Sustainability-Driven Supplier Selection: Insights from Supplier Life Value and Z-Numbers

Abstract

In recent years, the strategic selection of the most suitable supplier within the supply chain has garnered increasing attention. Incorporating vital criteria like sustainable development further complicates this decision-making process. Companies and manufacturing facilities recognize the pivotal role of suppliers in their overall success and aim for mutually advantageous partnerships. Establishing long-term relationships with suppliers can yield benefits for both parties. However, supplier selection is intricate, often transpiring within an environment of limited information. Consequently, evaluating and selecting organizational suppliers necessitate methodologies yielding more dependable and pragmatic results due to the uncertainties inherent in expert judgments. This study introduces Supplier Life Cycle Value (SLV) criteria for extended partnerships with suppliers and sustainability metrics for selecting “industrial equipment suppliers”. The Hierarchical Best-Worst Method (HBWM) is then applied to determine Sustainable Supplier Life Value (SSLV) criteria weights. Subsequently, employing the PROMETHEE-GAIA approach, suppliers are systematically ranked and comprehensively analyzed. To account for the inherent uncertainty in expert judgments, this study incorporates fuzzy numbers enriched with probability and reliability parameters (Z-Numbers) by introducing novel verbal spectra for supplier evaluation. This facilitates more effective decision making in supplier management. The findings underscore the significance of considering the supplier’s longevity beyond economic metrics, emphasizing the importance of sustained supplier participation. Moreover, the varying outcomes across definite and fuzzy scenarios, accounting for reliability (Z-Numbers), underscore the impact of data uncertainty on decision making. Given that fuzzy numbers incorporating reliability (Z-Numbers) encompass the confidence probability within the unclear number, they offer a more robust and realistic representation of real-world scenarios.

1. Introduction

The success of supply chain activities is pivotal for sustained economic and societal growth. Among these activities, the management of purchasing decisions, such as procuring raw materials and components, holds the utmost importance. A lack of due diligence in these decisions can result in substantial financial losses [1]. As in most industries, raw materials and components necessary for creating products or services constitute a significant portion of the overall cost [2]. The decision-making processes for selecting the optimal supplier capable of delivering supplies at the most favorable prices without compromising quality continue to be a subject of sustained research. The heightened emphasis on environmental concerns and the sustainability of supply chains has introduced an added dimension to the selection criteria, necessitating the choice of suppliers who align with sustainable practices while upholding standards of quality and competitiveness [3]. Anka Butnario emphasized that although many criteria can enhance the decision-making process related to suppliers, it is imperative to establish a comprehensive framework encompassing pertinent sustainability criteria. This framework facilitates the incorporation of sustainability aspects into the decision-making process. For instance, numerous studies have employed composite indices with a sustainable development perspective. However, it is acknowledged that they may need to be fully evolved or efficacious in assessing all sustainability dimensions, including social, economic, and environmental [4]. It is paramount to establish a robust system for delineating supplier selection criteria rooted in sustainability principles and determining their respective significance. In this context, the application of weighting is regarded as a fitting mechanism. This approach highlights the most pertinent criteria and streamlines the process of selecting the optimal supplier [5]. Incorporating non-contradictory criteria into the selection process and aligning them with the organization’s business strategies serves to optimize weightings and enhance the anticipated value derived from supplier selection [6]. Indeed, in the practical realm, numerous competitive factors necessitate careful consideration when selecting suppliers. Hence, it becomes imperative to employ a method that enables the application of comparable criteria for evaluating all potential suppliers, thereby facilitating the prioritization of those that align with the established criteria. Moreover, employing a supplier selection methodology streamlines the process of quantitatively and qualitatively assessing a range of potential suppliers, simplifying the task of identifying and comparing their respective merits. This approach not only aids in prioritizing suppliers based on their adherence to the criteria, but also contributes to value creation [7]. Furthermore, the process of supplier selection entails many quantitative and qualitative considerations. Quantitative aspects encompass cost, profit, risk, and project completion time. However, accurately gauging these factors can often take time and effort, leading to uncertainty in the decision-making process. Conversely, qualitative elements tied to social, environmental, functional, technical, and technological contexts can only be articulated through linguistic variables [8]. As frequently encountered, the criteria used to evaluate suppliers tend to be ambiguous. Consequently, employing qualitative, verbal, and multidisciplinary decision-making approaches is highly beneficial [9]. The intricacy of decision making regarding technical matters encompasses quantitative and qualitative parameters and criteria. This complexity renders it particularly challenging to arrive at unequivocal and conclusive judgments in a subjective and non-discrete manner, consequently adding to overall uncertainty [10]. Adopting fuzzy logic is considered an approach that aids in comprehending uncertainties and surmounting specific challenges inherent in the decision-making process within uncertain environments. It allows for the incorporation of input from both experts and technical authorities to the greatest extent possible [11]. To accurately mirror real-world scenarios, it is imperative to incorporate probability and uncertainty within fuzzy numbers [12]. Z-numbers are a subset of fuzzy sets that consider the probability levels within fuzzy numbers. This is essential as the expert judgments and linguistic spectra of experts inherently involve uncertainties [13]. In light of the preceding arguments, the objective of this study is to use fuzzy logic to examine criteria for supplier selection based on sustainability principles. Among the innovations of this research, the following are noteworthy:

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Identification and Categorization of Crucial Sustainability Sub-Criteria and Supplier Life Cycle Value Calculation: This research pioneers in discerning and classifying pivotal sustainability sub-criteria while introducing a novel method for evaluating the life cycle value of suppliers. This significant advancement lays the foundation for a more comprehensive and nuanced understanding of supplier selection.

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Introduction of a Novel Verbal Spectrum for Z-Number Utilizing Scholarly Literature: This study introduces a fresh approach in the form of a verbal spectrum for Z-Number application, drawing insights from rigorous scientific research. This innovation enhances the precision and applicability of Z-Numbers in the context of supplier selection, providing a more refined tool for decision making.

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Application of Z-HBWM for Sustainability Sub-Criteria Weighting with Expert-Generated Questionnaires: The utilization of the Z-HBWM approach for weighting sustainability sub-criteria, facilitated by questionnaires crafted and validated by experts and professors, represents a significant stride forward. This method incorporates expert perspectives effectively and ensures a robust and data-driven assessment of the sub-criteria.

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Employment of Z-PROMETHEE for Supplier Ranking via Expert-Generated Questionnaires: The article pioneers using the Z-PROMETHEE approach for supplier ranking, leveraging the insights gathered from meticulously crafted questionnaires administered by experts and professors. This methodological innovation adds a layer of objectivity and rigor to the supplier evaluation process, ensuring more accurate and reliable rankings.

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Integration of Z-HBWM-PROMETHEE Combined Approach for Supplier Ranking with Consideration of Importance Weights: This study introduces a groundbreaking combined approach, merging the Z-HBWM and Z-PROMETHEE methodologies for supplier ranking. This innovation is marked by its incorporation of importance weights, providing a comprehensive framework that considers the relative significance of various criteria. This approach stands out as a robust and holistic method for supplier evaluation.

In sum, this research introduces a series of innovative methodologies and techniques that collectively advance the field of supplier selection. These innovations not only enhance the precision and comprehensiveness of the evaluation process but also provide a robust framework reflective of real-world complexities.

This article is structured into six distinct parts. Section 1 serves as an introduction, providing an overview of the research, including its general description, objectives, rationale, underlying assumptions, identified gaps, and innovative contributions. Moving on, Section 2 delves into theoretical principles and presents a thorough literature review. This encompasses defining critical terms related to supplier selection, sustainable development, and fuzzy uncertainty. Following this, the chapter reviews prior literature on supplier selection and ranking, culminating in a table of relevant articles. The gaps in existing research and the proposed innovations in the current study are then elucidated based on this literature review. Section 3 focuses on the research methodology, initiating the identification, classification, and screening of the sustainability sub-criteria informed by expert opinions. A new verbal range of the Z-Number approach is introduced, setting the stage for the subsequent introduction of the Z-HBWM approach to weighting. This methodology extends to calculating sub-criteria for sustainability and the supplier life cycle value, as well as the Z-PROMETHEE approach for ranking suppliers. Section 4 and Section 5 encompass solving a numerical example and its related analysis. This phase involves the preparation of questionnaires for scoring sub-criteria screening, sub-criteria weighting, and supplier rating. Relevant data is collected through expert interviews and questionnaires. The article then navigates through the application of the Z-HBWM and Z-PROMETHEE approaches, with ensuing analyses, including comparisons with deterministic and fuzzy modes. Finally, Section 6 encapsulates the conclusion and presents future suggestions. It summarizes the findings concisely, highlights model innovations, and provides recommendations for future research endeavors.

2. Literature Review

2.1. Supplier Selection & Importance

The supplier selection process constitutes a strategic decision that wields substantial influence over a company’s competitive edge. This significance is further amplified when companies venture into new markets and seek out potential suppliers. In such cases, the supplier selection decision becomes even more critical, directly impacting the company’s ability to establish a strong foothold and competitive position in unfamiliar territories. Furthermore, the surge in outsourcing activities in recent years has intensified the focus on supplier selection and evaluation. The significance of this process lies in its potential to foster enduring partnerships between buyers and suppliers, thereby conferring a distinct competitive advantage within the industry. This strategic approach not only streamlines operations but also bolsters the company’s position in the market, ensuring a sustainable edge over competitors [14]. The supplier evaluation process’s primary objective is to mitigate risk and optimize total value, which translates to maximizing total profit for buyers. In today’s economy, where the quality and quantity of products hold paramount importance, supplier selection is pivotal in the procurement planning of every factory. This underscores the critical need for a rigorous and well-informed supplier assessment and selection approach. As global prosperity and economic well-being continue to rise, individuals are experiencing an improvement in their living standards and an increase in wealth. However, this progress has led to some businesses prioritizing their economic gains without due regard for their social responsibilities and environmental equilibrium. This pursuit, driven by ambition and avarice, has raised concerns about the ethical implications of such practices. It underscores the importance of adopting a balanced and responsible approach to business operations in an era of increasing affluence [15]. Henceforth, green supply chain considerations and the broader spectrum of sustainable supply chain practices are poised to assume an increasingly significant role in supplier selection. This shift reflects a growing recognition of the imperative to align procurement processes with environmental and sustainability objectives [16]. Suppliers encompass companies and institutions responsible for fulfilling their clients’ procurement requirements, including raw materials and intermediate goods utilized in creating and producing products and services. It is imperative for suppliers to consistently and punctually provide client organizations with the necessary products, parts, and materials. This ensures the client organization can sustain its presence in the marketplace [17]. Supplier selection encompasses the thorough review, assessment, and choice of suppliers to integrate into a company’s supply chain. The primary objective of this process is to mitigate risks associated with the procurement process, such as quality concerns or delays in deliveries. Additionally, it aims to foster robust and collaborative relationships between buyers and suppliers. This ensures a smoother operation and contributes to the supply chain’s overall effectiveness [18]. In research about supplier selection, a variety of models have been employed. Some studies rely on a singular model [19,20,21], while others adopt a combination of different models. This diversity in approach underscores the complexity of supplier selection and highlights the need for tailored strategies based on specific contexts and objectives [22,23,24]. Acknowledging that no single model can assert its superiority over others in all contexts is crucial. Each model possesses its strengths and weaknesses, and its suitability depends on various factors, including the specific problem at hand, the criteria being considered, the availability of reliable data, and decision makers’ preferences. Individual methods for evaluating and selecting suppliers often encompass mathematical, artificial intelligence (AI)-based, or a fusion of mathematical and AI-based techniques. These approaches offer various tools, including data envelopment analysis, multi-objective planning, hierarchical analysis, and neural networks, all of which enhance the effectiveness of decision-making processes. Genovese et al. contend that supplier selection encompasses the comprehensive assessment and evaluation of a pool of suppliers to rank systematically and ultimately choose the most fitting candidate. This process is vital for sustaining the overall efficiency of the supply chain system [25]. Indeed, selecting the right supplier constitutes a pivotal decision with far-reaching implications for sustainability within the supply chain [26]. The selection process is intricate, as it demands a thorough consideration of diverse criteria to arrive at a well-informed decision. Beyond economic factors, integrating criteria pertaining to social and sustainability dimensions is imperative for successfully implementing sustainable supply-chain management. These aspects encompass both internal and external considerations. Internal social criteria encompass critical employment practices, including compensation, human resource policies, and occupational safety and health committees. On the other hand, external social criteria extend to interactions with local associations and contractual partners, including suppliers, customers, and non-governmental organizations (NGOs). This comprehensive approach ensures that all sustainable supply-chain management facets are appropriately addressed [2]. When excessive criteria are employed, pinpointing sustainability becomes a formidable task. Moreover, management decisions related to sustainable supply chains entail navigating diverse viewpoints and shared interests, making it infeasible to condense all metrics into a singular unit of measurement. During the criteria selection process, organizations consider various considerations, including alignment with key objectives, measurability, data accessibility, and the reliability of information sources. Following the selection of criteria, a crucial step involves their prioritization for practical application. This strategic approach ensures that resources are efficiently allocated and attention is focused on the most critical aspects of sustainable supply-chain management [27].

2.2. Supplier Selection by Decision Approach

Considering the multitude of approaches to supplier selection explored in prior research, our focus will be on reviewing articles within sustainable supplier selection, specifically those employing the HBWM, PROMETHEE, and Z-Number methodologies, given their pronounced significance. Through this targeted examination, we aim to identify and address research gaps pertinent to these specific approaches. Chen et al. (2010) introduced an innovative framework for assessing logistics suppliers, considering both quantitative and qualitative aspects. They combined a multi-criteria decision-making method, merging the Prometheus technique with the maximum standard deviation approach. This integrated method effectively evaluated and categorized logistics suppliers based on their performance, marking significant progress in supplier selection strategies within the logistics industry [28]. Senvar et al. (2014) [19] presented a fuzzy Prometheus-based solution for selecting the most suitable supplier. This method stands out for its user-friendly interface and ability to account for fuzzy or uncertain elements in decision making. Notably, it boasts versatility, applicable not only in supplier selection but also in resolving various selection challenges across service and industrial sectors [16]. Rezaei et al. (2015) unveiled a novel approach to supplier development, integrating supplier components as a crucial factor in formulating effective strategies. This conceptual model optimizes resource allocation, ensuring more efficient use of limited resources. The authors leveraged the BMW method in implementing their innovative model [20]. Rezaei et al. (2016) introduced a life cycle-focused approach to supplier selection, incorporating conventional and environmental criteria using the BMW method. Their model encompasses three key stages: pre-selection, selection, and integration, offering a comprehensive framework for evaluating suppliers. Especially pertinent in the edible oil industry, this method elevates supplier selection within the food supply chain. In parallel, in 2016, Rezaei et al. unveiled a novel technique based on PROMETHEE, facilitating a transparent ranking of environmentally conscious suppliers in the food industry [14]. Sari and Timur (2016) investigated supplier selection employing the ANP multi-criteria decision-making method in conjunction with the Taguchi function, subsequently conducting a comparative analysis with the PROMETHEE method. Their endeavor aimed to furnish a model for intricate supplier selection challenges, yielding comparatively more precise outcomes than conventional and non-traditional approaches. This approach signifies a significant stride towards enhancing the efficacy of supplier selection processes [29]. Agakishiyev (2016) introduced an innovative approach to address supplier selection in a fuzzy environment, employing the Z-number technique [30]. In 2018, Lo et al. [31] presented an integrated model for selecting a green supplier and allocating orders. They applied the fuzzy TOPSIS multi-criteria decision-making method and fuzzy multi-objective linear programming to tackle this challenge effectively. The study’s findings highlight the model’s capability to assess the performance of green suppliers and enhance the allocation of orders to preferred suppliers [1]. Jabbarova (2017) delved into the intricacies of supplier selection amid uncertain circumstances, employing the concept of Z-numbers. This approach represents a strategic consideration of uncertainty in the supplier selection process, offering valuable insights for decision making in complex environments [32]. Gupta and Barua (2017) introduced a model addressing uncertainty in selecting green suppliers for small and medium-sized enterprises, employing fuzzy TOPSIS and BWM methodologies. The study’s findings demonstrate that the framework outlined in this paper applies to various organizations, facilitating the recurrent selection of suppliers. This model contributes to sustainable supplier selection in diverse business settings [22]. Krishnan Kumar et al. (2017) introduced an innovative method utilizing the PROMETHEE technique within an intuitive fuzzy framework for supplier selection, taking into account language preferences. Their proposed model involves two pivotal steps: firstly, integrating the LBA operator collector to incorporate a decision-makers language preference directly; secondly, applying the IFSP method to extend the findings of the PROMETHEE method within an IFS environment, thereby enabling the ranking of preferred options. This approach provides a resilient solution for supplier selection, with heightened attention to linguistic preferences [33]. Cheraghalipour et al. (2018) endeavored to introduce a straightforward framework for supplier selection in the agricultural industry, employing WBWM methods and multi-criteria decision-making techniques. The criteria considered for supplier selection encompass aspects like product delivery, pricing, services, training resources, demeanor, quality, and communication systems. This proposed model also exhibits the requisite comprehensiveness for application in diverse industries [15]. Tian et al. (2018) emphasized the importance of green supplier selection, especially in the Chinese food and agriculture industries, as a critical component of the green supply chain. Their study examined a range of criteria, covering financial aspects like transportation costs, delivery, and service metrics, including production capacity, quality factors such as product excellence and operational control, and environmental management system criteria. The research offers a valuable contribution towards advancing sustainability practices within these sectors [34]. Badi and Ballem (2018) investigated supplier selection using the combined ROUGH BWM-MAIRCA model, specifically focusing on the Libyan pharmaceutical industry [35]. Aboutorab et al. (2018) introduced an innovative model for multi-criteria decision making in supplier selection. This model enhanced the BMW method in a fuzzy environment and integrated the Z-number technique. Through comparisons with traditional methods, they validated the effectiveness of their approach. These studies provide significant advancements in supplier selection methodologies [36]. Liu et al. (2019) introduced a hybrid model employing the BWM and AQM methods (option queuing methods) to address supplier selection in uncertain conditions within a two-dimensional fuzzy environment. Their focus on sustainable supply-chain management led to the creation of a framework that aligns with governmental regulations while enhancing public awareness. This innovative approach contributes to more effective and responsible supplier selection practices [21]. Hashemkhani et al. (2019) introduced a structured framework for sustainable supplier selection, employing the BWM-COCOSO hybrid method. Their evaluation encompassed economic factors like product price and financial and production capacity alongside social considerations such as human rights, social responsibilities, and environmental criteria, including pollution, resource consumption, and fuel usage. Notably, this model found application in Iran’s Alborz steel industry, demonstrating its adaptability and relevance in specific industrial contexts [23]. Wu et al. (2019) combined a type II multi-criteria fuzzy decision-making method with the BWM technique to address green supplier selection. Focusing on business expansion, they aimed to develop a model conducive to an environmentally responsible supply chain. This research represents a significant step towards enhancing sustainability within supplier selection processes [37]. Darvishi et al. (2020) proposed a comprehensive model for supplier selection, incorporating economic, social, and environmental considerations. Their approach involved the application of multi-criteria decision-making methods, specifically leveraging the triangular fuzzy TOPSIS method alongside the BWM approach. The study conducted at Khuzestan Steel Company with the input of 20 expert managers contributed to a notable progression in supplier selection methodologies. This research is pivotal in enhancing sustainable procurement practices [24]. Reza Hosseini et al. (2020) introduced an innovative hierarchical model for uncertain sustainable supplier selection, employing the Z-number technique. By prioritizing sustainability criteria, they outlined a clear step-by-step numerical example to elucidate their model. Additionally, utilizing the Z-number technique enhanced data reliability and bolstered the credibility of the obtained results. This research contributes significantly to advancing sustainable procurement strategies [38]. Tavana et al. (2021) introduced a novel fuzzy green supplier selection model for sustainable reverse logistics in green supply-chain management. The model employs a hierarchical fuzzy best-worst method (HFBWM) for determining green criteria weights and integrates complex evaluation methods like Shannon’s fuzzy entropy. Hybrid models incorporating COPRAS, MULTIMOORA, and TOPSIS techniques are utilized for supplier ranking, demonstrating the effectiveness of even simpler methods like fuzzy COPRAS and MOORA. The study is validated through a case study in the asphalt manufacturing industry, affirming its practical applicability [39]. Masoomi et al.’s 2022 study addresses the crucial Supplier Selection Problem (SSP) in Renewable Supply-Chain Management (RSCM). They advocate for choosing a green supplier to promote sustainability, optimize resource consumption, and reduce environmental impact. The authors propose a comprehensive evaluation framework, combining FBWM, COPRAS, and WASPAS techniques and demonstrate its applicability through a case study in Iran’s renewable energy supply chain. This approach aids decision-makers in differentiating and assessing green suppliers in local and global markets, concluding with a comparative analysis of the framework’s strengths and limitations [40]. Shang et al. (2022) introduce a novel method for Sustainable Supplier Selection (SSS) to address shortcomings in traditional approaches. They combine subjective and objective weightings effectively, enhancing the robustness of the process. The study’s contributions lie in its innovative approach, the application of fuzzy MULTIMOORA in SSS, and the improved reference point method [41]. Li Zhong Tong et al.’s (2022) study highlights the vital role of small and medium-sized enterprises (SMEs) in national production and employment. SMEs face operational challenges in supplier dealings due to their limited scale, funding, and bargaining power. The paper introduces a tailored sustainable supplier selection framework for SMEs, employing an extended PROMETHEE Ⅱ method, which considers product/service capability, cooperation level, and risk factors [42]. Nazari-Shirkouhi et al. (2023) emphasize the importance of resilient supplier selection in mitigating disruptions in today’s uncertain business environment, especially in pharmaceutical companies. The proposed approach integrates traditional and resilient criteria using Z-number data analysis and artificial neural networks, offering a vital tool for maintaining competitiveness. Through a case study, the method proves effective in navigating uncertainties and ensuring timely pharmaceutical production and distribution [43].

Table 1. Summary of articles.

Reference	Selection Criteria				Uncertainty					Weighting Approach					Ranking Approach
	Economic	Social	Environment	Supplier Life Value	Triangle Fuzzy	Intuitionistic Fuzzy Set	Fuzzy Type 2	ROUGH Number	Z-Number	Maximum Deviation Method	Shannon Entropy	ANP	Best Worst Method	Hierachial Best Worst Method	CoCoSo	COPRAS	WASPAS	MULTIMOORA	AQM	MAIRCA	VIKOR	TOPSIS	hierarchical Model: Z-Number	DEA	PROMETHEE	PROMETHEE-GAIA
[28]		√			√					√															√
[19]	√	√			√																				√
[20]	√												√
[14]			√
[29]			√									√													√
[30]									√
[21]	√		√		√																	√
[32]	√								√														√
[22]			√		√								√									√
[44]						√																			√
[15]	√												√								√
[34]	√		√		√								√									√
[35]			√					√					√							√
[45]	√	√							√				√
[21]		√					√						√						√
[23]	√	√	√										√		√
[16]			√				√						√								√
[24]	√	√	√		√								√									√
[38]	√	√	√						√														√
[39]			√		√								√			√		√				√
[40]			√		√								√			√	√
[41]	√	√	√		√						√		√					√
[42]	√																								√
[43]	√								√															√
paper	√	√	√	√					√					√												√

summarizes past research.

The following section examines the research gaps and compares the approaches and concepts used most in previous articles with the techniques and ideas of this research. Several compelling reasons drive the selection of the HBWM approach in this research. First and foremost, it significantly reduces the number of required questionnaires compared to AHP and ANP methods. This reduction to just questionnaires conserve valuable time and results in a more efficient data collection process. With fewer questionnaires, experts can make more stable and reliable comparisons, leading to enhanced criteria-weighting accuracy.

Furthermore, the streamlined process decreases the number of iterations needed for hierarchical decisions, promoting efficiency and resource conservation. Overall, the HBWM approach balances accuracy and resource efficiency, making it a practical choice for this study. Conversely, the utilization of the PROMETHEE-GAIA approach is underpinned by its exceptional accuracy in multi-criteria decision making. PROMETHEE offers precise results, making it accessible and reliable for professionals and non-professionals. Its user-friendliness is critical, ensuring decision-makers can readily understand and utilize the method effectively. PROMETHEE empowers decision-makers by allowing them to create a preferential function, granting them direct influence over project outcomes. This approach also adeptly handles criteria with different measurement scales, eliminating the need for scaling matching. The integration of GAIA enhances efficiency, enabling options to be ranked based on both positive and negative aspects and providing more comprehensive evaluations. PROMETHEE’s compatibility with the GAIA graphic method allows for effectively visualizing option differences, improving decision-makers’ understanding. It can analyze quantitative and qualitative data, reducing compensatory effects and facilitating focused and precise decision making. Lastly, adopting the Z-Number approach in this study is motivated by its adeptness in expressing ambiguity and uncertainty, particularly in dealing with the probability of fuzzy numbers. This feature is invaluable when confronting real-world situations characterized by prevalent uncertainty. Additionally, the Z-Number approach offers simplicity in mathematical calculations compared to other approaches like Fuzzy Type 2 and the Intuitionistic Fuzzy Set, saving time and mitigating computational errors. Its real-world applicability is enhanced by its incorporation of the probability of fuzzy numbers, aligning it more closely with practical industrial applications and decision-making processes. The research innovations and gaps identified in this study include the following:

• Identification and Categorization of Sustainability Sub-criteria: This study identifies, categorizes, and filters the sustainability sub-criteria specific to the zinc metal industry. This focused approach ensures that only relevant criteria are considered in the evaluation process.
• Consideration of Supplier Life Cycle Value Criteria: This study introduces and takes into account the value criteria associated with the life cycle of suppliers. This emphasizes the importance of establishing long-term partnerships with suppliers, aligning with sustainable practices.
• Weighting of Sustainable Life Cycle Value Sub-criteria: The Z-HBWM approach is utilized for weighting the sub-criteria related to sustainable life cycle value. This approach employs fuzzy-probabilistic Z-data gathered through expert questionnaires, providing a robust and comprehensive assessment.
• Supplier Ranking using Z-PROMETHEE Approach: This study employs the Z-PROMETHEE approach for ranking suppliers. This method utilizes fuzzy-probabilistic Z-data collected from relevant experts. It ensures a thorough evaluation of suppliers based on sustainability criteria.
• Integration of Z-Number Information in Decision Making: This study leverages Z-Number information in both the HBWM and PROMETHEE decision-making approaches. This integration brings the decision-making process closer to real-world conditions, enhancing the reliability of outcomes.

3. Methodology

This research is divided into four main parts, which are as follows (Figure 1):

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Step 1: Identify, Categorize, and Filter Sustainability Criteria: This step focuses on identifying and organizing the key sustainability criteria that will be used to evaluate potential suppliers. The Fernandez Sanchez and Rodriguez Lopez approach is employed to systematically identify, categorize, and filter the sustainability criteria. This approach likely provides a structured framework for evaluating various aspects of supplier sustainability, such as environmental impact, social responsibility, and economic viability.

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Step 2: Calculate Supplier’s Life Cycle Value: This step involves assessing the value of a supplier’s life cycle, which encompasses various stages from procurement to disposal. The calculation of the supplier’s life cycle value is conducted as a critical criterion in the supplier selection process. This metric likely involves analyzing factors like product quality, reliability, and total cost of ownership over the entire life cycle.

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Step 3: Introduce the Z-HBWM Approach for Weighting Criteria: This step introduces the Z-HBWM (Z-Number Hierarchical Best Worst Method) approach to assign weights to the identified criteria. The Z-HBWM method involves using Z-Numbers, which are a specialized form of fuzzy numbers with additional probabilistic information. This approach likely enhances the ability to handle ambiguity and uncertainty in the weighting process. The introduction of a new lexical range for Z-Numbers suggests that specific linguistic terms or descriptors are used to define the degrees of membership and non-membership associated with the Z-Numbers.

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Step 4: Introduce the Z-PROMETHEE Approach for Supplier Ranking: This step introduces the Z-PROMETHEE approach to rank sustainable suppliers based on the weighted criteria. The Z-PROMETHEE method likely leverages Z-Numbers to incorporate additional uncertainty and ambiguity considerations in the ranking process. Defining a new word spectrum for Z-Numbers suggests that specific linguistic terms or descriptors are used to assess the preferences and outranking relationships between suppliers.

Figure 1. Methodology process.

Figure 1. Methodology process.

3.1. Identifying, Categorizing, and Filtering Sustainability Sub-Criteria and Calculating Supplier Life Cycle Value

While sustainability is a crucial factor in supplier selection, social and environmental criteria have been underutilized in this process. This section aims to identify and compile criteria and criteria that hold significance in choosing sustainable suppliers. Additionally, it focuses on evaluating the supplier’s lifespan value, signifying a commitment to long-term partnership with the organization.

3.1.1. Steps of Filtering Sustainability Criteria to Select Suppliers

In this study, a methodology akin to the one proposed by Fernandez Sanchez and Rodriguez Lopez was employed to establish the sustainability criteria and criteria for supplier selection [46]. This methodology encompasses three stages. An initial set of criteria is identified based on previous research and studies. However, not all of these criteria may be feasible, as some may lead to increased costs and longer delivery times for the supplier. Moreover, specific criteria might need to be physically or technically viable. Hence, screening and filtering these sustainability criteria is necessary before analyzing and assessing supplier sustainability.

Furthermore, considering too many criteria can lead to a high analysis cost and make the requirements complex to comprehend. Conversely, crucial metrics may only be noticed if more requirements are considered, potentially resulting in significant oversights. To prioritize the identified criteria, input from experts well-versed in procurement and supplier selection was solicited. The ensuing steps outline the process employed to identify these criteria using this method:

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Step 1 Identifying Criteria: In the initial phase, a comprehensive review of studies pertaining to supplier selection criteria and criteria for evaluating supplier sustainability was conducted. This process yielded an initial list of criteria and criteria crucial for sustainability in supplier selection.

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Step 2 Categorizing the Criteria: Subsequently, the criteria identified in the previous step were systematically classified. The table below illustrates the framework for sustainable assessment, which encompasses the three pillars of sustainability: economic, social, and environmental. Employing this same framework, the indices outlined in Table 2 were organized within the structure of sustainable assessment.

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Step 3 Filtering Criteria and Seeking Expert Opinions: During this phase, the opinions and insights of experts are leveraged to scrutinize each of the identified criteria. This process aids in filtering out criteria that may be irrelevant or impractical for this study.

Table 2. Classification of sustainability criteria.

Reference	Criteria	Dimensions
[47]	Quality of products	Economical
[48]	Sales price of products
[49]	Delivery time
[48]	Geographical location of the supplier
[48]	Production capacity
[50]	Supplier performance
[51]	Supplier profitability
[52]	Terms of payment
[50]	Warranty and after-sales service
[53]	Supplier inventory management
[54]	Customer relation management
[50]	Supplier risk and crisis management
[47]	Experience and background
[55]	Capacity and capability of e-commerce
[56]	Variety of products
[53]	Pay attention to social responsibilities	Social
[57]	Compliance of employment contracts with international and governmental laws
[26]	Respect for government policies
[58]	Occupational safety and health management systems
[59]	Work methods and competencies
[60]	Local shopping degree
[58]	The level of attention paid to working children and participation in charitable activities
[61]	Creating employment
[26]	Employee benefits and rights
[26]	Note the rights of stakeholders
[53]	Green design of products and use of environmentally friendly materials	Environmental
[53]	Environmental protection policy
[62]	Environmental competencies
[26]	Green R&D
[2]	Pollution rate and control
[26]	Green product production
[26]	Resource consumption and the use of non-renewable energy
[26]	Electrical and electronic waste
[63]	Product recyclability
[26]	Green supply-chain management
[53]	Green packaging

Table 2. Classification of sustainability criteria.

Reference	Criteria	Dimensions
[47]	Quality of products	Economical
[48]	Sales price of products
[49]	Delivery time
[48]	Geographical location of the supplier
[48]	Production capacity
[50]	Supplier performance
[51]	Supplier profitability
[52]	Terms of payment
[50]	Warranty and after-sales service
[53]	Supplier inventory management
[54]	Customer relation management
[50]	Supplier risk and crisis management
[47]	Experience and background
[55]	Capacity and capability of e-commerce
[56]	Variety of products
[53]	Pay attention to social responsibilities	Social
[57]	Compliance of employment contracts with international and governmental laws
[26]	Respect for government policies
[58]	Occupational safety and health management systems
[59]	Work methods and competencies
[60]	Local shopping degree
[58]	The level of attention paid to working children and participation in charitable activities
[61]	Creating employment
[26]	Employee benefits and rights
[26]	Note the rights of stakeholders
[53]	Green design of products and use of environmentally friendly materials	Environmental
[53]	Environmental protection policy
[62]	Environmental competencies
[26]	Green R&D
[2]	Pollution rate and control
[26]	Green product production
[26]	Resource consumption and the use of non-renewable energy
[26]	Electrical and electronic waste
[63]	Product recyclability
[26]	Green supply-chain management
[53]	Green packaging

3.1.2. Supplier Life-Cycle Value (SLV)

One effective strategy for achieving high-quality products and services involves establishing a collaborative environment with suppliers. This collaborative approach aims to elevate the quality standards of the suppliers to align with those of the core organization. It entails forging long-term committed partnerships between two or more organizations and pooling resources for the mutual pursuit of specific business goals and objectives. The advent of new technologies in supply-chain management has underscored the significance of establishing robust partnerships with suppliers. Historically, companies often viewed their suppliers skeptically and even as potential competitors. This mistrust led to low levels of loyalty and conscientiousness towards suppliers.

Consequently, suppliers needed to be more specific about the longevity of their relationship with an organization. Procurement and supply departments unwittingly played a role in perpetuating this issue. Recognizing the pivotal role of enduring supplier partnerships in today’s competitive market landscape, this section introduces the Supplier Life Cycle Value Index. This index is developed based on the Customer Life Cycle Value Model (CLV) and is tailored to establish long-term supplier partnerships. Calculating a supplier’s life cycle value draws inspiration from Customer Relationship Management (CRM) principles. This study delves into an extensive literature review on the value of customer life, encompassing both quantitative and qualitative sub-criteria for assessment. Subsequently, novel sub-criteria for evaluating and selecting suppliers are defined and presented. These sub-criteria are grounded in the definitions outlined in customer-centric literature. They are discerned from both retrospective and forward-looking perspectives for supplier evaluation. Determining these sub-criteria involved a meticulous review of supplier criteria literature and a thorough exploration of customer life value measurement literature. Common sub-criteria were eliminated, and those with meaningful implications for suppliers were selected and defined among the remaining ones. These sub-criteria delineated as sub-criteria within the context of supplier value, are presented in

Table 3. Classification of Supplier Life-Cycle Value criteria.

Time	Reference	Definition	Index Name
Retrospective	[64]	A loyal supplier pertains to those who have consistently furnished raw materials to the organization over an extended period, demonstrating a willingness to accommodate any sacrifices, costs, or limitations necessitated by the partnership for the sake of maintaining a long-term affiliation with the organization. Additionally, they refrain from favoring competitors of the organization when fulfilling supply commitments.	The level of supplier loyalty
	[65]	Employee satisfaction is a favorable internal sentiment that reflects the degree to which an organization’s anticipations regarding performance, financial aspects, behavioral conduct, and other relevant factors in the employment partnership have been fulfilled. This assessment is based on the experience gained from the supplier.	Satisfaction of the supplier
	[66]	The level of discord, divergence in ideas, misalignment of values, and objectives in the professional association between the supplier and buyer is referred to as “relationship conflict”. When there is a discrepancy in goals and values, it can lead one party to obstruct the other’s access to resources or impede a crucial activity for the sake of its own advancement.	Disagreement on goals and values
	[64]	The supplier’s collaboration with the organization in adhering to the workflow results in expedited operations. Consequently, if the supplier is unwilling to cooperate with the organization, it can lead to delays in addressing tasks and related issues.	The willingness of the supplier to work with the organization
	[66]	The frequency of disputes with the supplier and the tensions arising from factors like having unrealistic expectations can be quantified as “conflict incidents”. As the number of such incidents escalates between the supplier and the buyer, it intensifies the overall stress levels within their professional relationship.	Number of times arguing with the supplier
futuristic	[64]	This indicates a prospect of sustained collaboration with the supplier in the foreseeable future. Anticipations for enduring cooperation with the supplier can be rooted in the supplier’s demonstrated loyalty through numerous past collaborations, particularly the most recent one.	The potential for long-term communication in the future
	[64]	The anticipated profit margin from the acquisition (taking into account the purchase price, potential supplier-provided discounts, etc.) and the volume of future raw material purchases from the supplier are crucial factors to consider.	Expected profitability of the supplier’s lifetime in the future
	[64]	The proportionate expenses projected to be borne by the supplier in the future are a significant consideration. These costs encompass various elements such as expenses related to faulty items and wastage, shipping costs contingent on the supplier’s location, and costs stemming from delayed supply, including potential losses in opportunities, diminished credibility with the organization’s customers, and decreased organizational efficiency. Additionally, there are costs linked to the process of identifying and onboarding new suppliers if the current supplier lacks the potential for a sustained long-term relationship in the future.	Expected cost of the supplier’s lifetime

.

These criteria are filtered by designing a questionnaire and interview in the direction of the company’s strategic goals, and in addition to the filtered sustainability criteria, the final criteria are selected to rank suppliers.

3.2. Z-Number and the Linguistic Scale with Probabilities

In a paper in 2011, Zadeh expounded the concept of Z-numbers in regards to uncertain variable $X$ which were of real values [67]. We have also delineated a number of Z-numbers concepts in the following sections. According to Kang et al. 2012’s [68] proposed fuzzy model, for any specific Z-number with the pair of $Z=\left(A,B\right)$, $\tilde{B}=\left\{(x, {\mu }_{\tilde{B}}(x))|x ϵ \left[\mathrm{0,1}\right]\right\}$ denotes the fuzzy certainty value of and ${\mu }_{\tilde{B}}\left(x\right)$ represents the membership function [69]. The crisp centroid (center of gravity) value of $\tilde{B}$ is defined using Equation (1) as follows:

$$\gamma =\frac{\int x{\mu }_{\tilde{B}}\left(x\right)d_x}{\int {\mu }_{\tilde{B}}\left(x\right)d_x}$$

(1)

If $\tilde{B}~ TrFS \left[a,b,c\right],$ then the centroid defuzzification of this set is $\frac{a+b+c}{3}$. If $\overline{A}~ TrFS \left[d,e,f\right]$ the fuzzy Z-number is presented according to Equation (2) as follows:

$$A^{\prime }=\left[\sqrt{\gamma }d,\sqrt{\gamma }e,\sqrt{\gamma }f\right]=\left[d^{\prime }, e^{\prime }, f^{\prime }\right]$$

(2)

Table 4. Z-number corresponding probabilities [38].

Linguistic Variable	Corresponding Probabilities	$\mathit{\gamma }$	$\sqrt{\mathit{\gamma }}$
Unlikely	(0.1, 0.2, 0.3)	0.2	0.45
Fairly Impossible	(0.3, 0.4, 0.5)	0.4	0.63
Weak	(0.4, 0.5, 0.6)	0.5	0.71
Maybe	(0.5, 0.6, 0.7)	0.6	0.77
Likely	(0.7, 0.8, 0.9)	0.8	0.89
Most Likely	(0.8, 0.9, 1)	0.9	0.95
Certainly	(1, 1, 1)	1	1

demonstrates the ultimate fuzzy Z-numbers’ corresponding probabilities as well as the Z-numbers’ corresponding probabilities.

The table above along with $A^{\prime }=\left[\sqrt{\gamma }d,\sqrt{\gamma }e,\sqrt{\gamma }f\right]=\left[d^{\prime }, e^{\prime }, f^{\prime }\right]$ were used to construct the Linguistic Preference Scale and the Linguistic Scoring Scale with probabilities, which are the main contributions of this paper, and are elaborated on in the following sections.

3.3. Z-HBWM

The Best–Worst Decision Method (BWM) is a multidisciplinary decision-making approach initially introduced by Rezaei in 2015. Subsequently, in 2016, he further extended the model in another publication. Rezaei’s original model operated within a deterministic framework [70]. However, Zhao et al. conducted an examination of the BWM model within a fuzzy environment and successfully addressed this model’s implementation in such a context [71]. This was accomplished by offering various illustrative examples. The incorporation of fuzzy numbers serves to mitigate any uncertainties arising from respondents’ statements. The steps of HBWM technique in fuzzy and Z-Number conditions are as follows:

Step (1): Identifying the Best and Worst Criteria (Most and Least Important): This step can be accomplished through expert opinions or by employing the fuzzy Delphi method.

Step (2): Conducting Pairwise Comparisons between the best criterion and all other criteria, and between the worst criterion and all other criteria: In this step, pairwise comparisons can be conducted using any fuzzy scale.

Table 5. Linguistic preferences scale [72].

Linguistic	Acronym	Fuzzy Number	Scale of Fuzzy Number
Perfect	(P)	$\tilde{9}$	(8, 9, 10)
Absolute	(A)	$\tilde{8}$	(7, 8, 9)
Very Good	(VG)	$\tilde{7}$	(6, 7, 8)
Fairly Good	(FG)	$\tilde{6}$	(5, 6, 7)
Good	(G)	$\tilde{5}$	(4, 5, 6)
Preferable	(P)	$\tilde{4}$	(3, 4, 5)
Not Bad	(NB)	$\tilde{3}$	(2, 3, 4)
Weak Advantage	(WA)	$\tilde{2}$	(1, 2, 3)
Equal	(E)	$\tilde{1}$	(1, 1, 1)

demonstrates the linguistic preference scale numbers that would subsequently be used for paired comparisons in the following step:

Table 6. Table of preferential–probabilistic verbal spectrum.

Linguistic Phrase	Acronym	Certainty	Fuzzy Number	Z-Number
Perfect-Certainly	(PE-C)	$\tilde{9}$	(8, 9, 10)	(8.000, 9.000, 10.000)
Absolute-Certainly	(A-C)	$\tilde{8}$	(7, 8, 9)	(7.000, 8.000, 9.000)
Very Good-Certainly	(VG-C)	$\tilde{7}$	(6, 7, 8)	(6.000, 7.000, 8.000)
Fairly Good-Certainly	(FG-C)	$\tilde{6}$	(5, 6, 7)	(5.000, 6.000, 7.000)
Good-Certainly	(G-C)	$\tilde{5}$	(4, 5, 6)	(4.000, 5.000, 6.000)
Preferable-Certainly	(P-C)	$\tilde{4}$	(3, 4, 5)	(3.000, 4.000, 5.000)
Not Bad-Certainly	(NB-C)	$\tilde{3}$	(2, 3, 4)	(2.000, 3.000, 4.000)
Weak Advantage-Certainly	(WA-C)	$\tilde{2}$	(1, 2, 3)	(1.000, 2.000, 3.000)
Equal-Certainly	(E-C)	$\tilde{1}$	(1, 1, 1)	(1.000, 1.000, 1.000)
Perfect-Most Likely	(PE-ML)	$\tilde{9}$	(8, 9, 10)	(7.584, 8.532, 9.480)
Absolute-Most Likely	(A-ML)	$\tilde{8}$	(7, 8, 9)	(6.636, 7.584, 8.532)
Very Good-Most Likely	(VG-ML)	$\tilde{7}$	(6, 7, 8)	(5.688, 6.636, 7.584)
Fairly Good-Most Likely	(FG-ML)	$\tilde{6}$	(5, 6, 7)	(4.740, 5.688, 6.636)
Good-Most Likely	(G-ML)	$\tilde{5}$	(4, 5, 6)	(3.792, 4.740, 5.688)
Preferable-Most Likely	(P-ML)	$\tilde{4}$	(3, 4, 5)	(2.844, 3.792, 4.740)
Not Bad-Most Likely	(NB-ML)	$\tilde{3}$	(2, 3, 4)	(1.896, 2.844, 3.792)
Weak Advantage-Most Likely	(WA-ML)	$\tilde{2}$	(1, 2, 3)	(0.948, 1.896, 2.844)
Equal-Most Likely	(E-ML)	$\tilde{1}$	(1, 1, 1)	(0.948, 0.948, 0.948)
Perfect-Likely	(PE-L)	$\tilde{9}$	(8, 9, 10)	(7.152, 8.046, 8.940)
Absolute-Likely	(A-L)	$\tilde{8}$	(7, 8, 9)	(6.258, 7.152, 8.046)
Very Good-Likely	(VG-L)	$\tilde{7}$	(6, 7, 8)	(5.364, 6.258, 7.152)
Fairly Good-Likely	(FG-L)	$\tilde{6}$	(5, 6, 7)	(4.470, 5.364, 6.258)
Good-Likely	(G-L)	$\tilde{5}$	(4, 5, 6)	(3.576, 4.470, 5.364)
Preferable-Likely	(P-L)	$\tilde{4}$	(3, 4, 5)	(2.682, 3.576, 4.470)
Not Bad-Likely	(NB-L)	$\tilde{3}$	(2, 3, 4)	(1.788, 2.682, 3.576)
Weak Advantage-Likely	(WA-L)	$\tilde{2}$	(1, 2, 3)	(0.894, 1.788, 2.682)
Equal-Likely	(E-L)	$\tilde{1}$	(1, 1, 1)	(0.894, 0.894, 0.894)
Perfect-Maybe	(PE-M)	$\tilde{9}$	(8, 9, 10)	(6.192, 6.966, 7.740)
Absolute-Maybe	(A-M)	$\tilde{8}$	(7, 8, 9)	(5.418, 6.192, 6.966)
Very Good-Maybe	(VG-M)	$\tilde{7}$	(6, 7, 8)	(4.644, 5.418, 6.192)
Fairly Good-Maybe	(FG-M)	$\tilde{6}$	(5, 6, 7)	(3.870, 4.644, 5.418)
Good-Maybe	(G-M)	$\tilde{5}$	(4, 5, 6)	(3.096, 3.870, 4.644)
Preferable-Maybe	(P-M)	$\tilde{4}$	(3, 4, 5)	(2.322, 3.096, 3.870)
Not Bad-Maybe	(NB-M)	$\tilde{3}$	(2, 3, 4)	(1.548, 2.322, 3.096)
Weak Advantage-Maybe	(WA-M)	$\tilde{2}$	(1, 2, 3)	(0.774, 1.548, 2.322)
Equal-Maybe	(E-M)	$\tilde{1}$	(1, 1, 1)	(0.774, 0.774, 0.774)
Perfect-Weak	(PE-W)	$\tilde{9}$	(8, 9, 10)	(5.656, 6.363, 7.070)
Absolute-Weak	(A-W)	$\tilde{8}$	(7, 8, 9)	(4.949, 5.656, 6.363)
Very Good-Weak	(VG-W)	$\tilde{7}$	(6, 7, 8)	(4.242, 4.949, 5.656)
Fairly Good-Weak	(FG-W)	$\tilde{6}$	(5, 6, 7)	(3.535, 4.242, 4.949)
Good-Weak	(G-W)	$\tilde{5}$	(4, 5, 6)	(2.828, 3.535, 4.242)
Preferable-Weak	(P-W)	$\tilde{4}$	(3, 4, 5)	(2.121, 2.828, 3.535)
Not Bad-Weak	(NB-W)	$\tilde{3}$	(2, 3, 4)	(1.414, 2.121, 2.828)
Weak Advantage-Weak	(WA-W)	$\tilde{2}$	(1, 2, 3)	(0.707, 1.414, 2.121)
Equal-Weak	(E-W)	$\tilde{1}$	(1, 1, 1)	(0.707, 0.707, 0.707)
Perfect-Fairly Impossible	(PE-FI)	$\tilde{9}$	(8, 9, 10)	(5.056, 5.688, 6.320)
Absolute-Fairly Impossible	(A-FI)	$\tilde{8}$	(7, 8, 9)	(4.424, 5.056, 5.688)
Very Good-Fairly Impossible	(VG-FI)	$\tilde{7}$	(6, 7, 8)	(3.792, 4.424, 5.056)
Fairly Good-Fairly Impossible	(FG-FI)	$\tilde{6}$	(5, 6, 7)	(3.160, 3.792, 4.424)
Good-Fairly Impossible	(G-FI)	$\tilde{5}$	(4, 5, 6)	(2.528, 3.160, 3.792)
Preferable-Fairly Impossible	(P-FI)	$\tilde{4}$	(3, 4, 5)	(1.896, 2.528, 3.160)
Not Bad-Fairly Impossible	(NB-FI)	$\tilde{3}$	(2, 3, 4)	(1.264, 1.896, 2.528)
Weak Advantage-Fairly Impossible	(WA-FI)	$\tilde{2}$	(1, 2, 3)	(0.632, 1.264, 1.896)
Equal-Fairly Impossible	(E- FI)	$\tilde{1}$	(1, 1, 1)	(0.632, 0.632, 0.632)
Perfect-Unlikely	(PE-U)	$\tilde{9}$	(8, 9, 10)	(3.576, 4.023, 4.470)
Absolute-Unlikely	(A-U)	$\tilde{8}$	(7, 8, 9)	(3.129, 3.576, 4.023)
Very Good-Unlikely	(VG-U)	$\tilde{7}$	(6, 7, 8)	(2.682, 3.129, 3.576)
Fairly Good-Unlikely	(FG-U)	$\tilde{6}$	(5, 6, 7)	(2.235, 2.682, 3.129)
Good-Unlikely	(G-U)	$\tilde{5}$	(4, 5, 6)	(1.788, 2.235, 2.682)
Preferable-Unlikely	(P-U)	$\tilde{4}$	(3, 4, 5)	(1.341, 1.788,2.235)
Not Bad-Unlikely	(NB-U)	$\tilde{3}$	(2, 3, 4)	(0.894, 1.341, 1.788)
Weak Advantage-Unlikely	(WA-U)	$\tilde{2}$	(1, 2, 3)	(0.447, 0.894, 1.341)
Equal-Unlikely	(E-U)	$\tilde{1}$	(1, 1, 1)	(0.447, 0.447, 0.447)

demonstrates the linguistic preference scale with probabilities, which are the fuzzy numbers corresponding to Z-Numbers in probabilistic conditions. As is evident, this scale is similar to the F-HBWM scale, except for the fact that the probability of fuzzy numbers’ certainty is considered in calculation. The following $A^{\prime }=\left[\sqrt{\gamma }d,\sqrt{\gamma }e,\sqrt{\gamma }f\right]=\left[d^{\prime }, e^{\prime }, f^{\prime }\right]$ computes the linguistic scale of Z-Numbers.

Symbols and descriptions of the HBWM approach using the Z-Number information are given below.

Set	$j\in C=\left\{\mathrm{1,2},\dots ,n\right\}$	Criteria
	$k\in C_k=\left\{\mathrm{1,2},\dots ,m\right\}$	Sub-Criteria
Parameters	${\tilde{p}}_{Bj}^Z=\sqrt{\gamma }\times {\tilde{p}}_{Bj}=\sqrt{\gamma }\times \left(p_{Bj}^l, p_{Bj}^m, p_{Bj}^u\right)$ $=(p_{Bj}^{\prime l},p_{Bj}^{\prime m},p_{Bj}^{\prime u})$	Fuzzy preferential preference of the best criterion over the j criterion
	${\tilde{p}}_{jW}^Z=\sqrt{\gamma }\times {\tilde{p}}_{jW}=\sqrt{\gamma }\times \left(p_{jW}^l, p_{jW}^m, p_{jW}^u\right)$ $=(p_{jW}^{\prime l},p_{jW}^{\prime m},p_{jW}^{\prime u})$	Fuzzy preferential preference of the number one criterion over the worst criterion
	${\tilde{p}}_{Bk}^{jZ}=\sqrt{\gamma }\times {\tilde{p}}_{Bk}^j=\sqrt{\gamma }\times \left(p_{Bk}^{jl}, p_{Bk}^{jm}, p_{Bk}^{ju}\right)$ $=(p_{Bk}^{\prime jl}, p_{Bk}^{\prime jm}, p_{Bk}^{\prime ju})$	Preferential fuzzy preference of the best sub-criterion on the k sub-criterion of the j criterion
	${\tilde{p}}_{kW}^{jZ}=\sqrt{\gamma }\times {\tilde{p}}_{kW}^j=\sqrt{\gamma }\times \left(p_{kW}^{jl}, p_{kW}^{jm},p_{kW}^{ju} \right)$ $=(p_{kW}^{\prime jl}, p_{kW}^{\prime jm},p_{kW}^{\prime ju})$	The fuzzy preferred priority of the k sub-criterion over the worst sub-criterion of the j criterion
Variables	$w_B$	Weight of the best criteria
	$w_j$	Criterion weight j
	$w_W$	Weight of the worst criteria
	$w_B^j$	The weight of the best sub-criterion on the criterion j
	$w_k^j$	The weight of the k sub-criterion on the j criterion
	$w_W^j$	The weight of the worst sub-criterion on the criterion j
	$G{w}_k^j$	The total weight of the sub-criterion k for the criterion j

The general model of the best–worst hierarchical method in terms of Z-number data is as follows in Equations (3)–(10):

$$Min{\Psi }^L+\sum _j{\Psi }_j^L$$

(3)

$S.T:$

$$\left|w_B-{\tilde{p}}_{Bj}^Z{w}_j\right|\le {\Psi }^L, \forall j$$

(4)

$$\left|w_j-{\tilde{p}}_{jW}^Z{w}_W\right|\le {\Psi }^L,\forall j$$

(5)

$$\left|w_B^j-{\tilde{p}}_{Bk}^{jZ}{w}_k^j\right|\le {\Psi }_j^L, \forall j$$

(6)

$$\left|w_k^j-{\tilde{p}}_{kW}^{jZ}{w}_W^j\right|\le {\Psi }_j^L, \forall j$$

(7)

$$G{w}_k^j={w_{j}w}_k^j , \forall k$$

(8)

$$\sum _j{w}_j=1 ,w_j\ge 0$$

(9)

$$\sum _j{w}_k^j=1 ,w_k^j\ge 0$$

(10)

Step (3): Constructing a Fuzzy BWM Model: This stage involves computing the factors within the nonlinear programming model’s weightings.

Step (4): Utilize optimization software like Lingo or GAMS 46.1.0 to solve the model. This process yields the weights of the criteria. These importance weights for the value criteria of the sustainable life cycle of suppliers, obtained through solving the model, are as follows. They will be employed in the subsequent step for ranking suppliers Equation (11):

$$W=[W_1,W_2,\dots ,W_j,\dots ,W_n]$$

(11)

Jiménez et al. 2007 proposed a more effective way for solving linear programming problems where some of the coefficients were in the form of fuzzy numbers. In this method, they benefited from the mutual participation of decision-makers throughout the decision-making process [73]. Equations (12)–(19) are calculated as follows:

$$Min{\Psi }^L+\sum _j{\Psi }_j^L$$

(12)

$S.T:$

$$\left|w_B-(\alpha \times E_2^{{\tilde{p}}_{Bj}^Z}+(1-\alpha )\times E_1^{{\tilde{p}}_{Bj}^Z}{)w}_j\right|\le {\Psi }^L,\forall j$$

(13)

$$\left|w_j-(\alpha \times E_2^{{\tilde{p}}_{jw}^Z}+(1-\alpha )\times E_1^{{\tilde{p}}_{jw}^Z})w_w\right|\le {\Psi }^L,\forall j$$

(14)

$$\left|w_B^j-(\alpha \times E_2^{{\tilde{p}}_{Bk}^{jZ}}+(1-\alpha )\times E_1^{{\tilde{p}}_{Bk}^{jZ}})w_k^j\right|\le {\Psi }_j^L,\forall j$$

(15)

$$\left|w_k^j-(\alpha \times E_2^{{\tilde{p}}_{kw}^{jZ}}+(1-\alpha )\times E_1^{{\tilde{p}}_{kw}^{jZ}})w_w^j\right|\le {\Psi }_j^L,\forall j$$

(16)

$$G{w}_k^j={w_{j}w}_k^j , \forall k$$

(17)

$$\sum _j{w}_j=1 ,w_j\ge 0$$

(18)

$$\sum _j{w}_k^j=1 ,w_k^j\ge 0$$

(19)

3.4. Z-PROMETHEE

This method is employed to assess and prioritize discrete options, aiding in selecting the best option based on various criteria, even when these criteria possess different measurement scales. Unlike some methods, the PROMETHEE technique cannot compensate for one criterion’s weakness against another’s strength. Therefore, the ideal option should obtain the minimum value across all requirements. Furthermore, the PROMETHEE method accommodates criteria with differing measurement scales, eliminating the need for scale matching. It defines six distinct functions based on the information and scale of the requirements. This versatility is a notable strength for the decision-maker. In particular, six types of preference functions are proposed, as outlined in

Table 7. Preference functions for the Prometheus method.

Type and Parameters	Function	The Shape	Description
PROMOTHEE Ι (Normal criteria) Parameter: ----	$P(d)=\left\{\begin{array}{c}\begin{array}{cc}0& \mathrm{d}\le 0\end{array}\\ \begin{array}{cc}1& \mathrm{d}>0\end{array}\end{array}\right.$		If the values of the two options are equal, there will be no difference between them.
PROMETHE II (U-shaped criterion) Parameter: $q$	$P(d)=\left\{\begin{array}{c}\begin{array}{cc}0& \mathrm{d}\le \mathrm{q}\end{array}\\ \begin{array}{cc}1& \mathrm{d}>\mathrm{q}\end{array}\end{array}\right.$		If the values of the two options are in the range d ≤ q, there will be no difference between them.
PROMETHEE III (V-shaped criterion) Parameter: $p$	$P(d)=\left\{\begin{array}{c}\begin{array}{cc}0& d \le 0\end{array}\\ \begin{array}{cc}d/p& 0<d \le p\end{array}\\ \begin{array}{cc}1& d>p\end{array}\end{array}\right.$		Priority by changing values in the range $0<d\le p$ changes linearly. If the difference is greater than p, an alternative option is considered.
PROMETHEE IV (Surface criteria) Parameters: $q, p$	$P(d)=\left\{\begin{array}{c}\begin{array}{cc}0& d \le q\end{array}\\ \begin{array}{cc}1/2& q<d \le p\end{array}\\ \begin{array}{cc}1& d>p\end{array}\end{array}\right.$		If the values of the two options are in the range d ≤ q, there will be no difference between them. If the scores of the two options are in the range q < d ≤ p, there will be a relative advantage between them. If the scores of the two options are in the range d > p, there will be an advantage between them.
PROMETHEE V (Linear criteria) Parameters: $q, p$	$P(d)=\left\{\begin{array}{c}\begin{array}{cc}0& \mathrm{d}\le \mathrm{q}\end{array}\\ \begin{array}{cc}\frac{d-q}{p-q}& \mathrm{d}>\mathrm{q}\end{array}\end{array}\right.$		If the values of the two options are in the range d ≤ q, there will be no difference between them. If the values of the two options are in the range q < d ≤ p, there will be a linear advantage between them. If the scores of the two options are in the range d > p, there will be an advantage between them.
PROMETHEE VI (Gaussian criterion) $s$ parameter	$P(d)=1-e^{\left(\frac{d^2}{{2s}^2}\right)}$		Priority based on the equation increases with the difference between the options.

. Empirical evidence suggests that these six functions are generally practical for a wide range of real-world problems. However, it is important to note that there is no strict requirement to use these specific preference functions. The decision-maker can choose their preferred preference function based on the particular context of the decision.

In the above functions, the parameters $q$ and $p$ and $s$ are the indifference threshold, the superiority threshold, and the intermediate value threshold between p and q, respectively. In other words, q is the largest difference that the decision-maker can ignore when comparing the two options.

PROMETHEE Approach in Fuzzy and Z-NUMBER Conditions

In this study, the PROMETHEE II method is extended to accommodate the fuzzy-probabilistic environment (Z-Numbers), resulting in the development of a Z-PROMETHEE II model. This enhancement aims to yield more effective outcomes when dealing with ambiguous and uncertain information. The methodology comprises two key components: determining the weights of the branches and ranking the suppliers. The steps involved in this approach are outlined as follows:

Step (1) The initial step involves constructing a decision matrix, which, in this method, assumes an index-option format. It is akin to the decision matrix used in other methods such as TOPSIS or VIKOR. The values are then put into the decision matrix using a fuzzy scale ranging from $\tilde{1}$ (Very Low) to $\tilde{9}$ (Very High), which could be seen in

Table 8. Linguistic scoring scale [74].

Linguistic Phrase	Acronym	Fuzzy Number	Scale of Fuzzy Number
Very High	(VH)	$\tilde{9}$	(7, 9, 9)
High	(H)	$\tilde{7}$	(5, 7, 9)
Medium	(M)	$\tilde{5}$	(3, 5, 7)
Low	(L)	$\tilde{3}$	(1, 3, 5)
Very Low	(VL)	$\tilde{1}$	(1, 1, 3)

.

For Z-Numbers, the linguistic scoring scale is defined according to the aforementioned Table 8 and corresponding fuzzy Z-Numbers under probabilistic circumstances. The result is a linguistic scoring scale with probabilities as mentioned in

Table 9. Table of scoring-probabilistic verbal spectrum.

Linguistic Phrase	Acronym	Deterministic	Fuzzy Number	Z-Number
Very High-Certainly	(VH-C)	$\tilde{9}$	(7, 9, 9)	(7.000, 9.000, 9.000)
High-Certainly	(H-C)	$\tilde{7}$	(5, 7, 9)	(5.000, 7.000, 9.000)
Medium-Certainly	(M-C)	$\tilde{5}$	(3, 5, 7)	(3.000, 5.000, 7.000)
Low-Certainly	(L-C)	$\tilde{3}$	(1, 3, 5)	(1.000, 3.000, 5.000)
Very Low-Certainly	(VL-C)	$\tilde{1}$	(1, 1, 3)	(1.000, 1.000, 3.000)
Very High-Most Likely	(VH-ML)	$\tilde{9}$	(7, 9, 9)	(6.636, 8.532, 8.532)
High-Most Likely	(H-ML)	$\tilde{7}$	(5, 7, 9)	(4.740, 6.636, 8.532)
Medium-Most Likely	(M-ML)	$\tilde{5}$	(3, 5, 7)	(2.844, 4.740, 6.636)
Low-Most Likely	(L-ML)	$\tilde{3}$	(1, 3, 5)	(0.948, 2.844, 4.740)
Very Low-Most Likely	(VL-ML)	$\tilde{1}$	(1, 1, 3)	(0.948, 0.948, 2.844)
Very High-Likely	(VH-L)	$\tilde{9}$	(7, 9, 9)	(6.258, 8.046, 8.046)
High-Likely	(H-L)	$\tilde{7}$	(5, 7, 9)	(4.470, 6.258, 8.046)
Medium-Likely	(M-L)	$\tilde{5}$	(3, 5, 7)	(2.682, 4.470, 6.258)
Low-Likely	(L-L)	$\tilde{3}$	(1, 3, 5)	(0.894, 2.682, 4.470)
Very Low-Likely	(VL-L)	$\tilde{1}$	(1, 1, 3)	(0.894, 0.894, 2.682)
Very High-Maybe	(VH-M)	$\tilde{9}$	(7, 9, 9)	(5.418, 6.966, 6.966)
High-Maybe	(H-M)	$\tilde{7}$	(5, 7, 9)	(3.870, 5.418, 6.966)
Medium-Maybe	(M-M)	$\tilde{5}$	(3, 5, 7)	(2.322, 3.870, 5.418)
Low-Maybe	(L-M)	$\tilde{3}$	(1, 3, 5)	(0.774, 2.322, 3.870)
Very Low-Maybe	(VL-M)	$\tilde{1}$	(1, 1, 3)	(0.774, 0.774, 2.322)
Very High-Weak	(VH-W)	$\tilde{9}$	(7, 9, 9)	(4.949, 6.363, 6.363)
High-Weak	(H-W)	$\tilde{7}$	(5, 7, 9)	(3.535, 4.949, 6.363)
Medium-Weak	(M-W)	$\tilde{5}$	(3, 5, 7)	(2.121, 3.535, 4.949)
Low-Weak	(L-W)	$\tilde{3}$	(1, 3, 5)	(0.707, 2.121, 3.535)
Very Low-Weak	(VL-W)	$\tilde{1}$	(1, 1, 3)	(0.707, 0.707, 2.121)
Very High-Fairly Impossible	(VH-FI)	$\tilde{9}$	(7, 9, 9)	(4.424, 5.688, 5.688)
High-Fairly Impossible	(H-FI)	$\tilde{7}$	(5, 7, 9)	(3.160, 4.424, 5.688)
Medium-Fairly Impossible	(M-FI)	$\tilde{5}$	(3, 5, 7)	(1.896, 3.160, 4.424)
Low-Fairly Impossible	(L-FI)	$\tilde{3}$	(1, 3, 5)	(0.632, 1.896, 3.160)
Very Low-Fairly Impossible	(VL-FI)	$\tilde{1}$	(1, 1, 3)	(0.632, 0.632, 1.896)
Very High-Unlikely	(VH-U)	$\tilde{9}$	(7, 9, 9)	(3.129, 4.023, 4.023)
High-Unlikely	(H-U)	$\tilde{7}$	(5, 7, 9)	(2.235, 3.129, 4.023)
Medium-Unlikely	(M-U)	$\tilde{5}$	(3, 5, 7)	(1.341, 2.235, 3.129)
Low-Unlikely	(L-U)	$\tilde{3}$	(1, 3, 5)	(0.447, 1.341, 2.235)
Very Low-Unlikely	(VL-U)	$\tilde{1}$	(1, 1, 3)	(0.447, 0.447, 1.341)

. The scale is similar to the F-DEA scale, except that the probability of fuzzy number certainty is considered.

Step (2) is to convert qualitative criteria into quantitative ones using the fuzzy-probabilistic scoring verbal spectrum table of Table 9 Equation (20):

$${\tilde{X}}^z={\left[{\tilde{x}}_{ij}^z\right]}_{m\times n} {\tilde{x}}_{ij}^z=\left(a_{ij},b_{ij},c_{ij}\right) i=1, 2, \dots , m j=1, 2, \dots , n$$

(20)

Step (3) is to obtain the weights of the importance of the criteria from methods such as Shannon entropy, AHP method or BWM. In the previous stage of this study, through the HBWM approach, the weights of importance of the value criteria of suppliers’ stable life cycle in different scenarios (definite, fuzzy, and Z-Number) were obtained using Equation (21):

$$W=[W_1,W_2,\dots ,W_j,\dots ,W_n]$$

(21)

Step (4) is to normalize the fuzzy-probability decision matrix as follows Equations (22)–(24):

$${\tilde{r}}_{ij}^z=\left(\frac{a_{ij}}{c_j^{+}},\frac{b_{ij}}{c_j^{+}},\frac{c_{ij}}{c_j^{+}}\right) c_j^{+}=\underset{i}{\max }{c}_{ij}$$

(22)

$${\tilde{r}}_{ij}^z=\left(\frac{a_j^{-}}{c_{ij}},\frac{a_j^{-}}{b_{ij}},\frac{a_j^{-}}{a_{ij}}\right){ a}_j^{-}=\underset{i}{\min }{a}_{ij}$$

(23)

$${\tilde{R}}^z={\left[{\tilde{r}}_{ij}^z\right]}_{m\times n} j=1, 2, \dots , n i=1, 2, \dots , m$$

(24)

Step (5) is to pair the options for each criterion: In this step, the options should be compared with each other in pairs in relation to each criterion. In this research, the following preference function has been used Equations (25)–(30):

$$P_j({\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z)=\left\{\begin{array}{c}\begin{array}{cc}0& \tilde{\mathrm{d}}\le 0\end{array}\\ \begin{array}{cc}1& \tilde{\mathrm{d}}>0\end{array}\end{array}\right.$$

(25)

If

$${\tilde{r}}_{ij}^z=(a_1^{\prime },a_2^{\prime },a_3^{\prime }), {\tilde{r}}_{kj}^z=(b_1^{\prime },b_2^{\prime },b_3^{\prime })$$

(26)

then:

$$\begin{array}{l}{\tilde{P}}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^Z={\tilde{r}}_{ij}^z-{\tilde{r}}_{kj}^z=\left(a_1^{\prime },a_2^{\prime },a_3^{\prime }\right)-\left(b_1^{\prime },b_2^{\prime },b_3^{\prime }\right)=\left(a_1^{\prime },a_2^{\prime },a_3^{\prime }\right)+\left(-b_3^{\prime },-b_2^{\prime },-b_1^{\prime }\right)=\\ \left(a_1^{\prime }-b_3^{\prime },a_2^{\prime }-b_2^{\prime },a_3^{\prime }-b_1^{\prime }\right)=\left({pl}_{AB}^{\prime },{pm}_{AB}^{\prime },{pu}_{AB}^{\prime }\right)\end{array}$$

(27)

$${pl}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }=\left\{\begin{array}{c}\begin{array}{cc}0& {pl}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }\le 0\end{array}\\ \begin{array}{cc}{a}_1^{\prime }-b_3^{\prime }& {pl}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }>0\end{array}\end{array}\right.$$

(28)

$${pm}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }=\left\{\begin{array}{c}\begin{array}{cc}0& {pm}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }\le 0\end{array}\\ \begin{array}{cc}{a}_2^{\prime }-b_2^{\prime }& {pm}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }>0\end{array}\end{array}\right.$$

(29)

$${pu}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }=\left\{\begin{array}{c}\begin{array}{cc}0& {pu}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }\le 0\end{array}\\ \begin{array}{cc}{a}_3^{\prime }-b_1^{\prime }& {pu}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^{\prime }>0\end{array}\end{array}\right.$$

(30)

Step (6) Construct the options preference matrix: In this step, we form the options preference matrix using the following relationships Equations (31) and (32):

$${\tilde{\pi }}^z\left(a,b\right)=\sum _{j=1}^k{\tilde{P}}_{{\tilde{r}}_{ij}^z,{\tilde{r}}_{kj}^z}^Z\times w_j$$

(31)

$${\tilde{\pi }}^z\left(b,a\right)=\sum _{j=1}^k{\tilde{P}}_{{\tilde{r}}_{kj}^z,{\tilde{r}}_{ij}^z}^Z\times w_j$$

(32)

Step (7) Calculate ${\tilde{\Phi }}^{+z}$ and ${\tilde{\Phi }}^{-z}$ (fuzzy Prometheus 1): For example (${\tilde{\Phi }}^{+z}$) is the sum of preference a over the other options and (${\tilde{\Phi }}^{-z}$) The sum is another preference of option over option a. The higher ${\tilde{\Phi }}^{+z}$ and the lower ${\tilde{\Phi }}^{-z}$ the better option a becomes, and vice versa Equations (33) and (34):

$${\tilde{\Phi }}^{+z}=\frac{1}{m-1}\sum _{\begin{array}{c}b\in A\\ b\ne a\end{array}}^m{\tilde{\pi }}^z\left(a,b\right)$$

(33)

$${\tilde{\Phi }}^{-z}=\frac{1}{m-1}\sum _{\begin{array}{c}b\in A\\ b\ne a\end{array}}^m{\tilde{\pi }}^z\left(b,a\right)$$

(34)

Step (8) Calculate the net sum of the current ${\tilde{\Phi }}^z$ (fuzzy Prometheus 2): In this step, we calculate the sum of the net current of the fuzzy Prometheus using following Equation (35) as follows:

$${\tilde{\Phi }}^z=\left({\tilde{\Phi }}^{+z}\right)-\left({\tilde{\Phi }}^{-z}\right)=({\Phi }_l^z,{\Phi }_m^z,{\Phi }_u^z)$$

(35)

Step (9) Disposing of positive, negative and empty currents and ranking the options: In this step, we first store the fuzzy numbers using the following formula (weighted average) and then the options can be based on the sum of the net flow. This ranked Equation (36) was calculated as follows:

$${\tilde{\Phi }}^z=({\Phi }_l^z,{\Phi }_m^z,{\Phi }_u^z)⟹{\tilde{\Phi }}^{z:def}=\frac{{\Phi }_l^z+2\times {\Phi }_m^z+{\Phi }_u^z}{4}$$

(36)

The Fuzzy Promethee method, which is a combination of fuzzy logic and the Prometheus method, is more flexible and accurate. In real-life situations, it is often difficult to gather clear data to correctly define the problem and make optimal decisions. The use of fuzzy sets gives the decision-maker the ability to define the problem in ambiguous situations that are more realistic.

The utilization of Z-Numbers, HBWM (Z-number-based hybrid best-worst method), and PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) in this study’s methodology is underpinned by several key advantages that distinguish them from other alternatives in the realm of fuzzy numbers, weight determination methods, and decision-making methods. Firstly, Z-Numbers offer a robust framework for handling uncertainties and imprecise information inherent in real-world decision-making processes. Unlike traditional fuzzy numbers, Z-Numbers incorporate additional information regarding the reliability and probability of the underlying fuzzy values, thus providing a more nuanced representation of uncertainty. This added granularity enables more accurate and reliable decision-making outcomes. Secondly, the HBWM technique, particularly when augmented with Z-Numbers, stands out for its ability to capture both positive and negative preferences simultaneously. By leveraging the best–worst scaling approach, HBWM allows decision-makers to explicitly identify the most and least preferred options within a given set of alternatives, thus facilitating more informed and nuanced evaluations. Additionally, the PROMETHEE method offers a flexible and intuitive framework for aggregating preferences across multiple criteria and generating comprehensive rankings of alternatives. PROMETHEE’s pairwise comparison approach enables decision-makers to systematically evaluate the relative importance of different criteria and assess the overall attractiveness of each alternative based on these preferences. Furthermore, PROMETHEE’s graphical representation of outranking relationships provides stakeholders with visual insights into the decision-making process, enhancing transparency and facilitating consensus-building. Overall, the combined use of Z-Numbers, HBWM, and PROMETHEE in this study’s methodology offers a powerful and versatile toolkit for addressing the complexities and uncertainties inherent in sustainable supply-chain management and supplier evaluation, thereby yielding more robust and actionable decision support.

4. Case Study

In this section, we implement the proposed methodology in a real-world case. This selection will be based on a set of sustainability and life cycle value criteria meticulously assessed by a senior expert. The company aims to filter and evaluate these suppliers, ultimately ranking them according to these criteria. The initial phase involves presenting and elucidating the value criteria of sustainable life, meticulously vetted by the senior expert. Subsequently, in the second part, we conduct pairwise comparisons to assign scores to the dimensions of sustainable life value and the respective criteria, employing the HBWM approach. The expert uses a preferential verbal spectrum, factoring in probabilities (refer to Table 6). Moving on to the third part, we proceed with scoring the suppliers based on the value criteria of sustainable life. The senior expert employs the verbal scoring spectrum, considering the proposed probabilities (refer to Table 9). Following this, the suppliers are ranked utilizing the Z-HBWM-PROMETHEE approach. Furthermore, we compare the importance weights and rankings output obtained through the HBWM-PROMETHEE and F-HBWM-PROMETHEE approaches. This combined approach is evaluated and contrasted across various deterministic, fuzzy, and Z-Number scenarios.

4.1. Filtering Sustainability Sub-Criteria and Identifying Final Subscales of Sustainable Life Cycle Value to Rank Suppliers

As outlined in the methodology section, our initial endeavor was to identify and categorize the value criteria associated with the sustainable life cycle. These criteria play a crucial role in evaluating the performance of suppliers, considering both sustainability and life cycle value dimensions. In this section, the value criteria pertaining to the stable life cycle are meticulously filtered based on the insights provided by the senior expert at Company. Please refer to

Table 10. Filtered criteria of suppliers’ sustainable life cycle value.

Sustainability Dimension	Code	Sustainability Index	Explanation
Economical	E1	Product selling price	The product cost, inclusive of VAT and applicable taxes.
	E2	Delivery time	The lead time for the product to reach the customer or consumer.
	E3	Product quality	Meet the customer’s requirements for product quality and efficiency.
	E4	Warranty and after-sales service	Provide customers with assurance through product warranties and guarantees.
Social	S1	Customer Orientation	Ensuring customer satisfaction to enhance loyalty levels.
	S2	Pay attention to the human rights of employees	Upholding employees’ rights, promoting freedom of expression, and preventing discrimination.
	S3	Participate in charitable activities	Organizing and hosting public and charitable events.
Environmental	G1	Pollution generation rate	The origin of pollution lies in emissions from industries and factories, agricultural practices, the combustion of fossil fuels, mining activities, and similar sources. These give rise to various forms of pollution including air pollution, water pollution, land pollution, and others.
	G2	Ability to recycle product components	Employing alternatives to plastic that are recyclable or biodegradable, such as paper, etc.
	G3	Use of environmentally friendly materials	Utilizing environmentally-friendly materials that do not cause harm to the environment.
	G4	The rate of use of non-renewable energy	Minimize the reliance on non-renewable energy sources like fossil fuels and groundwater, considering their prolonged production or replacement process, and the imperative to conserve for future needs.
Life-Cycle Value	V1	The potential for long-term relationships in the future	This implies establishing a sustained partnership with the supplier in the years ahead. Anticipating long-term collaboration with the supplier is grounded in their demonstrated loyalty through a substantial history of collaborations, particularly the most recent one.
	V2	Expected profitability of the supplier’s lifetime in the future	The projected profit margin from the purchase, taking into account factors like the purchase price, anticipated discounts from the supplier, etc., along with the volume of raw material to be procured from the supplier in the future.
	V3	The cost of waiting for a supplier in the future	The proportionate expenses we anticipate the supplier will incur in the future encompass various aspects. These may comprise the expenses related to defective items and waste, shipping costs contingent on the supplier’s location, costs arising from untimely supply, including potential losses in opportunities, diminished credibility with our customers, and decreased organizational efficiency. Additionally, there are costs associated with the process of identifying and onboarding new suppliers if the current supplier lacks the potential for a sustained long-term relationship in the future.
	V4	The level of supplier loyalty	A loyal supplier pertains to those who have consistently supplied raw materials to the organization over an extended period. They demonstrate a willingness to accommodate any sacrifices, costs, or limitations necessitated by the relationship in order to maintain a long-term affiliation with the organization. Additionally, when fulfilling supply commitments, they refrain from favoring the organization’s competitors over the organization itself.
	V5	Satisfaction of the supplier	Employee satisfaction is an internal sentiment that reflects the degree to which an organization’s expectations regarding performance, financial matters, behavioral conduct, and other relevant aspects in the employment relationship have been met. This assessment is based on the experience gained from the supplier.
	V6	The willingness of the supplier to work with the organization	The supplier’s collaboration with the organization in adhering to the process results in expedited work. Consequently, if the supplier is unwilling to cooperate with the organization, it can lead to delays in handling matters and similar challenges.

for a comprehensive introduction and description of the filtered criteria.

4.2. Results from HBWM Numerical Example in Terms of Certainty, Fuzzy, and Z-Number

In this section, the expert first deals with pairwise comparisons between the best and worst dimensions of the sustainability life cycle value compared to other dimensions of the purpose of the problem (

Table 11. Verbal pairwise comparisons between dimensions of sustainability.

Dimension	Best	Worse
Economic	(E-C)	(FG-M)
Social	(FG-W)	(E-C)
Environment	(NB-ML)	(WA-W)
SLV	(WA-M)	(NB-M)

) based on the verbal spectrum of preferences (Table 4).

Then, pairwise comparisons between the best and worst economic criteria relative to other criteria (

Table 12. Verbal pairwise comparisons between economic criteria.

Relative to Economic	Best	Worse
Product selling price	(E-C)	(A-W)
Delivery time	(FG-ML)	(E-C)
Product quality	(WA-W)	(NB-M)
Warranty and after-sales service	(NB-ML)	(WA-W)

), pairwise comparisons between the best and worst lifetime value criteria relative to other criteria (

Table 13. Verbal pairwise comparisons between life value criteria.

Relative to SLV	Best	Worse
The potential for long-term relationships in the future	(E-C)	(PE-W)
Expected profitability of the supplier’s lifetime in the future	(E-C)	(A-M)
The cost of waiting for a supplier in the future	(WA-M)	(G-ML)
The level of supplier loyalty	(WA-M)	(P-W)
Satisfaction of the supplier	(NB-M)	(NB-M)
The willingness of the supplier to work with the organization	(PE-W)	(E-C)

), pairwise comparisons between the best and the worst social criteria compared to other criteria (

Table 14. Verbal pairwise comparisons between social criteria.

Relative to Social	Best	Worse
Customer Orientation	(E-C)	(P-M)
Pay attention to the human rights of employees	(P-M)	(E-C)
Participate in charitable activities	(WA-W)	(WA-L)

), pairwise comparisons between the best and worst environmental criteria compared to other criteria (

Table 15. Verbal pairwise comparisons between environmental criteria.

Relative to Environment	Best	Worse
Pollution generation rate	(E-C)	(P-ML)
Recyclability	(WA-W)	(WA-M)
Use of environmentally friendly materials	(WA-W)	(WA-ML)
The rate of use of non-renewable energy	(P-L)	(E-C)

) are discussed.

Then, using the aforementioned quantified pairwise comparison tables and the Z-HBWM mathematical model, we derive the weights signifying the importance of the dimensions and the value indices of the sustainable life cycle. These calculations were performed using GAMS 46.1.0 software.

Table 16. Weights of importance of sustainable life cycle value criteria under the Z-Number scenario.

Resilient Sustainable Dimension	Z-Number	Resilient Sustainable Criteria	Z-Number	Z-Number towards the Goal
Economic	0.460	E1	0.426	0.195
		E2	0.078	0.035
		E3	0.335	0.154
		E4	0.161	0.074
Social	0.110	S1	0.470	0.051
		S2	0.160	0.017
		S3	0.370	0.040
Environment	0.167	G1	0.349	0.058
		G2	0.274	0.045
		G3	0.274	0.045
		G4	0.103	0.017
SLV	0.263	V1	0.258	0.067
		V2	0.258	0.067
		V3	0.185	0.048
		V4	0.139	0.036
		V5	0.118	0.031
		V6	0.042	0.011

presents the significant dimensions and value indices of the sustainability life cycle in the context of the Z-Number scenario.

4.3. Results from PROMETHEE Numerical Example in Terms of Certainty, Fuzzy and Z-Number

In this section, the expert assesses qualitative criteria for suppliers using the decision matrix and the verbal scoring range, taking into account probability (as defined by the Z-Number scoring verbal spectrum in Table 9). Additionally, quantitative criteria based on historical data are utilized to calculate scores for suppliers, considering the worst-case scenario, probability, and best-case scenario (with a probability of 1). Subsequently, employing the Z-HBWM-PROMETHEE approach, suppliers are ranked based on the value of the sustainable life cycle. Furthermore, to conduct a comparative evaluation of suppliers, the F-HBWM-PROMETHEE approach is also employed and the results are juxtaposed with those obtained using the Z-HBWM-PROMETHEE method. The decision

Table 17. Scoring decision matrix based on Z-Number verbal spectrum.

Supplier	Economic				Social			Environment				SLV
	E1	E2	E3	E4	S1	S2	S3	G1	G2	G3	G4	V1	V2	V3	V4	V5	V6
SU (1)	C	C	VL-ML	L-L	L-ML	L-L	L-L	H-ML	H-C	L-L	M-ML	M-L	L-C	VL-L	VL-W	L-L	L-W
SU (2)	C	C	H-L	H-ML	M-ML	H-ML	VH-ML	VH-C	M-ML	L-L	VL-L	H-L	L-ML	M-ML	L-ML	H-L	M-L
SU (3)	C	C	VL-ML	VL-L	L-L	VL-L	H-L	M-C	L-L	M-C	VH-L	VL-ML	L-ML	M-L	M-L	L-ML	M-L
SU (4)	C	C	VH-L	M-ML	H-ML	H-L	L-L	M-L	M-C	L-ML	H-L	H-ML	H-L	H-C	VH-W	L-ML	H-C
SU (5)	C	C	VL-L	M-L	H-L	H-ML	VL-L	M-ML	VH-C	M-W	M-L	VH-C	VL-L	VH-ML	H-L	M-ML	L-W
SU (6)	C	C	M-L	L-ML	L-L	VL-L	VL-L	H-L	VL-L	VH-W	L-ML	M-ML	VH-L	H-L	VH-W	VL-C	VL-ML
SU (7)	C	C	H-L	L-ML	M-L	VH-L	VH-L	VL-ML	VH-L	M-L	L-W	L-C	H-W	VH-W	VH-L	VL-W	M-L
SU (8)	C	C	L-L	M-L	M-L	H-L	H-ML	VH-L	H-L	M-L	H-W	VH-L	VH-L	M-ML	M-ML	L-L	H-L
SU (9)	C	C	H-L	H-L	L-ML	H-L	L-ML	VL-C	M-L	L-C	M-L	VH-L	M-W	H-C	L-L	VH-C	M-L
SU (10)	C	C	M-L	H-L	M-ML	M-L	M-ML	L-W	VH-L	VH-L	H-ML	M-ML	L-C	M-L	H-ML	M-L	M-L

for scoring qualitative criteria, incorporating the verbal spectrum scoring from Table 9 with consideration of probability, has been completed by the expert. Additionally, quantitative criteria have been incorporated based on prior data.

Finally, the net positive currents and net negative currents of each option are calculated based on them, and at the end of the fuzzy value, the final net is stored (

Table 18. Table of positive flows, negative flows, and net flows of each option.

Supplier	${\tilde{\mathit{\Phi }}}^{+\mathit{z}}$	${\tilde{\mathit{\Phi }}}^{+\mathit{z}}$	${\tilde{\mathit{\Phi }}}^{\mathit{z}}$	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank
SU (1)	(0.004, 0.052, 0.17)	(0.049, 0.231, 0.471)	(−0.045, −0.178, −0.3)	10	−0.157
SU (2)	(0.022, 0.143, 0.357)	(0.015, 0.105, 0.301)	(0.0069, 0.0381, 0.0557)	4	0.077
SU (3)	(0.02, 0.094, 0.287)	(0.046, 0.227, 0.414)	(−0.025, −0.132, −0.127)	9	−0.107
SU (4)	(0.043, 0.207, 0.456)	(0.006, 0.07, 0.231)	(0.0362, 0.1368, 0.2249)	1	0.140
SU (5)	(0.009, 0.088, 0.234)	(0.045, 0.183, 0.36)	(−0.036, −0.095, −0.125)	8	−0.091
SU (6)	(0.02, 0.114, 0.274)	(0.018, 0.152, 0.346)	(0.0022, −0.037, −0.071)	7	−0.049
SU (7)	(0.038, 0.201, 0.407)	(0.009, 0.097, 0.288)	(0.0295, 0.1045, 0.1194)	3	0.079
SU (8)	(0.022, 0.123, 0.285)	(0.021, 0.128, 0.331)	(0.0013, −0.005, −0.046)	6	−0.027
SU (9)	(0.037, 0.223, 0.459)	(0.008, 0.074, 0.255)	(0.0297, 0.1488, 0.2036)	2	0.086
SU (10)	(0.01, 0.128, 0.366)	(0.008, 0.106, 0.297)	(0.0016, 0.0217, 0.0681)	5	0.051

).

5. Discussion

In this section, we examine and contrast the outcomes of the HBWM and HBWM-PROMETHEE approaches across distinct states—definite, fuzzy, and Z-Number. The findings reveal that the results exhibit more significant variability as we transition from solid to vague and Z-Number states. This suggests that in the presence of data uncertainty, the results become more dependable in

Table 19. Weights of importance of the dimensions and value criteria of sustainable life cycle under different scenarios.

Resilient Sustainable Dimension	Deterministic	Fuzzy	Z-Number	Resilient Sustainable Criteria	Deterministic	Fuzzy	Z-Number	Definite towards the Goal	Fuzzy towards the Goal	Z-Number towards the Goal
Economic	0.5	0.519	0.460	E1	0.5	0.48	0.426	0.25	0.249	0.195
				E2	0.083	0.083	0.078	0.041	0.043	0.035
				E3	0.25	0.266	0.335	0.125	0.138	0.154
				E4	0.167	0.171	0.161	0.083	0.088	0.074
Social	0.083	0.084	0.110	S1	0.571	0.55	0.470	0.047	0.046	0.051
				S2	0.143	0.145	0.160	0.011	0.012	0.017
				S3	0.286	0.305	0.370	0.023	0.025	0.040
Environment	0.167	0.162	0.167	G1	0.444	0.421	0.349	0.074	0.068	0.058
				G2	0.222	0.234	0.274	0.037	0.037	0.045
				G3	0.222	0.234	0.274	0.037	0.037	0.045
				G4	0.111	0.111	0.103	0.018	0.017	0.017
SLV	0.25	0.236	0.263	V1	0.3	0.286	0.258	0.075	0.067	0.067
				V2	0.274	0.276	0.258	0.068	0.065	0.067
				V3	0.15	0.159	0.185	0.037	0.037	0.048
				V4	0.143	0.144	0.139	0.035	0.033	0.036
				V5	0.1	0.102	0.118	0.025	0.024	0.031
				V6	0.033	0.033	0.042	0.008	0.007	0.011

and depicted in Figure 1 and Figure 2, the outcomes derived from the weights underscore the significance of the dimensions of the sustainable life cycle value and the corresponding value indices, observed under different scenarios. These findings will exert a substantial influence on our decision-making process.

As depicted in Figure 2 and Figure 3, it is evident that the economic dimension and the life cycle value consistently hold substantial weight and significance across various scenarios. This underscores the crucial importance of considering long-term partnerships with suppliers in the supply chain when selecting suppliers, alongside sustainability considerations.

In essence, with the inherent uncertainties in experts’ opinions and mental judgments, transitioning from definite conditions to fuzzy conditions, and further to fuzzy conditions with associated probabilities (Z-Number), allows us to approach a more realistic representation of the real world and yield more reliable outcomes. Therefore, under consideration of the probability of confidence in the experts’ fuzzy opinions, Z-Number conditions offer results that are closer to reality and more dependable. In this segment of the model, the outcomes of the PROMETHEE approach under various scenarios with the parameter $q$ set at 0 are detailed in

Table 20. Ranking of suppliers under different scenarios.

Supplier	Scenario: Definitive		Scenario: Fuzzy		Scenario: Z-Number
	$\mathit{\Phi }$	Rank	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank
SU (1)	−0.20635	10	−0.176	10	−0.157	10
SU (2)	0.042404	4	0.035	4	0.077	4
SU (3)	−0.11631	9	−0.105	9	−0.107	9
SU (4)	0.163403	2	0.134	1	0.140	1
SU (5)	−0.10964	8	−0.088	8	−0.091	8
SU (6)	−0.04694	7	−0.036	7	−0.049	7
SU (7)	0.102206	3	0.090	3	0.079	3
SU (8)	−0.01798	6	−0.014	6	−0.027	6
SU (9)	0.168086	1	0.133	2	0.086	2
SU (10)	0.021126	5	0.028	5	0.051	5

.

According to Table 20, it is evident that in the fuzzy scenario and Z-Number scenario, the values of net flows underwent changes, yet they did not impact the ranking of suppliers. This suggests that, in this specific case study, the decision remained unaffected. However, in scenarios involving uncertainty (both fuzzy and Z-Number), there were discrepancies in suppliers’ rankings. Given that the real world often entails uncertainties in experts’ opinions and judgments, and fuzzy and fuzzy–probabilistic data encapsulate this realm of uncertainty, the outcomes derived from these fuzzy scenarios are deemed to be more reliable and closely aligned with reality. Moving forward, we will delve into an analysis of the results from the Z-HBWM-PROMETHEE approach. Figure 4 shows the orientation of the criteria towards suppliers and shows in which criteria each supplier has more orientation and power.

Also, in Figure 5, the positive and negative currents of the criteria for each supplier are specified.

By changing the parameter $q=0$ to $q=0.5$ and $q=0.7$, we examine the results in

Table 21. Ranking of suppliers under different preference parameter.

Supplier	$\mathit{q}=0$		$\mathit{q}=0.5$		$\mathit{q}=0.7$
	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank
SU (1)	−0.157	10	−0.091	9	−0.0279	7
SU (2)	0.077	4	0.066	2	0.048	1
SU (3)	−0.107	9	−0.099	10	−0.054	10
SU (4)	0.140	1	0.084	1	0.047	2
SU (5)	−0.091	8	−0.053	8	−0.029	8
SU (6)	−0.049	7	−0.047	7	−0.033	9
SU (7)	0.079	3	0.066	3	0.017	5
SU (8)	−0.027	6	−0.006	6	−0.017	6
SU (9)	0.086	2	0.049	4	0.022	4
SU (10)	0.051	5	0.031	5	0.026	3

.

According to the table, it can be seen that by changing the parameter of preference difference, the ranking of suppliers will also change, so the definition of a suitable preference parameter will be very important and will influence our decision. In the following, by eliminating the value criteria of the supplier’s life cycle, we will rank the suppliers only by considering the sustainability criteria (

Table 22. Ranking of suppliers under different criteria.

Supplier	Scenario: Z-Number Criteria: Sustainable Life Cycle Value		Scenario: Z-Number Criteria: Sustainability		Scenario: Z-Number Criteria: Life Cycle Value
	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank	${\mathit{\Phi }}^{\mathit{d}\mathit{e}\mathit{f}}$	Rank
SU (1)	−0.157	10	−0.147	10	−0.010	8
SU (2)	0.077	4	0.079	3	−0.002	6
SU (3)	−0.107	9	−0.055	6	−0.003	7
SU (4)	0.140	1	0.117	1	−0.052	10
SU (5)	−0.091	8	−0.091	9	0.022	2
SU (6)	−0.049	7	−0.061	7	−0.001	5
SU (7)	0.079	3	0.099	2	0.012	4
SU (8)	−0.027	6	−0.063	8	−0.020	9
SU (9)	0.086	2	0.070	4	0.036	1
SU (10)	0.051	5	0.053	5	0.016	3

):

According to Figure 6, it is evident that alterations in decision-making criteria significantly impact the ranking of suppliers. For instance, Supplier (4) emerges as the top-performing supplier when sustainability criteria are taken into account. However, if only life value criteria are considered, it is ranked last. This highlights the remarkable superiority of Supplier (4) in terms of sustainability, such that when these criteria are prioritized, it is deemed the foremost supplier. On the contrary, Supplier (5) is initially ranked similarly to Supplier (2) based on the life value index. However, when sustainability criteria are factored in, Supplier (5) plummets to the 9^th, and final, position. This indicates that, although Supplier (5) excels in life value criteria, it exhibits significant weaknesses in sustainability criteria, leading to a substantial decline in its ranking when these factors are taken into consideration.

Indeed, our study underscores the critical role of carefully chosen criteria in the supplier selection process. The correct selection of indicators aligned with the organization’s goals and strategic plans significantly influences the identification of the right supplier. Our findings, particularly through the PROMETHEE approach, illustrate that certain suppliers may excel in specific indicators while performing less favorably in others. This highlights the significance of determining the types of indicators and prioritizing them according to their relevance to the organization’s objectives. By understanding the strengths and weaknesses of suppliers across various criteria, decision-makers can make more informed choices that align with their strategic priorities and ultimately enhance their supply chain performance.

The application of the proposed approach in this research case study has yielded insightful conclusions, shaping our understanding of crucial aspects in supplier evaluation. The key findings and results are succinctly summarized as follows:

▪

Uncertainty and Fuzzy Approaches: This study underscores the profound impact of uncertainties in expert judgments on the conclusions, decisions, and rankings of suppliers. To address this challenge, advocacy for fuzzy approaches is emphasized by providing a method to consider uncertainty in expert opinions. The incorporation of Z-Numbers is suggested, offering a valuable means to account for the reliability and probability of fuzzy numbers. The Z-HBWM-PROMETHEE method emerges as a reliable approach, particularly noted for its efficacy in real-world situations.

▪

Preferential Threshold Parameter: This text highlights the influential role of the preferential threshold parameter in determining supplier rankings. The flexibility to adjust this parameter is acknowledged as a crucial factor that can lead to changes in how suppliers are ranked.

▪

Decision-Making Criteria: This study emphasizes the critical role of selecting relevant criteria in decision making. Notably, the text highlights the observed variations in supplier rankings when sustainability criteria were considered compared to when criteria related to the supplier’s life cycle value were emphasized. This underscores the importance of aligning chosen criteria with the organization’s overall goals and objectives.

▪

Significance of Supplier Life Cycle Value Criteria: The introduction of value indices related to a supplier’s life cycle emerges as a noteworthy contribution. These criteria, introduced as novel elements in the research, underscore the substantial importance companies place on considering the long-term vision and establishing enduring partnerships with their suppliers.

In summary, this research accentuates the critical consideration of uncertainty in supplier evaluations, the pivotal role of specific parameters in rankings, the importance of judicious criteria selection, and the value attributed to long-term partnerships with suppliers. These insights not only contribute valuable guidance for organizations seeking to enhance their supplier evaluation processes but also signify the innovative aspects introduced by the proposed methodology in the realm of sustainable supply-chain management.

In a variety of countries, including some of European countries, the supplier selection procedure is performed according to their sustainability rankings. So, our proposed approach may offer some benefits to the following categories:

▪

Tax incentives and green economy;

▪

Supplier selection award procedure;

▪

Sustainable development aligned with UN’s sustainable development goals;

▪

Sustainable procurement and purchasing;

▪

Boosting environmental sustainability.

Our research offers a novel approach that integrates Z-Numbers, HBWM, and PROMETHEE methodologies to enhance the effectiveness and robustness of supplier selection processes. By leveraging Z-Numbers, we address uncertainties inherent in supplier evaluation, providing a more realistic representation of decision-makers’ preferences and uncertainties. Additionally, the HBWM method allows for the integration of subjective expert judgments with objective data, enabling a comprehensive assessment of supplier performance. Furthermore, PROMETHEE facilitates the aggregation of multi-criteria decision making, enabling decision-makers to rank suppliers based on diverse criteria while considering their preferences and priorities. This integrated approach contributes to advancing the field by providing a systematic and effective framework for sustainable supplier selection, thereby supporting organizations in making informed decisions to enhance their supply chain sustainability and resilience.

6. Conclusions

The recent surge in supply chain sustainability has become a pivotal focus in supply-chain management research. Acknowledging the critical role of suppliers within the supply chain, it is evident that they hold substantial sway in fostering sustainability. Selecting suppliers is the foundational step in forming a supply chain, underscoring the paramount importance of reasonable criteria. Collaborative efforts with suppliers are crucial in achieving superior product and service quality, aligning their standards with the core organization. Establishing long-term partnerships between organizations serves as a means to optimize resource utilization and attain specific business objectives. Historically, suppliers were often viewed with apprehension, almost as competitors, leading to low levels of loyalty and reliability. This sentiment resulted in uncertainty regarding future relations with suppliers.

Procurement and supply departments contributed to perpetuating this issue. Given this backdrop, the present study aims to make informed selections of industrial equipment suppliers for the Zinc Ingot Production Company, utilizing the proposed Z-HBWM-PROMETHEE combined decision-making approach. This study emphasizes the meticulous selection of sustainability criteria and the delineation of value criteria related to the supplier’s life cycle, particularly emphasizing long-term partnerships. However, this study acknowledges its limitations, including constraints on the number of experts and the senior expert’s time availability for answering questionnaires. Despite these limitations, this study emphasizes the value of the expert’s input. This study’s primary focus lies in introducing and applying the Z-HBWM-PROMETHEE approach, offering companies a valuable tool to effectively rank and assess their suppliers, particularly emphasizing sustainability and long-term partnership criteria. This approach provides a strategic tool for companies to select suppliers that align with their objectives and prospects. This study embarked on a comprehensive process to identify and validate effective sustainability criteria for supplier selection.

A structured methodology was employed to systematically identify, classify, and prioritize these criteria into three distinct dimensions: economic, social, and environmental. Expert opinions were sought to validate the significance of each identified criterion. Additionally, value criteria about the lifespan of suppliers were defined, recognizing the importance of fostering long-term partnerships. The Z-HBWM method was applied to ascertain the weight and importance of each dimension and index relating to sustainability and lifespan value. The results revealed that the weights of the filtered criteria fell within a narrow range in the Z-Number mode, underscoring the criticality of these criteria. The dimension associated with the supplier’s lifespan held the highest weight after the economic dimension, indicating a strong inclination toward establishing enduring partnerships. This study also notes that alterations in the selected criteria can lead to shifts in supplier rankings and outcomes, emphasizing the pivotal role of appropriate criteria in the supplier selection process.

The main goal in evaluating and selecting suppliers for sustainable supply chains is to find partners that closely align with the company’s sustainability goals. Given the pivotal role of suppliers in ensuring the sustainability and resilience of supply chains, assessing their sustainability performance becomes a critical starting point for achieving supply chain sustainability. Establishing enduring partnerships with suitable suppliers can bring about a myriad of benefits for companies. In this research, a systematic approach was employed. Firstly, sustainability and life cycle value criteria were diligently identified, classified, and refined. This process aimed to streamline the selection of suppliers for filter press products, aligning with the goals of the case study. Subsequently, the Z-HBWM approach was utilized to assign weights, indicating the significance of these criteria. Notably, it was evident that criteria related to the supplier’s lifespan, making up 25% of the weightage, held considerable importance alongside sustainability criteria. These criteria were anticipated to wield a substantial influence on the rankings of suppliers. Finally, in the third step, the Z-HBWM-PROMETHEE approach was applied. This involved factoring in the assigned weights related to the sustainable life cycle criteria within the decision-making matrix, ultimately determining supplier rankings. This comprehensive approach enables a thorough evaluation and selection of suppliers grounded in a nuanced understanding of sustainability and long-term value.

The proposed approach in this study not only contributes valuable insights to the current research landscape but also opens avenues for future exploration and practical applications. The provided text encapsulates a roadmap for future research endeavors aimed at refining and advancing the proposed approach outlined in this study. It underscores the significance of exploring alternative methodologies, such as Z-DEMATEL, and extending the application of the proposed approach to diverse industrial contexts to enrich its generalizability and applicability. Moreover, the integration of supplementary criteria, along with the exploration of uncertainty models and preference functions, is highlighted as crucial steps toward enhancing the methodology’s comprehensiveness and effectiveness in supplier evaluation. Additionally, the text emphasizes the importance of addressing potential biases stemming from limitations like expert availability, suggesting strategies to mitigate these biases for stronger, more reliable findings. Furthermore, the suggestion to incorporate comparative experiments with various fuzzy numbers and decision-making methods aims to strengthen the study’s robustness and provide deeper insights into sustainable supply-chain management and supplier evaluation methodologies. Overall, these future proposals aim to elevate the proposed approach, marking significant progress in advancing the field and addressing critical challenges in sustainable supply-chain management.