Green Supplier Evaluation and Selection Based on Bi-Directional Shapley Choquet Integral in Interval Intuitive Fuzzy Environment

Abstract

The evaluation and selection of green suppliers is an important way for enterprises to maintain sustainable development and help them reduce costs and increase efficiency. This paper proposes a multi-criteria decision-making (MCDM) model in an interval intuitive fuzzy environment. This model uses interval-valued intuitive uncertainty language number (IVIULN) to describe expert evaluation of qualitative indices. Expert weights are determined through expert social networks, and an improved aggregation operator is proposed to aggregate the evaluation information. The proposed operator can ensure the stability of the results even in the case of extreme values. Subsequently, considering a large number of mutually related indices, a novel teaching-learning-based optimization (NTLBO) algorithm is used to identify the value of $\lambda$-fuzzy measures. This algorithm improves the teaching stage and proposes the idea of teaching students in accordance with their aptitude, introduces precision parameters, and adds a self-study stage. It has been verified by numerical examples that it is far superior to commonly used heuristic algorithms in terms of algorithm accuracy and run time. Finally, the alternatives are ranked by bi-direction Shapley–Choquet integral. The model’s effectiveness is demonstrated through a case study. This paper also examines the impact of key parameters on the results through sensitivity analysis.

1. Introduction

In the contemporary business landscape, enterprises face a dual challenge: intensifying supply chain competition and increasingly stringent environmental regulations. Against this backdrop, coupled with the persistent deterioration of ecological conditions, enterprises are compelled to not only evaluate traditional supply chain impacts but also implement operational practices that enhance sustainability outcomes [1]. Green supply chain management integrates environmental factors into corporate supply chain strategies. That is, it considers three goals of economy, environment, and society at the same time [2]. Although there is no consensus definition of a green supply chain [3], considerable research show that green supply chain is rapidly penetrating into the field of supply chain management. Research contents that are highly related to “sustainability”, such as the evaluation and selection of green suppliers, will play an increasingly important role in the field of supply chain management [4]. Green suppliers are key participants in green supply chain management. These suppliers focus on energy conservation, emission reduction, and resource recycling during production to minimize their environmental impact [5].

The evaluation and selection of green suppliers is a classic MCDM problem. MCDM problems have accumulated considerable research and achieved rich results [6]. Among them, the widely used and more classic methods include AHP [7], ANP [8], PROMETHEE [9], TOPSIS [10], and various other methods, such as DEMATEL [11], etc. To make the decision-making results more reliable, some studies combine the advantages of the above methods [12,13]. However, in green supplier evaluation, several critical challenges and methodological gaps persist. First, traditional fuzzy sets (e.g., triangular or trapezoidal fuzzy numbers) often fail to comprehensively capture the uncertainty inherent in expert evaluations. Second, current methods for determining expert weights predominantly rely on professional status, neglecting the dynamic influence of social interactions within expert networks. Trust propagation and consensus-building among experts significantly impact decision reliability. Third, existing aggregation operators, such as weighted arithmetic or geometric averages, exhibit instability when encountering extreme evaluation values or imbalanced expert weights. Fourth, heuristic algorithms for identifying fuzzy measures (e.g., genetic algorithms) often suffer from suboptimal convergence rates or local minima issues, limiting their precision. Therefore, when faced with the actual decision-making process, such methods cannot effectively solve the following difficulties.

• How to reasonably describe evaluation information? In green supply chain management, there are many uncertain factors [14]. When evaluating and selecting green suppliers, the uncertainty of evaluation information poses a major challenge to research. During the evaluation process, the specific information of the indices is related to personal feelings, environment, etc. Therefore, when describing the evaluation information, the fuzziness of human perception should be considered [15].
• How to determine the expert weight? When experts are scoring and evaluating, due to their different experiences, positions, and research fields, it is necessary to use aggregation operators for information fusion in the decision-making process [16]. It is crucial to determine the weights of experts reasonably. Experts can reach a higher level of consensus through conversation and discussion when expressing their preferences. In the process of group decision-making, considering such social relationships will lead to more reasonable results [17].
• How to identify the importance of indices? Determining the importance of indices is difficult and vital [18]. Due to the differences in the index sets and the evaluation objectives, the same index may have different importance. In addition, there are complex interrelationships among the indices. Rational methods need to be proposed to identify and define them [19].
• How to improve aggregation operators and heuristic algorithms? This paper needs to describe the uncertainty in the expert evaluation process as completely as possible. This requires the development of aggregation operators tailored to the characteristics of the evaluation tools, ensuring that the results of the aggregation operators are reasonable and stable. At the same time, the paper requires a more accurate heuristic algorithm to better assist in measuring the importance of the indices.

Considering the practical circumstances of the evaluation process, it is essential to establish an effective multi-criteria decision-making model to address potential challenges encountered in the research. This paper addresses the evaluation and selection of green suppliers with the following methodological characteristics.

• Interval-valued intuitive uncertainty language numbers (IVIULNs) are employed to represent expert semantic information. This fuzzy numerical framework preserves the completeness of expert evaluation data by simultaneously capturing linguistic uncertainty and interval-valued intuitionistic characteristics.
• For expert consensus decision-making, a trust propagation network is utilized to adjust expert weights within the group through their social relationships, ensuring balanced influence in collaborative evaluations.
• To address interactions among indices, the paper uses λ-fuzzy measures and a two-level indicator system, where fuzzy measures for index sets are derived by an improved heuristic algorithm.
• An improved aggregation operator is proposed to enhance result stability even in the case of extreme values.
• The ranking process integrates Choquet integrals with bi-directional weight propagation and generalized Shapley functions, reinforcing the rationality and stability of outcomes.

The remainder of this paper is organized as follows. In Section 2, a review and summary of literature related to fuzzy sets, expert social networks, fuzzy measures, and fuzzy integral are conducted. In Section 3, the basic definitions and preliminaries of the methods used in the paper are provided. Section 4 introduces MCDM models based on interval-valued intuitive uncertainty language number, λ-fuzzy measure, and bi-directional Shapley–Choquet integral. In Section 5, we validate the effectiveness of the proposed model using a case study and conduct discussions and sensitivity analyses. Finally, Section 6 presents a conclusion and provides suggestions and directions for future research.

2. Literature Review

2.1. Multi-Criteria Decision-Making

Decision-making arises from individuals’ evaluations of alternatives, which are typically grounded in decision-makers’ preferences, experiences, and empirical data [20]. The evaluation process often requires balancing multiple interrelated or independent perspectives or indices [21]. In research, MCDM is widely adopted to address such challenges. MCDM refers to evaluating alternative solutions or options by selecting a preferred alternative or ranking alternatives from optimal to inferior [22]. Currently, MCDM methods have been extensively applied across disciplines, particularly in sustainable engineering [23].

MCDM research inevitably involves qualitative data, leading to the early adoption of qualitative approaches such as the Delphi method, which remains prevalent for establishing expert consensus [24]. To derive more scientifically robust conclusions, numerous hybrid qualitative-quantitative methods have been developed, including the AHP, factor analysis, fuzzy comprehensive evaluation, data envelopment analysis (DEA), and the technique for order of preference by similarity to ideal solution (TOPSIS). Recent studies continue to refine these approaches. In supplier evaluation under subjective environments, a rough set based DEMATEL-ANP integration method enhances the objectivity of assessment processes by quantifying implicit interdependencies within decision networks [25]. For green supplier performance analysis, a dynamic multi-objective optimization framework integrates ratio analysis with random forest algorithms to quantify systematic deviations between green supplier performance and industry benchmarks [26]. In internet industry supplier selection, the empirical application of a hybrid fuzzy TOPSIS-ELECTRE methodology in Chinese internet supplier evaluations demonstrates the superiority of this integrated approach under uncertain conditions through comparative analysis with conventional fuzzy ELECTRE methods [27].

2.2. Fuzzy Set

Fuzzy theory constitutes an integral component of uncertainty methodologies, enabling systematic uncertainty measurement through well-defined frameworks and operational procedures [28]. When quantifying uncertain parameters, such measurements can be approximated through managerial judgments, intuition, and empirical expertise, which are formally characterized as “fuzzy variables” defined over possibility spaces [29].

The fuzzy set theory is often used when interpreting and making decisions on fuzzy evaluation information. Fuzzy sets were first proposed by Zadeh [30], and then Zadeh and several scholars such as Kalmanson [31] expanded this field and gradually applied it to specific problems. When applying fuzzy sets, the biggest challenge is to determine the membership degree of elements to the set [32]. Triangular fuzzy numbers and trapezoidal fuzzy numbers are two common methods [33,34]. Directly defining membership and non-membership degrees to represent uncertain information is another common approach [35,36].

The core idea of fuzzy multi-criteria decision-making methods is constructing a function to represent the membership degree of decision information or directly defining its membership degree. Based on intuitive fuzzy sets (IFS), interval-valued intuitive fuzzy sets extended fuzzy sets to a more general form, where interval-valued fuzzy linguistic sets can be used to represent the uncertainty of membership and non-membership degree [37]. To enhance the flexibility of fuzzy information representation, the application of interval-values is combined with Pythagorean fuzzy sets and Fermatean fuzzy sets [38,39]. Based on interval-valued intuitive fuzzy sets and intuitive uncertain language sets, the interval-valued intuitive uncertain language number is proposed, which includes six elements to explain and evaluate the language and can cover more uncertain information [40].

2.3. Expert Social Network and Evaluation Information Aggregation

Some studies have viewed green supplier evaluation and selection as a group decision-making problem. Group decision-making aims to find solutions that are collectively satisfactory based on the preference information given by decision makers [41]. Because of the conflicts between and divergent views of decision makers, group decision-making requires a consensus-reaching process to improve group consensus [42]. During this process, decision makers may modify their initial opinions, and this process will undergo multiple iterations until a collective opinion with a high group consensus is reached [43].

The development of expert social networks provides a way to enhance group consensus [44,45]. When considering social networks, experts can have more opportunities for communication and their opinions will also be influenced by other experts [46].

After collecting the decision information provided by the expert group, these different types of preference information need to be aggregated to represent the collective preferences of the expert group. For representing expert information through fuzzy sets, various aggregation operators based on weighted average (WA) and weighted geometry (WG) have been widely used for the aggregation of fuzzy information [47]. The existing aggregation operators include PFWA for Pythagorean fuzzy sets [48], normalized hesitant fuzzy weighted arithmetic/geometric operators for hesitant fuzzy languages [49], etc. However, as the basis for these aggregation operators, the results obtained by WA and GA may be inconsistent. When it has extreme evaluation values or extreme expert weights, this situation will be even more severe [50].

2.4. Fuzzy Measures and Fuzzy Integrals

Traditional MCDM methods tend to assume that indices are independent of each other. Obviously, such assumptions are unreasonable when facing practical problems. Compared with the weights of indices, fuzzy measures are also able to indicate the importance of indices [51]. In addition, fuzzy measures are a non-additivity metric, which gives them the ability to represent the relationships between indices. Fuzzy measures have various forms, such as quasi-measures [52], belief measures [53], complex fuzzy measures [54], $k$-additive fuzzy measures [55], 2-additive fuzzy measure [56], and $\lambda$-fuzzy measures. Among them, $\lambda$-fuzzy measures are most widely used in MCDM problems.

Fuzzy integrals are effective tools for dealing with uncertain problems. Usually, fuzzy measures and fuzzy integrals are also used in combination. In fuzzy integrals, the Choquet integral is more suitable for dealing with MCDM problems [57]. To represent the interactions within ordered index sets, the generalized Shapley–Choquet integral is proposed to solve multi-criteria group decision-making problems [58]. To verify the rationality of the CI results, exponent Choquet integral (ECI) and reverse Choquet integral (RCI) are developed [59]. ECI transforms the calculation of the weighted sum into an exponential product. If CI is understood as interpreting the importance of the elements in an ordered index set as the loss value of the elements leaving the index set, then the RCI can be interpreted as the gain value of the elements entering the index set. The sorting method that combines the results of CI and RCI is called the bi-directional Choquet integral. The proposal of this concept has inspired further research on bi-directional Choquet integrals [60,61].

2.5. Research Gap

Through a review and summary of existing literature, current research has extensively explored MCDM problems in uncertain environments. However, there are still some deficiencies in the current research.

• Determination of experts’ weights. Current methodologies predominantly determine expert weights through social status and professional influence. However, this approach exhibits over reliance on the subjective assessments of decision-makers.
• Determination of indices’ fuzzy measures. In an evaluation system, there are interactions among the various evaluation indices. When determining the indices’ weights, this relationship must be recognized and appropriately represented. This is an aspect that most current studies overlook.
• Uncertainty expression. Expert evaluation is a key component of multi-criteria decision-making. During the evaluation process, differences in expert opinions and the inherent uncertainties should be fully addressed.
• Applicability of ranking methods. It is essential to identify appropriate methods to effectively incorporate the interactions among indices while ensuring the accuracy and rationality of the ranking results.
• Limitations of existing Methods. Current aggregation operators are not stable enough when dealing with extreme values. Existing heuristic algorithms still have room for improvement in terms of precision when solving fuzzy measures.

This paper aims to assist companies in constructing a multi-criteria decision-making model to assess the selection of suitable green suppliers, thus enhancing their green supply chain management level. Based on the existing literature, this paper presents several significant features and differences compared to prior studies. Based on a thorough review of existing research, this paper further innovates and improves upon the current methods, as shown in

Table 1. Methods and the problems addressed.

Method	Problem Addressed
Expert Social Network	Combine subjective and objective factors to determine expert weights.
$\lambda$-fuzzy Measure	Fully express the interactions among indices.
IVIULN	Fully represent uncertainty in the expert evaluation process.
Bi-direction Shapley-Choquet Integral	Enable ranking based on fuzzy measures, with more reasonable and stable results.
IVIULN-WAGA	Ensure result stability under extreme values.
NTLBO	Improve the precision of heuristic algorithms.

.

3. Preliminaries

To introduce bi-directional Shapley–Choquet integrals in interval-valued intuitive fuzzy environments, some related basic definitions are presented in this section, including interval intuitive fuzzy set, $\lambda$-fuzzy measure, and Choquet integral.

3.1. Interval-Valued Intuitive Uncertain Linguistic Sets

Definition 1.

(Ref. [40]). Let $X$ is a given domain ${\tilde{s}}_x\in \tilde{S}$. Then

$$\tilde{A}=\left\{〈x\left({\tilde{s}}_x,{\tilde{u}}_{\tilde{A}}\left(x\right),{\tilde{v}}_{\tilde{A}}\left(x\right)\right)〉\right\}$$

(1)

is called an interval-valued intuitive uncertainty language set (IVIULS).

Where ${\tilde{s}}_x=\left[s_{\theta \left(x\right)},s_{\tau \left(x\right)}\right]$ represents an uncertain linguistic variable, $\theta \left(x\right)$ and $\tau \left(x\right)$ are lower and upper bound of ${\tilde{s}}_x$, respectively. ${\tilde{u}}_{\tilde{A}}\left(x\right)$ and ${\tilde{v}}_{\tilde{A}}\left(x\right)$ represent the membership degree and non-membership degree of the element $x$ to ${\tilde{s}}_x$, respectively. They satisfy the following relations: ${\tilde{u}}_{\tilde{A}}:X\to D\left[0,1\right]$, ${\tilde{v}}_{\tilde{A}}:X\to D\left[0,1\right]$, $0\le \sup \left({\tilde{u}}_{\tilde{A}}\left(x\right)\right)+\sup \left({\tilde{v}}_{\tilde{A}}\left(x\right)\right)\le 1,x\in X$. For any $x\in X$, ${\tilde{u}}_{\tilde{A}}\left(x\right)$ and ${\tilde{v}}_{\tilde{A}}\left(x\right)$ are closed intervals, their lower and upper end points can be denoted by $u_{\tilde{A}}^L\left(x\right)$, $v_{\tilde{A}}^L\left(x\right)$, $u_{\tilde{A}}^U\left(x\right)$, and $v_{\tilde{A}}^U\left(x\right)$. Then, IVIULS can be represented as:

$$\tilde{A}=\left\{\left.〈x\left(\left[s_{\theta \left(x\right)},s_{\tau \left(x\right)}\right],[u_{\tilde{A}}^L\left(x\right),u_{\tilde{A}}^U\left(x\right)],\left[v_{\tilde{A}}^L\left(x\right),v_{\tilde{A}}^U\left(x\right)\right]\right)〉\right|x\in X\right\}$$

(2)

where $s_{\theta \left(x\right)},s_{\tau \left(x\right)}\in \tilde{S}$, $0\le u_{\tilde{A}}^U\left(x\right)+u_{\tilde{A}}^U\left(x\right)\le 1$, $u_{\tilde{A}}^L\left(x\right)\ge 0$, and $v_{\tilde{A}}^L\left(x\right)\ge 0$.

Definition 2.

(Ref. [40]). Suppose that $\tilde{A}=\left\{〈x\left({\tilde{s}}_x,{\tilde{u}}_{\tilde{A}}\left(x\right),{\tilde{v}}_{\tilde{A}}\left(x\right)\right)〉\right\}$ is an IVIULS. The 6-Tuple $〈\left[s_{\theta \left(x\right)},s_{\tau \left(x\right)}\right],\left[u_{\tilde{A}}^L\left(x\right),u_{\tilde{A}}^U\left(x\right)\right],\left[v_{\tilde{A}}^L\left(x\right),v_{\tilde{A}}^U\left(x\right)\right]〉$ can be viewed as an IVIULN. $\tilde{A}$ can be viewed as a collection of interval-valued intuitive uncertainty language variables, and it can be expressed as:

$$\tilde{A}=\left\{〈\left[s_{\theta \left(x\right)},s_{\tau \left(x\right)}\right],\left[u_{\tilde{A}}^L\left(x\right),u_{\tilde{A}}^U\left(x\right)\right],\left[v_{\tilde{A}}^L\left(x\right),v_{\tilde{A}}^U\left(x\right)\right]〉,x\in X\right\}$$

(3)

${\tilde{a}}_1=〈\left[s_{\theta \left({\tilde{a}}_1\right)},s_{\tau \left({\tilde{a}}_1\right)}\right],\left[u^L\left({\tilde{a}}_1\right),u^U\left({\tilde{a}}_1\right)\right],\left[v^L\left({\tilde{a}}_1\right),v^U\left({\tilde{a}}_1\right)\right]〉$ and ${\tilde{a}}_2=〈\left[s_{\theta \left({\tilde{a}}_2\right)},s_{\tau \left({\tilde{a}}_2\right)}\right],\left[u^L\left({\tilde{a}}_2\right),u^U\left({\tilde{a}}_2\right)\right],\left[v^L\left({\tilde{a}}_2\right),v^U\left({\tilde{a}}_2\right)\right]〉$ are two IVIULNs, $k\ge 0$. The operational rules about ${\tilde{a}}_1$ and ${\tilde{a}}_2$ can be defined as [62]:

$$\begin{array}{cc}\hfill {\tilde{a}}_1\oplus {\tilde{a}}_2& =〈\left[s_{\theta \left({\tilde{a}}_1\right)+\theta \left({\tilde{a}}_2\right)},\right.\left.s_{\tau \left({\tilde{a}}_1\right)+\tau \left({\tilde{a}}_2\right)}\right],\left[1-\left(1-u^L\left({\tilde{a}}_1\right)\right)\left(1-u^L\left({\tilde{a}}_2\right)\right)\right.,\hfill \\ & \left.1-\left(1-u^U\left({\tilde{a}}_1\right)\right)\left(1-u^U\left({\tilde{a}}_2\right)\right)\right],\left[v^L\left({\tilde{a}}_1\right)v^L\left({\tilde{a}}_2\right),v^U\left({\tilde{a}}_1\right)v^U\left({\tilde{a}}_2\right)\right]〉\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{a}}_1\otimes {\tilde{a}}_2& =〈\left[s_{\theta \left({\tilde{a}}_1\right)\times \theta \left({\tilde{a}}_2\right)},\right.\left.s_{\tau \left({\tilde{a}}_1\right)\times \tau \left({\tilde{a}}_2\right)}\right],\left[u^L\left({\tilde{a}}_1\right)u^L\left({\tilde{a}}_2\right),\right.\left.u^U\left({\tilde{a}}_1\right)u^U\left({\tilde{a}}_2\right)\right],\hfill \\ & \left[1-\left(1-v^L\left({\tilde{a}}_1\right)\right)\left(1-v^L\left({\tilde{a}}_2\right)\right)\right.,\left.1-\left(1-v^U\left({\tilde{a}}_1\right)\right)\left(1-v^U\left({\tilde{a}}_2\right)\right)\right]〉\hfill \end{array}$$

$$k{\tilde{a}}_1=〈\left[s_{k\times \theta \left({\tilde{a}}_1\right)},\left.s_{k\times \tau \left({\tilde{a}}_1\right)}\right]\right.,\left[1-{\left(1-u^L\left({\tilde{a}}_1\right)\right)}^k\right.,\left.1-{\left(1-u^U\left({\tilde{a}}_1\right)\right)}^k\right],\left[{\left(v^L\left({\tilde{a}}_1\right)\right)}^k,\left.{\left(v^U\left({\tilde{a}}_1\right)\right)}^k\right]\right.〉$$

$${\tilde{a}}_1^k=〈\left[s_{\theta {\left({\tilde{a}}_1\right)}^k},\right.\left.s_{\tau {\left({\tilde{a}}_1\right)}^k}\right],\left[{\left(u^L\left({\tilde{a}}_1\right)\right)}^k,\right.\left.{\left(u^U\left({\tilde{a}}_1\right)\right)}^k\right],\left[1-{\left(1-v^L\left({\tilde{a}}_1\right)\right)}^k,\left.1-{\left(1-v^U\left({\tilde{a}}_1\right)\right)}^k\right]\right.〉$$

Definition 3.

(Ref. [62]). ${\tilde{a}}_1=〈\left[s_{\theta \left({\tilde{a}}_1\right)},s_{\tau \left({\tilde{a}}_1\right)}\right],\left[u^L\left({\tilde{a}}_1\right),u^U\left({\tilde{a}}_1\right)\right],\left[v^L\left({\tilde{a}}_1\right),v^U\left({\tilde{a}}_1\right)\right]〉$ is an IVIULN. Its expected value is denoted by:

$$\begin{array}{cc}\hfill E\left({\tilde{a}}_1\right)& =\frac{1}{2}\left(\frac{u^L\left({\tilde{a}}_1\right)+u^U\left({\tilde{a}}_1\right)}{2}+1-\frac{v^L\left({\tilde{a}}_1\right)+v^U\left({\tilde{a}}_1\right)}{2}\right)\times \frac{\theta \left({\tilde{a}}_1\right)+\tau \left({\tilde{a}}_1\right)}{2}\hfill \\ & =\frac{\left(\theta \left({\tilde{a}}_1\right)+\tau \left({\tilde{a}}_1\right)\right)\times \left(u^L\left({\tilde{a}}_1\right)+u^U\left({\tilde{a}}_1\right)+2-v^L\left({\tilde{a}}_1\right)-v^U\left({\tilde{a}}_1\right)\right)}{8}\hfill \end{array}$$

(4)

The accuracy of ${\tilde{a}}_1$ can be expressed as:

$$\begin{array}{ll}\hfill T\left({\tilde{a}}_1\right)& ={\displaystyle \frac{\theta \left({\tilde{a}}_1\right)+\tau \left({\tilde{a}}_1\right)}{2}}\times \left({\displaystyle \frac{u^L\left({\tilde{a}}_1\right)+u^U\left({\tilde{a}}_1\right)}{2}}+{\displaystyle \frac{v^L\left({\tilde{a}}_1\right)+v^U\left({\tilde{a}}_1\right)}{2}}\right)\hfill \\ & =\left(u^L\left({\tilde{a}}_1\right)+u^U\left({\tilde{a}}_1\right)+v^L\left({\tilde{a}}_1\right)+v^U\left({\tilde{a}}_1\right)\right)\times {\displaystyle \frac{\theta \left({\tilde{a}}_1\right)+\tau \left({\tilde{a}}_1\right)}{4}}\hfill \end{array}$$

(5)

Let ${\tilde{a}}_1=〈\left[s_{\theta \left({\tilde{a}}_1\right)},s_{\tau \left({\tilde{a}}_1\right)}\right],\left[u^L\left({\tilde{a}}_1\right),u^U\left({\tilde{a}}_1\right)\right],\left[v^L\left({\tilde{a}}_1\right),v^U\left({\tilde{a}}_1\right)\right]〉$ and ${\tilde{a}}_2=〈\left[s_{\theta \left({\tilde{a}}_2\right)},s_{\tau \left({\tilde{a}}_2\right)}\right],\left[u^L\left({\tilde{a}}_2\right),u^U\left({\tilde{a}}_2\right)\right],\left[v^L\left({\tilde{a}}_2\right),v^U\left({\tilde{a}}_2\right)\right]〉$ be any two IVIULNs. The comparison rules are given as follows [62].

• If $E\left({\tilde{a}}_1\right)>E\left({\tilde{a}}_2\right)$, then ${\tilde{a}}_1>{\tilde{a}}_2$;
• If $E\left({\tilde{a}}_1\right)=E\left({\tilde{a}}_2\right)$, then
  • If $T\left({\tilde{a}}_1\right)>T\left({\tilde{a}}_2\right)$, then ${\tilde{a}}_1>{\tilde{a}}_2$;
  • If $T\left({\tilde{a}}_1\right)=T\left({\tilde{a}}_2\right)$, then ${\tilde{a}}_1={\tilde{a}}_2$.

3.2. λ-Fuzzy Measure and Choquet Integral

Definition 4.

(Ref. [63]). Suppose that the index set $L$ is nonempty and $P\left(L\right)$ is its power set. If the function $\mu :P\left(L\right)\to \left[0,1\right]$ satisfies following three axioms, then $\mu$ can be called a fuzzy measure:

• Boundness: $\mu \left(\varnothing \right)=0$ and $\mu \left(L\right)=1$.
• Monotonicity: $\forall B,C\in P\left(L\right)$, if $B\subseteq C$, then $\mu \left(B\right)\le \mu \left(C\right)$.
• Weak Continuity: If $\left\{A_i\right\}\subset P\left(L\right)$, $\left\{A_i\right\}$ increases or decreases monotonically, then $\underset{i\to \infty }{\lim }\mathsf{\mu }\left(A_i\right)=\mu \left(\underset{i\to \infty }{\lim }{A}_i\right)$.

Fuzzy measures can be used to express the importance of any subset of an index set. In multi-criteria decision-making problems, there may be interactions between multiple indices that do not satisfy additivity, which can be represented by an $\lambda$-fuzzy measure.

Definition 5.

(Ref. [63]). If fuzzy measure $\mu$ satisfies following axiom, then it is called $\lambda$-fuzzy measure.

If $A,B\in P\left(L\right)$ and $A\cap B\in \varnothing$, then $\exists \lambda \in \left[-1,\infty \right),\mu \left(A\cup B\right)=\mu \left(A\right)+\mu \left(B\right)+\lambda \mu \left(A\right)\mu \left(B\right)$.

According to Definition 5, let $L=\left\{l_1,l_2,\dots ,l_n\right\}$, which is a finite set. If the fuzzy density of the index $l_i$ is ${\mu }_i$, then ${\mu }_{\lambda }$ can be used to represent the $\lambda$-fuzzy measure of $L$. It could be calculated as follows:

$$\begin{array}{ll}{\mu }_{\lambda }\left(\left\{l_1,l_2,\dots ,l_j\right\}\right)& ={\displaystyle \sum _{i=1}^n{\mu }_i}+\lambda {\displaystyle \sum _{i_1=1}^{n-1}{\displaystyle \sum _{i_2=i_1+1}^n{\mu }_{i_1}{\mu }_{i_2}+\dots +{\lambda }^{j-1}{\mu }_1{\mu }_2\dots {\mu }_j}}\\ & ={\displaystyle \frac{1}{\lambda }}\left|{\displaystyle \prod _{i=1}^j\left(1+\lambda {\mu }_i\right)-1}\right|\end{array}$$

(6)

where $\lambda \in \left(-1,\infty \right)$.

Fuzzy measures are effective in reflecting the interactions between indices. The Choquet integral based on fuzzy measures is an effective and convenient tool for evaluation.

Definition 6.

(Ref. [64]). Let $f$ be a positive real-valued function on $L=\left\{l_1,l_2,\dots ,l_n\right\}$ and $\mu$ be a fuzzy measure on $L$. Then the Choquet integral of $f$ with regard to $\mu$ can be defined as follows:

$$C{I}_{\mathsf{\mu }}\left(f\left(x_{\left(1\right)}\right),f\left(x_{\left(2\right)}\right),\dots ,f\left(x_{\left(n\right)}\right)\right)={\displaystyle \sum _{i=1}^{n}f\left(x_{\left(i\right)}\right)\left(\mu \left(A_{\left(i\right)}\right)-\mu \left(A_{\left(i+1\right)}\right)\right)}$$

(7)

where $\left(i\right)$ expresses a permutation on $L$ such that $f\left(x_{\left(1\right)}\right)\le f\left(x_{\left(2\right)}\right)\le \dots \le f\left(x_{\left(n\right)}\right)$ and $A_{\left(i\right)}=\left\{x_{\left(i\right)},\dots ,x_{\left(n\right)}\right\}$ with $A_{\left(n+1\right)}=\varnothing$.

Based on the traditional Choquet integral, various deformations of the Choquet integral have been derived, which can lead to more reasonable conclusions for specific problems.

Definition 7.

(Ref. [59]). Let $f$ be a positive real-valued function on $L=\left\{l_1,l_2,\dots ,l_n\right\}$ and $\mu$ be a fuzzy measure on $L$. Then the exponent Choquet integral (ECI) of $f$ with regard to $\mu$ can be defined as follows:

$$EC{I}_{\mathsf{\mu }}\left(f\left(x_{\left(1\right)}\right),f\left(x_{\left(2\right)}\right),\dots ,f\left(x_{\left(n\right)}\right)\right)={\displaystyle \prod _{i=1}^{n}f{\left(x_{\left(i\right)}\right)}^{\left(\mu \left(A_{\left(i\right)}\right)-\mu \left(A_{\left(i+1\right)}\right)\right)}}$$

(8)

where $\left(i\right)$ is shown in Definition 6.

Definition 8.

(Ref. [59]). Let $f$ be a positive real-valued function on $L=\left\{l_1,l_2,\dots ,l_n\right\}$ and $\mu$ be a fuzzy measure on $L$. Then the reverse Choquet integral (RCI) of $f$ with regard to $\mu$ can be defined as follows:

$$RC{I}_{\mathsf{\mu }}\left(f\left(x_{\left(1\right)}\right),f\left(x_{\left(2\right)}\right),\dots ,f\left(x_{\left(n\right)}\right)\right)={\displaystyle \sum _{i=1}^{n}f\left(x_{\left(i\right)}\right)\left(\mu \left(D_{\left(i\right)}\right)-\mu \left(D_{\left(i-1\right)}\right)\right)}$$

(9)

where${\mathrm{ D}}_{\left(i\right)}=\left\{x_{\left(1\right)},\dots ,x_{\left(i\right)}\right\}$ with $D_{\left(0\right)}=\varnothing$.

4. Proposed Approach

In this section, a green supplier evaluation and selection model based on IVIULN and a bi-direction Shapley–Choquet integral is proposed. In the proposed model, IVIULN is used to evaluate the performance of alternatives in terms of economic, environment, etc., and the bi-direction Shapley–Choquet integral is used to rank the alternatives.

It is assumed that there are $m$ alternatives $S_i\left(i=1,2,\dots ,m\right)$, $n$ evaluation indices $C_j\left(j=1,2,\dots ,n\right)$, and $l$ decision experts $E_k\left(k=1,2,\dots l\right)$. The weight of the decision experts is ${\omega }_k$ and satisfies ${\omega }_k>0$, and ${\displaystyle {\sum }_{k=1}^l{\omega }_k=1}$, ${\omega }_k$ reflects the importance of the experts involved in the evaluation. The score of the decision-making expert $k$ on the alternative $i$ on the index $j$ is denoted by $P^k=\left[p_{ij}^k\right]$, which is an IVIULN, $p_{ij}^k=〈\left[s_{\theta \left(p_{ij}^k\right)},s_{\tau \left(p_{ij}^k\right)}\right],\left[u^L\left(p_{ij}^k\right),u^U\left(p_{ij}^k\right)\right],\left[v^L\left(p_{ij}^k\right),v^U\left(p_{ij}^k\right)\right]〉$. Based on the above notation, the details of the model and the calculation process will be explained in the following subsection.

4.1. IVIULN-WAGA Aggregation Operator

When aggregating fuzzy values, the use of weighted arithmetic average (WAA) or weighted geometric average (WGA) is usually considered. However, when certain evaluative values in IVIULNs tend to be extreme, aggregation via WAA or WGA operators can yield unreasonable results [65]. This phenomenon is illustrated below by three examples.

Example 1.

${\tilde{a}}_1=〈\left[1,2\right],\left[0.8,0.9\right],\left[0,0.1\right]〉$ and ${\tilde{a}}_2=〈\left[8,9\right],\left[0.1,0.2\right],\left[0,0.1\right]〉$ are two IVIULNs with the weights ${\omega }_1=0.5$ and ${\omega }_2=0.5$, respectively. According to the operations in Section 3.1, the aggregation result can be obtained:

$$\begin{array}{ll}{H}_{IVIULN-WA\left({\tilde{a}}_1,{\tilde{a}}_2\right)}& =〈\left[s_{0.5\times 1+0.5\times 8},s_{0.5\times 2+0.5\times 9}\right],\left[1-\sqrt{\left(1-0.8\right)\times \left(1-0.1\right)}\right.,\\ & \left.1-\sqrt{\left(1-0.9\right)\times \left(1-0.2\right)}\right],\left[\sqrt{0\times 0},\left.\sqrt{0.1\times 0.1}\right]\right.〉\\ & =〈\left[s_{4.5},s_{5.5}\right],\left[0.58,0.72\right],\left[0,0.1\right]〉\end{array}$$

$$\begin{array}{ll}{H}_{IVIULN-GA\left({\tilde{a}}_1,{\tilde{a}}_2\right)}& =〈\left[s_{\sqrt{1\times 8}},s_{\sqrt{2\times 9}}\right],\left[\sqrt{0.8\times 0.1},\sqrt{0.9\times 0.2}\right]\\ & \left[1-\sqrt{\left(1-0\right)\times \left(1-0\right)},\left.1-\sqrt{\left(1-0.1\right)\times \left(1-0.1\right)}\right]\right.\\ & =〈\left[s_3,s_{4.24}\right],\left[0.28,0.42\right],\left[0,0.1\right]〉\end{array}$$

Example 2.

Change the weights to ${\omega }_1=0.9$ and ${\omega }_2=0.1$; another result can be calculated as follows:

$$\begin{array}{ll}{H}_{IVIULN-WA\left({\tilde{a}}_1,{\tilde{a}}_2\right)}& =〈\left[s_{0.9\times 1+0.1\times 8},s_{0.9\times 2+0.1\times 9}\right],\left[1-{\left(1-0.8\right)}^{0.9}\times {\left(1-0.1\right)}^{0.1},\right.\\ & \left.1-{\left(1-0.9\right)}^{0.9}\times {\left(1-0.2\right)}^{0.1}\right],\left[0^{0.9}\times 0^{0.1},{0.1}^{0.9}\times {0.1}^{0.1}\right]〉\\ & =〈\left[s_{1.7},s_{2.7}\right],\left[0.77,0.88\right],\left[0,0.1\right]〉\end{array}$$

$$\begin{array}{ll}{H}_{IVIULN-GA\left({\tilde{a}}_1,{\tilde{a}}_2\right)}& =〈\left[s_{1^{0.9}\times 8^{0.1}},s_{2^{0.9}+9^{0.1}}\right],\left[{0.8}^{0.9}\times {0.1}^{0.1},{0.9}^{0.9}\times {0.2}^{0.1}\right],\\ & \left[1-{\left(1-0\right)}^{0.9}\times {\left(1-0\right)}^{0.1},1-{\left(1-0.1\right)}^{0.9}\times {\left(1-0.1\right)}^{0.1}\right]〉\\ & =〈\left[s_{1.23},s_{2.32}\right],\left[0.65,0.77\right],\left[0,0.1\right]〉\end{array}$$

Example 3.

Furthermore, by changing the weights to ${\omega }_1=0.1$ and ${\omega }_2=0.9$, another result can be calculated as follows:

$$\begin{array}{ll}{H}_{IVIULN-WA\left({\tilde{a}}_1,{\tilde{a}}_2\right)}& =〈\left[s_{0.1\times 1+0.9\times 8},s_{0.1\times 2+0.9\times 9}\right],\left[1-{\left(1-0.8\right)}^{0.1}\times {\left(1-0.1\right)}^{0.9}\right.,\\ & \left.1-{\left(1-0.9\right)}^{0.1}\times {\left(1-0.2\right)}^{0.9}\right],\left[0^{0.1}\times 0^{0.9},\left.{0.1}^{0.1}\times {0.1}^{0.9}\right]\right.〉\\ & =〈\left[s_{7.3},s_{8.3}\right],\left[0.23,0.35\right],\left[0,0.1\right]〉\end{array}$$

$$\begin{array}{ll}{H}_{IVIULN-GA\left({\tilde{a}}_1,{\tilde{a}}_2\right)}& =〈\left[s_{1^{0.1}\times 8^{0.9}},s_{2^{0,1}\times 9^{0,9}}\right],\left[{0.8}^{0.1}\times {0.1}^{0.9},{0.9}^{0.1}\times {0.2}^{0.9}\right]\\ & \left[1-{\left(1-0\right)}^{0.1}\times {\left(1-0\right)}^{0.9},\left.1-{\left(1-0.1\right)}^{0.1}\times {\left(1-0.1\right)}^{0.9}\right]\right.\\ & =〈\left[s_{6.50},s_{7.74}\right],\left[0.12,0.23\right],\left[0,0.1\right]〉\end{array}$$

It can be observed that given two IVIULNs, when they differ significantly but have equal weights, the results obtained by using WAA and WGA differ greatly, especially in the membership degree. Furthermore, even if a larger weight is given to the first IVIULN, there is still a significant difference in the results.

Therefore, the weighted geometric arithmetic average aggregation operator for interval-valued intuitive uncertain linguistic numbers (IVIULN-WAGA) is proposed to compensate for the deficiencies of WWA and WGA. This aggregation operator can be expressed in two forms, one of which is shown in the following equation:

$$\begin{array}{l}{H}_{IVIULN-WAGA1\left({\tilde{a}}_1,{\tilde{a}}_2,\dots ,{\tilde{a}}_n\right)}={\left({\displaystyle \sum _{j=1}^n{\omega }_j{\tilde{a}}_j}\right)}^k\otimes {\left({\displaystyle \prod _{j=1}^n{\tilde{a}}_j{\omega }_j}\right)}^{1-k}=〈\left[s_{{\left({\displaystyle \sum _{j=1}^n\left({\omega }_j\theta \left({\tilde{a}}_j\right)\right)}\right)}^k},s_{{\left({\displaystyle \sum _{j=1}^n\left({\omega }_j\tau \left({\tilde{a}}_j\right)\right)}\right)}^k}\right],\\ \left[{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-u^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k,\left.{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-u^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k\right]\right.,\left[1-{\left(1-{\displaystyle \prod _{j=1}^n{\left(v^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k,\right.\\ \left.1-{\left(1-{\displaystyle \prod _{j=1}^n{\left(v^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k\right]〉\otimes 〈\left[s_{{\left({\displaystyle \prod _{j=1}^n{\left(\theta \left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^{1-k}},s_{{\left({\displaystyle \prod _{j=1}^n\left(\tau \left({\tilde{a}}_j\right)\right){\omega }_j}\right)}^{1-k}}\right],\left[{\displaystyle \prod _{j=1}^n{\left(u^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j(1-k)}},\right.\\ \left.{\displaystyle \prod _{j=1}^n{\left(u^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j(1-k)}}\right],\left[1-{{\displaystyle \prod _{j=1}^n\left(1-v^L\left({\tilde{a}}_j\right)\right)}}^{{\omega }_j(1-k)},1-{{\displaystyle \prod _{j=1}^n\left(1-v^U\left({\tilde{a}}_j\right)\right)}}^{{\omega }_j(1-k)}\right]〉\\ =〈\left[s_{{\left({\displaystyle \sum _{j=1}^n\left({\omega }_j\theta \left({\tilde{a}}_j\right)\right)}\right)}^k{\left({\displaystyle \prod _{j=1}^n{\left(\theta \left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^{1-k}},\right.\left.s_{{\left({\displaystyle \sum _{j=1}^n\left({\omega }_j\tau \left({\tilde{a}}_j\right)\right)}\right)}^k{\left({\displaystyle \prod _{j=1}^n{\left(\tau \left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^{1-k}}\right],\left[{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-u^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k\right.\\ {\displaystyle \prod _{j=1}^n{\left(u^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j\left(1-k\right)}},\left.{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-u^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k{\displaystyle \prod _{j=1}^n{\left(u^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j\left(1-k\right)}}\right],\\ \left[1-{\left(1-{\displaystyle \prod _{j=1}^n{\left(v^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k\left({\displaystyle \prod _{j=1}^n{\left(1-v^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j\left(1-k\right)}}\right),\right.\\ \left.1-{\left(1-{\displaystyle \prod _{j=1}^n{\left(v^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^k\left({\displaystyle \prod _{j=1}^n{\left(1-v^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j\left(1-k\right)}}\right)\right]〉\end{array}$$

where $k$ is the adjustment coefficient.

The other form of aggregation operator is shown as follows (calculation process is omitted):

$$\begin{array}{l}{H}_{IVIULN-WAGA2\left({\tilde{a}}_1,{\tilde{a}}_2,\dots ,{\tilde{a}}_n\right)}=k{\displaystyle \sum _{j=1}^n{\omega }_j{\tilde{a}}_j}\oplus (1-k){\displaystyle \prod _{j=1}^n{\tilde{a}}_j{\omega }_j}=〈\left[s_{{\displaystyle \sum _{j=1}^n\left(k{\omega }_j\theta \left({\tilde{a}}_j\right)\right)}+(1-k){\displaystyle \prod _{j=1}^n\theta \left({\tilde{a}}_j\right){\omega }_j}},\right.\\ \left.s_{{\displaystyle \sum _{j=1}^n\left(k{\omega }_j\tau \left({\tilde{a}}_j\right)\right)}+(1-k){\displaystyle \prod _{j=1}^n\tau {\left({\tilde{a}}_j\right)}^{{\omega }_j}}}\right],\left[1-{\displaystyle \prod _{j=1}^n{\left(1-u^L\left({\tilde{a}}_j\right)\right)}^{k{\omega }_j}}{\left(1-{\displaystyle \prod _{j=1}^n{\left(u^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^{1-k}\right.,\\ \left.1-{\displaystyle \prod _{j=1}^n{\left(1-u^U\left({\tilde{a}}_j\right)\right)}^{k{\omega }_j}}{\left(1-{\displaystyle \prod _{j=1}^n{\left(u^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^{1-k}\right],\left[{\displaystyle \prod _{j=1}^n{\left(v^L\left({\tilde{a}}_j\right)\right)}^{k{\omega }_j}}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-v^L\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^{1-k},\right.\\ \left.{\displaystyle \prod _{j=1}^n{\left(v^U\left({\tilde{a}}_j\right)\right)}^{k{\omega }_j}}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-v^U\left({\tilde{a}}_j\right)\right)}^{{\omega }_j}}\right)}^{1-k}\right]〉\end{array}$$

Example 4.

${\tilde{a}}_1=〈\left[1,2\right],\left[0.8,0.9\right],\left[0,0.1\right]〉$ and ${\tilde{a}}_1=〈\left[1,2\right],\left[0.8,0.9\right],\left[0,0.1\right]〉$ are two IVIULNs with the weights ${\omega }_1=0.5$ and ${\omega }_2=0.5$, respectively; the adjustment coefficient $\lambda =0.5$. According to the operations in Section 3.1, the aggregation result can be obtained as follows (calculation process is omitted):

$$H_{IVIULN-WAGA1({\tilde{a}}_1,{\tilde{a}}_2)}=〈\left[s_{3.6},s_{4.8}\right],\left[0.40,0.55\right],\left[0,0.10\right]〉$$

$$H_{IVIULN-WAGA2({\tilde{a}}_1,{\tilde{a}}_2)}=〈\left[s_{3.7},s_{4.9}\right],\left[0.40,0.60\right],\left[0,0.10\right]〉$$

Example 5.

Change the weights to ${\omega }_1=0.9$ and ${\omega }_2=0.1$ and another result can be obtained:

$$H_{IVIULN-WAGA1({\tilde{a}}_1,{\tilde{a}}_2)}=〈\left[s_{1.4},s_{2.5}\right],\left[0.71,0.82\right],\left[0,0.10\right]〉$$

$$H_{IVIULN-WAGA2({\tilde{a}}_1,{\tilde{a}}_2)}=〈\left[s_{1.5},s_{2.5}\right],\left[0.71,0.83\right],\left[0,0.10\right]〉$$

Example 6.

Furthermore, by changing the weights to ${\omega }_1=0.1$ and ${\omega }_2=0.9$, another result can be calculated as follows:

$$H_{IVIULN-WAGA1({\tilde{a}}_1,{\tilde{a}}_2)}=〈\left[s_{6.9},s_{8.0}\right],\left[0.17,0.29\right],\left[0,0.10\right]〉$$

$$H_{IVIULN-WAGA2({\tilde{a}}_1,{\tilde{a}}_2)}=〈\left[s_{6.9},s_{8.0}\right],\left[0.18,0.29\right],\left[0,0.10\right]〉$$

Through the above three examples, it is evident that using the IVIULN-WAGA operator to aggregate IVIULNs yields similar aggregation results. It is believed that the proposed aggregation operator can achieve relatively stable aggregation results.

4.2. Expert Weights Based on Social Networks

In most studies, expert weights are often subjectively determined based on their experience, knowledge, status, etc., which is unreliable. Obviously, interpersonal relationships can influence experts’ evaluations, and this relationship can be clearly and effectively expressed in social networks. This section will introduce a method of calculating expert weights through social networks.

Fuzzy language number ${\sigma }_{ij}=〈t_{ij},f_{ij}〉$ can represent the trust relationship if $e_i$ and $e_j$ have a direct trust propagation. However, in practical situations, the social network relationship between experts is incomplete, so there may not be a direct trust propagation relationship between any two experts. This requires the introduction of a third expert who has a direct propagation relationship with these two experts to connect the incomplete trust relationship [17].

Definition 9.

(Ref. [50]). Trust propagation: Trust value ${\sigma }_{12}=〈t_{12},f_{12}〉$ indicates the propagation of trust from expert $e_1$ to expert $e_2$ and ${\sigma }_{23}=〈t_{23},f_{23}〉$ indicates the propagation of trust from expert $e_2$ to expert $e_3$. Thus, the propagation of trust from expert $e_1$ to expert $e_3$ can be calculated as follows:

$$TP\left({\sigma }_{12},{\sigma }_{23}\right)=〈{\displaystyle \frac{t_{12}\cdot t_{23}}{1+\left(1-t_{12}\right)\left(1-t_{23}\right)}},{\displaystyle \frac{f_{12}+f_{23}}{1+f_{12}\cdot f_{23}}}〉$$

(10)

When using fuzzy language number ${\sigma }_{ij}=〈t_{ij},f_{ij}〉$ to represent the trust propagation between experts, it is obvious that when experts have a higher membership degree and a lower non-membership degree, they should be given higher weights. Trust score is used to represent this relationship, which can be expressed as follows:

$$TS\left({\sigma }_{ij}\right)=t_{ij}-f_{ij}$$

(11)

Then, the degree to which the experts are trusted is obtained as follows:

$$O{T}_h={\displaystyle \frac{1}{k}}{\displaystyle \sum _{i=1}^{k}TS\left({\sigma }_{ih}\right)}$$

(12)

A higher $O{T}_h$ represents that $e_h$ has higher status in the expert group, and he should be given higher weights. By standardizing $OT$s, all expert weights can be obtained.

4.3. Fuzzy Measures of Index Sets Based on λ-Fuzzy Measure

Determining the importance of indices is a key issue in MCDM. There are interactions between indices, and the weight of the indices no longer satisfies additivity. In 1986, a fuzzy measure to measure any interactions between indices is proposed [66]. $\lambda$-fuzzy measure is a method developed based on this theory.

The value of the $\lambda$-fuzzy measure can be quickly and accurately calculated using heuristic algorithms. Tian et al. [63] compared multiple heuristic algorithms for solving λ, and the results showed that the improved TLBO algorithm performed better than other algorithms. The TLBO algorithm is a heuristic algorithm used to improve student performance by simulating the teaching process. This paper will further optimize the performance of the algorithm based on Tian et al. [63] and propose NTLBO to accurately identify the value of $\lambda$. The specific steps are as follows:

(1)

Define optimization problems and initialize parameters. Let $X=\left(g_1,g_2,\dots ,g_n,\lambda \right)$ be the solution vector, where $g_i$ represents the fuzzy measure of the index $i$.

The main parameters include population size P, number of iterations G, number of decision variables $n+1$, accuracy factors ${\mathsf{\delta }}_s$ and ${\mathsf{\delta }}_e$, and upper and lower limits of decision variables $\left(X^L,X^U\right)$. The upper and lower limit constraints for decision variables are specified as $0\le g_i\le 1, -1<\lambda <\infty$.

Based on Equation (6), the objective is to minimize the error between the fuzzy measures formulated by experts and the fuzzy measures calculated through $\lambda$-fuzzy measure. The fitness function is as follows:

$$f\left(X\right)={\displaystyle \sum _{A\in P\left(X\right)}\left|{\widehat{g}}_{\lambda }\left(A\right)-\frac{1}{\lambda }\left[{\displaystyle \prod \left(1+\lambda g_i\right)-1}\right]\right|}$$

(13)

According to the upper and lower limits, an initial population can be generated, and its values are randomly generated. For TLBO, the population size represents the number of students, the number of decision variables represents all the subjects that students need to learn, and the value of fitness function represents the overall performance of each student. The initialized population can be expressed as follows:

$$pop=\left[\begin{array}{cccc}{x}_1^1& x_2^1& \dots & x_{n+1}^1\\ x_1^2& x_2^2& \dots & x_{n+1}^2\\ ⋮& ⋮& \ddots & ⋮\\ x_1^P& x_2^P& \dots & x_{n+1}^P\end{array}\right]$$

(2)

Teaching stage. Select the student with the best performance (fitness) and the selected student will be the teacher in the current iteration. During the teaching stage, teaching outcomes will be reflected by the gap between the teacher and overall average grade. The teacher’s teaching contents may not always be correct, so the accuracy factors δ_s and δ_e are introduced. Their values would be adjusted according to current iteration $t$ and number of iterations G. After studying, the change in student grades is calculated by the following equation:

$$X_a^j=\mathsf{\delta }{X}_b^j+d^j$$

(14)

where $X_b^j$ and $X_a^j$ denote the solution vectors (grades) for the individual $j$ (student) before and after the teaching stage, respectively. The calculation method of $\mathsf{\delta }$ is as follows:

$$\mathsf{\delta }=\left({\mathsf{\delta }}_s-{\mathsf{\delta }}_e\right){\left({\displaystyle \frac{t}{G}}\right)}^2+\left({\mathsf{\delta }}_e-{\mathsf{\delta }}_s\right)\left({\displaystyle \frac{2t}{G}}\right)+{\mathsf{\delta }}_s$$

$d^j$ can be calculated using the following equation:

$$d^j=r\cdot \left(X^t-F^j\cdot M\right)$$

(15)

where $r$ refers to a random number with a range of $\left(0,1\right)$ and $M$ refers to overall average grades. Due to the different learning abilities and personal conditions of each student, the teacher can better help students improve their grades by teaching according to their aptitude. For students with fitness higher than average, it is necessary to learn as much knowledge as possible from the teacher’s teaching, and the value of $F^j$ is 1; for students with fitness lower than average, they can not only learn from the teacher’s teaching content, but also discover their strengths in other areas, therefore $F^j=round\left(1+r\right)$.

(3)

Mutual learning stage. Mutual learning allows two students to learn from each other and make progress together through mutual assistance. At this stage, students need to randomly select another student and learn from each other by comparing the differences between them. This stage can be expressed as follows:

$$X_a^j=\left\{\begin{array}{c}{X}_b^j+r\cdot \left(X^j-X^k\right), f\left(X^j\right)<f\left(X^k\right)\\ X_b^j+r\cdot \left(X^k-X^j\right), f\left(X^j\right)\ge f\left(X^k\right)\end{array}\right.$$

(16)

(4)

Self-learning stage. During the self-learning stage, students learn content on their own, thereby changing their performance. For the algorithm, this step improves its global search capability. The changes in student performance during the self-learning stage is referred to in the following equation:

$$X_a^j=X_b^j+r\cdot \left(X^U-X^L\right)$$

(17)

(5)

Terminate the algorithm. Continuously iterate through steps (2) to (4) and terminate the algorithm when the maximum iteration G is reached. Output the current best student, namely the fuzzy measures of all indices and the values of λ.

4.4. Bi-Direction Shapley-Choquet Integral

The Choquet integral can effectively handle the interrelation between indices. However, the results calculated by Equations (7) and (9) are not consistent [59]. To have an adequate description, it is considered necessary to combine the information from both integrals using a weighted average.

Definition 10.

For real valued function $f$ on fuzzy measure $\mu$, its BCI can be expressed as follows:

$$\begin{array}{ll}BC{I}_{\mu }\left(f\right)& =\kappa C{I}_{\mu }\left(f\right)+\left(1-\kappa \right)RC{I}_{\mu }\left(f\right)\\ & ={\displaystyle \sum _{i=1}^{n}f\left(x_{\sigma \left(i\right)}\right)\left(\kappa \left(\mu \left(A_{\sigma \left(i\right)}\right)-\mu \left(A_{\sigma \left(i+1\right)}\right)\right)+\left(1-\kappa \right)\left(\mu \left(D_{\sigma \left(i\right)}\right)-\mu \left(D_{\sigma \left(i-1\right)}\right)\right)\right)}\end{array}$$

(18)

Similarly, for $ECI$, the definition of $BECI$ can be obtained through Equations (8) and (9).

Definition 11.

For real valued function $f$ on fuzzy measure $\mu$, its BECI can be expressed as follows:

$$\begin{array}{ll}BEC{I}_{\mu }\left(f\right)& =EC{I}_{\mu }{\left(f\right)}^{\kappa }\cdot REC{I}_{\mu }{\left(f\right)}^{1-\kappa }\\ & ={\displaystyle \prod _{i=1}^{n}f{\left(x_{\sigma \left(i\right)}\right)}^{\kappa \left(\mu \left(A_{\sigma \left(i\right)}\right)-\mu \left(A_{\sigma \left(i+1\right)}\right)\right)+\left(1-\kappa \right)\left(\mu \left(D_{\sigma \left(i\right)}\right)-\mu \left(D_{\sigma \left(i-1\right)}\right)\right)}}\end{array}$$

(19)

$BCI$ and $BECI$ assign weights by considering the increase and loss values of adjacent ordered index sets. However, it cannot indicate the interactions among ordered index sets globally, which can be effectively represented by the generalized Shapley function. It can be represented by the following equation [67]:

$$GS{F}_T\left(\mu \right)={\displaystyle \sum _{S\subseteq N\backslash T}\frac{\left(n-t-s\right)!s!}{\left(n-t+1\right)!}}\left(\mu \left(S\cup T\right)-\mu \left(S\right)\right)$$

(20)

where $N$ represents the universal set and the $T$ represents the set of indices whose weights are to be calculated. $n$, $t$, and $s$ respectively represent the number of elements in the corresponding sets. When $t=1$, Equation (20) could be used to calculate the Shapley value.

Based on the above definitions, the generalized Shapley function can be further combined with $BCI$ or $BECI$.

Definition 12.

For real valued function $f$ on fuzzy measure $\mu$, $GSFBC{I}_{\mu }\left(f\right)$ can be defined as follows:

$$\begin{array}{l}GSFBC{I}_{GSF\left(\mu \right)}\left(f\right)=\kappa C{I}_{GSF\left(\mu \right)}\left(f\right)+\left(1-\kappa \right)RC{I}_{GSF\left(\mu \right)}\left(f\right)\\ ={\displaystyle \sum _{i=1}^{n}f\left(x_{\sigma \left(i\right)}\right)\left(\kappa \left(GS{F}_{A_{\sigma \left(i\right)}}\left(\mu \right)-GS{F}_{A_{\sigma \left(i+1\right)}}\left(\mu \right)\right)+\left(1-\kappa \right)\left(GS{F}_{D_{\sigma \left(i\right)}}\left(\mu \right)-GS{F}_{D_{\sigma \left(i-1\right)}}\left(\mu \right)\right)\right)}\end{array}$$

(21)

Definition 13.

For real valued function $f$ on fuzzy measure $\mu$, $GSFBEC{I}_{\mu }\left(f\right)$ can be defined as follows:

$$\begin{array}{ll}GSFBEC{I}_{GSF\left(\mu \right)}\left(f\right)& =EC{I}_{GSF\left(\mu \right)}{\left(f\right)}^{\kappa }\cdot REC{I}_{GSF\left(\mu \right)}{\left(f\right)}^{1-\kappa }\\ & ={\displaystyle \prod _{i=1}^{n}f{\left(x_{\sigma \left(i\right)}\right)}^{\kappa \left(GS{F}_{A_{\sigma \left(i\right)}}\left(\mu \right)-GS{F}_{A_{\sigma \left(i+1\right)}}\left(\mu \right)\right)+\left(1-\kappa \right)\left(GS{F}_{D_{\sigma \left(i\right)}}\left(\mu \right)-GS{F}_{D_{\sigma \left(i-1\right)}}\left(\mu \right)\right)}}\end{array}$$

(22)

It is important to note that in this equation, the values of the real-valued function should be considered to avoid the influence of special values of the function on the ranking results.

4.5. Steps and Process of MCDM Model

Before evaluating green suppliers, it is necessary to invite experts with high authority from relevant fields to form an evaluation expert group $E=\{e_1,e_2,\dots ,e_k\}$, with their respective weights being $w^e=\left\{w_1^e,w_2^e,\dots ,w_k^e\right\}$. The expert group will conduct a comprehensive evaluation of $n$ alternatives $G=\left\{G_1,G_2,\dots ,G_n\right\}$. The expert group will analyze the existing evaluation indices and form the final index system. The corresponding fuzzy measures are represented as $w^c=\left\{w_1^c,w_2^c,\dots ,w_m^c\right\}$, and for the index set $T$, its fuzzy measure is $w_T^c$. To accurately express the evaluation made by experts, IVIULN will be used to represent evaluation information and FN will be used to represent the trust propagation between experts. The specific process of the model is shown in Figure 1. The specific steps of the model process are summarized as follows:

Step 1: Construction of the index system. Determine the index system based on existing research and experts’ discussion. Considering the lack of measurement of the $\lambda$-fuzzy measure, the index system includes two levels. There is a mutual relationship between first-level indices, and second-level indices under each first-level index should also have redundant interactions.

Step 2: Evaluation information acquisition. Based on the historical performance of suppliers, determine the objective values of all quantifiable indices for the suppliers. After that, score the performance of alternative suppliers under different indices. During the scoring process, experts are allowed to discuss, and their opinions can be modified on the spot. The scoring results of experts are recorded through IVIULNs, and each expert needs to evaluate the performance of $n$ suppliers in $m$ indices.

Step 3: Expert weights obtained by social network. FN is used to represent the trust relationship between experts. FN is a fuzzy number, which can be expressed as $F=\left\{x,u\left(x\right),v\left(x\right)\right\}$, where $u\left(x\right)$ and $v\left(x\right)$ represent the membership and non-membership of $x$, respectively. In this process, it is necessary to first calculate the trust propagation relationship, and then then $OT$ of each expert can be obtained through Equations (10)–(12). We can solve the weight of each expert through standardized transformation.

Step 4: Solutions of fuzzy measures for index sets. Based on the fuzzy measures given by experts for partial index sets, the NTLBO algorithm is used to solve them. It can be foreseen that if experts make incorrect evaluations during the process of preset fuzzy measures, the results will have significant errors. In this case, new fuzzy measures should be pre-assigned. When the error is controlled within a reasonable range, the fuzzy measures of all index sets can be obtained through Equation (6).

Step 5: Evaluation information aggregation. For the subjective evaluation information obtained in Step 2 and Step 3, use IVIULN-WAGA1 or IVIULN-WAGA2 to aggregate the evaluation information. Calculate the expected value using Equation (4). To eliminate the effect of dimensions and ensure all results are positive, the expected value is normalized using min-max scaling to a range of 0.5 to 1. Then, the expected value $E^c=\left\{E_1^c,E_2^c,\dots ,E_n^c\right\}$ of all alternatives under the index $c$ can be calculated.

Step 6: Sorting and selection. This step is divided into two stages. First, it is necessary to examine the performance of the suppliers under each first-level index and then examine the overall performance of each supplier. Both stages can be obtained through Equations (21) and (22), where the value of the real value function $f$ is the expected value in step 5, and the fuzzy measures of index sets are obtained through step 4. According to the final scores of alternatives, select the best supplier.

Step 7: Discussion and sensitivity analysis. Discuss the results obtained from the model and analyze the sensitivity of the key parameters involved in the model.

5. Case Study

The proposed model will undergo empirical validation through a case study of a Chinese enterprise. Company A is an enterprise that provides digital services for government and enterprises. Its main business involves software service development, digital bidding and procurement, and the smart construction site. Currently, it has more than 20 regional branches. The regional branch located in Shanghai has recently taken over a project relating to a smart construction site. To achieve environmental protection goals, it needs to evaluate and select material suppliers. After preliminary screening, six alternative suppliers have been identified. There are a total of four experts participating in the evaluation, including the principal of the client, the manager of the project, the head of the project management department, and the liaison personnel of the PMO.

5.1. Weights of Experts

$R$ is a matrix composed of FNs representing direct trust propagation.

$$R=\left(\begin{array}{cccc}-& 〈0.9,0.1〉& -& -\\ 〈0.7,0.2〉& -& 〈0.9,0.1〉& 〈0.8,0.1〉\\ -& -& -& 〈0.7,0.3〉\\ 〈0.8,0.1〉& 〈0.6,0.3〉& 〈0.7,0.3〉& -\end{array}\right)$$

Except for the diagonal, if there is no FN at a corresponding position on the matrix, it indicates that there is no direct trust propagation between these two experts. For example, the principal of the client directly interacts with the project team members of Company A, so he has a direct trust propagation relationship with the manager of the project, while there is no trust propagation relationship with the head of the project management department and the liaison personnel of the PMO. To reveal the trust propagation relationship among all experts, experts with no direct trust propagation will form a trust propagation relationship through another expert. For example, to obtain a trust relationship from $e_1$ to $e_3$, the propagation path is $e_1\to e_2\to e_3$. The FN between $e_1$ and $e_3$ can be obtained from Equation (10).

$$TP\left({\sigma }_{12},{\sigma }_{23}\right)=〈{\displaystyle \frac{0.9\times 0.9}{1+\left(1-0.9\right)\left(1-0.9\right)}},{\displaystyle \frac{0.1\times 0.1}{1+0.1\times 0.1}}〉=〈0.80,0.20〉$$

Similarly, the complete trust propagation matrix can be obtained through calculation.

$$R=\left(\begin{array}{cccc}-& 〈0.9,0.1〉& 〈0.80,0.20〉& 〈0.71,0.20〉\\ 〈0.7,0.2〉& -& 〈0.9,0.1〉& 〈0.8,0.1〉\\ 〈0.53,0.39〉& 〈0.38,0.55〉& -& 〈0.7,0.3〉\\ 〈0.8,0.1〉& 〈0.6,0.3〉& 〈0.7,0.3〉& -\end{array}\right)$$

According to Equation (11), the trust score matrix is as follows:

$$TS=\left(\begin{array}{cccc}\text{-}& 0.8& 0.6& 0.51\\ 0.5& \text{-}& 0.8& 0.7\\ 0.14& -0.17& \text{-}& 0.4\\ 0.7& 0.3& 0.4& \text{-}\end{array}\right)$$

Then, $O{T}_h$ can be obtained:

$$OT=\left(0.47,0.31,0.6,0.54\right)$$

So, the final weight of experts is $w_1^e=0.24$, $w_2^e=0.16$, $w_3^e=0.31$, $w_4^e=0.28$. In this case, $e_3$ is the most trusted and has the highest weight.

5.2. Fuzzy Measures of Index Sets

By consulting experts and discussing and referring to previous research [68,69,70,71,72,73,74,75], an evaluation index system was designed, as shown in Figure 2.

The index system includes a total of 5 first-level indices and 26 s-level indices. To ensure the objectivity of the case data, relevant indices were quantified based on the historical performance of the alternative suppliers. For data that cannot be quantified, such as customer satisfaction, experts will assign subjective scores based on their evaluation and discussions with the suppliers. The indices that require quantification and their descriptions are shown in

Table 2. The quantitative indices, units, and descriptions.

Index	Unit	Description
Delivery Efficiency	$\%$	On-time delivery rate
Pollution and Emissions	$\mu \mathrm{g}/{\mathrm{m}}^3$	Air pollutants during production and construction
Environmental Protection Exception Handling	h	Average repair time
Supply Chain Exception Handling	h	Average repair time
Employee Training	$\%$	Quarterly average employee training compliance rate
Cost of Production	CNY	Unit product cost
Environmental Costs	CNY	Unit product environmental cost
Resource Consumption	$t$	Unit product energy consumption

. When calculating, it is necessary to handle them level by level. To save space, only the fuzzy measures of the second-level index sets under environmental protection are provided here to illustrate this process.

The initial preset fuzzy measures for the index sets by experts are shown in

Table 3. Initial fuzzy measures of environmental protection.

Set	Fuzzy Measure	Set	Fuzzy Measure
$\left\{c_{21}\right\}$	0.450	$\left\{c_{21},c_{22}\right\}$	0.518
$\left\{c_{22}\right\}$	0.118	$\left\{c_{21},c_{25}\right\}$	0.656
$\left\{c_{23}\right\}$	0.337	$\left\{c_{22},c_{24}\right\}$	0.230
$\left\{c_{24}\right\}$	0.132	$\left\{c_{25},c_{26}\right\}$	0.570
$\left\{c_{25}\right\}$	0.313	$\left\{c_{21},c_{22},c_{24}\right\}$	0.597
$\left\{c_{26}\right\}$	0.330	$\left\{c_{21},c_{23},c_{25},c_{26}\right\}$	0.933

.

Note that population size $P$, the number of iterations $G$, and the accuracy factors ${\mathsf{\delta }}_s$ and ${\mathsf{\delta }}_e$ are set to 500, 3000, 0.95, and 0.4, respectively. To compare the performance of the proposed algorithm, we compare relevant algorithms, mainly the TLBO algorithm, the ITLBO algorithm by Tian et al. [63], and the genetic algorithm. Each algorithm runs 20 times and records the results, as shown in the

Table 4. Comparison of different algorithms.

Algorithm	Average Error	Min Error	Average Run Time
TLBO	0.1194	0.0697	2.4138 s
ITLBO (Tian et al.) [63]	0.0693	0.0405	2.3816 s
GA	0.0601	0.0544	12.4641 s
NTLBO	0.0333	0.0293	2.6932 s

.

It can be observed that the proposed algorithm performs outstandingly and has significantly improved the performance of the existing TLBO. As one of the classic optimization algorithms, the genetic algorithm is much worse than NTLBO in terms of running time. The comparison results validate the effectiveness of the proposed algorithm.

Next, based on the fuzzy measures obtained from NTLBO operation $w_{\left\{c_{21}\right\}}^c=0.4487, w_{\left\{c_{22}\right\}}^c=0.1111, w_{\left\{c_{23}\right\}}^c=0.3369, w_{\left\{c_{24}\right\}}^c=0.1303, w_{\left\{c_{25}\right\}}^c=0.3189, w_{\left\{c_{26}\right\}}^c=0.3300$, and $\lambda =-0.7819$, we can calculate the fuzzy measures of all index sets, as shown in

Table 5. Fuzzy measures of index set.

Set	Fuzzy Measure	Set	Fuzzy Measure	Set	Fuzzy Measure
$\left\{c_{21},c_{22}\right\}$	0.5193	$\left\{c_{21},c_{23}\right\}$	0.6652	$\left\{c_{21},c_{24}\right\}$	0.5330
$\left\{c_{21},c_{25}\right\}$	0.6560	$\left\{c_{21},c_{26}\right\}$	0.6607	$\left\{c_{22},c_{23}\right\}$	0.4183
$\left\{c_{22},c_{24}\right\}$	0.2313	$\left\{c_{22},c_{25}\right\}$	0.4053	$\left\{c_{22},c_{26}\right\}$	0.4119
$\left\{c_{23},c_{24}\right\}$	0.4337	$\left\{c_{23},c_{25}\right\}$	0.5736	$\left\{c_{23},c_{26}\right\}$	0.5789
$\left\{c_{24},c_{25}\right\}$	0.4210	$\left\{c_{24},c_{26}\right\}$	0.4275	$\left\{c_{25},c_{26}\right\}$	0.5684
$\left\{c_{21},c_{22},c_{23}\right\}$	0.7176	$\left\{c_{21},c_{22},c_{24}\right\}$	0.5970	$\left\{c_{21},c_{22},c_{25}\right\}$	0.7093
$\left\{c_{21},c_{22},c_{26}\right\}$	0.7135	$\left\{c_{21},c_{23},c_{24}\right\}$	0.7276	$\left\{c_{21},c_{23},c_{25}\right\}$	0.8178
$\left\{c_{21},c_{23},c_{26}\right\}$	0.8212	$\left\{c_{21},c_{24},c_{25}\right\}$	0.7194	$\left\{c_{21},c_{24},c_{26}\right\}$	0.7236
$\left\{c_{21},c_{25},c_{26}\right\}$	0.8144	$\left\{c_{22},c_{23},c_{24}\right\}$	0.5065	$\left\{c_{22},c_{23},c_{25}\right\}$	0.6341
$\left\{c_{22},c_{23},c_{26}\right\}$	0.6389	$\left\{c_{22},c_{24},c_{25}\right\}$	0.4949	$\left\{c_{22},c_{24},c_{26}\right\}$	0.5008
$\left\{c_{22},c_{25},c_{26}\right\}$	0.6231	$\left\{c_{23},c_{24},c_{25}\right\}$	0.6456	$\left\{c_{23},c_{24},c_{26}\right\}$	0.6503
$\left\{c_{23},c_{25},c_{26}\right\}$	0.7536	$\left\{c_{24},c_{25},c_{26}\right\}$	0.6409	$\left\{c_{21},c_{22},c_{23},c_{24}\right\}$	0.7745
$\left\{c_{21},c_{22},c_{23},c_{25}\right\}$	0.8568	$\left\{c_{21},c_{22},c_{23},c_{26}\right\}$	0.8599	$\left\{c_{21},c_{22},c_{24},c_{25}\right\}$	0.7671
$\left\{c_{21},c_{22},c_{24},c_{26}\right\}$	0.7709	$\left\{c_{21},c_{22},c_{25},c_{26}\right\}$	0.8538	$\left\{c_{21},c_{23},c_{24},c_{25}\right\}$	0.8642
$\left\{c_{21},c_{23},c_{24},c_{26}\right\}$	0.8673	$\left\{c_{21},c_{23},c_{25},c_{26}\right\}$	0.9339	$\left\{c_{21},c_{24},c_{25},c_{26}\right\}$	0.8612
$\left\{c_{22},c_{23},c_{24},c_{25}\right\}$	0.6997	$\left\{c_{22},c_{23},c_{24},c_{26}\right\}$	0.7040	$\left\{c_{22},c_{23},c_{25},c_{26}\right\}$	0.7982
$\left\{c_{22},c_{24},c_{25},c_{26}\right\}$	0.6955	$\left\{c_{23},c_{24},c_{25},c_{26}\right\}$	0.8067	$\left\{\begin{array}{l}{c}_{21},c_{22},c_{23},\\ c_{24},c_{25}\end{array}\right\}$	0.8992
$\left\{\begin{array}{l}{c}_{21},c_{22},c_{23},\\ c_{24},c_{26}\end{array}\right\}$	0.9020	$\left\{\begin{array}{l}{c}_{21},c_{22},c_{24},\\ c_{25},c_{26}\end{array}\right\}$	0.9627	$\left\{\begin{array}{l}{c}_{21},c_{23},c_{24},\\ c_{25},c_{26}\end{array}\right\}$	0.8964
$\left\{\begin{array}{l}{c}_{21},c_{22},c_{23},\\ c_{25},c_{26}\end{array}\right\}$	0.9682	$\left\{\begin{array}{l}{c}_{22},c_{23},c_{24},\\ c_{25},c_{26}\end{array}\right\}$	0.8467	$\left\{\begin{array}{l}{c}_{21},c_{22},c_{23},\\ c_{24},c_{25},c_{26}\end{array}\right\}$	0.9940

.

5.3. Sorting and Results

The initial evaluation data are $EP=\left\{E{P}_1,E{P}_2,E{P}_3,E{P}_4\right\}$, representing the evaluation made by the experts on all second-level indices under environmental protection. The specific evaluation information is shown as follows. Among them, $c_{24}$ is an objective figure derived from the supplier’s historical performance. In addition, the input from all experts are IVIULNs. This paper employs a 9-level scale method to assess each supplier based on the constructed index system, where the $s_1$ and $s_9$ represent the worst and best performance of the suppliers evaluated under this index, respectively. In the evaluation process, the degree of hesitation in the experts’ evaluation language will be represented by membership and non-membership, and the uncertainty can be represented by the width of the interval and numerical values.

Based on evaluation information and the weights of four experts, aggregate evaluation information by IVIULN-WAGA1 or IVIULN-WAGA2. Furthermore, through Equation (4), the expected values can be obtained. Subsequently, the expected value and the objective evaluation value of $c_{24}$ are normalized using min-max scaling, with their range controlled between 0.5 and 1 to prevent the situation where all results are 0 when sorting using GSFBECI. Thus, when the adjustment coefficient $k=0.5$, IVIULN-WAGA1 is used for aggregation, and the normalized expected values of the six alternatives under six s-level environmental protection indices are as

Table 6. The normalized expected value after aggregation (IVIULN-WAGA1).

Supplier		${\mathit{c}}_{21}$	${\mathit{c}}_{22}$	${\mathit{c}}_{23}$	${\mathit{c}}_{24}$	${\mathit{c}}_{25}$	${\mathit{c}}_{26}$
Expected value	$G_1$	0.5128	0.6225	0.7840	1	0.9444	0.6284
	$G_2$	0.5559	0.8308	0.5792	0.8824	1	0.5106
	$G_3$	0.5	0.7510	0.6974	0.8088	0.8567	0.9234
	$G_4$	0.7319	0.5	0.5	0.8529	0.6123	0.6498
	$G_5$	1	0.9009	0.7698	0.6618	0.7230	1
	$G_6$	0.7678	1	1	0.5	0.5	0.5

.

$$E{P}_1=\left[\begin{array}{cccccc}〈\left[s_6,s_7\right],\left[0.65,0.75\right],\left[0.1,0.2\right]〉& 〈\left[s_5,s_6\right],\left[0.6,0.7\right],\left[0.1,0.2\right]〉& 〈\left[s_6,s_7\right],\left[0.55,0.75\right],\left[0.05,0.15\right]〉& 79& 〈\left[s_7,s_8\right],\left[0.7,0.8\right],\left[0.1,0.2\right]〉& 〈\left[s_5,s_6\right],\left[0.6,0.7\right],\left[0.2,0.3\right]〉\\ 〈\left[s_3,s_4\right],\left[0.75,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_6,s_7\right],\left[0.65,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_5,s_6\right],\left[0.5,0.6\right],\left[0.2,0.3\right]〉& 87& 〈\left[s_7,s_8\right],\left[0.4,0.5\right],\left[0.3,0.5\right]〉& 〈\left[s_2,s_3\right],\left[0.85,0.9\right],\left[0,0.1\right]〉\\ 〈\left[s_5,s_6\right],\left[0.6,0.7\right],\left[0.2,0.25\right]〉& 〈\left[s_5,s_6\right],\left[0.5,0.75\right],\left[0.2,0.25\right]〉& 〈\left[s_5,s_6\right],\left[0.3,0.4\right],\left[0.25,0.3\right]〉& 92& 〈\left[s_5,s_6\right],\left[0.65,0.7\right],\left[0.1,0.25\right]〉& 〈\left[s_7,s_8\right],\left[0.75,0.9\right],\left[0.05,0.1\right]〉\\ 〈\left[s_4,s_5\right],\left[0.55,0.6\right],\left[0.3,0.4\right]〉& 〈\left[s_3,s_4\right],\left[0.7,0.8\right],\left[0.1,0.2\right]〉& 〈\left[s_3,s_4\right],\left[0.4,0.6\right],\left[0.15,0.3\right]〉& 89& 〈\left[s_5,s_6\right],\left[0.5,0.7\right],\left[0.2,0.3\right]〉& 〈\left[s_5,s_6\right],\left[0.2,0.4\right],\left[0.4,0.55\right]〉\\ 〈\left[s_7,s_8\right],\left[0.7,0.8\right],\left[0.05,0.1\right]〉& 〈\left[s_7,s_8\right],\left[0.65,0.7\right],\left[0.2,0.25\right]〉& 〈\left[s_5,s_6\right],\left[0.6,0.7\right],\left[0.15,0.3\right]〉& 102& 〈\left[s_5,s_6\right],\left[0.75,0.8\right],\left[0.05,0.1\right]〉& 〈\left[s_8,s_9\right],\left[0.8,0.9\right],\left[0.05,0.1\right]〉\\ 〈\left[s_6,s_7\right],\left[0.65,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_7,s_8\right],\left[0.7,0.8\right],\left[0.1,0.15\right]〉& 〈\left[s_7,s_8\right],\left[0.55,0.85\right],\left[0.05,0.15\right]〉& 113& 〈\left[s_4,s_5\right],\left[0.6,0.75\right],\left[0.2,0.25\right]〉& 〈\left[s_3,s_4\right],\left[0.6,0.8\right],\left[0.15,0.2\right]〉\end{array}\right]$$

$$E{P}_2=\left[\begin{array}{cccccc}〈\left[s_6,s_7\right],\left[0.5,0.75\right],\left[0.2,0.25\right]〉& 〈\left[s_5,s_6\right],\left[0.7,0.8\right],\left[0.1,0.15\right]〉& 〈\left[s_6,s_7\right],\left[0.45,0.6\right],\left[0.2,0.3\right]〉& 79& 〈\left[s_6,s_7\right],\left[0.9,0.95\right],\left[0,0.05\right]〉& 〈\left[s_6,s_7\right],\left[0.5,0.6\right],\left[0.3,0.4\right]〉\\ 〈\left[s_4,s_5\right],\left[0.6,0.8\right],\left[0.1,0.2\right]〉& 〈\left[s_5,s_6\right],\left[0.6,0.7\right],\left[0.25,0.3\right]〉& 〈\left[s_5,s_6\right],\left[0.55,0.65\right],\left[0.2,0.25\right]〉& 87& 〈\left[s_7,s_8\right],\left[0.6,0.8\right],\left[0.1,0.2\right]〉& 〈\left[s_4,s_5\right],\left[0.9,0.95\right],\left[0,0.05\right]〉\\ 〈\left[s_4,s_5\right],\left[0.6,0.7\right],\left[0.25,0.3\right]〉& 〈\left[s_6,s_7\right],\left[0.45,0.6\right],\left[0.25,0.35\right]〉& 〈\left[s_4,s_5\right],\left[0.25,0.4\right],\left[0.2,0.4\right]〉& 92& 〈\left[s_6,s_7\right],\left[0.6,0.75\right],\left[0.15,0.2\right]〉& 〈\left[s_7,s_8\right],\left[0.45,0.65\right],\left[0.3,0.35\right]〉\\ 〈\left[s_7,s_8\right],\left[0.8,0.85\right],\left[0,0.05\right]〉& 〈\left[s_4,s_5\right],\left[0.75,0.85\right],\left[0.05,0.1\right]〉& 〈\left[s_4,s_5\right],\left[0.5,0.65\right],\left[0.15,0.25\right]〉& 89& 〈\left[s_5,s_6\right],\left[0.65,0.8\right],\left[0.15,0.2\right]〉& 〈\left[s_7,s_8\right],\left[0.65,0.7\right],\left[0.1,0.2\right]〉\\ 〈\left[s_7,s_8\right],\left[0.7,0.75\right],\left[0.1,0.2\right]〉& 〈\left[s_6,s_7\right],\left[0.6,0.75\right],\left[0.15,0.2\right]〉& 〈\left[s_6,s_7\right],\left[0.45,0.6\right],\left[0.2,0.3\right]〉& 102& 〈\left[s_6,s_7\right],\left[0.5,0.6\right],\left[0.3,0.4\right]〉& 〈\left[s_8,s_9\right],\left[0.7,0.85\right],\left[0.1,0.15\right]〉\\ 〈\left[s_6,s_7\right],\left[0.75,0.9\right],\left[0.05,0.1\right]〉& 〈\left[s_7,s_8\right],\left[0.7,0.8\right],\left[0.1,0.15\right]〉& 〈\left[s_7,s_8\right],\left[0.6,0.75\right],\left[0.15,0.2\right]〉& 113& 〈\left[s_5,s_6\right],\left[0.85,0.9\right],\left[0,0.05\right]〉& 〈\left[s_5,s_6\right],\left[0.7,0.8\right],\left[0.05,0.1\right]〉\end{array}\right]$$

$$E{P}_3=\left[\begin{array}{cccccc}〈\left[s_5,s_6\right],\left[0.25,0.4\right],\left[0.45,0.55\right]〉& 〈\left[s_5,s_6\right],\left[0.4,0.6\right],\left[0.3,0.35\right]〉& 〈\left[s_6,s_7\right],\left[0.5,0.6\right],\left[0.2,0.35\right]〉& 79& 〈\left[s_7,s_8\right],\left[0.65,0.7\right],\left[0.2,0.3\right]〉& 〈\left[s_7,s_8\right],\left[0.55,0.7\right],\left[0.2,0.3\right]〉\\ 〈\left[s_5,s_6\right],\left[0.75,0.8\right],\left[0.1,0.15\right]〉& 〈\left[s_5,s_6\right],\left[0.8,0.9\right],\left[0,0.05\right]〉& 〈\left[s_4,s_5\right],\left[0.55,0.6\right],\left[0.35,0.4\right]〉& 87& 〈\left[s_8,s_9\right],\left[0.9,0.95\right],\left[0,0.05\right]〉& 〈\left[s_4,s_5\right],\left[0.9,0.95\right],\left[0,0.05\right]〉\\ 〈\left[s_4,s_5\right],\left[0.6,0.75\right],\left[0.15,0.2\right]〉& 〈\left[s_5,s_6\right],\left[0.8,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_6,s_7\right],\left[0.75,0.8\right],\left[0.15,0.2\right]〉& 92& 〈\left[s_7,s_8\right],\left[0.9,0.95\right],\left[0,0.05\right]〉& 〈\left[s_7,s_8\right],\left[0.8,0.9\right],\left[0,0.1\right]〉\\ 〈\left[s_6,s_7\right],\left[0.75,0.85\right],\left[0.05,0.1\right]〉& 〈\left[s_3,s_4\right],\left[0.85,0.9\right],\left[0,0.1\right]〉& 〈\left[s_3,s_4\right],\left[0.6,0.8\right],\left[0.15,0.2\right]〉& 89& 〈\left[s_6,s_7\right],\left[0.65,0.7\right],\left[0.15,0.25\right]〉& 〈\left[s_6,s_7\right],\left[0.8,0.85\right],\left[0.1,0.15\right]〉\\ 〈\left[s_8,s_9\right],\left[0.8,0.9\right],\left[0,0.05\right]〉& 〈\left[s_6,s_7\right],\left[0.7,0.8\right],\left[0.15,0.2\right]〉& 〈\left[s_7,s_8\right],\left[0.65,0.7\right],\left[0.2,0.3\right]〉& 102& 〈\left[s_6,s_7\right],\left[0.55,0.75\right],\left[0.1,0.2\right]〉& 〈\left[s_7,s_8\right],\left[0.7,0.8\right],\left[0.15,0.2\right]〉\\ 〈\left[s_6,s_7\right],\left[0.75,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_6,s_7\right],\left[0.65,0.8\right],\left[0.05,0.15\right]〉& 〈\left[s_8,s_9\right],\left[0.75,0.8\right],\left[0.15,0.2\right]〉& 113& 〈\left[s_4,s_5\right],\left[0.75,0.8\right],\left[0.15,0.2\right]〉& 〈\left[s_5,s_6\right],\left[0.55,0.7\right],\left[0.15,0.25\right]〉\end{array}\right]$$

$$E{P}_4=\left[\begin{array}{cccccc}〈\left[s_5,s_6\right],\left[0.5,0.6\right],\left[0.3,0.4\right]〉& 〈\left[s_4,s_5\right],\left[0.55,0.7\right],\left[0.2,0.3\right]〉& 〈\left[s_5,s_6\right],\left[0.75,0.8\right],\left[0.1,0.2\right]〉& 79& 〈\left[s_6,s_7\right],\left[0.7,0.8\right],\left[0.05,0.1\right]〉& 〈\left[s_6,s_7\right],\left[0.45,0.6\right],\left[0.2,0.3\right]〉\\ 〈\left[s_5,s_6\right],\left[0.65,0.7\right],\left[0.15,0.25\right]〉& 〈\left[s_5,s_6\right],\left[0.6,0.75\right],\left[0.1,0.2\right]〉& 〈\left[s_4,s_5\right],\left[0.55,0.65\right],\left[0.3,0.35\right]〉& 87& 〈\left[s_6,s_7\right],\left[0.75,0.8\right],\left[0.1,0.2\right]〉& 〈\left[s_5,s_6\right],\left[0.45,0.65\right],\left[0.3,0.35\right]〉\\ 〈\left[s_5,s_6\right],\left[0.45,0.65\right],\left[0.3,0.35\right]〉& 〈\left[s_5,s_6\right],\left[0.7,0.75\right],\left[0.1,0.2\right]〉& 〈\left[s_5,s_6\right],\left[0.55,0.85\right],\left[0.05,0.15\right]〉& 92& 〈\left[s_6,s_7\right],\left[0.55,0.75\right],\left[0.1,0.2\right]〉& 〈\left[s_6,s_7\right],\left[0.7,0.85\right],\left[0.1,0.15\right]〉\\ 〈\left[s_6,s_7\right],\left[0.75,0.8\right],\left[0.15,0.2\right]〉& 〈\left[s_2,s_3\right],\left[0.8,0.9\right],\left[0,0.05\right]〉& 〈\left[s_3,s_4\right],\left[0.65,0.7\right],\left[0.2,0.3\right]〉& 89& 〈\left[s_6,s_7\right],\left[0.65,0.75\right],\left[0.2,0.25\right]〉& 〈\left[s_5,s_6\right],\left[0.75,0.85\right],\left[0.1,0.15\right]〉\\ 〈\left[s_7,s_8\right],\left[0.7,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_6,s_7\right],\left[0.65,0.75\right],\left[0.15,0.2\right]〉& 〈\left[s_6,s_7\right],\left[0.55,0.6\right],\left[0.3,0.4\right]〉& 102& 〈\left[s_6,s_7\right],\left[0.8,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_7,s_8\right],\left[0.75,0.8\right],\left[0.15,0.2\right]〉\\ 〈\left[s_5,s_6\right],\left[0.75,0.85\right],\left[0.1,0.15\right]〉& 〈\left[s_7,s_8\right],\left[0.65,0.7\right],\left[0.2,0.3\right]〉& 〈\left[s_6,s_7\right],\left[0.75,0.8\right],\left[0.15,0.2\right]〉& 113& 〈\left[s_5,s_6\right],\left[0.6,0.75\right],\left[0.2,0.25\right]〉& 〈\left[s_4,s_5\right],\left[0.6,0.8\right],\left[0.1,0.2\right]〉\end{array}\right]$$

When using IVIULN-WAGA2, the result is shown in

Table 7. The normalized expected value after aggregation (IVIULN-WAGA2).

Supplier		${\mathit{c}}_{21}$	${\mathit{c}}_{22}$	${\mathit{c}}_{23}$	${\mathit{c}}_{24}$	${\mathit{c}}_{25}$	${\mathit{c}}_{26}$
Expected value	$G_1$	0.5134	0.6206	0.7841	1	0.9337	0.6283
	$G_2$	0.5549	0.8434	0.5792	0.8824	1	0.5216
	$G_3$	0.5	0.7499	0.6992	0.8088	0.8424	0.9387
	$G_4$	0.7450	0.5	0.5	0.8529	0.5865	0.6538
	$G_5$	1	0.9003	0.7698	0.6618	0.6954	1
	$G_6$	0.7632	1	1	0.5	0.5	0.5

.

When $\kappa =0.5$, according to Equations (21) and (22), the GSFBCI and GSFBECI of all alternatives can be obtained, as shown in

Table 8. Suppliers’ scores in environmental protection.

Supplier	IVIULN-WAGA1		IVIULN-WAGA2
	GSFBCI	GSFBECI	GSFBCI	GSFBECI
$G_1$	0.7104	0.6949	0.7085	0.6935
$G_2$	0.6751	0.6569	0.6779	0.6600
$G_3$	0.7227	0.7088	0.7234	0.7092
$G_4$	0.6375	0.6342	0.6372	0.6331
$G_5$	0.8646	0.8602	0.8593	0.8537
$G_6$	0.7028	0.6783	0.7016	0.6772

.

The results show that after using different aggregation methods and different Choquet integrals, the ranking of all suppliers in terms of environmental protection is $G_5>G_3>G_1>G_6>G_2>G_4$. This indicates that the aggregation method and sorting calculation proposed in this paper can achieve more stable results. Meanwhile, it is believed that Supplier $G_5$ has the most outstanding environmental performance, while Supplier $G_4$ has the worst performance in this area.

By using the above method to calculate the scores of other first-level indices, the performance of all alternatives under each first-level index can be obtained. To save space, the data displayed here are taken as an example of GSFBCI aggregated by IVIULN-WAGA1, as shown in Figure 3.

According to the radar chart, $G_1$ performs relatively evenly in all indices, while suppliers $G_4$, $G_5$, and $G_6$ have obvious strengths and weaknesses. In terms of economic control, $G_2$ receives significantly higher score, but performs poorly in other indices. $G_3$ performs well in all aspects, especially in the index of supply chain risk and economic control. Interestingly, the pentagon formed by $G_3$ completely covers $G_2$.

Finally, using the obtained CI score as the expected value, GSFBCI and GSFBECI were recalculated. The results obtained using different aggregation methods and CI calculation methods are shown in

Table 9. Suppliers’ final scores.

Supplier			${\mathit{G}}_1$	${\mathit{G}}_2$	${\mathit{G}}_3$	${\mathit{G}}_4$	${\mathit{G}}_5$	${\mathit{G}}_6$
IVIULN-WAGA1	GSFBCI	GSFBCI	0.7232	0.7217	0.8090	0.7422	0.7485	0.7025
		GSFBECI	0.7190	0.7150	0.8028	0.7342	0.7331	0.6958
	GSFBECI	GSFBCI	0.7089	0.7021	0.7938	0.7323	0.7392	0.6852
		GSFBECI	0.7046	0.6954	0.7870	0.7245	0.7239	0.6784
IVIULN-WAGA2	GSFBCI	GSFBCI	0.7217	0.7220	0.8122	0.7439	0.7456	0.7020
		GSFBECI	0.7176	0.7152	0.8057	0.7362	0.7306	0.6955
	GSFBECI	GSFBCI	0.7074	0.7029	0.7966	0.7335	0.7362	0.6844
		GSFBECI	0.7032	0.6961	0.7896	0.7258	0.7214	0.6778

.

5.4. Discussion

All results categorize the suppliers into two distinct categories, with suppliers $G_3, G_4$, and $G_5$ scoring higher and suppliers $G_1, G_2$, and $G_6$ scoring lower. It can be observed that the results of the second ranking using GSFBCI indicate $G_3>G_5>G_4$, while GSFBECI concludes $G_3>G_4>G_5$. This is because when calculating CI, it is necessary to arrange the indices in a certain order. By observing the radar chart, it is found that suppliers $G_4$ and $G_5$ each have their own strengths and weaknesses. However, when calculating GSFBECI, their strengths and weaknesses each correspond to specific exponents, and the calculation method of power increases the gap between the two. Similarly, it can be observed that under different aggregation operators, subtle differences in the ranking results also arise. Specifically, the score differences between suppliers $G_1$ and $G_2$ are smaller when using the IVIULN-WAGA2 aggregation compared to the IVIULN-WAGA1. As a result, when aggregating with IVIULN-WAGA2, the two rankings using GSFBCI conclude $G_2>G_1>G_6$, while in all other cases, the results are $G_1>G_2>G_6$. This may be due to the fact that when using IVIULN-WAGA2, if any expert evaluates the lower point of non-membership degree ${\tilde{v}}_{\tilde{A}}\left(x\right)$ as 0, according to the aggregation equation, regardless of how other experts evaluate non-membership degree, the results will be 0. However, supplier $G_2$ has more experts who give the lower point of the non-membership degree as 0, so it will have higher expected values after aggregation. Regardless of the aggregation method and CI calculation method chosen, all results prove that Supplier $G_3$ is the optimal supplier in this case.

5.5. Sensitivity Analysis

To further analyze the impact of parameters on the sorting results, sensitivity analysis will be conducted on the key parameters in the model in this section to examine how they will affect the results.

Figure 4 shows the changes in the scores of each alternative as the parameters $k$ change when $\kappa =0.5$. The CI calculation method is GSFBCI through IVIULN-WAGA1 aggregation.

Observing Figure 4, it can be seen that with the variation of parameter $k$, the scores of the alternative suppliers do not change significantly overall. When $k$ is around $k=0.8$, the scores of supplier $G_5$ and $G_4$ and supplier $G_2$ and $G_1$ are very close, and their rankings will change. This figure indicates that the WAA aggregation operator tends to believe $G_5$ is preferred over $G_4$ and $G_1$ is superior to $G_2$, whereas the WGA operator exhibits the opposite preference.

To examine the impact of parameter $\kappa$ on the results and to analyze the score changes of suppliers $G_1$ and $G_2$ as discussed in Section 5.4, Figure 5 is plotted. It can be observed from Figure 5 that as the parameter $\kappa$ changes, the suppliers’ overall scores remain relatively stable. Specifically, the scores of suppliers $G_1$ and $G_2$ are very close, and by comparing the specific score data, it is evident that near $\kappa =0.8$, the ranking of supplier $G_1$ and supplier $G_2$ has changed.

Next, the expert weights will be modified to $w_1^e=0.1$, $w_2^e=0.01$, $w_3^e=0.74$, $w_4^e=0.15$, to examine the robustness of the model under extreme conditions. The results can be seen in

Table 10. Suppliers’ final scores under extreme weight values.

Supplier			${\mathit{G}}_1$	${\mathit{G}}_2$	${\mathit{G}}_3$	${\mathit{G}}_4$	${\mathit{G}}_5$	${\mathit{G}}_6$
IVIULN-WAGA1	GSFBCI	GSFBCI	0.7386	0.7301	0.8119	0.7718	0.7488	0.7100
		GSFBECI	0.7352	0.7246	0.8069	0.7664	0.7308	0.7052
	GSFBECI	GSFBCI	0.7233	0.7107	0.7967	0.7600	0.7426	0.6912
		GSFBECI	0.7196	0.7051	0.7918	0.7545	0.7247	0.6865
IVIULN-WAGA2	GSFBCI	GSFBCI	0.7378	0.7314	0.8140	0.7746	0.7463	0.7095
		GSFBECI	0.7343	0.7258	0.8091	0.7693	0.7288	0.7047
	GSFBECI	GSFBCI	0.7223	0.7133	0.7986	0.7620	0.7399	0.6908
		GSFBECI	0.7187	0.7075	0.7937	0.7564	0.7225	0.6861

.

Based on the results, it is not difficult to observe that under extreme expert weights, the model proposed in this paper should have strong robustness. Except for the discrepancies in the rankings of $G_1$ and $G_5$ due to different sorting methods, the ranking results of other suppliers are quite stable.

5.6. Suggestions for Countermeasures

The green supplier evaluation and selection model provides an important decision-making tool for enterprises in green supply chain management. This paper offers the following management insights and suggestions for managers:

• This research comprehensively evaluates the economic and environmental factors of green suppliers, enabling enterprises to balance cost-effectiveness while prioritizing suppliers’ performance in energy conservation and emission reduction.
• When evaluating and selecting green suppliers, enterprises should recognize potential synergistic effects or conflicting relationships among indices and fully account for interactions between evaluation indices.
• In interpreting expert evaluation language, inherent uncertainty must be identified while ensuring the stability of aggregation operators for evaluation language sets under extreme scenarios.
• The expert social network proposed in this study effectively quantifies trust relationships and mitigates individual experiential biases. The improved heuristic algorithm significantly enhances computational efficiency and can be applied to complex settings.
• The application of bi-directional Shapley–Choquet integrals quantifies interactions between indices, assisting enterprises in identifying suppliers that focus solely on single-aspect improvements.

In the current business environment, characterized by green transition and uncertainty, enterprises must establish scientific green supply chain management systems. By holistically considering economic and environmental factors, emphasizing interactions among indices, optimizing information aggregation operator, and employing quantitative calculations, this study offers insights and recommendations to help enterprises select and manage green suppliers while advancing sustainable development goals.

6. Conclusions, Limitations and Future Research

6.1. Conclusions

This study develops a multi-criteria decision-making model using IVIULN and bi-directional Shapley–Choquet integrals to evaluate and select green suppliers in uncertain environments. The proposed mathematical model can be employed in scenarios where expert information and evaluation criteria are unknown, while comprehensively considering the interactions among indices. The research demonstrates the following theoretical and practical contributions.

• This paper uses IVIULN to handle the uncertainty in the process of evaluation. An aggregation operator based on IVIULN is proposed, which compensates for the shortcomings of existing aggregation operators and improves the stability of evaluation results.
• This paper combines fuzzy measures with bi-directional Shapley–Choquet integrals, fully revealing the interrelationships between indices. At the same time, the establishment of a second level index system compensates for the shortcomings of the λ-fuzzy measure, which can only represent a set of fuzzy measures of indices.
• This paper improves the TLBO algorithm by introducing the idea of teaching students in accordance with their aptitude and incorporating a self-learning stage. This approach improves the algorithm’s global search capability. At the same time, the teaching stage is improved.
• A universal MCDM model is proposed, which helps decision makers achieve effective and reasonable evaluations. This model also hopes to better promote the development of green suppliers, thereby standardizing and supervising the construction of a good full process green supply chain.

At the same time, the paper also points out that when different aggregation operators or sorting methods are used for calculations, the results can be affected. This requires experts to pay attention to the lower bounds of membership/non-membership during the scoring process. Similarly, when calculating the GSFBECI, attention should also be paid to the selection of the real-valued functions. Sensitivity analysis shows that the adjustment coefficient $k$ and $\kappa$ might have an impact on the results. In cases where the data are extremely close, the determination of the adjustment coefficient should be approached with caution.

6.2. Limitations

This paper has effectively improved the existing MCDM models. However, the model proposed in this study still has certain limitations. First, as mentioned earlier, the aggregation operators proposed in this study still have certain drawbacks. Specifically, when the lower limit of the membership/non-membership degree is 0, the evaluation results obtained by the aggregation operators IVIULN-WAGA1/IVIULN-WAGA2 will be underestimated/overestimated. When the overall conditions of two candidate alternatives are extremely close, this situation may lead to different results from the two aggregation operators. Second, the paper should explore further interaction and integration with other fields, such as the combination of game theory and MCDM models. Additionally, the paper focuses only on the upstream part of the supply chain. However, green supply chain management should be viewed as a holistic concept, and future research should take a comprehensive approach, considering the entire supply chain. Overall, the model proposed in this study has made an effective attempt at evaluating and selecting green suppliers. However, to further enhance the accuracy and practical significance of the model, the aforementioned drawbacks must be addressed.

6.3. Future Research

Green supply chain management has become an essential component of sustainable development phase for enterprises. Similarly, the process of green development does not merely encompass the raw material procurement stage. In the future, the proposed model could be extended to comprehensive evaluation and optimization of the entire green supply chain, thereby achieving further enhancement of the enterprise’s supply chain management and sustainability capabilities. Moreover, the integration of game theory and MCDM models can be considered to further advance research in the field of green supply chain management [76]. Additionally, in the process of supply chain management, the performance of suppliers is influenced by a variety of dynamic factors. Technologies such as machine learning could be integrated with the model. By learning from the historical performance of suppliers, these technologies can predict future trends and incorporate the prediction results into the supplier evaluation process. Green supply chain management involves the integration of multiple disciplines, such as economics and environmental science. By assessing the economic and environmental performance data of suppliers, the scientific validity and practical applicability of the model can be further enhanced.