Intuitionistic Linguistic EDAS Method with New Score Function: Case Study on Evaluating Universities’ Innovation and Entrepreneurship Education

Abstract

Intuitionistic linguistic numbers (ILNs) describe expert evaluation information by representing semantic assessment values and reflecting the confidence level and hesitation of decision-makers. ILNs are widely used to handle uncertain and incomplete information. The Evaluation Based on Distance from Average Solution (EDAS) method selects the optimal solution based on the distance of each alternative from the average solution, making it suitable for multi-attribute decision-making with conflicting attributes. This study proposes a new scoring function for ILNs and develops an evaluation method combining ILNs with EDAS (IL-EDAS). Experts’ evaluations of each alternative’s indices are expressed using ILNs, and the EDAS method ranks the alternatives to select the optimal solution. We apply this method to assess innovation and entrepreneurship education capabilities in universities, and compare the results with those from other methods to verify their applicability and practicality.

1. Introduction

In the natural environment, many phenomena’s attribute information is difficult to describe numerically, whereas linguistic evaluation [1] can better express this situation. Linguistic multi-attribute decision-making problems have recently become a research hotspot in decision analysis [2]. However, linguistic evaluation [1] implies a confidence level of 1 in the evaluation information, not reflecting the decision-maker’s distrust and hesitation. Atanassov [3], based on fuzzy set membership degrees, added non-membership degrees, proposing intuitionistic fuzzy numbers and intuitionistic fuzzy sets, which more delicately reflect fuzzy information. However, when using intuitionistic fuzzy numbers to express decision information, the membership and non-membership elements in the hesitation domain are not specific, leading to certain limitations in practical applications. Wang and Li [4] extended intuitionistic fuzzy numbers by adding membership and non-membership degrees to linguistic evaluation, defining intuitionistic linguistic numbers (ILNs) and intuitionistic linguistic sets (ILSs) and related operational rules. Unlike intuitionistic fuzzy numbers, ILNs add a linguistic evaluation value, making the membership and non-membership degrees relative to a specific linguistic scale. Therefore, ILNs are more specific and flexible in characterizing and describing decision information.

Currently, multi-attribute decision-making methods for intuitionistic linguistic information have attracted widespread attention from scholars. For the intuitionistic linguistic information aggregation operators, Wang and Li [4] proposed the intuitionistic linguistic weighted arithmetic averaging (ILWAA) operator and the intuitionistic linguistic weighted geometric averaging (ILWGA) operator. Since the ILWAA and ILGWA operators only perform weighted aggregation based on the importance of the ILNs themselves, their application scope is limited. Wang et al. [5,6] suggested first ranking the ILNs according to specific requirements and then weighting them based on their positions. Several groundbreaking operators have been introduced in the field of intuitionistic linguistic aggregation. Initially, the Intuitionistic Linguistic Ordered Weighted Averaging (ILOWA) operator, the Intuitionistic Linguistic Hybrid Aggregation (ILHA) operator, the Intuitionistic Linguistic Ordered Weighted Geometric (ILOWG) operator, and the Intuitionistic Linguistic Hybrid Geometric (ILHG) operator were defined. Building on this foundation, Liu [7] contributed to the theoretical framework with the introduction of the Intuitionistic Linguistic Generalized Weighted Average (ILGWA) operator, the Intuitionistic Linguistic Generalized Dependent Ordered Weighted Average (ILGDOWA) operator, and the Intuitionistic Linguistic Generalized Dependent Hybrid Weighted Average (ILGDHWA) operator. Following this, Liu and Wang [8] expanded the scope by introducing the Intuitionistic Linguistic Power Generalized Weighted Average (ILPGWA) operator and the Intuitionistic Linguistic Power Generalized Ordered Weighted Average (ILPGOWA) operator, providing more nuanced tools for handling complex intuitionistic linguistic information. To better reflect the interrelationships between attributes during data integration, Liu et al. [9] proposed the intuitionistic linguistic Bonferroni mean (ILBM) operator and the intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator. Liu and Li [10] extended the partitioned Heronian mean (PHM) operator to the intuitionistic linguistic environment, proposing the intuitionistic linguistic partitioned Heronian mean (ILPHM) operator and the intuitionistic linguistic weighted partitioned Heronian mean (ILWPHM) operator. In optimization problems, the optimal choice may change with the decision-makers’ risk attitudes. Gao et al. [11] defined an intuitionistic linguistic generalized ordered weighted utility averaging-hyperbolic absolute risk aversion (IL-GOWUA-HARA) operator to represent DMs’ risk preferences.

The distance measurement between ILNs significantly compares the information carried by ILNs. Wang and Li [4] defined the Hamming distance between two ILNs. Su et al. [12] extended this to the normalized Hamming distance between two intuitionistic linguistic sets. They combined the normalized Hamming distance with the ordered weighted averaging (OWA) operator and proposed the intuitionistic linguistic ordered weighted averaging distance (ILOWAD) operator. Liu et al. [13] combined the normalized Hamming distance with the OWA weighted average (OWAWA) operator and proposed the intuitionistic linguistic OWAWA (ILOWAWA) operator. Based on linguistic scale functions, Yu et al. [14] defined new comparison and operational rules for ILNs and the generalized distance between any two ILNs. They also proposed the intuitionistic linguistic TODIM evaluation method based on the generalized distance.

The comparison methods for ILNs have also attracted scholars’ research interest, as these methods can transform ILNs into precise numbers, thereby determining the ranking of alternatives. Wang and Li [4] defined the compromise expected value of ILNs and, based on this compromise expected value and the relationship between membership and non-membership degrees, proposed the score function and accuracy function for ILNs. Moreover, various types of score and accuracy functions for ILNs have been proposed from different perspectives in the literature [4,5,6,7,14]. However, these comparison methods have shortcomings; in some cases, they cannot distinguish between different ILNs or result in unreasonable rankings. This paper proposes a new comparison method for ILNs to improve existing methods’ deficiencies.

Scholars have also focused on integrating ILNs with other fuzzy concepts. For example, Zhou et al. [15] proposed intuitionistic hesitant linguistic numbers. Ashraf et al. [16] proposed picture fuzzy linguistic numbers. Liu et al. [17] introduced intuitionistic linguistic rough numbers. Guo et al. [18] proposed intuitionistic grey linguistic numbers.

Keshavarz Ghorabaee et al. [19,20,21] proposed the Evaluation based on Distance from Average Solution (EDAS) method. The EDAS method aims to determine the positive and negative distances of each decision alternative from the average solution. By integrating these two distance values, a comprehensive evaluation value for each decision alternative is obtained, which is then used to rank the alternatives. The stability, effectiveness, and simplicity of the calculation process of this method have led to its rapid development in recent years. Some studies have extended the EDAS method used in multi-attribute decision-making to the domain of fuzzy numbers. They collect evaluation information using fuzzy numbers and integrate the calculation rules of fuzzy numbers with the logic of the EDAS method to ultimately select the optimal solution. For example, in the interval-valued neutrosophic environment [22], in the interval-valued intuitionistic trapezoidal fuzzy set environment [23], in the 2-tuple Linguistic Pythagorean fuzzy set environment [24], and the probabilistic linguistic term set environment [25], the evaluation of online live course platforms in the D numbers environment [26], and the selection of remote education video conferencing tools in the spherical fuzzy sets environment [27], and so forth. Torkayesh et al. [28] conducted a literature review on the development and application scenarios of the EDAS method.

The combination of fuzzy numbers and multi-attribute decision methods has also been used in many educational fields, such as evaluations of e-learning platforms [29], information literacy assessment of teachers [30], teaching audit and evaluation [31,32], determining the best teaching method [33], evaluation of Open and Distance Education Websites [34].

In recent years, many universities worldwide have begun transitioning to an education model driven by innovation and entrepreneurship, making the evaluation of such education crucial. To assess the overall outcomes of university innovation and entrepreneurship education, Vesper and Gartner [35] developed the “Seven-Factor Evaluation Method”. This method evaluates universities’ entrepreneurial education across seven dimensions: the availability of entrepreneurship courses, the publication of entrepreneurial textbooks or monographs by faculty, the social impact of the university, alumni participation in innovation and entrepreneurship education, alumni opting for self-employment, the innovativeness of alumni entrepreneurial projects, and the entrepreneurial extension activities of university scholars. For evaluating entrepreneurship courses, Fayolle et al. [36], based on the Theory of Planned Behavior, proposed two paradigms: process factor evaluation and impact evaluation. The CIPP model, proposed by Stufflebeam [37], is a classic method in process evaluation, emphasizing dynamic evaluation during program implementation and characterized by its systematic, comprehensive, and operational features. Ge and Liu [38] introduced the CIPP educational evaluation model into the study of entrepreneurship education capability evaluation. ILNs can precisely represent expert opinions, while the EDAS method demonstrates advantages in selecting alternatives with conflicting attributes. In addressing a multi-stage evaluation problem, we can integrate the strengths of various methods at each stage to achieve the optimal solution. Hence, this study combines ILNs and the EDAS method to construct a novel multi-stage, multi-attribute decision-making method, termed the IL-EDAS method, and validates it by evaluating university innovation and entrepreneurship education.

The structure of this paper is organized as follows: Section 2 briefly reviews the relevant definitions and theorems of ILNs. In Section 3, we propose a new score function for ILNs to overcome the deficiencies of other definitions. Section 4 presents the methodology and computational steps of the proposed IL-EDAS method. Section 5 provides a specific case study on the evaluation of innovation and entrepreneurship education in universities to verify the feasibility of the proposed method. Section 6 concludes the paper.

2. Preliminaries

In this section, we introduce some basic concepts, such as Linguistic term sets (LTSs), linguistic scale functions (LSF), and so on.

2.1. Linguistic Term Sets (LTSs)

In some multi-attribute group decision-making environments, researchers have found that using qualitative evaluation methods to handle uncertain information is often more flexible and effective than quantitative evaluation methods. For example, when experts express the abilities of a candidate, they usually use natural language terms such as “good”, “average”, and “poor” rather than precise numerical values. Therefore, to standardize linguistic evaluations, Zadeh [1] proposed the LTSs concept.

Definition 1 

([1,39]). Suppose $S=\left\{s_t|t=0,1,\cdots ,2T\right\}$ is a linguistic term set, where $s_t$ is a linguistic variable, and it has the following properties:

(1)

Orderliness: If $\alpha >\beta$ , then $s_{\alpha }>s_{\beta }$;

(2)

Negation operator: $neg(s_{\alpha })>s_{2T-\alpha }$;

where $2T+1$ is the granularity of the linguistic term set. For example, if $T=2$, the granularity of the linguistic term set is 5, and the linguistic term set is defined as $S=\{s_0=very poor, s_1=poor, s_2=medium,s_3=good, s_4=very good\}$.

In addition, to preserve the original decision information as much as possible, Xu [40] extended the discrete linguistic term set to a continuous form  $\overline{S}=\left\{s_t|t\in [0, 2T]\right\}$. If $s_{\alpha }\notin S$, then $s_{\alpha }$ is a linguistic term generated during the operation and has no actual meaning. To convert the semantic values contained in linguistic terms into numerical values, Wang et al. [41] defined linguistic scale function (LSF).

Definition 2 

([41]). Suppose $S=\left\{s_t|t=0,1,\cdots ,2T\right\}$  is a linguistic term set, ${\sigma }_i$ is a non-negative real number. Then the LSF $f$, which conducts the mapping from $s_i$ to  ${\sigma }_i$ $(i=0,1,\cdots ,2T)$, can be defined as follows:

$$f: s_{i }\to {\sigma }_i (i=0,1,\cdots ,2T)$$

(1)

where $0\le {\sigma }_0{<\sigma }_1<\cdots <{\sigma }_{2T}$, meaning the function $f$ is monotonically increasing with respect to the indices of $s_{i }$.

A representative linguistic scale function is $f\left(s_i\right)={\sigma }_i={\displaystyle \frac{i}{2T}}$, where $i=0,1,\cdots ,2T$ and ${\sigma }_i\in [0,1]$.

2.2. Intuitionistic Fuzzy Sets (IFSs)

Zadeh [42] proposed the concept of fuzzy numbers to better handle information’s fuzziness and uncertainty. Atanassov [3] extended the traditional fuzzy set and proposed the IFSs, which simultaneously consider the membership degree, non-membership degree, and hesitation degree of uncertain information, thereby more effectively managing information uncertainty.

Definition 3 

([3]). Suppose $X$ is a non-empty set, then the set

$$A=\{\langle x,{\mu }_A(x),{\nu }_A(x)\rangle \left|x\in X\right\}$$

(2)

is an intuitionistic fuzzy set, where, the functions ${\mu }_A\left(x\right):X\to [0,1]$, ${\nu }_A\left(x\right):X\to [0,1]$ satisfy ${0\le \mu }_A\left(x\right)+{\nu }_A\left(x\right)\le 1, \forall x\in X$. Here, ${\mu }_A\left(x\right) and {\nu }_A\left(x\right)$ represent the membership degree and non-membership degree of element $x\in X$. Additionally, ${{\pi }_A\left(x\right)=1-\mu }_A\left(x\right)-{\nu }_A\left(x\right)$ is referred to as the hesitation degree of element $x\in X$.

To facilitate the expression of decision information, Xu [43] defined an intuitionistic fuzzy number (IFN) as $\alpha =〈{\mu }_{\alpha }\left(x\right),{\nu }_{\alpha }\left(x\right)〉$, where ${\mu }_{\alpha }\left(x\right)\in [0,1]$, ${\nu }_{\alpha }\left(x\right)\in [0,1]$, ${0\le \mu }_{\alpha }\left(x\right)+{\nu }_{\alpha }\left(x\right)\le 1$.

2.3. The Intuitionistic Linguistic Sets (ILSs)

By integrating LTSs with IFSs, Wang and Li [4] introduced ILSs, thereby enhancing the comprehensiveness of information representation.

Definition 4 

([4]). Let $s_{\theta (x)}\in S$, $X$ be a nonempty set of the universe, $S=\left\{s_{\alpha }|\alpha \in [0, 2T]\right\}$ be a continuous linguistic term set (LTS), then the intuitionistic linguistic set (ILS) in $X$ is proposed as follows:

$$A=\left\{〈x,{[s_{\theta \left(x\right)},\mu }_A\left(x\right),{\nu }_A\left(x\right)]〉|x\in X\right\}$$

(3)

where ${\mu }_A\left(x\right):X\to [0,1]$, ${\nu }_A\left(x\right):X\to [0,1]$ respectively represent the degree to which an element belongs to and does not belong to the linguistic evaluation value $s_{\theta (x)}$, satisfying ${0\le \mu }_A\left(x\right)+{\nu }_A\left(x\right)\le 1, \forall x\in X$. The value ${{\pi }_A\left(x\right)=1-\mu }_A\left(x\right)-{\nu }_A\left(x\right)$ represents the hesitation or uncertainty regarding the element $x$ belonging to the linguistic evaluation value $s_{\theta (x)}$.

For example, if an evaluator considers the importance $s_3$ of an indicator $x$ with a certainty degree of $0.85$, a non-certainty degree of $0.1$, and an uncertainty degree of $0.05$, this evaluation can be represented as an intuitionistic linguistic number (ILN) $〈s_3,0.85,0.1〉$.

The operations of ILNs based on LSF are defined as follows:

Definition 5 

([14]). For any two ILNs ${\beta }_1=〈s_{\theta \left({\beta }_1\right)}, \mu \left({\beta }_1\right), \nu \left({\beta }_1\right)〉$ and ${\beta }_2=〈s_{\theta \left({\beta }_2\right)}, \mu \left({\beta }_2\right), \nu \left({\beta }_2\right)〉$,  and $f$ and $f^{-1}$ an LSF and its inverse function respectively. For convenience, let $f_1=f(s_{\theta \left({\beta }_1\right)})$ and $f_2=f(s_{\theta \left({\beta }_2\right)})$.

Then the operational laws are defined as follows:

$${neg(\beta }_1)=〈{f^{-1}(f(s}_{2T})-f_1), \nu \left({\beta }_1\right), \mu \left({\beta }_1\right)〉;$$

$${\beta }_1⨁{\beta }_2=\langle f^{-1}(f_1+f_2), {\displaystyle \frac{f_1\mu \left({\beta }_1\right)+f_2\mu \left({\beta }_2\right)}{f_1+f_2}}, {\displaystyle \frac{f_1\nu \left({\beta }_1\right)+f_2\nu \left({\beta }_2\right)}{f_1+f_2}}\rangle ;$$

$${\beta }_1⨂{\beta }_2=〈f^{-1}(f_1{f}_2), \mu \left({\beta }_1\right)\mu \left({\beta }_2\right),\nu \left({\beta }_1\right)\nu \left({\beta }_2\right)〉;$$

$${\lambda \beta }_1=〈f^{-1}(\lambda f_1), \mu \left({\beta }_1\right), \nu \left({\beta }_1\right)〉, \lambda \ge 0;$$

$${{\beta }_1}^{\lambda }=〈f^{-1}({\left(f_1\right)}^{\lambda }){, (\mu \left({\beta }_1\right))}^{\lambda }, {(\nu \left({\beta }_1\right))}^{\lambda }〉, \lambda \ge 0;$$

Theorem 1 

([14]). For any two ILNs ${\beta }_1=〈s_{\theta \left({\beta }_1\right)}, \mu \left({\beta }_1\right), \nu \left({\beta }_1\right)〉$ and ${\beta }_2=〈s_{\theta \left({\beta }_2\right)}, \mu \left({\beta }_2\right), \nu \left({\beta }_2\right)〉$, then the following properties are true:

(1)

${\beta }_1⨁{\beta }_2={\beta }_2⨁{\beta }_1$;

(2)

${(\beta }_1⨁{\beta }_2)⨁{\beta }_3={{\beta }_1⨁(\beta }_2⨁{\beta }_3)$;

(3)

$\lambda {(\beta }_1⨁{\beta }_2)=\lambda {\beta }_1⨁{\lambda \beta }_2$;

(4)

${\lambda }_1{\beta }_1⨁{\lambda }_2{\beta }_1=\left({\lambda }_1+{\lambda }_2\right){\beta }_1, {\lambda }_1, {\lambda }_2\ge 0$;

For more operational rules and detailed discussions, please refer to [14].

3. New Comparison Method for ILNs

Several scholars have proposed comparison methods for intuitionistic linguistic numbers. Representative methods are as follows:

For an ILN  $\beta =〈s_{\theta \left(\beta \right)}, \mu \left(\beta \right), \nu \left(\beta \right)〉$, the score function $S(\beta )$ and the accuracy function $H(\beta )$ of $\beta$ are defined in

Table 1. The definitions of the score function $S(\beta )$ and the accuracy function $H(\beta )$.

$\mathit{S}(\mathit{\beta })$	$\mathit{H}(\mathit{\beta })$
$S_1(\beta )={\displaystyle \frac{\theta \left(\beta \right)(1+\mu \left(\beta \right)-\nu \left(\beta \right))}{2}}·(\mu \left(\beta \right)-\nu \left(\beta \right))$	$H_1(\beta )={\displaystyle \frac{\theta \left(\beta \right)\left(1+\mu \left(\beta \right)-\nu \left(\beta \right)\right)}{2}}·(\mu \left(\beta \right)+\nu \left(\beta \right))$	[4]
$S_2\left(\beta \right)={\displaystyle \frac{\theta \left(\beta \right)}{2T}}·{\displaystyle \frac{1+\mu \left(\beta \right)-\nu \left(\beta \right)}{2}}$	$H_2(\beta )={\displaystyle \frac{\theta \left(\beta \right)}{2T}}·(\mu \left(\beta \right)+\nu \left(\beta \right))$	[7]
$S_3(\beta )=\theta \left(\beta \right)·(\mu \left(\beta \right)-\nu \left(\beta \right))$	$H_3(\beta )=\theta \left(\beta \right)·(\mu \left(\beta \right)+\nu \left(\beta \right))$	[5]
$S_4(\beta )=\theta \left(\beta \right)·(1+\mu \left(\beta \right)-\nu \left(\beta \right))$	$H_4(\beta )=\theta \left(\beta \right)·(1-\mu \left(\beta \right)-\nu \left(\beta \right))$	[6]

.

The order relationship for any two ILNs ${\beta }_1$ and ${\beta }_2$ be defined as follows [4,5,6,7]:

If $S\left({\beta }_1\right)>S\left({\beta }_2\right)$, then ${\beta }_1>{\beta }_2$;

If $S\left({\beta }_1\right)=S\left({\beta }_2\right)$, then

If $H\left({\beta }_1\right)>H\left({\beta }_2\right), {\beta }_1>{\beta }_{2 }$;

If $H\left({\beta }_1\right)=H\left({\beta }_2\right), {\beta }_1={\beta }_2$.

Yu et al. [14] pointed out that in the above definitions, the impact of membership degree and non-membership degree exceeds that of linguistic terms. For example, ${\beta }_1=〈s_1, 0.8, 0.1〉$ and ${\beta }_2=〈s_6, 0.1, 0.9〉$, according to the above definition, both have ${\beta }_1>{\beta }_{2 }$ which is obviously counterintuitive. For a more detailed discussion, please refer to [14].

To enhance the impact of linguistic terms and overcome the deficiencies of the above definitions, Yu et al. [14] proposed the following score function and accuracy function based on linguistic scale functions:

Definition 6

([14]). Let ${\beta }_i=〈s_{\theta \left({\beta }_i\right)}, \mu \left({\beta }_i\right), \nu \left({\beta }_i\right)〉 (i=1, 2,\cdots ,n)$ be a collection of ILNs. When $d=\underset{i,j=1,2,\cdots ,n}{\mathit{max}}\left\{\left|\theta \left({\beta }_i\right)-\theta \left({\beta }_j\right)\right|\right\}$, the score function $S({\beta }_i)$ and the accuracy function $H({\beta }_i)$ of ${\beta }_i$ are defined as follows:

$$S({\beta }_i)={(f_i^{*})}^d·(1+{\displaystyle \frac{1+\mu \left({\beta }_i\right)-\nu \left({\beta }_i\right)}{2}}),$$

$$H({\beta }_i)={(f_i^{*})}^d·(\mu \left({\beta }_i\right)+\nu \left({\beta }_i\right))$$

where $f_i^{*}=f^{*}(s_{\theta \left({\beta }_i\right)})$.

The order relationship for any two ILNs ${\beta }_1$ and ${\beta }_2$ can be defined as follows:

If $S\left({\beta }_1\right)>S\left({\beta }_2\right)$, then ${\beta }_1>{\beta }_2$;

If $S\left({\beta }_1\right)=S\left({\beta }_2\right)$, then

If $H\left({\beta }_1\right)>H\left({\beta }_2\right), {\beta }_1>{\beta }_{2 }$;

If $H\left({\beta }_1\right)=H\left({\beta }_2\right), {\beta }_1={\beta }_2$.

We found that the above comparison rules for ILNs might still yield counterintuitive results. Here, we provide an example.

Example 1.

Let LTS $S=\left\{s_t|t=0,1,\cdots ,2T\right\}=\{s_0,s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8\}$ and the linguistic scale function is $f\left(s_i\right)={\displaystyle \frac{i}{8}} ,$ for $i=0,1,2,\cdots ,8$. Suppose ${\beta }_1=〈s_7, 0.45, 0〉$, ${\beta }_2=〈s_8, 0.5, 0.5〉$ and ${\beta }_3=〈s_8, 0.2, 0.1〉$. Then $d=1$, $S\left({\beta }_1\right)=1.5094$, $S\left({\beta }_2\right)=1.5$, and $S\left({\beta }_3\right)=1.55$. Therefore, ${\beta }_3>{\beta }_{1 }>{\beta }_2$. This result is inconsistent with our intuition. Intuitively, when we choose a stronger evaluation term, it is accompanied by a higher intuitionistic judgment score, because the linguistic term scores of ILNs ${\beta }_2$ and ${\beta }_3 are both s_8$, Therefore, in the evaluator’s mind, their scores should not be lower than ${\beta }_1$. In this case, a more effective comparison method of ILNs needs to be proposed.

Definition 7.

For an ILN $\beta =〈s_{\theta \left(\beta \right)}, \mu \left(\beta \right), \nu \left(\beta \right)〉$, the score function and accuracy function of $\beta$  are defined as follows:

$$S(\beta )=\left\{\begin{array}{r}{\displaystyle \frac{f\left(s_{\theta \left(\beta \right)}\right)+f\left(s_{\theta \left(\beta \right)-1}\right)}{2}}+{\displaystyle \frac{(\mu \left(\beta \right)-\nu \left(\beta \right))·(f\left(s_{\theta \left(\beta \right)}\right)-f\left(s_{\theta \left(\beta \right)-1}\right))}{2}}, \theta \left(\beta \right)\ge 1\\ {\displaystyle \frac{f\left(s_{\theta \left(\beta \right)}\right)}{2}}+{\displaystyle \frac{(\mu \left(\beta \right)-\nu \left(\beta \right))·f\left(s_{\theta \left(\beta \right)}\right)}{2}}, \theta \left(\beta \right)<1\end{array}\right.$$

(4)

$$H(\beta )=f\left(s_{\theta \left(\beta \right)}\right)·(\mu \left(\beta \right)+\nu \left(\beta \right)),$$

(5)

The order relationship for any two ILNs ${\beta }_1$  and ${\beta }_2$  can be defined as follows:

If $S\left({\beta }_1\right)>S\left({\beta }_2\right)$, then ${\beta }_1>{\beta }_2;$

If $S\left({\beta }_1\right)=S\left({\beta }_2\right)$, then

If $H\left({\beta }_1\right)>H\left({\beta }_2\right), {\beta }_1>{\beta }_{2 };$

If $H\left({\beta }_1\right)=H\left({\beta }_2\right), {\beta }_1={\beta }_2$.

In the above definition, we divide the scoring function of ILN $\beta =〈s_{\theta \left(\beta \right)}, \mu \left(\beta \right), \nu \left(\beta \right)〉$ into two parts. The first part is a benchmark score corresponding to its linguistic term, denoted as ${\displaystyle \frac{f\left(s_{\theta \left(\beta \right)}\right)+f\left(s_{\theta \left(\beta \right)-1}\right)}{2}},$ $\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \theta \left(\beta \right)\ge 1$; ${\displaystyle \frac{f\left(s_{\theta \left(\beta \right)}\right)}{2}}$, $\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \theta \left(\beta \right)<1$. The second part involves adjusting this benchmark score based on the difference between support $\mu \left(\beta \right)$ and opposition $\nu \left(\beta \right)$. When the support degree of ILN $\beta$ is greater than the opposition degree, the benchmark score corresponding to the linguistic term receives a positive incentive. Conversely, when the support degree is less than the opposition degree, the benchmark score corresponding to the linguistic term incurs a negative penalty.

Let’s use this method to operate Example 1. For ${\beta }_1=〈s_7, 0.45, 0〉$, ${\beta }_2=〈s_8, 0.5, 0.5〉$ and ${\beta }_3=〈s_8, 0.2, 0.1〉$, $S\left({\beta }_1\right)=0.8406$, $S\left({\beta }_2\right)=0.9375$, and $S\left({\beta }_3\right)=0.9438$. Therefore, ${\beta }_3>{\beta }_{2 }>{\beta }_1$.

Compared to the comparison rule results ${\beta }_3>{\beta }_{1 }>{\beta }_2$ of [14], ${\beta }_3$ still has the highest score. Although ${\beta }_{2 }$ and ${\beta }_3$ have the same linguistic term $s_8$, the difference between membership and non-membership degrees is greater for ${\beta }_3$, while ${\beta }_1$ has the lowest score. Thus, this result is relatively more reasonable.

4. The EDAS Model with Intuitionistic Linguistic Information

In this section, we propose the EDAS method under intuitionistic linguistic information. Based on the newly proposed ILNs comparison rules, the EDAS method is used to evaluate and rank various alternatives.

The EDAS method, introduced by Keshavarz Ghorabaee et al. [19,20,21] in 2015, addresses multi-attribute decision-making problems with conflicting attributes. This method calculates the average value of each alternative and measures differences from this average as Positive Distance and Negative Distance. The optimal alternative typically has the maximum PDA and minimum NDA. In this section, we apply the EDAS method to decision-making problems in an intuitionistic linguistic context. After obtaining each expert’s intuitionistic linguistic evaluation matrix, we calculate the comprehensive linguistic evaluation matrix and normalize the data. We compute the arithmetic average of each attribute’s evaluations across all alternatives to derive average evaluation results. The proposed score function then calculates each attribute’s positive and negative distances for each alternative from these average results. We derive weighted positive and negative distances for each alternative based on attribute weights. Finally, we standardize the results, rank the alternatives, and identify the optimal choice.

Suppose there are m alternatives $\{A_1, A_2,\cdots ,A_m\}$, n attributes $\{C_1, C_2,\cdots ,C_n\}$ and k experts $\{E_1, E_2,\cdots ,E_k\}$. Let $\{{\omega }_1, {\omega }_2,\cdots ,{\omega }_n\}$ and $\left\{{\theta }_1, {\theta }_2,\cdots ,{\theta }_k\right\}$ be the weighting vector of attributes and experts respectively which satisfy ${\omega }_i\in [0, 1]$, ${\theta }_i\in [0, 1]$ and ${\sum }_{i=1}^n{\omega }_i=1$, ${\sum }_{i=1}^k{\theta }_i=1$.

The IL-EDAS method is implemented through the following steps:

Step 1. Construct the initial evaluation matrix with intuitionistic linguistic information $R^s={\left[{\beta }_{ij}^s\right]}_{m\times n}$, $i=1,2,\cdots ,m, j=1,2,\cdots ,n, s=1,2,\cdots ,k,$ where ${\beta }_{ij}^s$ represents the evaluation of alternative $i$ by expert $\mathrm{s}$ based on attribute $j$.

Step 2: Use the operational rules of intuitionistic linguistic information to calculate the weighted average ${\beta }_{ij}={\oplus }_{s=1}^k{{\theta }_s\beta }_{ij}^s$ of the experts’ evaluations, resulting in the comprehensive intuitionistic linguistic evaluation matrix $R={\left[{\beta }_{ij}\right]}_{m\times n}$.

Step 3: Normalize the comprehensive evaluation matrix $R={\left[{\beta }_{ij}\right]}_{m\times n}$ to obtain the normalized matrix $R^N={\left[{\beta }_{ij}^N\right]}_{m\times n}$.

For benefit attributes:

$${\beta }_{ij}^N={\beta }_{ij}, i=1,2,\cdots ,m,j=1,2,\cdots ,n,$$

(6)

For cost attributes:

$${\beta }_{ij}^N={neg(\beta }_{ij}), i=1,2,\cdots ,m,j=1,2,\cdots ,n,$$

(7)

Step 4: Calculate the average values of all alternatives ${\mathrm{A}}_{\mathrm{i}}$ based on the operational rules of intuitionistic linguistic information to obtain the average value matrix.

$$AV={\left[{AV}_j\right]}_{1\times n}={[{\displaystyle \frac{\sum _{i=1}^m{\beta }_{ij}^N}{m}}]}_{1\times n}.$$

(8)

Step 5: Determine the positive distance (PDA) and negative distance (NDA) matrices of each alternative from the average solution using the newly proposed score function formula Equation (4).

$$PDA={\left[{PDA}_{ij}\right]}_{m\times n}, NDA={\left[{NDA}_{ij}\right]}_{m\times n}.$$

$${PDA}_{ij}={\displaystyle \frac{max(0, (S\left({\beta }_{ij}^N\right)-S({AV}_j)))}{S({AV}_j)}},$$

(9)

$${NDA}_{ij}={\displaystyle \frac{max(0, (S\left({AV}_j\right)-S\left({\beta }_{ij}^N\right)))}{S({AV}_j)}}.$$

(10)

Step 6: Calculate the weighted forward distance (the weighted sum of PDA) ${WSP}_i$ and the reverse weighted distance (the weighted sum of NDA) ${WSN}_i$ of each alternative based on the attribute weight vector $\{{\omega }_1, {\omega }_2,\cdots ,{\omega }_n\}$ and the positive and negative distance matrices obtained in the previous step.

$${WSP}_i={\sum }_{j=1}^n{\omega }_j{PDA}_{ij}, {WSN}_i={\sum }_{j=1}^n{\omega }_j{NDA}_{ij}$$

(11)

Step 7. The results of Equation (11) can be normalized as:

$${WSP}_i^N={\displaystyle \frac{{WSP}_i}{max({WSP}_i)}}, {WSN}_i^N=1-{\displaystyle \frac{{WSN}_i}{\mathit{max}\left({WSN}_i\right) }}.$$

(12)

Step 8: Calculate the final evaluation score ${FES}_i$ based on ${WSP}_i^N$ and ${WSN}_i^N$, and rank the alternatives accordingly ${FES}_i$. The larger the value of ${FES}_i$, the better the alternative.

$${FES}_i={\displaystyle \frac{1}{2}}({WSP}_i^N+{WSN}_i^N).$$

(13)

5. Empirical Example and Comparative Analysis

5.1. Empirical Example: Ranking Entrepreneurship Education Ability of Universities

Ge and Liu [38] introduced the CIPP educational evaluation model to entrepreneurship education ability evaluation. They constructed an evaluation index system for the innovation and entrepreneurship education ability of universities from four aspects: context evaluation, input evaluation, process evaluation, and product evaluation. Therefore, in this section, the intuitionistic linguistic EDAS method will be used to provide an effective solution for evaluating universities’ innovation and entrepreneurship education ability based on the evaluation index system of Ge and Liu [38]. Explanations for the criteria considered in the study are given in

Table 2. The evaluation criteria of Ge and Liu [38].

CIPP	Criteria	Type	Description
Context evaluation	Entrepreneurial Environment Fundamental Ability ${\mathrm{C}}_1$	Benefit	It involves a regional environment, knowledge foundation, and technological foundation.
Input evaluation	Entrepreneurial Resource Allocation Ability ${\mathrm{C}}_2$	Benefit	It involves faculty input, funding input, and organizational support.
Process evaluation	Entrepreneurial Process Action Ability ${\mathrm{C}}_3$	Benefit	It involves entrepreneurship courses, entrepreneurship projects, and practical platforms.
Product evaluation	Entrepreneurial Outcome Performance Ability ${\mathrm{C}}_4$	Benefit	It involves quality improvement, entrepreneurship outcomes, and social benefits.

.

There are five universities $\{A_1, A_2,\cdots ,A_5\}$ to be evaluated with four criteria $\{C_1, C_2,C_3,C_4\}$ by three experts $\{E_1, E_2,E_3\}$ (whose weighting vector $\theta =(0.4, 0.3, 0.3))$. A brief description of each criterion is given in Table 1. Assume that the weight vector of these criteria is $\omega =(0.22, 0.25, 0.23, 0.30)$. Let S = $\left\{s_t|t=0,1,\cdots ,2T\right\}$ = $\left\{s_0,s_1,s_2,s_3,s_4,s_5,s_6\right\}$ = $\{extremely poor, very poor, poor, medium, good, very good, extremely good\}$ be a given LTS in which the elements are not mutually exclusive. In addition, the linguistic scale function is $f\left(s_i\right)={\displaystyle \frac{i}{6}} ,$ for $i=0,1,2,\cdots ,6$.

Step 1. Construct the evaluation matrix $R^s={\left[{\beta }_{ij}^s\right]}_{m\times n}$, $i=1,2,3,4,5, j=1,2,3,4, s=1,2,3$. The evaluation matrix is listed in

Table 3. Intuitionistic linguistic evaluation information by $E_1$.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$
${ A}_1$	$〈s_4, 0.6, 0.3〉$	$〈s_5, 0.7, 0.2〉$	$〈s_4, 0.9, 0.1〉$	$〈s_5, 0.5, 0.4〉$
${ A}_2$	$〈s_6, 0.5, 0.4〉$	$〈s_5, 0.6, 0.4〉$	$〈s_5, 0.5, 0.4〉$	$〈s_5, 0.8, 0.1〉$
${ A}_3$	$〈s_4, 0.5, 0.4〉$	$〈s_4, 0.7, 0.3〉$	$〈s_5, 0.7, 0.1〉$	$〈s_4, 0.8, 0.2〉$
${ A}_4$	$〈s_5, 0.6, 0.3〉$	$〈s_6, 0.5, 0.4〉$	$〈s_4, 0.6, 0.3〉$	$〈s_5, 0.7, 0.1〉$
${ A}_5$	$〈s_5, 0.7, 0.2〉$	$〈s_5, 0.8, 0.1〉$	$〈s_4, 0.5, 0.5〉$	$〈s_4, 0.8, 0.1〉$

,

Table 4. Intuitionistic linguistic evaluation information by ${ E}_2$.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$
${ A}_1$	$〈s_5, 0.7, 0.3〉$	$〈s_4, 0.9, 0.1〉$	$〈s_6, 0.7, 0.2〉$	$〈s_4, 0.6, 0.4〉$
${ A}_2$	$〈s_4, 0.9, 0.1〉$	$〈s_4, 0.8, 0.2〉$	$〈s_4, 0.9, 0.1〉$	$〈s_5, 0.6, 0.4〉$
${ A}_3$	$〈s_5, 0.9, 0.1〉$	$〈s_4, 0.8, 0.1〉$	$〈s_5, 0.7, 0.3〉$	$〈s_5, 0.6, 0.2〉$
${ A}_4$	$〈s_6, 0.6, 0.4〉$	$〈s_4, 0.6, 0.4〉$	$〈s_4, 0.8, 0.2〉$	$〈s_6, 0.6, 0.3〉$
${ A}_5$	$〈s_4, 0.7, 0.3〉$	$〈s_6, 0.7, 0.3〉$	$〈s_5, 0.8, 0.1〉$	$〈s_4, 0.7, 0.3〉$

and

Table 5. Intuitionistic linguistic evaluation information by ${ E}_3$.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$
${ A}_1$	$〈s_5, 0.6, 0.3〉$	$〈s_5, 0.8, 0.1〉$	$〈s_5, 0.8, 0.1〉$	$〈s_5, 0.8, 0.1〉$
${ A}_2$	$〈s_5, 0.7, 0.2〉$	$〈s_5, 0.8, 0.2〉$	$〈s_5, 0.7, 0.3〉$	$〈s_4, 0.8, 0.1〉$
${ A}_3$	$〈s_4, 0.7, 0.3〉$	$〈s_4, 0.7, 0.3〉$	$〈s_6, 0.6, 0.3〉$	$〈s_5, 0.6, 0.4〉$
${ A}_4$	$〈s_5, 0.7, 0.3〉$	$〈s_4, 0.8, 0.1〉$	$〈s_4, 0.8, 0.1〉$	$〈s_5, 0.8, 0.1〉$
${ A}_5$	$〈s_5, 0.8, 0.1〉$	$〈s_5, 0.7, 0.2〉$	$〈s_5, 0.7, 0.3〉$	$〈s_5, 0.6, 0.4〉$

.

Step 2. Use the operational rules of intuitionistic linguistic information to calculate the weighted average of the experts’ evaluations ${\beta }_{ij}={\oplus }_{s=1}^3{{\theta }_s\beta }_{ij}^s$, resulting in the comprehensive intuitionistic linguistic evaluation matrix $R={\left[{\beta }_{ij}\right]}_{5\times 4}$ (

Table 6. The fused values by using a weighted average.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$
${ A}_1$	$〈s_{4.6}, 0.6326, 0.300〉$	$〈s_{4.7}, 0.7830, 0.1426〉$	$〈s_{4.9}, 0.7960, 0.1367〉$	$〈s_{4.7}, 0.6213, 0.3043〉$
${ A}_2$	$〈s_{5.1}, 0.6529, 0.2706〉$	$〈s_{4.7}, 0.7149, 0.2851〉$	$〈s_{4.7}, 0.6660, 0.2915〉$	$〈s_{4.7}, 0.7362, 0.1957〉$
${ A}_3$	$〈s_{4.3}, 0.6953, 0.2674〉$	$〈s_4, 0.7300, 0.2400〉$	$〈s_{5.3}, 0.6660, 0.2245〉$	$〈s_{4.6}, 0.6696, 0.2652〉$
${ A}_4$	$〈s_{5.3}, 0.6283, 0.3340〉$	$〈s_{4.8}, 0.6000, 0.3250〉$	$〈s_4, 0.7200, 0.2100〉$	$〈s_{5.3}, 0.6943, 0.1679〉$
${ A}_5$	$〈s_{4.7}, 0.7319, 0.1936〉$	$〈s_{5.3}, 0.7377, 0.1962〉$	$〈s_{4.6}, 0.6630, 0.3043〉$	$〈s_{4.3}, 0.7023, 0.2605〉$

).

Step 3. Normalize the comprehensive evaluation matrix $R={\left[{\beta }_{ij}\right]}_{5\times 4}$ to obtain the matrix $R^N={\left[{\beta }_{ij}^N\right]}_{5\times 4}$. For all attributes that are beneficial, normalization is not needed.

Step 4. According to Table 6, we can obtain the value of average value matrix $AV={\left[{AV}_j\right]}_{1\times 4}={[{\displaystyle \frac{\sum _{i=1}^m{\beta }_{ij}^N}{m}}]}_{1\times 4}$ of the four attributes:

$${ AV}_{1\times 4}=\left(〈s_{4.8}, 0.6666, 0.2746〉, 〈s_{4.7}, 0.7128, 0.2370〉,〈s_{4.7}, 0.7017, 0.2327〉,〈s_{4.72}, 0.6848, 0.2364〉\right)$$

(14)

Step 5. Determine the positive distance (PDA) and negative distance (NDA) matrices of each alternative from the average solution using the score function formula Equation (4).

$$PDA=\left(\begin{array}{cccc}0& 0.0185& 0.0666& 0\\ 0.0657& 0& 0& 0.0058\\ 0& 0& 0.1322& 0\\ 0.1004& 0& 0& 0.1393\\ 0& 0.1425& 0& 0\end{array}\right)$$

$$NDA=\left(\begin{array}{cccc}0.0510& 0& 0& 0.0193\\ 0& 0.0053& 0.0107& 0\\ 0.1072& 0.1561& 0& 0.0320\\ 0& 0.0001& 0.1533& 0\\ 0.0059& 0& 0.0350& 0.0952\end{array}\right)$$

Step 6. Calculate the values of ${ WSP}_i$ and ${ WSN}_i$ (see

Table 7. WSP and WSN in five universities.

	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$
$WSP$	0.0199	0.0162	0.0304	0.0639	0.0356
$WSN$	0.0170	0.0038	0.0722	0.0353	0.0379

) according to Equation (11):

Step 7. The results of Table 7 can be normalized as follows (see

Table 8. ${WSP}^N$ and ${WSN}^N$ in five universities.

	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$
${WSP}^N$	0.3114	0.2535	0.4757	1	0.5571
${WSN}^N$	0.7645	0.9473	0	0.5111	0.4751

) according to Equation (12):

Step 8. Calculate the Final Evaluation Score ${ FES}_i$ according to Equation (13). Rank the alternatives based on ${ FES}_i$. The larger the value of ${ FES}_i$, the better the alternative.

The Final Evaluation Score:

$${ FES}_1=0.5380, { FES}_2=0.6004, { FES}_3=0.2379, { FES}_4=0.7556, { FES}_5=0.5161$$

Thereby, we consider that university $A_4\succ { A}_2\succ { A}_1\succ { A}_5\succ { A}_3$, in which the symbol $‘\succ ’$ means ‘superior’.

5.2. Comparative Analysis

5.2.1. Using the IL-WAA Operator to Solve the Case

Now we compare our proposed IL-EDAS method with the IL-WAA operator [4]. According to the results of Table 6 and attributes weighting vector $\omega =(0.22, 0.25, 0.23, 0.30)$, we can obtain the final intuitionistic fuzzy evaluation number for each alternative in

Table 9. The score function value matrix corresponds to the comprehensive evaluation matrix.

	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$
IL-WAA [4]	$〈s_{4.724}, 0.7056, 0.2232〉$	$〈s_{4.788}, 0.6956, 0.2568〉$	$〈s_{4.545}, 0.6873, 0.2492〉$	$〈s_{4.876}, 0.6602, 0.2542〉$	$〈s_{4.707}, 0.7099, 0.2375〉$

.

${ S(A}_1)=0.7442, { S(A}_2)=0.7512, { S(A}_3)=0.7107, { S(A}_4)=0.7632, { S(A}_5)=0.7405$

Thus, the ranking order is ${ A}_4\succ { A}_2\succ { A}_1\succ { A}_5\succ { A}_3$.

5.2.2. Using the TOPSIS Method to Solve the Case

TOPSIS [44] is a ranking method that calculates the proximity to the ideal solution. The basic idea is to rank a finite number of alternatives based on their closeness to the optimal solution, providing a relative evaluation of their advantages and disadvantages. Due to its simple principle and practical convenience, it has been applied in multiple fields. According to the results of Table 6, we use the score function formula Equation (4) to obtain the score matrix corresponding to the comprehensive evaluation matrix:

$$SC={\left[s_{ij}\right]}_{5\times 4}=\left(\begin{array}{cccc}0.7111& 0.7534& 0.7883& 0.7264\\ 0.7985& 0.7358& 0.7312& 0.7450\\ 0.6690& 0.6242& 0.8368& 0.7170\\ 0.8245& 0.7396& 0.6258& 0.8439\\ 0.7449& 0.8451& 0.7132& 0.6702\end{array}\right)$$

Since all indicators are maximization indicators, we normalize them using the formula $b_{ij}={\displaystyle \frac{s_{ij}-min(s_{ij})}{\mathit{max}\left(s_{ij}\right)-min(s_{ij})}}$ to obtain the normalized score matrix:

$$B={\left[b_{ij}\right]}_{5\times 4}=\left(\begin{array}{cccc}0.3934& 0.5849& 0.7429& 0.4627\\ 0.7890& 0.5052& 0.4844& 0.5469\\ 0.2028& 0& 0.9624& 0.4201\\ 0.9067& 0.5224& 0.0072& 0.9946\\ 0.5464& 1& 0.4038& 0.2082\end{array}\right)$$

Multiply the attribute weights $\omega =(0.22, 0.25, 0.23, 0.30)$ with the corresponding normalized score matrix $B={\left[b_{ij}\right]}_{5\times 4}$ to obtain the weighted matrix $C={\left[c_{ij}\right]}_{5\times 4}$, where $c_{ij}={\omega }_j{b}_{ij}$.

$$C={\left[c_{ij}\right]}_{5\times 4}=\left(\begin{array}{cccc}0.0865& 0.1462& 0.1709& 0.1388\\ 0.1736& 0.1263& 0.1114& 0.1641\\ 0.0446& 0& 0.2214& 0.1260\\ 0.1995& 0.1306& 0.0017& 0.2984\\ 0.1202& 0.2500& 0.0929& 0.0625\end{array}\right)$$

Determine the positive ideal solution $X^{+}$ and the negative ideal solution $X^{-}$. The calculation results are:

$$X^{+}=\left\{\underset{i}{\mathit{max}}{c}_{i1},\underset{i}{\mathit{max}}{c}_{i2},\underset{i}{\mathit{max}}{c}_{i3},\underset{i}{\mathit{max}}{c}_{i4}\right\}=\{0.1995, 0.2500, 0.2214, 0.2984\}$$

$$X^{-}=\left\{\underset{i}{\mathit{min}}{c}_{i1},\underset{i}{\mathit{min}}{c}_{i2},\underset{i}{\mathit{min}}{c}_{i3},\underset{i}{\mathit{min}}{c}_{i4}\right\}=\{0.0446, 0, 0.0017, 0.0625\}$$

Calculate the Euclidean distance between each evaluation sample and the positive and negative ideal solutions using formulas Equations (15) and (16):

$$D_i^{+}=\sqrt{{\sum }_{j=1}^4{(c_{ij}-X_j^{+})}^2}$$

(15)

$$D_i^{-}=\sqrt{{\sum }_{j=1}^4{(c_{ij}-X_j^{-})}^2}$$

(16)

Calculate the closeness coefficient using formula Equation (17). Rank the alternatives based on the closeness coefficient, where a higher value indicates a better alternative. The obtained results are presented in

Table 10. The closeness coefficient in five universities.

	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$
$D_i^{+}$	0.2271	0.2147	0.3409	0.2500	0.2801
$D_i^{-}$	0.2400	0.2344	0.2287	0.3110	0.2766
$D_i$	0.5138	0.5219	0.4015	0.5544	0.4969

.

$$D_i={\displaystyle \frac{D_i^{-}}{D_i^{+}+D_i^{-}}}$$

(17)

Thus, the ranking order is ${ A}_4\succ { A}_2\succ { A}_1\succ { A}_5\succ { A}_3$.

To validate the effectiveness of this method, we compared it with the IL-WAA operator and the TOPSIS method. It can be seen that the ranking results are entirely consistent, and the results obtained by the IL-EDAS method were more sensitive than those obtained by the other two methods (see Figure 1).

ILN not only contains the evaluator’s language item evaluation information but also uses an intuitive fuzzy number to indicate the degree of certainty of the language item. Therefore, it contains more comprehensive information, and defining a reasonable and interpretable score function is significant. The above analysis shows that the ranking results obtained by the IL-WAA operator and IL-TOPSIS methods are the same as those obtained by the IL-EDAS method when ILN collects the evaluation information, which means the score function we defined for ILN is feasible and meaningful. However, when different methods are used, the values of their fractional functions are different. The results obtained by the IL-EDAS method are more sensitive, which is related to the characteristics of these methods:

(1)

The IL-WAA operator method can ultimately retain the language item information and intuitional fuzzy number information of the aggregated ILNs. However, it does not consider the conflicting relationship between the scheme’s attributes.

(2)

The IL-TOPSIS method considers the distance between each scheme and the positive and negative ideal solutions but insufficiently considers the difference between attributes.

(3)

The IL-EDAS method selects the average value of all schemes under each attribute as the decision standard, calculates the positive and negative distances between each scheme and the decision standard, and finally determines the optimal scheme according to the comprehensive situation of the positive and negative distances.

6. Conclusions

This paper proposes a new score function for ILNs combined with the EDAS method to solve multi-criteria decision-making problems with uncertain information. Intuitionistic linguistic information is used to collect expert evaluations, and the extended EDAS method is employed to rank the alternatives. The method is applied to evaluate universities’ innovation and entrepreneurship education ability. The proposed method’s implementation steps and specific application details are thoroughly introduced. To verify the stability and practicality of the method, we compared the results with the IL-WAA operator and the TOPSIS method. The experimental results demonstrate the feasibility of the proposed method.

This method can be applied to other fields, such as AI-enabled evaluations and sustainability assessments of new energy sources. Additionally, intuitionistic linguistics can be further explored in uncertain and incomplete information environments, for instance, by combining it with other fuzzy concepts for better expansion.