right)(1-{\Psi }_2)+({\mathfrak{a}}_5+{\mathfrak{a}}_6){\Psi }_2}{4}}.$$

(38)

As described, the proposed approach accurately calculates the crisp values of various POFNs while considering their structural characteristics, without the computational complexities of other methods. These crisp values can be utilized in operations involving pentagonal and hexagonal Z-numbers. Moreover, they have significant applications in various stages of decision-making approaches that require the crispification of POFNs.

3.3. The Proposed PZ-LMAW

The LMAW approach, proposed by Pamučar et al. [59], unlike most MCDM methods, possesses the capability to calculate the weights of decision-making criteria and prioritize the alternatives of the problem simultaneously and integrally. The LMAW approach has so far been developed in various forms to address uncertainty [60,61,62]. In this study, the LMAW approach is also proposed to address the uncertainty present in the decision-making space, utilizing PFNs, Z-numbers, and the concepts of EI and EV to tackle the SES problem.

The proposed approach follows steps similar to the traditional LMAW method, with the distinction that the required data are gathered and analyzed based on the characteristics of linear symmetric PFNs, as described in Section 3.1. Additionally, the crispification process required to convert Z-PFNs to regular PFNs and PFNs to crisp values adheres to the concepts outlined in Section 3.2. Since only the part of the LMAW approach related to obtaining the weights of decision-making criteria is utilized in this study, the steps for applying the proposed PZ-LMAW approach are summarized as follows.

Step 1. In this step, the decision-making criteria $C=\left\{C_1,C_2,\dots ,C_n\right\}$ are identified and determined. Subsequently, the group of experts $\mathcal{Y}=\left\{{\mathcal{y}}_1,{\mathcal{y}}_2,\dots ,{\mathcal{y}}_n\right\}$ is selected. Each expert evaluates the criteria using two predefined linguistic scales. The first scale corresponds to the fuzzy restriction function of Z-numbers, and the second scale represents the fuzzy reliability function of Z-numbers, both of which consist of PFNs. Each expert evaluates the criteria by assigning the highest value on the linguistic scale to the most important criterion and the lowest value to the less important criteria. Additionally, using the fuzzy reliability function of Z-numbers, each expert expresses their level of confidence in their preferences.

After evaluating the criteria using predefined linguistic scales, the fuzzy constraint vector ${\tilde{\mathcal{A}}}^{\mathcal{y}}=\left\{{\tilde{\mathfrak{a}}}_{C1}^{\mathcal{y}},{\tilde{\mathfrak{a}}}_{C2}^{\mathcal{y}},\dots ,{\tilde{\mathfrak{a}}}_{Cn}^{\mathcal{y}}\right\}$ and the fuzzy reliability vector ${\tilde{\mathcal{B}}}^{\mathcal{y}}=\left\{{\tilde{\mathfrak{b}}}_{C1}^{\mathcal{y}},{\tilde{\mathfrak{b}}}_{C2}^{\mathcal{y}},\dots ,{\tilde{\mathfrak{b}}}_{Cn}^{\mathcal{y}}\right\}$ are formed. Here, ${\tilde{\mathfrak{a}}}_{C1}^{\mathcal{y}}$ and ${\tilde{\mathfrak{b}}}_{C1}^{\mathcal{y}}$ are the values assigned by expert $\mathcal{y}$ to criterion $j$, which have pentagonal membership functions.

Table 2. Linguistic terms and membership function equivalents of PFN restriction and reliability functions.

Linguistic Constraint	PFN Membership Function	Linguistic Reliability	PFN Membership Function
Extremely low (EL)	(1, 1, 1, 1, 1; 0.7)	Absolutely reliable (AR)	(0.8, 0.85, 0.9, 0.95, 1; 0.7)
Very slight (VS)	(1, 1.25, 1.5, 1.75, 2; 0.7)	Strongly reliable (SR)	(0.7, 0.75, 0.8, 0.85, 0.9; 0.7)
Slight (S)	(1.5, 1.75, 2, 2.25, 2.5; 0.7)	Very highly reliable (VR)	(0.6, 0.65, 0.7, 0.75, 0.8; 0.7)
Moderately low (ML)	(2, 2.25, 2.5, 2.75, 3; 0.7)	Highly reliable (HR)	(0.5, 0.55, 0.6, 0.65, 0.7; 0.7)
Equal (E)	(2.5, 2.75, 3, 3.25, 3.5; 0.7)	Fairly reliable (FR)	(0.4, 0.45, 0.5, 0.55, 0.6; 0.7)
Moderately high (MH)	(3, 3.25, 3.5, 3.75, 4; 0.7)	Weakly reliable (WR)	(0.3, 0.35, 0.4, 0.45, 0.5; 0.7)
High (H)	(3.5, 3.75, 4, 4.25, 4.5; 0.7)	Very weakly reliable (VW)	(0.2, 0.25, 0.3, 0.35, 0.4; 0.7)
Very high (VH)	(4, 4.25, 4.5, 4.75, 5; 0.7)	Strongly unreliable (SU)	(0.1, 0.15, 0.2, 0.25, 0.3; 0.7)
Extremely high (EH)	(4.5, 4.75, 5, 5, 5; 0.7)	Absolutely unreliable (AU)	(0, 0, 0, 0, 0.2; 0.7)

presents the linguistic scales, including the fuzzy constraints and fuzzy reliability scales, along with their corresponding pentagonal membership functions. Figure 6 illustrates the linguistic scale for the fuzzy constraint functions of Z-numbers, represented under symmetric PFN membership functions.

Step 2. Conducting the Z-transformation operation. In this section, the experts’ preferences regarding decision-making criteria, expressed as pentagonal Z-numbers $Z_j^{\mathcal{y}}=\left[\left(\tilde{\mathcal{A}}\right),\left(\tilde{\mathcal{B}}\right)\right]=\left[\left({\mathfrak{a}}_{Cj}^{\mathcal{A}},{\mathfrak{a}}_{Cj}^{\mathcal{A}},{\mathfrak{a}}_{Cj}^{\mathcal{A}},{\mathfrak{a}}_{Cj}^{\mathcal{A}},{\mathfrak{a}}_{Cj}^{\mathcal{A}};\Psi \right),\left({\mathfrak{b}}_{Cj}^{\mathcal{B}},{\mathfrak{b}}_{Cj}^{\mathcal{B}},{\mathfrak{b}}_{Cj}^{\mathcal{B}},{\mathfrak{b}}_{Cj}^{\mathcal{B}},{\mathfrak{b}}_{Cj}^{\mathcal{B}};\Psi \right)\right],j=1.2\dots ,n$, are converted into conventional PFNs. To transform Z-numbers into conventional fuzzy numbers in decision-making approaches, a commonly followed process is employed, which is described below [60,63,64]. In this process, the deterministic value of the fuzzy reliability function is first obtained, and the derived weight is then applied to the fuzzy constraint function. Here, the defuzzification process of the fuzzy reliability function for pentagonal Z-numbers is performed using Equation (33).

Step 2.1. If $\tilde{\mathcal{A}}$ represents the fuzzy constraint function of a pentagonal Z-number and $\tilde{\mathcal{B}}$ represents the fuzzy reliability function of a pentagonal Z-number, the deterministic value of $\tilde{\mathcal{B}}$, denoted as $EV\left(\tilde{\mathcal{B}}\right)$, is obtained using Equation (33). This is because $\tilde{\mathcal{B}}$ is a symmetric linear PFN (refer to Definition 11).

Step 2.2. Apply the deterministic value of the reliability function $EV\left(\tilde{\mathcal{B}}\right)$ to the fuzzy constraint function $\tilde{\mathcal{A}}$ using Equation (39):

$${\tilde{Z}}^{EV}=\left\{〈x,{\mu }_{{\tilde{\mathcal{A}}}^{EV}}(x)〉|{\mu }_{{\tilde{\mathcal{A}}}^{EV}}(x)=EV{\mu }_{\tilde{\mathcal{A}}}(x)\right\}.$$

(39)

Step 2.3. Transform the weighted pentagonal Z-number into a conventional PFN according to Equations (40) and (41):

$$\tilde{Z}=\left\{〈x,{\mu }_{\tilde{Z}}(x)〉|{\mu }_{\tilde{Z}}(x)={\mu }_{\tilde{\mathcal{A}}}({\displaystyle \frac{x}{\sqrt{EV}}})\right\},$$

(40)

$${\tilde{Z}}^{\prime }=\sqrt{EV}.\tilde{\mathcal{A}}=\left(\sqrt{EV}.{\mathfrak{a}}_1,\sqrt{EV}.{\mathfrak{a}}_2,\sqrt{EV}.{\mathfrak{a}}_3,\sqrt{EV}.{\mathfrak{a}}_4,\sqrt{EV}.{\mathfrak{a}}_5;\Psi \right).$$

(41)

After completing the Z-transformation operations, the preference vectors for each expert are obtained in the form ${\tilde{\mathcal{P}}}^{\mathcal{y}}=\left({\widetilde{{\rm Y}}}_{C1}^{\mathcal{y}},{\widetilde{{\rm Y}}}_{C2}^{\mathcal{y}}\dots ,{\widetilde{{\rm Y}}}_{Cn}^{\mathcal{y}}\right)$.

Step 3. Determining the fuzzy anti-ideal point (${\widetilde{\mathsf{{\rm Y}}}}_{AIP}$). The ${\widetilde{\mathsf{{\rm Y}}}}_{AIP}$ is a fuzzy value smaller than the smallest element of the priority vector obtained in Step 2 and can be calculated as ${\widetilde{\mathsf{{\rm Y}}}}_{AIP}={\displaystyle \frac{{\widetilde{{\rm Y}}}_{min}^{\mathcal{y}}}{S}},S=3$ [59].

Step 4. Obtaining the fuzzy relation vector. The fuzzy relation vector, denoted as ${\tilde{\mathcal{R}}}^{\mathcal{y}}=\left({\tilde{\eta }}_{C1}^{\mathcal{y}},{\tilde{\eta }}_{C2}^{\mathcal{y}}\dots ,{\tilde{\eta }}_{Cn}^{\mathcal{y}}\right)$, is calculated using Equation (42). This vector represents the relationship between the elements of the fuzzy priority vector and the fuzzy absolute anti-ideal point ${\widetilde{\mathsf{{\rm Y}}}}_{AIP}$:

$${\tilde{\mathsf{\eta }}}_{Cn}^{\mathcal{y}}={\displaystyle \frac{{\widetilde{{\rm Y}}}_{Cn}^{\mathcal{y}}}{{\widetilde{{\rm Y}}}_{AIP}}}=\left({\displaystyle \frac{{\widetilde{{\rm Y}}}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_1\right)}}{{\widetilde{{\rm Y}}}_{AIP}^{\left({\mathfrak{a}}_5\right)}}},{\displaystyle \frac{{\widetilde{{\rm Y}}}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_2\right)}}{{\widetilde{{\rm Y}}}_{AIP}^{\left({\mathfrak{a}}_4\right)}}},{\displaystyle \frac{{\widetilde{{\rm Y}}}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_3\right)}}{{\widetilde{{\rm Y}}}_{AIP}^{\left({\mathfrak{a}}_3\right)}}},{\displaystyle \frac{{\widetilde{{\rm Y}}}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_4\right)}}{{\widetilde{{\rm Y}}}_{AIP}^{\left({\mathfrak{a}}_2\right)}}},{\displaystyle \frac{{\widetilde{{\rm Y}}}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_5\right)}}{{\widetilde{{\rm Y}}}_{AIP}^{\left({\mathfrak{a}}_1\right)}}};min(\Psi ,{\Psi }^{\prime })\right).$$

(42)

Step 5. Calculating the criteria weight coefficients. The weight coefficients of the criteria for each expert are calculated using Equation (43):

$$\begin{gathered} {\tilde{\mathcal{W}}}_j^{\mathcal{y}}=\left({\displaystyle \frac{ln\left({\tilde{\eta }}_{Cn}^{\mathcal{y}}\right)}{ln\left({\prod }_{j=1}^n{\tilde{\eta }}_{Cn}^{\mathcal{y}}\right)}}\right), \\ =\left({\displaystyle \frac{ln\left({\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)}{ln\left({\prod }_{j=1}^n{\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)}},{\displaystyle \frac{ln\left({\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)}{ln\left({\prod }_{j=1}^n{\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)}},{\displaystyle \frac{ln\left({\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)}{ln\left({\prod }_{j=1}^n{\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)}},{\displaystyle \frac{ln\left({\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)}{ln\left({\prod }_{j=1}^n{\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)}},{\displaystyle \frac{ln\left({\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)}{ln\left({\prod }_{j=1}^n{\tilde{\eta }}_{Cn}^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)}}\right). \end{gathered}$$

(43)

Step 6. Aggregation. In this step, the weight coefficient vectors obtained for each expert are aggregated using Equations (44) and (45), which employ the pentagonal fuzzy Dombi aggregation (PFDA) operators [65,66,67]. This process calculates the final weight coefficient vector as follows:

$$PFDA({\overline{\tilde{\mathcal{W}}}}_j)=\left\{\begin{array}{c}{\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}={\displaystyle \frac{\sum _{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)}{1+{\left\{{\sum }_{j=1}^n{\mathcal{S}}_j{\left(\frac{1-f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)}{f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)}\right)}^p\right\}}^{1/p}}}\\ {\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_2\right)}={\displaystyle \frac{\sum _{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)}{1+{\left\{{\sum }_{j=1}^n{\mathcal{S}}_j{\left(\frac{1-f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)}{f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)}\right)}^p\right\}}^{1/p}}}\\ {\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_3\right)}={\displaystyle \frac{\sum _{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)}{1+{\left\{{\sum }_{j=1}^n{\mathcal{S}}_j{\left(\frac{1-f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)}{f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)}\right)}^p\right\}}^{1/p}}}\\ {\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_4\right)}={\displaystyle \frac{\sum _{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)}{1+{\left\{{\sum }_{j=1}^n{\mathcal{S}}_j{\left(\frac{1-f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)}{f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)}\right)}^p\right\}}^{1/p}}}\\ {\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}={\displaystyle \frac{\sum _{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)}{1+{\left\{{\sum }_{j=1}^n{\mathcal{S}}_j{\left(\frac{1-f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)}{f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)}\right)}^p\right\}}^{1/p}}}\end{array}\right.$$

(44)

where ${\mathcal{S}}_j$ represents the weights of the decision-makers, $p>0$ ensures the non-negativity of the numbers, and ${\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}$ to ${\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}$ correspond to the first through fifth components of the PFN (${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4,{\mathfrak{a}}_5;\mathsf{\Psi })$, respectively, for each expert $\mathcal{y}$ and criterion $j$:

$$f\left({\mathcal{W}}_j^{\mathcal{y}}\right)=\left\{\begin{array}{c}f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)={\displaystyle \frac{\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)}{{\sum }_{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_1\right)}\right)}}\\ f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)={\displaystyle \frac{\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)}{{\sum }_{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_2\right)}\right)}}\\ f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)={\displaystyle \frac{\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)}{{\sum }_{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_3\right)}\right)}}\\ f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)={\displaystyle \frac{\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)}{{\sum }_{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_4\right)}\right)}}\\ f\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)={\displaystyle \frac{\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)}{{\sum }_{j=1}^n\left({\mathcal{W}}_j^{\mathcal{y}\left({\mathfrak{a}}_5\right)}\right)}}\end{array}\right.$$

(45)

Step 7. Determining the deterministic weights $\left({\mathcal{W}}_j\right)$ of decision-making criteria. In this step, the deterministic weights of the decision-making criteria can be calculated using Equation (33).

3.4. The Proposed PZ-WASPAS

The WASPAS approach is an effective MCDM technique proposed by Zavadskas et al. [68]. Due to its combination of two well-known and effective MCDM methods, namely the weighted sum model (WSM) and the weighted product model (WPM), this approach has been extensively utilized in decision-making problems [69,70,71]. Various extensions of the WASPAS method exist in the literature, each developed for specific reasons and under certain conditions. Most of the developments in the WASPAS method have aimed to enhance its applicability in addressing decision-making problems under uncertainty, such as the fuzzy WASPAS and Z-WASPAS approaches [72,73].

In this study, we aim to propose an extended WASPAS approach based on asymmetric linear PFNs, considering the concepts discussed in Section 3.1, as well as the proposed method for the crispification process of symmetric and asymmetric linear POFNs in Section 3.2. The incorporation of asymmetric linear PFNs into the WASPAS approach enhances the flexibility of this method for dealing with environments characterized by uncertainty and demonstrates its capability in computations involving various types of POFNs. The PZ-LMAW approach, described in the previous section, was employed to determine the weights of decision-making criteria, while the proposed PZ-WASPAS approach will be used for the final ranking of suppliers. The steps for implementing the proposed PZ-WASPAS approach are outlined as follows.

Step 1. First, a group of experts is selected to evaluate the alternatives to the problem (in this case, the suppliers). Subsequently, the alternatives to the problem are identified and determined. The expert group must possess sufficient knowledge of the nature of the decision-making problem and the status of the alternatives. Additionally, the alternatives should be chosen from among the most significant and accessible options.

Step 2. The evaluation of the alternatives is carried out by the experts. Using two predefined linguistic scales, the experts assess the alternatives. In this process, each alternative is evaluated by each expert with respect to each criterion.

Table 3. Linguistic terms and membership function equivalents of asymmetric linear PFN restriction and reliability functions.

Linguistic Constraint	PFN Membership Function	Crisp	Linguistic Reliability	PFN Membership Function
Extremely low (EL)	(1, 1, 1, 1, 1; 0.6, 0.8)	$\cong 1$	Absolutely reliable (AR)	(0.8, 0.85, 0.9, 0.95, 1; 0.6, 0.8)
Very low (VL)	(1, 1, 2, 3, 4; 0.6, 0.8)	$\cong 2$	Strongly reliable (SR)	(0.7, 0.75, 0.8, 0.85, 9; 0.6, 0.8)
Low (L)	(1, 2, 3, 4, 5; 0.6, 0.8)	$\cong 3$	Very highly reliable (VR)	(0.6, 0.65, 0.7, 0.75, 0.8; 0.6, 0.8)
Slightly below average (SB)	(2, 3, 4, 5, 6; 0.6, 0.8)	$\cong 4$	Highly reliable (HR)	(0.5, 0.55, 0.6, 0.65, 0.7; 0.6, 0.8)
Average (A)	(3, 4, 5, 6, 7; 0.6, 0.8)	$\cong 5$	Fairly reliable (FR)	(0.4, 0.45, 0.5, 0.55, 0.6; 0.6, 0.8)
Slightly above average (SA)	(4, 5, 6, 7, 8; 0.6, 0.8)	$\cong 6$	Weakly reliable (WR)	(0.3, 0.35, 0.4, 0.45, 0.5; 0.6, 0.8)
High (H)	(5, 6, 7, 8, 9; 0.6, 0.8)	$\cong 7$	Very weakly reliable (VW)	(0.2, 0.25, 0.3, 0.35, 0.4; 0.6, 0.8)
Very high (VH)	(6, 7, 8, 9, 9; 0.6, 0.8)	$\cong 8$	Strongly unreliable (SU)	(0.1, 0.15, 0.2, 0.25, 0.3; 0.6, 0.8)
Extremely high (EH)	(7, 8, 9, 10, 10; 0.6, 0.8)	$\cong 9$	Absolutely unreliable (AU)	(0, 0, 0, 0, 0.2; 0.6, 0.8)

presents the linguistic scales (constraint and reliability functions of Z-numbers) and their corresponding linear asymmetric PFNs for evaluating the alternatives to the decision-making problem. Figure 7 depicts the fuzzy constraint functions and the membership functions of the corresponding asymmetric linear PFNs used for evaluating the alternatives to the decision-making problem.

The first linguistic scale represents the constraint function $\left(\tilde{\mathcal{A}}\right)$ of the Z-number (the preference of expert $\mathcal{y}$ regarding the status of alternative $i$ with respect to criterion $j$), while the second scale represents the reliability function $\left(\tilde{\mathcal{B}}\right)$ of the Z-number (the degree of reliability of expert $\mathcal{y}$ in his/her preference regarding the status of alternative $i$ with respect to criterion $j$). Since the numerical counterparts of both predefined linguistic scales are asymmetric linear PFNs, the initial fuzzy decision matrix ${\tilde{\mathcal{D}}}_{ij}^{\mathcal{y}}$ for each expert is constructed according to Equation (46):

$$\begin{gathered} {\tilde{\mathcal{D}}}_{ij}^{\mathcal{y}}=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}{\tilde{\mathcal{d}}}_{11}\\ {\tilde{\mathcal{d}}}_{21}\end{array}& \begin{array}{c}{\tilde{\mathcal{d}}}_{12}\\ {\tilde{\mathcal{d}}}_{22}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\dots \\ \dots \end{array}& \begin{array}{c}{\tilde{\mathcal{d}}}_{1n}\\ {\tilde{\mathcal{d}}}_{2n}\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{c}⋮\\ {\tilde{\mathcal{d}}}_{m1}\end{array}& \begin{array}{c}⋮\\ {\tilde{\mathcal{d}}}_{m2}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\ddots \\ \dots \end{array}& \begin{array}{c}⋮\\ {\tilde{\mathcal{d}}}_{mn}\end{array}\end{array}\end{array}\right], \\ where,{\tilde{\mathcal{d}}}_{ij}=\left[\left(\tilde{\mathcal{A}}\right),\left(\tilde{\mathcal{B}}\right)\right] \\ =\left[\left({\mathfrak{a}}_{ij}^{\mathcal{A}},{\mathfrak{a}}_{ij}^{\mathcal{A}},{\mathfrak{a}}_{ij}^{\mathcal{A}},{\mathfrak{a}}_{ij}^{\mathcal{A}},{\mathfrak{a}}_{ij}^{\mathcal{A}};{\Psi }_1,{\Psi }_2\right),\left({\mathfrak{b}}_{ij}^{\mathcal{B}},{\mathfrak{b}}_{ij}^{\mathcal{B}},{\mathfrak{b}}_{ij}^{\mathcal{B}},{\mathfrak{b}}_{ij}^{\mathcal{B}},{\mathfrak{b}}_{ij}^{\mathcal{B}};{\Psi }_1,{\Psi }_2\right)\right], \\ i=\mathrm{1,2}\dots ,m,j=\mathrm{1,2}\dots ,n. \end{gathered}$$

(46)

Step 3. In this step, the elements of the initial decision matrix ${\tilde{\mathcal{D}}}_{ij}^{\mathcal{y}}$, which are asymmetric pentagonal Z-numbers, undergo the Z-transformation process to form matrix ${\tilde{X}}_{ij}^{\mathcal{y}}$ with regular asymmetric PFNs. To achieve this, the crisp values of the reliability functions of the asymmetric pentagonal Z-numbers are calculated using Equation (36). Subsequently, the Z-transformation is performed using Equations (39) through (41). Finally, the decision matrix with elements consisting of regular asymmetric PFNs is obtained, as shown in Equation (47):

$$\begin{gathered} {\tilde{X}}_{ij}^{\mathcal{y}}=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}{\tilde{x}}_{11}\\ {\tilde{x}}_{21}\end{array}& \begin{array}{c}{\tilde{x}}_{12}\\ {\tilde{x}}_{22}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\dots \\ \dots \end{array}& \begin{array}{c}{\tilde{x}}_{1n}\\ {\tilde{x}}_{2n}\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{c}⋮\\ {\tilde{x}}_{m1}\end{array}& \begin{array}{c}⋮\\ {\tilde{x}}_{m2}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\ddots \\ \dots \end{array}& \begin{array}{c}⋮\\ {\tilde{x}}_{mn}\end{array}\end{array}\end{array}\right], \\ where,{\tilde{x}}_{ij}=\left({\mathfrak{a}}_{1\left(ij\right)},{\mathfrak{a}}_{2\left(ij\right)},{\mathfrak{a}}_{3\left(ij\right)},{\mathfrak{a}}_{4\left(ij\right)},{\mathfrak{a}}_{5\left(ij\right)};{\Psi }_1,{\Psi }_2\right),i=\mathrm{1,2}\dots ,m,j=\mathrm{1,2}\dots ,n. \end{gathered}$$

(47)

Step 4. Aggregation. In this step, the decision matrices ${\tilde{X}}_{ij}^{\mathcal{y}}$, obtained for each expert, are aggregated to form the unified fuzzy decision matrix ${\tilde{U}}_{ij}$. For this purpose, the PFDA operator (Equations (44) and (45)) is utilized.

Step 5. Normalization. In this step, the elements of the unified decision matrix ${\tilde{U}}_{ij}$ are normalized using Equation (48) to obtain the normalized unified decision matrix ${\tilde{\overline{U}}}_{ij}$:

$$\begin{gathered} {\tilde{\overline{u}}}_{ij}\left\{\begin{array}{c}\left({\displaystyle \frac{{\mathfrak{a}}_{1\left(ij\right)}}{{\mathfrak{a}}_{5\left(ij\right)}^{\oplus }}},{\displaystyle \frac{{\mathfrak{a}}_{2\left(ij\right)}}{{\mathfrak{a}}_{5\left(ij\right)}^{\oplus }}},{\displaystyle \frac{{\mathfrak{a}}_{3\left(ij\right)}}{{\mathfrak{a}}_{5\left(ij\right)}^{\oplus }}},{\displaystyle \frac{{\mathfrak{a}}_{4\left(ij\right)}}{{\mathfrak{a}}_{5\left(ij\right)}^{\oplus }}},{\displaystyle \frac{{\mathfrak{a}}_{5\left(ij\right)}}{{\mathfrak{a}}_{5\left(ij\right)}^{\oplus }}};{\Psi }_1,{\Psi }_2\right),{\mathfrak{a}}_{5\left(ij\right)}^{\oplus }=max{\mathfrak{a}}_{5\left(ij\right)},forj\in B\\ \left({\displaystyle \frac{{\mathfrak{a}}_{1\left(ij\right)}^{\ominus }}{{\mathfrak{a}}_{5\left(ij\right)}}},{\displaystyle \frac{{\mathfrak{a}}_{1\left(ij\right)}^{\ominus }}{{\mathfrak{a}}_{4\left(ij\right)}}},{\displaystyle \frac{{\mathfrak{a}}_{1\left(ij\right)}^{\ominus }}{{\mathfrak{a}}_{3\left(ij\right)}}},{\displaystyle \frac{{\mathfrak{a}}_{1\left(ij\right)}^{\ominus }}{{\mathfrak{a}}_{2\left(ij\right)}}},{\displaystyle \frac{{\mathfrak{a}}_{1\left(ij\right)}^{\ominus }}{{\mathfrak{a}}_{1\left(ij\right)}}};{\Psi }_1,{\Psi }_2\right),{\mathfrak{a}}_{1\left(ij\right)}^{\ominus }=min{\mathfrak{a}}_{1\left(ij\right)},forj\in C\end{array}\right. \\ where{\mathfrak{a}}_{1\left(ij\right)}\ne 0. \end{gathered}$$

(48)

Step 6. In this step, the values ${\left({\tilde{Q}}_i^S\right)}_{WSM}$ and ${\left({\tilde{Q}}_i^P\right)}_{WPM}$ are calculated based on Equations (49) and (50). For this purpose, the concepts and operators mentioned in Definition 8 can be utilized:

$${\tilde{Q}}_i^S=\sum _{j=1}^n{\mathcal{W}}_j{\tilde{\overline{u}}}_{ij},$$

(49)

$${\tilde{Q}}_i^P=\prod _{j=1}^n{{\tilde{\overline{u}}}_{ij}}^{{\mathcal{W}}_j}.$$

(50)

Step 7. Calculation of the ${\tilde{Q}}_i$ index. Using Equation (51) and adjusting the combination parameter $\beta$, the ${\tilde{Q}}_i$ index is calculated. By tuning the parameter $\beta$, the influence of ${\tilde{Q}}_i^S$ and ${\tilde{Q}}_i^P$ on the final ranking of the alternatives can be adjusted. If $\beta =0.5$ is selected, the contributions of these values to the final ranking of the alternatives will be equal.

$${\tilde{Q}}_i=\beta .{\tilde{Q}}_i^S+\left(1-\beta \right).{\tilde{Q}}_i^P,0\le \beta \le 1.$$

(51)

Step 8. Final ranking of alternatives. In this step, the alternatives of the decision-making problem are ranked based on the higher values of the $Q_i$ index. To perform the crispification process for the ${\tilde{Q}}_i$ index, Definition 12 and Equation (36) can be utilized.

4. Case Study and Results

A real-world case study concerning the sustainable–smart SES in the home appliance supply chain was conducted to evaluate the applicability of the proposed approach. The company in question is active in the design, production, and distribution of a wide range of home appliances and is recognized as a leading brand in this field. Some of the company’s customized products, developed based on its in-house expertise, include various types of televisions, washing machines, dishwashers, and refrigerators. Additionally, receiving multiple international certifications, standards, and awards is among the notable achievements of this company. Since the company has requested to remain anonymous, we are unable to disclose its name or provide further details.

Recently, with technological advancements and the emergence of novel technologies, the company’s overarching policies have shifted towards the smartening of the supply chain and the adoption of innovative technologies such as AI, the IoT, big data, AI-based robotics, and more. Company policymakers emphasize smartening supply chain processes while maintaining sustainability and fulfilling corporate social responsibilities. On the other hand, the role of suppliers as key players and the first link in the supply chain is crucial for achieving the macro policies of smartening supply chain processes [8]. SSCs, due to their feasibility in achieving economic, environmental, and social benefits, have gradually become a primary strategy for companies to enhance sustainable development [10]. Given that the company works with a wide range of suppliers in various fields such as metal part manufacturing, painting, electronic chips, and more, it has initiated the evaluation of existing and potential sustainable smart suppliers in specific and limited areas. Since advanced and novel technologies among the company’s target suppliers have either been recently implemented or are used on a limited scale, the process of evaluating smart sustainable suppliers is faced with significant uncertainty. Therefore, the company evaluates and selects sustainable smart suppliers using the proposed PZ-LMAW and PZ-WASPAS approaches, leveraging POFNs and Z-numbers to address the extensive uncertainties present in the decision-making process. The comprehensive research framework is presented in Figure 8.

To carry out the process of evaluating and selecting sustainable smart suppliers, a group of experts comprising three members was formed. Two of the members are managers from the aforementioned company, and the third is an academic expert. The details and background of the expert group members are presented in

Table 4. Background and specifications of the expert panel members.

Experts	Education		Specialized Area	Field	Position	Experience
	Ph.D.	MA
E(A)	✓		Industrial engineering	Industry	Chief procurement officer	20+
E(B)		✓	Production planning	Industry	Production manager	25+
E(C)	✓		Industrial management	Academic	Faculty member	15+

.

After reviewing the literature on SES from the perspectives of sustainability and smartness and utilizing the opinions of experts, six evaluation criteria were identified and established for this purpose. The criteria for evaluating sustainable smart suppliers, along with the supplier requirements for each criterion, are shown in

Table 5. Criteria for smart SES.

Sustainable Smart SES Criteria	Supplier Requirements	References
C1: Employee development in a smart environment	Suppliers foster employee development by providing training in smart technologies, offering flexible work arrangements, and creating opportunities for professional growth.	[11,25,26]
C2: Green and smart logistics and manufacturing	Suppliers adopt smart technologies and advanced management practices to enhance energy efficiency, reduce emissions, and implement green logistics through solutions like smart electric vehicles and big data optimization.	[11,17,18]
C3: Waste reduction using smart technologies	Suppliers employ AI-driven recycling technologies to optimize material identification, sorting, and resource recovery, reducing waste contamination and improving sustainability in the supply chain.	[19,20,21]
C4: Cost reduction using smart technologies	Suppliers leverage smart technologies to reduce costs across the supply chain by employing IoT for real-time machine monitoring and using AI and big data analytics to minimize unplanned downtime and defects.	[22,23]
C5: Smart working environment	Suppliers create safe and smart working environments by utilizing tools like VR and AR for remote machine operations, ensuring employee safety and workplace health.	[16,24]
C6: Smart delivery to customer	Suppliers utilize smart tools to ensure timely product delivery by forecasting customer needs with AI (e.g., machine learning) and developing efficient production plans through reinforcement learning.	[10,11]

. The criteria were selected based on various dimensions of sustainability, including economic, environmental, and social aspects, with an emphasis on supplier smartness. Subsequently, four potential suppliers were chosen to participate in the evaluation and selection process, with the company intending to sign cooperation contracts with the top two suppliers. In the following sections, the SES criteria and the final prioritization of suppliers under uncertainty will be elaborated using the proposed PZ-LMAW and PZ-WASPAS approaches.

4.1. Calculation of the Importance of SES Criteria

In this section, the importance of SES criteria is calculated by the expert group using the proposed PZ-LMAW approach, as previously described. The steps for calculating the weights of the decision-making criteria are as follows.

Step 1. Each expert evaluates the criteria using the linguistic terms provided in Table 2. The experts’ preferences include two distinct dimensions: the fuzzy constraint function and the fuzzy reliability function of Z-numbers. The fuzzy constraint represents the expert’s preference regarding the importance of the criteria, while the fuzzy reliability indicates the expert’s confidence in their preference.

Table 6. Symmetric Z-PFN-based experts’ preferences regarding the evaluation criteria.

Experts	C1	C2	C3	C4	C5	C6
	$\tilde{\mathcal{A}},\tilde{\mathcal{B}}$
E(A)	ML, VR	H, SR	MH, SR	EH, SR	H, HR	ML, VR
E(B)	S, AR	EH, AR	VH, SR	H, AR	MH, SR	E, AR
E(C)	VS, VR	MH, SR	S, SR	VH, AR	H, AR	ML, SR

presents the results of the decision-making criteria evaluations conducted by the expert group.

Step 2. After collecting the experts’ preferences, which are expressed as symmetric pentagonal Z-numbers, the Z-transformation process is carried out using Equations (33), (39), and (40). This transformation converts the experts’ preferences into standard symmetric PFNs. As a result, the preference vectors ${\tilde{\mathcal{P}}}^{\mathcal{y}}=\left({\widetilde{\mathsf{{\rm Y}}}}_{C1}^{\mathcal{y}},{\widetilde{\mathsf{{\rm Y}}}}_{C2}^{\mathcal{y}}\dots ,{\widetilde{\mathsf{{\rm Y}}}}_{Cn}^{\mathcal{y}}\right)$ were obtained for each expert, as shown in

Table 7. Z-transformation operation results and preference vectors for each expert.

		C1	C2	C3	C4	C5	C6
E(A)	$EV\left(\tilde{\mathcal{B}}\right)$	0.7	0.8	0.8	0.8	0.6	0.7
	${\tilde{\mathcal{P}}}^{\mathcal{y}}$	(1.67, 1.88, 2.09, 2.3, 2.5; 0.7)	(3.13, 3.35, 3.57, 3.8, 4.02; 0.7)	(2.68, 2.9, 3.13, 3.35, 3.57; 0.7)	(4.02, 4.24, 4.47, 4.47, 4.47; 0.7)	(2.71, 2.9, 3.09, 3.29, 3.48; 0.7)	(1.67, 1.88, 2.09, 2.3, 2.5; 0.7)
E(B)	$EV\left(\tilde{\mathcal{B}}\right)$	0.9	0.9	0.8	0.9	0.8	0.9
	${\tilde{\mathcal{P}}}^{\mathcal{y}}$	(1.42, 1.66, 1.89, 2.13, 2.37; 0.7)	(4.26, 4.5, 4.74, 4.74, 4.74; 0.7)	(3.57, 3.8, 4.02, 4.24, 4.47; 0.7)	(3.32, 3.55, 3.79, 4.03, 4.26; 0.7)	(2.68, 2.9, 3.13, 3.35, 3.57; 0.7)	(2.37, 2.6, 2.84, 3.08, 3.32; 0.7)
E(C)	$EV\left(\tilde{\mathcal{B}}\right)$	0.7	0.8	0.8	0.9	0.9	0.8
	${\tilde{\mathcal{P}}}^{\mathcal{y}}$	(0.83, 1.04, 1.25, 1.46, 1.67; 0.7)	(2.68, 2.9, 3.13, 3.35, 3.57; 0.7)	(1.34, 1.56, 1.78, 2.01, 2.23; 0.7)	(3.79, 4.03, 4.26, 4.5, 4.74; 0.7)	(3.32, 3.55, 3.79, 4.03, 4.26; 0.7)	(1.78, 2.01, 2.23, 2.45, 2.68; 0.7)

.

Step 3. In this step, the numerical value of the fuzzy anti-ideal point is calculated for each preference vector and each expert. To achieve this, the algebraic operations of symmetrical PFNs described in Definition 7 are utilized. The fuzzy anti-ideal points are obtained as follows:

E_(A): ${\widetilde{\mathsf{{\rm Y}}}}_{AIP}=$ (0.55, 0.62, 0.69, 0.76, 0.83; 0.7)

E_(B): ${\widetilde{{\rm Y}}}_{AIP}=$ (0.47, 0.55, 0.63, 0.71, 0.79; 0.7)

E_(C): ${\widetilde{\mathsf{{\rm Y}}}}_{AIP}=$ (0.27, 0.34, 0.41, 0.48, 0.55; 0.7)

Step 4. Using Equation (42) and applying the algebraic operations of symmetric PFNs, the fuzzy relation vectors are calculated for each expert.

Table 8. Fuzzy relation vectors obtained for each expert.

Experts	C1	C2	C3	C4	C5	C6
	${\tilde{\mathcal{R}}}^{\mathcal{y}}:\left({\tilde{\mathsf{\eta }}}_{\mathit{C}1}^{\mathcal{y}},{\tilde{\mathsf{\eta }}}_{\mathit{C}2}^{\mathcal{y}}\dots ,{\tilde{\mathsf{\eta }}}_{\mathit{C}\mathit{n}}^{\mathcal{y}}\right)$
E(A)	(2, 2.45, 3, 3.66, 4.5; 0.7)	(3.74, 4.37, 5.13, 6.05, 7.21; 0.7)	(3.2, 3.79, 4.48, 5.34, 6.41; 0.7)	(4.81, 5.53, 6.41, 7.12, 8.01; 0.7)	(3.24, 3.78, 4.44, 5.24, 6.24; 0.7)	(2, 2.45, 3, 3.66, 4.5; 0.7)
E(B)	(1.8, 2.33, 3, 3.85, 5; 0.7)	(5.4, 6.33, 7.5, 8.57, 10; 0.7)	(4.52, 5.34, 6.36, 7.67, 9.42; 0.7)	(4.2, 5, 6, 7.28, 9; 0.7)	(3.39, 4.08, 4.94, 6.06, 7.54; 0.7)	(3, 3.66, 4.5, 5.57, 7; 0.7)
E(C)	(1.5, 2.14, 3, 4.2, 6; 0.7)	(4.81, 5.95, 7.48, 9.62, 12.82; 0.7)	(2.4, 3.2, 4.27, 5.77, 8.01; 0.7)	(6.8, 8.26, 10.2, 12.92, 17; 0.7)	(5.95, 7.28, 9.07, 11.56, 15.3; 0.7)	(3.2, 4.12, 5.34, 7.05, 9.62; 0.7)

presents the fuzzy relation vectors for each expert.

Step 5. Using the fuzzy values of the relation vectors and applying Equation (44), the fuzzy weight coefficients of the evaluation criteria are calculated, as detailed below:

$$\begin{gathered} {\tilde{\mathcal{W}}}_1^{\mathcal{y}}:{\tilde{\mathcal{W}}}_2^{\mathcal{y}}: \\ \begin{array}{c}{E}_{\left(A\right)}:\\ E_{\left(B\right)}:\\ E_{\left(C\right)}:\end{array}\left[\begin{array}{c}\left(0.06,0.11,0.12,0.16,0.22;0.7\right)\\ \left(0.04,0.09,0.11,0.15,0.21;0.7\right)\\ \left(0.02,0.07,0.1,0.15,0.23;0.7\right)\end{array}\right]\begin{array}{c}{E}_{\left(A\right)}:\\ E_{\left(B\right)}:\\ E_{\left(C\right)}:\end{array}\left[\begin{array}{c}\left(0.12,0.18,0.18,0.23,0.29;0.7\right)\\ \left(0.13,0.19,0.2,0.24,0.3;0.7\right)\\ \left(0.11,0.17,0.18,0.24,0.33;0.7\right)\end{array}\right] \\ {\tilde{\mathcal{W}}}_3^{\mathcal{y}}:{\tilde{\mathcal{W}}}_4^{\mathcal{y}}: \\ \begin{array}{c}{E}_{\left(A\right)}:\\ E_{\left(B\right)}:\\ E_{\left(C\right)}:\end{array}\left[\begin{array}{c}\left(0.1,0.16,0.17,0.21,0.28;0.7\right)\\ \left(0.12,0.18,0.18,0.23,0.29;0.7\right)\\ \left(0.06,0.11,0.13,0.19,0.26;0.7\right)\end{array}\right]\begin{array}{c}{E}_{\left(A\right)}:\\ E_{\left(B\right)}:\\ E_{\left(C\right)}:\end{array}\left[\begin{array}{c}\left(0.14,0.21,0.21,0.25,0.31;0.7\right)\\ \left(0.11,0.17,0.18,0.22,0.29;0.7\right)\\ \left(0.13,0.21,0.21,0.27,0.36;0.7\right)\end{array}\right] \\ {\tilde{\mathcal{W}}}_5^{\mathcal{y}}:{\tilde{\mathcal{W}}}_6^{\mathcal{y}}: \\ \begin{array}{c}{E}_{\left(A\right)}:\\ E_{\left(B\right)}:\\ E_{\left(C\right)}:\end{array}\left[\begin{array}{c}\left(0.1,0.16,0.17,0.21,0.27;0.7\right)\\ \left(0.09,0.15,0.16,0.2,0.26;0.7\right)\\ \left(0.12,0.19,0.2,0.26,0.35;0.7\right)\end{array}\right]\begin{array}{c}{E}_{\left(A\right)}:\\ E_{\left(B\right)}:\\ E_{\left(C\right)}:\end{array}\left[\begin{array}{c}\left(0.06,0.11,0.12,0.16,0.22;0.7\right)\\ \left(0.08,0.14,0.15,0.19,0.25;0.7\right)\\ \left(0.08,0.14,0.15,0.21,0.29;0.7\right)\end{array}\right] \end{gathered}$$

Step 6. The fuzzy weight coefficients of criteria, calculated for each expert, are aggregated using the pentagonal fuzzy Dombi aggregation (PFDA) operators, as shown in Equations (44) and (45). This process yields an integrated fuzzy weight coefficient vector.

Step 7. Since the fuzzy weight coefficients of the decision-making criteria are calculated based on symmetric PFNs, it is necessary to determine their precise crisp values considering the properties of symmetric PFNs. For this purpose, Equation (33) is used to perform the crispification process.

The integrated fuzzy weight coefficients and the final crisp weight coefficients of the decision-making criteria are presented in

Table 9. Aggregated fuzzy weights and final deterministic weights of decision-making criteria.

	C1	C2	C3	C4	C5	C6
${\overline{\tilde{\mathcal{W}}}}_j$	(0.04, 0.09, 0.11, 0.16, 0.22; 0.7)	(0.12, 0.18, 0.19, 0.24, 0.31; 0.7)	(0.08, 0.14, 0.16, 0.21, 0.28; 0.7)	(0.13, 0.19, 0.2, 0.25, 0.32; 0.7)	(0.11, 0.17, 0.17, 0.22, 0.29; 0.7)	(0.07, 0.12, 0.14, 0.19, 0.25; 0.7)
${\mathcal{W}}_j$	0.126	0.212	0.179	0.222	0.196	0.160

.

4.2. Prioritization of Smart Logistic Suppliers

After calculating the weights of the decision-making criteria, the prioritization of suppliers, considering the importance of these criteria, becomes the focus. Therefore, in this section, the final prioritization of suppliers is conducted using the proposed PZ-WASPAS approach based on asymmetric PFNs. The algebraic operations for asymmetric PFNs and their crispification process are detailed in Definitions 8 and 12, respectively. The steps for applying the proposed PZ-WASPAS approach and the results obtained are outlined in the following sections.

Step 1. In this step, an expert group is formed, and its members are identified to gather their preferences regarding the suppliers. Subsequently, the potential suppliers to be included in the evaluation process are determined. Here, four potential suppliers have been selected, and the company intends to enter into short-term contracts with two suppliers upon completion of the evaluation.

Step 2. In this step, each expert independently evaluates the suppliers. During this process, each supplier is assessed based on each criterion. For example, the first supplier is evaluated across criteria one to six, and the other suppliers are similarly assessed for each criterion. Since the experts’ evaluations are based on asymmetric pentagonal Z-numbers, each expert expresses the degree of importance for each supplier in each criterion (fuzzy constraint function) and then indicates their confidence in their preference (fuzzy reliability function) using a predefined linguistic scale.

Table 10. Asymmetric Z-PFN-based experts’ preferences regarding the importance of each supplier for each criterion.

Experts	Suppliers	C1	C2	C3	C4	C5	C6
		$\tilde{\mathcal{A}},\tilde{\mathcal{B}}$
E(A)	S(1)	H, SR	A, FR	SB, VR	A, FR	SB, VR	SA, VR
	S(2)	H, AR	L, FR	SA, SR	SB, SR	A, HR	L, SR
	S(3)	SA, SR	EH, AR	SA, FR	EH, AR	H, SR	SA, SR
	S(4)	A, FR	VH, AR	SA, AR	H, AR	SB, FR	H, AR
E(B)	S(1)	SB, VR	H, HR	SB, HR	A, VR	SA, SR	H, SR
	S(2)	L, FR	A, HR	SA, FR	SA, FR	A, HR	SB, SR
	S(3)	SA, VR	VH, AR	A, VR	VH, AR	H, SR	VH, SR
	S(4)	SB, FR	SA, FR	SA, HR	EH, SR	VH, SR	SA, VR
E(C)	S(1)	SA, FR	A, HR	SA, FR	H, SR	H, SR	A, SR
	S(2)	VH, AR	SB, VR	A, FR	L, FR	H, FR	VH, SR
	S(3)	H, AR	EH, AR	H, AR	EH, AR	VH, SR	SA, VR
	S(4)	SA, SR	VH, AR	SA, SR	SA, VR	H, FR	A, VR

presents the evaluation results of the suppliers for each criterion.

Thus, the initial fuzzy decision matrix ${\tilde{\mathcal{D}}}_{ij}^{\mathcal{y}}$ is obtained for each expert, where each element is an asymmetric pentagonal Z-number.

Step 3. The elements of matrix ${\tilde{\mathcal{D}}}_{ij}^{\mathcal{y}}$, which are asymmetric pentagonal Z-numbers, require Z-transformation to form matrix ${\tilde{X}}_{ij}^{\mathcal{y}}$ with regular asymmetric PFNs. Since asymmetric PFNs have specific structural properties, such as two weights $\left({\Psi }_1,{\Psi }_2\right)$, the Z-transformation process must be compatible with these numbers. For this purpose, Equation (36), which is specific to the crispification process of asymmetric PFNs, is used. Subsequently, the Z-transformation process is performed using Equations (39) to (41).

Table 11. The results of the Z-transformation operations for asymmetric Z-PFNs in SES.

Criteria	Experts	Suppliers
		S(1)	S(2)	S(3)	S(4)
C1	E(A)	(4.48, 5.38, 6.28, 7.17, 8.07; 0.6, 0.8)	(4.75, 5.7, 6.65, 7.61, 8.56; 0.6, 0.8)	(3.58, 4.48, 5.38, 6.28, 7.17; 0.6, 0.8)	(2.13, 2.84, 3.55, 4.26, 4.97; 0.6, 0.8)
	E(B)	(1.67, 2.51, 3.35, 4.19, 5.03; 0.6, 0.8)	(0.71, 1.42, 2.13, 2.84, 3.55; 0.6, 0.8)	(3.35, 4.19, 5.03, 5.87, 6.71; 0.6, 0.8)	(1.42, 2.13, 2.84, 3.55, 4.26; 0.6, 0.8)
	E(C)	(2.84, 3.55, 4.26, 4.97, 5.68; 0.6, 0.8)	(5.7, 6.65, 7.61, 8.56, 8.56; 0.6, 0.8)	(4.75, 5.7, 6.65, 7.61, 8.56; 0.6, 0.8)	(3.58, 4.48, 5.38, 6.28, 7.17; 0.6, 0.8)
C2	E(A)	(2.13, 2.84, 3.55, 4.26, 4.97; 0.6, 0.8)	(0.71, 1.42, 2.13, 2.84, 3.55; 0.6, 0.8)	(6.65, 7.61, 8.56, 9.51, 9.51; 0.6, 0.8)	(5.7, 6.65, 7.61, 8.56, 8.56; 0.6, 0.8)
	E(B)	(3.88, 4.66, 5.44, 6.22, 7; 0.6, 0.8)	(2.33, 3.11, 3.88, 4.66, 5.44; 0.6, 0.8)	(5.7, 6.65, 7.61, 8.56, 8.56; 0.6, 0.8)	(2.84, 3.55, 4.26, 4.97, 5.68; 0.6, 0.8)
	E(C)	(2.33, 3.11, 3.88, 4.66, 5.44; 0.6, 0.8)	(1.67, 2.51, 3.35, 4.19, 5.03; 0.6, 0.8)	(6.65, 7.61, 8.56, 9.51, 9.51; 0.6, 0.8)	(5.7, 6.65, 7.61, 8.56, 8.56; 0.6, 0.8)
C3	E(A)	(1.67, 2.51, 3.35, 4.19, 5.03; 0.6, 0.8)	(3.58, 4.48, 5.38, 6.28, 7.17; 0.6, 0.8)	(2.84, 3.55, 4.26, 4.97, 5.68; 0.6, 0.8)	(3.8, 4.75, 5.7, 6.65, 7.61; 0.6, 0.8)
	E(B)	(1.55, 2.33, 3.11, 3.88, 4.66; 0.6, 0.8)	(2.84, 3.55, 4.26, 4.97, 5.68; 0.6, 0.8)	(2.51, 3.35, 4.19, 5.03, 5.87; 0.6, 0.8)	(3.11, 3.88, 4.66, 5.44, 6.22; 0.6, 0.8)
	E(C)	(2.84, 3.55, 4.26, 4.97, 5.68; 0.6, 0.8)	(2.13, 2.84, 3.55, 4.26, 4.97; 0.6, 0.8)	(4.75, 5.7, 6.65, 7.61, 8.56; 0.6, 0.8)	(3.58, 4.48, 5.38, 6.28, 7.17; 0.6, 0.8)
C4	E(A)	(2.13, 2.84, 3.55, 4.26, 4.97; 0.6, 0.8)	(1.79, 2.69, 3.58, 4.48, 5.38; 0.6, 0.8)	(6.65, 7.61, 8.56, 9.51, 9.51; 0.6, 0.8)	(4.75, 5.7, 6.65, 7.61, 8.56; 0.6, 0.8)
	E(B)	(2.51, 3.35, 4.19, 5.03, 5.87; 0.6, 0.8)	(2.84, 3.55, 4.26, 4.97, 5.68; 0.6, 0.8)	(5.7, 6.65, 7.61, 8.56, 8.56; 0.6, 0.8)	(6.28, 7.17, 8.07, 8.97, 8.97; 0.6, 0.8)
	E(C)	(4.48, 5.38, 6.28, 7.17, 8.07; 0.6, 0.8)	(0.71, 1.42, 2.13, 2.84, 3.55; 0.6, 0.8)	(6.65, 7.61, 8.56, 9.51, 9.51; 0.6, 0.8)	(3.35, 4.19, 5.03, 5.87, 6.71; 0.6, 0.8)
C5	E(A)	(1.67, 2.51, 3.35, 4.19, 5.03; 0.6, 0.8)	(2.33, 3.11, 3.88, 4.66, 5.44; 0.6, 0.8)	(4.48, 5.38, 6.28, 7.17, 8.07; 0.6, 0.8)	(1.42, 2.13, 2.84, 3.55, 4.26; 0.6, 0.8)
	E(B)	(3.58, 4.48, 5.38, 6.28, 7.17; 0.6, 0.8)	(2.33, 3.11, 3.88, 4.66, 5.44; 0.6, 0.8)	(4.48, 5.38, 6.28, 7.17, 8.07; 0.6, 0.8)	(5.38, 6.28, 7.17, 8.07, 8.07; 0.6, 0.8)
	E(C)	(4.48, 5.38, 6.28, 7.17, 8.07; 0.6, 0.8)	(3.55, 4.26, 4.97, 5.68, 6.39; 0.6, 0.8)	(5.38, 6.28, 7.17, 8.07, 8.07; 0.6, 0.8)	(3.55, 4.26, 4.97, 5.68, 6.39; 0.6, 0.8)
C6	E(A)	(3.35, 4.19, 5.03, 5.87, 6.71; 0.6, 0.8)	(0.89, 1.79, 2.69, 3.58, 4.48; 0.6, 0.8)	(3.58, 4.48, 5.38, 6.28, 7.17; 0.6, 0.8)	(4.75, 5.7, 6.65, 7.61, 8.56; 0.6, 0.8)
	E(B)	(4.48, 5.38, 6.28, 7.17, 8.07; 0.6, 0.8)	(1.79, 2.69, 3.58, 4.48, 5.38; 0.6, 0.8)	(5.38, 6.28, 7.17, 8.07, 8.07; 0.6, 0.8)	(3.35, 4.19, 5.03, 5.87, 6.71; 0.6, 0.8)
	E(C)	(2.69, 3.58, 4.48, 5.38, 6.28; 0.6, 0.8)	(5.38, 6.28, 7.17, 8.07, 8.07; 0.6, 0.8)	(3.35, 4.19, 5.03, 5.87, 6.71; 0.6, 0.8)	(2.51, 3.35, 4.19, 5.03, 5.87; 0.6, 0.8)

presents the elements of matrix ${\tilde{X}}_{ij}^{\mathcal{y}}$, which are regular asymmetric PFNs.

Step 4. Using Equations (44) and (45), which are PFDA operators, the experts’ fuzzy preferences regarding the importance of each supplier in each criterion are aggregated to form the unified fuzzy decision matrix ${\tilde{U}}_{ij}$.

Table 12. The unified fuzzy decision matrix.

Criteria	Suppliers
	S(1)	S(2)	S(3)	S(4)
C1	(2.56, 3.47, 4.33, 5.18, 6.02; 0.6, 0.8)	(1.67, 2.91, 3.99, 5, 5.82; 0.6, 0.8)	(3.81, 4.71, 5.61, 6.51, 7.4; 0.6, 0.8)	(2.06, 2.87, 3.66, 4.44, 5.21; 0.6, 0.8)
C2	(2.59, 3.38, 4.15, 4.92, 5.68; 0.6, 0.8)	(1.23, 2.1, 2.92, 3.73, 4.52; 0.6, 0.8)	(6.3, 7.26, 8.21, 9.17, 9.17; 0.6, 0.8)	(4.27, 5.15, 6.03, 6.9, 7.32; 0.6, 0.8)
C3	(1.88, 2.71, 3.51, 4.3, 5.09; 0.6, 0.8)	(2.72, 3.5, 4.27, 5.04, 5.81; 0.6, 0.8)	(3.12, 3.97, 4.81, 5.65, 6.48; 0.6, 0.8)	(3.47, 4.34, 5.21, 6.08, 6.95; 0.6, 0.8)
C4	(2.75, 3.59, 4.41, 5.24, 6.06; 0.6, 0.8)	(1.29, 2.21, 3.05, 3.86, 4.66; 0.6, 0.8)	(6.3, 7.26, 8.21, 9.17, 9.17; 0.6, 0.8)	(4.49, 5.42, 6.34, 7.26, 7.95; 0.6, 0.8)
C5	(2.73, 3.72, 4.66, 5.58, 6.49; 0.6, 0.8)	(2.63, 3.41, 4.19, 4.96, 5.72; 0.6, 0.8)	(4.75, 5.65, 6.55, 7.45, 8.07; 0.6, 0.8)	(2.56, 3.47, 4.33, 5.16, 5.82; 0.6, 0.8)
C6	(3.36, 4.26, 5.16, 6.05, 6.94; 0.6, 0.8)	(1.61, 2.75, 3.8, 4.79, 5.63; 0.6, 0.8)	(3.93, 4.83, 5.72, 6.61, 7.28; 0.6, 0.8)	(3.31, 4.21, 5.11, 5.99, 6.88; 0.6, 0.8)

presents the values of the unified fuzzy decision matrix.

Step 5. Using Equation (48) and considering whether the criteria are of benefit or cost type, the values of the unified fuzzy decision matrix are normalized to obtain the normalized unified fuzzy decision matrix. Here, all evaluation criteria are of the benefit type.

Table 13. The normalized unified fuzzy decision matrix.

Criteria	Suppliers
	S(1)	S(2)	S(3)	S(4)
C1	(0.346, 0.468, 0.585, 0.7, 0.812; 0.6, 0.8)	(0.225, 0.393, 0.539, 0.674, 0.786; 0.6, 0.8)	(0.514, 0.636, 0.757, 0.878, 1; 0.6, 0.8)	(0.279, 0.388, 0.494, 0.599, 0.704; 0.6, 0.8)
C2	(0.283, 0.368, 0.452, 0.536, 0.619; 0.6, 0.8)	(0.134, 0.229, 0.319, 0.406, 0.492; 0.6, 0.8)	(0.687, 0.791, 0.896, 1, 1; 0.6, 0.8)	(0.465, 0.562, 0.657, 0.752, 0.798; 0.6, 0.8)
C3	(0.271, 0.389, 0.505, 0.619, 0.732; 0.6, 0.8)	(0.392, 0.503, 0.614, 0.725, 0.835; 0.6, 0.8)	(0.449, 0.571, 0.692, 0.812, 0.932; 0.6, 0.8)	(0.5, 0.625, 0.75, 0.875, 1; 0.6, 0.8)
C4	(0.3, 0.391, 0.481, 0.571, 0.66; 0.6, 0.8)	(0.141, 0.241, 0.332, 0.421, 0.508; 0.6, 0.8)	(0.687, 0.791, 0.896, 1, 1; 0.6, 0.8)	(0.49, 0.591, 0.692, 0.791, 0.867; 0.6, 0.8)
C5	(0.338, 0.461, 0.578, 0.692, 0.804; 0.6, 0.8)	(0.326, 0.423, 0.519, 0.614, 0.709; 0.6, 0.8)	(0.588, 0.7, 0.811, 0.923, 1; 0.6, 0.8)	(0.317, 0.43, 0.536, 0.639, 0.721; 0.6, 0.8)
C6	(0.461, 0.586, 0.709, 0.832, 0.953; 0.6, 0.8)	(0.221, 0.378, 0.521, 0.658, 0.773; 0.6, 0.8)	(0.54, 0.664, 0.786, 0.909, 1; 0.6, 0.8)	(0.455, 0.579, 0.702, 0.824, 0.945; 0.6, 0.8)

shows the values of the unified fuzzy decision matrix.

Step 6. At this stage, the indices ${\tilde{Q}}_i^S$ and ${\tilde{Q}}_i^P$ are calculated based on the algebraic operations of asymmetric PFNs and using Equations (49) and (50).

Step 7. Using Equation (51), the fuzzy index ${\tilde{Q}}_i$ is calculated for ranking the suppliers. At this stage, by setting the combining parameter $\beta$, the fuzzy index ${\tilde{Q}}_i$ is derived based on a combination of the values from the WSM and WPM approaches.

Step 8. After calculating the fuzzy index ${\tilde{Q}}_i$, it is necessary to determine the crisp value of this parameter, which represents the fuzzy score of each supplier. Since this index is an asymmetric PFN, Equation (36) is used for the crispification process to compute the precise value of this index based on the structural properties of asymmetric PFNs.

Table 14. The values of the fuzzy scores, crisp scores, and the final ranking of the suppliers.

Suppliers	${\tilde{\mathit{Q}}}_{\mathit{i}}^{\mathit{S}}$	${\tilde{\mathit{Q}}}_{\mathit{i}}^{\mathit{P}}$	${\tilde{\mathit{Q}}}_{\mathit{i}}$	${\mathit{Q}}_{\mathit{i}}$	Order
S(1)	(0.359, 0.478, 0.594, 0.708, 0.822; 0.6, 0.8)	(0.29, 0.398, 0.505, 0.613, 0.722; 0.6, 0.8)	(0.324, 0.438, 0.549, 0.661, 0.772; 0.6, 0.8)	0.560	3
S(2)	(0.258, 0.385, 0.504, 0.62, 0.728; 0.6, 0.8)	(0.187, 0.303, 0.413, 0.521, 0.624; 0.6, 0.8)	(0.222, 0.344, 0.459, 0.57, 0.676; 0.6, 0.8)	0.466	4
S(3)	(0.645, 0.769, 0.893, 1.016, 1.082; 0.6, 0.8)	(0.553, 0.674, 0.796, 0.919, 0.987; 0.6, 0.8)	(0.599, 0.722, 0.844, 0.967, 1.035; 0.6, 0.8)	0.846	1
S(4)	(0.467, 0.588, 0.707, 0.824, 0.922; 0.6, 0.8)	(0.384, 0.499, 0.613, 0.726, 0.821; 0.6, 0.8)	(0.425, 0.543, 0.66, 0.775, 0.872; 0.6, 0.8)	0.667	2

presents the values of the indices ${\tilde{Q}}_i^S$, ${\tilde{Q}}_i^P$, and ${\tilde{Q}}_i$, the crisp score $Q_i$, and the final ranking of the suppliers.

5. Validation of the Results

In this section, sensitivity analysis is conducted to validate and assess the robustness of the results obtained from the proposed PZ-LMAW and PZ-WASPAS approaches.

First, sensitivity analysis based on the WASPAS measure, which includes the WSM and WPM approaches, is performed in Section 5.1. For this purpose, the final ranking of suppliers is adjusted for different levels of $\beta$ within the range of 0 to 1, and the impact of varying $\beta$ levels on the final ranking of suppliers is analyzed. Second, in Section 5.2, using a logical pattern approach, the influence of different evaluation criteria under various scenarios is analyzed to assess their effect on supplier rankings. Finally, in Section 5.3, a comparative analysis is conducted to compare the results of the proposed approach with other extensions of the WASPAS method.

5.1. Sensitivity Analysis Based on the WASPAS Measure

Using Equation (51) and adjusting the level of the combining parameter $\beta$ within the range of 0 to 1, the impact of different $\beta$ levels on the final supplier ranking is determined. The closer $\beta$ is to 1, the greater the influence of the ${\tilde{Q}}_i^S$ index on the ranking. Conversely, the closer $\beta$ is to 0, the ${\tilde{Q}}_i^P$ index will have a more significant impact on the final ranking. Furthermore, if $\beta$ is set to 0.5, the influence of both indices on the final ranking will be equal.

Here, the fuzzy indices ${\tilde{Q}}_i^S$ and ${\tilde{Q}}_i^P$ have been calculated at different levels of $\beta$. Given that these indices utilize the structure of asymmetric PFNs, Equation (36) has been employed for the precise defuzzification of these values throughout the sensitivity analysis.

Table 15. Changes in the final ranking of suppliers based on different β levels during the sensitivity analysis.

$\mathit{\beta }$-Level	Suppliers				Order
	S(1)	S(2)	S(3)	S(4)
$\beta =0$	0.516	0.421	0.798	0.620	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.1$	0.525	0.430	0.807	0.629	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.2$	0.534	0.439	0.817	0.639	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.3$	0.542	0.448	0.827	0.648	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.4$	0.551	0.457	0.836	0.657	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.5$	0.560	0.466	0.846	0.667	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.6$	0.569	0.475	0.855	0.676	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.7$	0.577	0.484	0.865	0.685	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.8$	0.586	0.493	0.874	0.695	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =0.9$	0.595	0.502	0.884	0.704	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
$\beta =1$	0.604	0.511	0.893	0.713	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)

presents the final supplier rankings based on various levels of the combining parameter $\beta$. The score and status of each supplier at different $\beta$ levels are illustrated in Figure 9. As the results indicate, the final ranking of suppliers remains unchanged despite variations in $\beta$ levels and adjustments to the ${\tilde{Q}}_i^S$ and ${\tilde{Q}}_i^P$ indices. This demonstrates that the results exhibit a high level of consistency and robustness.

5.2. Sensitivity Analysis Based on the Logical Pattern

Various approaches for performing sensitivity analysis on the results of MCDM methodologies are discussed in the literature [74]. Most sensitivity analysis approaches examine changes in the weights of criteria and variations in the ranking of alternatives, assessing the stability and robustness of the results under different conditions.

Here, we aim to use the logical pattern sensitivity analysis approach proposed by Keshavarz-Ghorabaee et al. [75]. In the logical pattern approach, sets are created based on the number of criteria in the decision-making problem. Therefore, in this study, where six criteria are considered for evaluating suppliers, six sets are formed to perform the sensitivity analysis. Then, by applying the Equation $LP={\sum }_{j=1}^{n}j={\displaystyle \frac{n(n+1)}{2}}$, where $n$ is the number of criteria, a value for normalization in the logical pattern (LP) is obtained. Then, by dividing $j=\mathrm{1,2},\dots ,n$ by the value of the LP, the generated weight vector is obtained [75]. Here, the generated weight vector is calculated as $LP=\left\{0.048,0.096,0.143,0.191,0.238,0.286\right\}$. Subsequently, in each of the six considered sets, one criterion is assigned the highest weight, another is given the lowest weight, and non-repeating intermediate weights are assigned to the remaining criteria. This process results in six sets with the described characteristics, as shown in

Table 16. The sets of criteria weights during sensitivity analysis.

	C1	C2	C3	C4	C5	C6	Order
Set 1	0.048	0.096	0.143	0.191	0.238	0.286	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
Set 2	0.096	0.143	0.191	0.238	0.286	0.048	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
Set 3	0.143	0.191	0.238	0.286	0.048	0.096	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
Set 4	0.191	0.238	0.286	0.048	0.096	0.143	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
Set 5	0.238	0.286	0.048	0.096	0.143	0.191	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)
Set 6	0.286	0.048	0.096	0.143	0.191	0.238	S(3) $\succ$ S(4) $\succ$ S(1) $\succ$ S(2)

. Figure 10 illustrates the allocation of the generated weights to each set.

Using the logical pattern approach, the effects of changes in criterion weights on the final ranking of suppliers are assessed more efficiently. As shown in Table 16, the final ranking of suppliers remains unchanged across all scenarios and sets under consideration. Although the number of criteria, the number of alternatives, and the method of eliciting expert preferences can influence sensitivity analysis, the results indicate that the proposed approach is robust, consistent, and capable of delivering reliable outcomes.

Moreover, despite the use of both symmetric and asymmetric PFNs and the implementation of specific algebraic operations throughout the SES process, the results align closely with the initial preferences of the experts. This confirms the efficiency and effectiveness of the proposed approach in providing accurate and dependable results.

To observe changes in the final ranking of suppliers based on different weights of evaluation criteria in the considered sets, Spearman’s rank correlation coefficient can be used. This coefficient measures the correlation between two variables [76].

If the ranking of suppliers is performed using the generated and allocated weights for the six sets, and the results indicate a high similarity in the final ranking, the coefficient will approach 1. The results of the Spearman correlation coefficient calculations for the final ranking of suppliers in all six sets are presented in

Table 17. Spearman correlation coefficient values between sensitivity analysis sets.

	Set 1	Set 2	Set 3	Set 4	Set 5	Set 6
Set 1	1	0.9942	0.9905	0.9919	0.9954	0.9965
Set 2	0.9942	1	0.9928	0.9983	0.9922	0.9812
Set 3	0.9905	0.9928	1	0.9976	0.9771	0.9567
Set 4	0.9919	0.9983	0.9976	1	0.9848	0.9690
Set 5	0.9954	0.9922	0.9771	0.9848	1	0.9965
Set 6	0.9965	0.9812	0.9567	0.9690	0.9965	1

. Additionally, Figure 11 illustrates the correlation coefficient values between each set and the others.

As indicated, the correlation coefficient exceeds 0.9 in all cases, demonstrating a very strong relationship among the considered sets. Therefore, it can be concluded that changes in the criteria weights have minimal impact on the final ranking of suppliers. This confirms the robustness of the proposed PZ-LMAW and PZ-WASPAS approaches against variations in criteria weights.

5.3. Comparative Analysis

In this section, to validate the results obtained from the proposed approach, the score values and the final prioritization of sustainable–smart suppliers were compared with other approaches, including asymmetric linear pentagonal WASPAS (ALP-WASPAS), WASPAS with normalization (WASPAS-N) [77], optimized WASPAS (O-WASPAS) [78], and classic WASPAS, considering identical expert preferences. Since the proposed approach extends the WASPAS method under asymmetric linear pentagonal fuzzy Z-numbers, comparing the results with other fuzzy extensions of WASPAS is not feasible due to the nature of the data. Therefore, the comparison between approaches was conducted using the crisp values of expert preferences, as presented in Table 3.

Compared with the ALP-WASPAS approach, the reliability functions of pentagonal fuzzy Z-numbers were disregarded, and only the constraint functions of pentagonal fuzzy Z-numbers were considered. Despite structural differences from the proposed approach, the WASPAS-N and O-WASPAS approaches yielded similar results. The supplier prioritization results throughout the comparative analysis are presented in

Table 18. Results of the comparative analysis in the prioritization of sustainable–smart suppliers.

Suppliers	PZ-WASPAS		ALP-WASPAS		WASPAS-N		O-WASPAS		Classic WASPAS
	${\mathit{Q}}_{\mathit{i}}$	Order	${\mathit{Q}}_{\mathit{i}}$	Order	${\mathit{Q}}_{\mathit{i}}$	Order	${\mathit{Q}}_{\mathit{i}}$	Order	${\mathit{Q}}_{\mathit{i}}$	Order
S(1)	0.560	3	0.697	3	0.775	3	0.230	3	0.230	3
S(2)	0.466	4	0.600	4	0.700	4	0.209	4	0.208	4
S(3)	0.846	1	0.905	1	1.000	1	0.297	1	0.296	1
S(4)	0.667	2	0.786	2	0.882	2	0.262	2	0.262	2
$\rho$	-		0.994		0.995		0.996		0.995

.

The last row of Table 18 displays the Spearman correlation coefficients between the results of the proposed PZ-WASPAS approach and other approaches. As observed, the correlation coefficients in the examined cases are all above 0.8. Despite variations in supplier scores across the methods used in the comparative analysis, the final prioritization remained unchanged, thereby confirming the validity of the results obtained from the proposed PZ-WASPAS approach.

6. Discussion

In this section, we aim to delve deeper into the results and contributions of this research. From an application perspective, the findings of this study provide valuable insights into the problem of evaluating and selecting sustainable–smart suppliers, which can be expanded upon or utilized in other research within this domain. Since the concept of the SSC is still an emerging one, limited research has been conducted on it to date [11]. Although evaluating suppliers based on their sustainability and smartness may require identifying and employing different criteria across various industries and companies, the contribution of each study in this field is inherently valuable. In this study, six criteria were considered for evaluating sustainable–smart suppliers in a real-world case study. These criteria include employee development in a smart environment C₁, green and smart logistics and manufacturing C₂, waste reduction using smart technologies C₃, cost reduction using smart technologies C₄, smart working environment C₅, and smart delivery to customer C₆. Criteria C₄, C₂, and C₅ were identified as the top three criteria, with weights of 0.222, 0.212, and 0.196, respectively. Criteria C₃, C₆, and C₁ ranked fourth to sixth, with weights of 0.179, 0.160, and 0.126, respectively. In comparison with other studies in the literature on sustainable smart suppliers, which are admittedly limited in this area, a relative similarity is observed in the criteria identified as highly important. In most studies conducted in this domain, criteria related to cost reduction using smart technologies, waste reduction through smart technologies, green and smart technologies, and managerial measures for enhancing sustainable and smart frameworks are recognized as key criteria for SES [11,15,27]. Among the four potential suppliers considered in this study, suppliers S₍₃₎ and S₍₄₎, which demonstrated the best performance in simultaneously meeting sustainable and smart requirements, were identified as the top suppliers.

In the results validation section, two approaches were employed to evaluate the outcomes of the proposed method: sensitivity analysis based on the WASPAS measure and the logical pattern approach. The findings indicated that, in the generated scenarios of both sensitivity analysis approaches, despite modifications to the weights of decision-making criteria, the supplier ranking remained unchanged. This confirms that the proposed approach provides reliable, stable, and robust results. Additionally, the comparative analysis demonstrated that the proposed approach delivers reliable outcomes compared to other extensions of the WASPAS method.

From a methodological perspective, this study provides significant contributions to the literature on uncertainty and MCDM. POFNs, such as PFNs and HFNs, can more flexibly address the uncertainty present in decision-making processes. However, due to the structure of these numbers, which can be symmetric or asymmetric and linear or nonlinear, specific operators and calculations are required to handle them correctly. The explanation of algebraic operations of POFNs considering their symmetric and asymmetric structures and characteristics, the suggestion of an integrated and efficient approach for calculating the membership degree of symmetric and asymmetric POFNs, the suggestion of an efficient approach for formulating the $\mathcal{a}$-cuts of POFNs, and the introduction of a proposed method for the crispification process of POFNs based on their nature and structure are among the other contributions of this research.

As highlighted in the literature, there are significant gaps in the computations and operators of POFNs, particularly for asymmetric POFNs [49,50,56], some of which were addressed in this study. Furthermore, the application of POFNs as fuzzy constraint functions and fuzzy reliability functions within the Z-number framework is another contribution of this research. According to the proposed crispification process for symmetric and asymmetric POFNs, the Z-transformation process was seamlessly implemented, ultimately leading to the development of the PZ-LMAW and PZ-WASPAS approaches based on symmetric and asymmetric POFNs. In the proposed approaches, algebraic operations and defuzzification processes were conducted based on the structure and nature of POFNs, such as their symmetry (${\mathsf{\Psi }}_1={\mathsf{\Psi }}_2$) or asymmetry (${\mathsf{\Psi }}_1\ne {\mathsf{\Psi }}_2$), to achieve accurate and unique results from utilizing these numbers in the proposed MCDM approaches.

7. Conclusions

The SSCs have introduced new capabilities into the business landscape by leveraging advanced smart technologies. The adoption of emerging and efficient technologies such as AI, big data, and the IoT across various supply chain links has significantly enhanced the flexibility, efficiency, and productivity of traditional supply chains. As the first link in supply chains, suppliers play a critical and indispensable role in achieving the organization’s overarching visions and strategies, a role that becomes even more vital in the context of SSCs.

In the issue of SES, organizations assess potential new members for integration into their supply chains and establish partnerships with new collaborators. Alongside the drive to achieve smart principles in business processes, economic, environmental, and social concerns also play a significant role, bringing sustainable SSCs into focus. The primary objective of this study was to evaluate and select sustainable suppliers within SSCs. To achieve this, an efficient and flexible decision support framework was proposed to operate effectively in the context of uncertainty. The proposed approach, utilizing the concepts of POFNs, EI and EV, and Z-numbers, effectively addressed the high degree of uncertainty present in the decision-making environment. Furthermore, efficient and robust methods were introduced for computing membership functions, formulating $\mathcal{a}$-cuts, and performing crispification processes for both symmetric and asymmetric POFNs. Ultimately, leveraging the concepts and characteristics of symmetric and asymmetric POFNs, the decision-making approaches PZ-LMAW and PZ-WASPAS were developed and employed. These approaches were utilized to determine the importance of criteria for evaluating sustainable smart suppliers and to prioritize potential suppliers. In the proposed framework, Z-number fuzzy constraint functions and Z-number fuzzy reliability functions were developed based on the membership functions of symmetric and asymmetric POFNs. This development significantly enhanced the ability to address the high level of uncertainty inherent in the decision-making process.

The PZ-LMAW approach, developed based on symmetric linear PFNs and Z-numbers, was able to not only cover a high level of uncertainty in the decision-making environment but also take into account the reliability of expert preferences and incorporate them into the decision-making process. In addition, the proposed PZ-WASPAS approach significantly assisted decision-makers in prioritizing suppliers within the space of asymmetric linear PFNs, considering the uncertainty in the decision-making environment. The development of the PZ-WASPAS approach, based on asymmetric linear PFNs and Z-numbers, was also capable of incorporating expert preferences in the prioritization of suppliers.

To evaluate the applicability and feasibility of the proposed approach, a real-world case study was conducted to assess sustainable smart suppliers in the home appliance supply chain. The results were analyzed, revealing that the criteria cost reduction using smart technologies, green and smart logistics and manufacturing, and smart working environment were the most significant for evaluating sustainable suppliers in SSC. Suppliers three and four, which demonstrated the highest capabilities in these criteria, were identified as the top-performing suppliers. To validate the outcomes of the proposed approach, a sensitivity analysis was conducted on the main results. The findings confirmed that the results derived from the proposed approach exhibited substantial stability and robustness.

Certain limitations influenced the execution of this study. Evaluating suppliers in terms of their smart capabilities based on the documentation provided and their capabilities faced several challenges. Some potential suppliers had either recently adopted advanced and smart technologies or were in the process of completing and upgrading their smart technologies, making their precise evaluation difficult. Additionally, due to the time-intensive nature of gathering expert preferences regarding the importance of criteria and the ranking of each supplier for each criterion, some sessions were conducted in a compressed format, which might have impacted this study’s results. Furthermore, the SES process was carried out within a specific domain and production line of the company. To achieve the expected objectives, suppliers in other divisions of the company should also be evaluated based on sustainability and smartness principles.

Numerous suggestions can be envisioned for future research. As the concept of SSCs is still emerging in SCM, evaluating smart suppliers from both practical and theoretical perspectives requires further investigation. Suppliers in various industries with distinct characteristics can be assessed in terms of their smartness. Additionally, principles such as resilience, greenness, circularity, and others can be simultaneously considered when evaluating smart suppliers. The proposed approach in this study can be applied to other decision-making problems under uncertainty, and the results can be analyzed. Based on the concepts explained in this research, other types of POFNs can also be developed and evaluated in alternative MCDM approaches. Furthermore, advancing the proposed methods for calculating the EI and EV in the crispification process, computing membership functions, and formulating $\mathcal{a}$-cuts for nonlinear POFNs would be valuable contributions. Finally, the development of other approaches for the crispification process of symmetric and asymmetric linear and nonlinear POFNs, such as area-based approaches that can preserve the inherent characteristics of these numbers, is another suggestion for future research.