An Integrated Decision-Making Model Based on Plithogenic-Neutrosophic Rough Number for Sustainable Financing Enterprise Selection

Abstract

Due to the continuous improvement of people’s awareness of sustainable development, sustainable financing enterprise selection (SFES) has gradually become a hotspot in the field of multi-criteria group decision-making (MCGDM). In the environment of increasing risk factors, how to accurately and objectively select the optimal enterprise for financing is still pending. Thus, this paper proposes an integrated plithogenic-neutrosophic rough number (P-NRN) information aggregation decision model. The model is adapted to group decision-making by taking advantages of plithogenic set operations in handling uncertainty and vagueness and the merit of NRN in eliminating imprecision and subjectivity of decision-makers (DM) in evaluating information boundaries. Then, this paper develops an MCGDM framework based on the weight determination techniques and complex proportional assessment (COPRAS). Moreover, by extending the similarity measure theory and the maximizing deviation method, the weights of DMs and risk criteria are derived, respectively. After obtaining the results of P-NRN information aggregation and weight evaluation, we apply COPRAS to conduct alternative ranking and select the optimal one. The proposed model is successfully implemented in a real case of financing enterprise selection, and comparisons with five representative tools from three decision-making phases are performed to verify the superiority of the model in dealing with uncertainty and subjectivity.

1. Introduction

As an innovative financing service to alleviate the financial difficulties faced by small and medium-sized enterprises (SMEs), supply chain finance (SCF) can not only help SMEs obtain loans with guaranteed attributes, but also expand the profit model of the entire supply chain (SC) to avoid potential bankruptcy crisis [1]. However, with the outbreak of COVID-19, the traditional SCF development model has suffered strong internal and external shocks. There is growing interest among practitioners and academics in optimize traditional SCF model from the perspective of sustainable development, that is, seeking an effective balance based on the triple bottom line (TBL) (economic, social and environmental) [2,3]. Financial institutions have also developed new product categories and effective financing evaluation mechanisms to achieve sustainable coordination while meeting current needs [4]. Sustainable supply chain finance (SSCF) solutions can improve the performance of the entire SC by incentivizing the sustainable capability of manufacturers, suppliers and retailers, reducing barriers to sustainable supply chain management practices [5]. Given the current resource-constrained market and the highly competitive globalization of SCs, the successful development of enterprise’s SSCF is particularly important to ensure effective SC management and risk aversion. McDermott et al. [6] hold the view that on the basis of sustainable behavior, banks should provide SMEs with more financial resources to incentivize their sustainable participation. Appropriate financing enterprise selection can effectively increase the production vitality of the industry, reduce unnecessary maintenance costs, and weaken the harmfulness of risks spreading throughout the supply chain. For the managers of SSCF business, how to select the optimal enterprise from a set of alternatives according to some criteria for financing cooperation with the least negative impact is considered as one of the most critical issues that need to be addressed to improve the performance of SSCF. Therefore, the SFES problem can be regarded as a multi-criteria decision-making (MCDM) problem. Since a single DM is not sufficient to meet the challenges of all situations in an increasingly complex environment, many selection processes take place in the form of MCGDM [7,8].

With the increasing complexity of objective things and the limitations of human cognition, more and more ambiguous and uncertain information appears in the process of describing and evaluating things. Under this uncertainty, it is difficult for DMs to express cognitive preferences in precise evaluation terms. A more realistic approach may tend to introduce linguistic assessments with interval values rather than exact numerical values to deal with uncertainty and ambiguity in decision-making [9,10]. Consequently, a series of powerful strategies have emerged to obtain the evaluation information of DMs with uncertain consideration [11,12,13]. Nevertheless, these tools can only be applied to describe the uncertainty based on the sets of membership degree and non-membership degree. In view of this, the neutrosophic set was introduced by Smarandache [14] based on the generalization of the above sets, which explicitly quantifies the indeterminacy while considering the influence of truth-membership and falsity-membership. Neutrosophic set logic can be used to represent mathematical models of uncertainty, ambiguity, imprecision, and can pre-digest undefined, incomplete, contradictory information of real-world problems [15]. In order to obtain the inherent information characteristics among criteria, Smarandache developed the plithogenic set [16] on the basis of the neutrosophic set, and took additional consideration of the degree of appurtenance and contradiction when integrating criteria evaluation information, which is of great significance for improving the accuracy of DMs’ subjective judgments and reducing ambiguity. Since then, many scholars have used this theory to solve MCGDM problems and achieved good results [17,18].

Fuzzy set theory or its extensions can be integrated with a series of aggregation operators [19,20,21] from a holistic perspective to form a more efficient and objective framework to deal with uncertain information. However, these information integration strategies require pre-definition of different parameters, which may involve the cognitive biases of DMs and additional subjective judgments brought by the complex computational processes [22]. In addition, many studies only involve numerical integration of plithogenic set information, which makes it difficult to comprehensively consider all aspects of uncertain factors. In view of these shortcomings, some RN-based theories are proposed to further enhance the expression of uncertainty and make up for the lack of objectivity in the process of information aggregation by simply using fuzzy set related theory [23,24]. Inspired by rough set theory [25], RN is endowed with the ability to express uncertainty by defining upper and lower limits to determine a flexible rough boundary interval [26]. More importantly, RNs have the advantage of relying only on the original evaluation data, without any new assumptions and redefinition of parameters and auxiliary information in subsequent processing. Considering this ability to objectively aggregate DMs’ assessment information, RNs have been used in combination with various traditional fuzzy theories to deal with the problem of information aggregation in uncertain MCCGM [27,28,29]. Many scholars have also made attempts to deepen the understanding of the concepts of neutrosophic and rough set extension. Yang et al. [30] proposed single valued neutrosophic RNs on the basis of single valued neutrosophic sets and RNs, and established the corresponding algorithm to solve the decision-making problem. Mondal and Pramanik [31] extended RNs to the neutrosophic environments, constructed an integrated neutrosophic RN tri-complex model for dealing with indeterminate information, and proved some of its properties and characteristics. Akram et al. [32] developed a neutrosophic rough digraph by applying a neutrosophic rough hybrid model to graph theory.

P-NRN is characterized by flexibility and objectivity in expressing complex, ambiguous, and uncertain decision-making information, which greatly expands the research depth of the information aggregation stage in MCGDM problems. However, to obtain reasonable and scientific decision results, two stages need to be carried out: the first is determining the weight information of criteria and DMs, and the other is obtaining the final evaluation results and ranking alternatives. In terms of criterion weight acquisition, many advances have been made based on the integration of RN and FN related theory, such as RN-AHP [22], RN-BWM [33] and interval rough integrated clouds-statistical variance [34]. Another useful tool to objectively reflect the amount of numerical information contained in each criterion is the maximizing deviation method optimization model (MDMOM) [35]. This method is an extension of maximizing deviation method [36], which can determine the corresponding weights according to the difference of performance values among various criteria. Sahin et al. [37] and Wei [38] applied MDMOM to intuitionistic and neutrosophic environments, respectively, to solve the MCGDM problem with incomplete weight information. Compared with AHP and BWM, MDMOM has the advantage of greatly reducing the adverse effect of decision subjectivity while fully retaining the original decision information. For another kind of weight, that is, the acquisition of DM weight, some studies ignore the characteristic differences of DMs in terms of knowledge background, professional ability, and assume that they have the same degree of influence on the determination of alternatives, which is obviously not in line with the actual value [10,23,39]. In view of this, Ye [40] proposed the extended similarity measure (ESM), which can determine the weight of DM according to the closeness degree of each DM’s evaluation value to the ideal interval. As an effective weight determination tool, ESM has been widely used in combination with various fuzzy theories to solve MCGDM problems [41,42].

However, on this basis, it is necessary and crucial to choose an appropriate decision-making technique to prioritize the alternatives. Many classical ranking approaches have been developed by researches to solve the MCGDM problems with uncertain information, such as TOPSIS (technique in order of preference by similarity to ideal solution) [17], VIKOR [22], and COPRAS (complex proportional assessment) [43]. In addition, compared with methods such as SAW (simple additive weighting), TODIM, and DEMATEL, the COPRAS method has the advantages of simple operation, being less time-consuming, and simultaneously able to consider the proportions of different characteristic criteria [44]. Therefore, through the application of COPRAS, the optimal financing alternative can be obtained by DMs according to the relative importance (weight) and utility function of the criteria information, which is an effective supplement to the decision-making strategy under uncertainty. Wei et al. [45] and Mishra et al. [46] extended the COPRAS-based decision-making model to the neutrosophic 2-tuple linguistic and intuitionistic fuzzy environment, respectively, to deal with the imprecision and vagueness that may arise in real-life.

1.1. Motivation

(1) Although fuzzy theory has been extensively applied as an effective tool to describe evaluation information with uncertainty, in some studies [18,47], the aggregation of information is completely based on the subjective preferences of DMs, which ignores the objective relationship between criterion information in the alternative. Therefore, it is necessary to integrate the advantages of various developed uncertainty theories to address the emerging gap between objective evaluations and uncertain subjective environment. (2) Until now, there have been few studies on COPRAS-based decision-making frameworks in which criterion values of alternatives are aggregated by P-NRN. Considering the respective strengths of plithogenic sets and NRNs in dealing with imprecision and uncertainty, it is necessary to combine them to construct a P-NRN for aggregating evaluation information. (3) Considering the fact that the objective evaluation information about decision criteria and the difference of DM’s subjective characteristics has significant impact on the processing effect of uncertain information, it is imperative that both the objectivity and subjectivity of criteria and DMs are taken into account simultaneously under the P-NRN environment in order to make the assessment results more accurate and convincing. (4) Due to the limitations of the techniques and models, the abundance of alternatives, and the ambiguity of evaluation, some previous studies have failed to effectively deal with the vague and inconsistent information that usually exists in the decision-making process of sustainable financing [4,48].

1.2. Objective

To fill current research gaps, enhance the representation of uncertainty, and eliminate the adverse effects of subjectivity, an NRN-based MCGDM model that utilizes the advantages of plithogenic set operators, NRN, ESM, MDMOM, and COPRAS method is proposed in this paper to deal with the financing enterprise selection problem. Compared with models in [17,22,23,30], this paper can not only express the characteristics of uncertainty in the decision-making process and eliminate potentially imprecise cognition, but can also consider the subjective and objective information of criteria, the preference and importance difference of DMs.

1.3. Novelty

The contributions of this paper can be summarized as follows:

(1)

A novel integrated P-NRN is introduced to express and aggregate the evaluation information of DMs, in order to obtain an objective and comprehensive evaluation result. We also propose the construction process and integration properties of the P-NRN.

(2)

The difference characteristics of DMs and the relative importance of criteria are measured by ESM and MDMOM methods, avoiding the influence of completely subjective or objective evaluation on the accuracy of decision-making. It is the first time that the integration of ESM-MDMOM in an extended NRN environment was introduced.

(3)

A P-NRN-based COPRAS is presented to determine the ranking of alternatives and to select the optimal one, which can fully express their relative significance and utility degree, effectively characterizing uncertainty and subjectivity.

(4)

The validity and applicability of the proposed COPRAS-based approach is examined using a real case study concerning the selection of SSCF financing enterprise. The results of comparative analysis verified that the proposed approach has superior performance.

The rest of this study is structured as follows. The definitions of the plithogenic set, P-NRN, and COPRAS, as well as some basic operational laws are briefly reviewed in Section 2. In Section 3, the general steps of the novel proposed decision-making framework are presented on the basis of P-NRN, weight determination methods, and COPRAS, whose effectiveness and applicability are demonstrated by a numerical example of financing enterprise selection in Section 4. Section 5 compares and discusses the relationship and differences between the proposed approach and another five representative MCGDM methods in detail, while highlighting its superiority. The last section gives the concluding remarks and points out an outlook for future research directions.

2. Preliminaries

Since the plithogenic set and NRN can handle uncertain information well and improve the accuracy of decision-making, this paper mainly proposes a novel model for group decision-making which extends the NRN under plithogenic environment to form a concept of P-NRN. In this section, we briefly review some basic concepts of the plithogenic set, NRN, and COPRAS method that are applied in the next sections.

2.1. Plithogenic Set

The plithogenic set, proposed by Smarandache [16] and generally represented as $(p,a,V,d,c)$, is a generalization of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic set. The elements of the plithogenic set are characterized by one or more criteria $A=\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m\}$, and each criterion may have four or more values $V=\{v_1,v_2,\dots ,v_n\}$ (membership for crisp set and fuzzy set; membership and non-membership for intuitionistic fuzzy set; membership, non-membership and indeterminacy for neutrosophic set).

The degree of appurtenance $d(x,v)$ may be: a fuzzy degree, intuitionistic fuzzy degree, or neutrosophic degree of appurtenance to the plithogenic set, that is, for criterion value $v$, the appurtenance degree function is $\forall x\in P,d:P\times V\to P({[0,1]}^Z)$, where $d(x,v)$ is a subset of ${[0,1]}^Z$ and $P({[0,1]}^Z$ is the power set of ${[0,1]}^Z$, of which $z=1$ for fuzzy, $z=2$ for intuitionistic fuzzy, and $z=3$ for neutrosophic degree of appurtenance. The general form of criterion value contradiction degree function can be represented as $c_F:V\times V\to [0,1]$ for fuzzy, $c_{IF}:V\times V\to {[0,1]}^2$ for intuitionistic fuzzy, and $c_N:V\times V\to {[0,1]}^3$ for neutrosophic criterion value contradiction function.

Definition 1.

[17] Let $\tilde{a}=(a_1,a_2,a_3)$and$\tilde{b}=(b_1,b_2,b_3)$be two plithogenic sets. The intersection operation of plithogenic is:

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}((a_{i1},a_{i2},a_{i3}),1\le i\le n)\wedge p((b_{i1},b_{i2},b_{i3}),1\le i\le n)=\\ ((a_{i1}{\wedge }_F{b}_{i1},\frac{1}{2}(a_{i2}{\wedge }_F{b}_{i2})+\frac{1}{2}(a_{i2}{\vee }_F{b}_{i2}),a_{i3}{\vee }_F{b}_{i3})),1\le i\le n,\end{array}$$

(1)

The plithogenic union operation is:

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}((a_{i1},a_{i2},a_{i3}),1\le i\le n)\vee p((b_{i1},b_{i2},b_{i3}),1\le i\le n)=\\ ((a_{i1}{\vee }_F{b}_{i1},\frac{1}{2}(a_{i2}{\wedge }_F{b}_{i2})+\frac{1}{2}(a_{i2}{\vee }_F{b}_{i2}),a_{i3}{\wedge }_F{b}_{i3})),1\le i\le n,\end{array}$$

(2)

where,

$$\begin{array}{l}{a}_{i1}\wedge {pb}_{i1}=\left[1-c({\upsilon }_D,{\upsilon }_1)\right].t_{norm}({\upsilon }_D,{\upsilon }_1)+c({\upsilon }_D,{\upsilon }_1).t_{conorm}({\upsilon }_D,{\upsilon }_1),\\ a_{i1}\vee {pb}_{i1}=\left[1-c({\upsilon }_D,{\upsilon }_1)\right].t_{conorm}({\upsilon }_D,{\upsilon }_1)+c({\upsilon }_D,{\upsilon }_1).t_{norm}({\upsilon }_D,{\upsilon }_1),\\ t_{norm}={\wedge }_{F}b=ab,\hspace{0.33em}{t}_{conorm}a{V}_{F}b=a+b-ab.\end{array}$$

(3)

2.2. Neutrosophic Rough Number

RN is developed on the basis of the rough set theory proposed by Pawlak [25], and can efficiently process subjective decision information. In the philosophy of rough set, approximations are the most critical tools to deal with the uncertainty and ambiguity of identification of things caused by insufficient information [26]. Similarly, an RN is also composed of lower and upper limits, extending the single representation of a crisp number and equivalence relation to an approximation with upper and lower values, thus defining a rough boundary interval to represent imprecise and vague information. The acquisition process of RNs merely relies on the original evaluation data without any prior knowledge, thus it can effectively capture the real perception of DMs based on expectations and experience, and aggregate them into an objective and consistent group judgement [49].

As explained in Section 2.1, the plithogenic set is an extension of the neutrosophic set, that is, it additionally considers the appurtenance degree function and contradiction degree function. NRN [50] is a generalization of fuzzy RN [10] and intuitionistic rough fuzzy number [51], while in this section we will extend the NRN by combining the plithogenic set.

Definition 2.

[30,52] Suppose$Y$is a non-null set and  $R$ is an equivalence relation on  $Y$. A neutrosophic set in  $Y$ is represented by  $C$, with the membership function  $T_C$, indeterminacy function  $I_{C }$ and non-membership function  $F_C$. The tuple  $\left(Y,R\right)$ is called a neutrosophic approximation space of  $C$ , with the lower and the upper approximations denoted by  $\underset{¯}{Apr}\left(C\right)$ and$\overline{Apr}\left(C\right),$ respectively, which are defined as follows:

$$\underset{¯}{Apr}(C)=〈<x,T_{\underset{¯}{Apr}(C)}(x),I_{\underset{¯}{Apr}(C)}(x),F_{\underset{¯}{Apr}(C)}(x)>/Y\in {[x]}_R,x\in Y〉,$$

(4)

$$\overline{Apr}(C)=〈<x,T_{\overline{Apr}(C)}(x),I_{\overline{Apr}(C)}(x),F_{\overline{Apr}(C)}(x)>/Y\in {[x]}_R,x\in Y〉,$$

(5)

where

$$\begin{gathered} T_{\underset{¯}{Apr}(C)}(x)={\wedge }_Y\in {[x]}_R{T}_C(Y),I_{\underset{¯}{Apr}(C)}(x)={\wedge }_Y\in {[x]}_R{I}_C(Y),F_{\underset{¯}{Apr}(A)}(x)={\wedge }_Y\in {[x]}_R{F}_C(Y), \\ {T}_{\overline{Apr}(C)}(x)={\vee }_Y\in {[x]}_R{T}_C(Y),I_{\overline{Apr}(C)}(x)={\vee }_Y\in {[x]}_R{I}_C(Y),F_{\overline{Apr}(A)}(x)={\vee }_Y\in {[x]}_R{F}_C(Y). \end{gathered}$$

Therefore, we can obtain $0\le T_{\underset{¯}{Apr}(C)}(x)+I_{\underset{¯}{Apr}(C)}(x)+F_{\underset{¯}{Apr}(C)}(x)\le 3$ and $0\le T_{\overline{Apr}(C)}(x)+I_{\overline{Apr}(C)}(x)+F_{\overline{Apr}(C)}(x)\le 3$. Symbols $\wedge$ and $\vee$ denote “max” and “min” operators, respectively. $\underset{¯}{Apr}(C)$ and $\overline{Apr}(C)$ are two neutrosophic sets in $C$.

Next, it is not difficult to obtain the neutrosophic set mapping, $\underset{¯}{Apr},\overline{Apr}$:$Apr(Y)\to Apr(C)$ represent the lower and upper rough NS approximation operators, and $(\underset{¯}{Apr}(C),\overline{Apr}(C))$ is called the rough neutrosophic set in $\left(Y,R\right)$.

Then we can continue to explore NRN. There are $p$ classes denoted by $C=\left\{\left.C_i\right|i=1,2,\dots p\right\}$, ordered as $C_1<C_2<\dots <C_n.$ $C_i=(C_{iT},C_{iI},C_{iF})$ is a neutrosophic number. Each $A_i$ can be expressed by an NRN as $NRN(C_i)=[\underset{¯}{NRN}(C_i),\overline{NRN}(C_i)]$, which is determined by its lower limit $\underset{¯}{NRN}(C_i)$ and upper limit $\overline{NRN}(C_i)$ as follows:

$$\underset{¯}{NRN}(C_{iT})=\frac{1}{N_{LT}}{\displaystyle \sum _{i=1}^{N_{LT}}Y\in }\underset{¯}{Apr}(C_{iT}),$$

(6)

$$\underset{¯}{NRN}(C_{iI})=\frac{1}{N_{LI}}{\displaystyle \sum _{i=1}^{NLI}Y\in }\underset{¯}{Apr}(C_{iI}),$$

(7)

$$\underset{¯}{NRN}(C_{iF})=\frac{1}{N_{LF}}{\displaystyle \sum _{i=1}^{NLF}Y\in }\underset{¯}{Apr}(C_{iF}),$$

(8)

$$\overline{NRN}(C_{iT})=\frac{1}{N_{UT}}{\displaystyle \sum _{i=1}^{NUT}Y\in }\overline{Apr}(C_{iT}),$$

(9)

$$\overline{NRN}(C_{iI})=\frac{1}{N_{UI}}{\displaystyle \sum _{i=1}^{NUI}Y\in }\overline{Apr}(C_{iI}),$$

(10)

$$\overline{NRN}(C_{iF})=\frac{1}{N_{UF}}{\displaystyle \sum _{i=1}^{NUF}Y\in }\overline{Apr}(C_{iF}),$$

(11)

where $N_{LT}$, $N_{LI}$, and $N_{LF}$ are the numbers of elements included in $\underset{¯}{Apr}(C_{iT})$, $\underset{¯}{Apr}(C_{iI})$, and $\underset{¯}{Apr}(C_{iF})$, respectively. $N_{UT}$, $N_{UI}$, and $N_{UF}$ are the numbers of elements contained in $\overline{Apr}(C_{iT})$, $\overline{Apr}(C_{iI})$, and $\overline{Apr}(C_{iF})$, respectively. For convenience, the NRN of $C_i$ is expressed as $[C_i]=[\underset{¯}{C_i},\overline{C_i}]$.

Definition 3.

[52] Suppose $[a_1]=[\underset{¯}{a_1},\overline{a_1}]$and$[a_2]=[\underset{¯}{a_2},\overline{a_2}]$are two NRNs with the condition  $\underset{¯}{a_1},\overline{a_1},\underset{¯}{a_2},\overline{a_2}>0$ , and  $\alpha >0$ is a real number. Then we can obtain the operations of NRNs as follows:

$$\alpha \times [a_1]=[\alpha \times \underset{¯}{a_1},\alpha \times \overline{a_1}],\hspace{0.33em}[a_1]/[a_2]=[\underset{¯}{a_1}/\overline{a_2},\overline{a_1}/\underset{¯}{a_2}],$$

(12)

$$[a_1]+[a_2]=[\underset{¯}{a_1}+\underset{¯}{a_2},\overline{a_1}+\overline{a_2}],\hspace{0.33em}[a_1]\times [a_2]=[\underset{¯}{a_1}\times \underset{¯}{a_2},\overline{a_1}\times \overline{a_2}].$$

(13)

Based on the above definition and discussion, we can combine the plithogenic set with the NRN to construct the P-NRN and apply the intersection operation, union operation of plithogenic as shown in Equations (1)–(3) on NRN, that simultaneously takes into account the function of contradiction degree and appurtenance.

2.3. COPRAS

As an efficient MCGDM problem-solving technique, the method of complex proportional assessment (COPRAS) was first proposed by Zavadskas et al. [44]. Since then, it has been gradually applied in various fields [43,53]. COPRAS assumes a direct and proportional relationship of the significance of researched subjects on a system of attributes adequately describing the decision variants and on the degree of utility and weights under the presence of mutually conflicting attributes [28,54]. In this method, the relative importance (weight) and utility function of the benefit (positive ideal solutions) and cost (negative ideal solutions) criteria are the basis for ranking and selecting alternatives.

3. Methods

In this section, three MCGDM methods (ESM, MDMOM, and COPRAS) are employed in order to evaluate risk-based enterprises’ SSCF performance. These methods are based on the P-NRN in order to increase the accuracy of the decision-making procedure and handle the vagueness of information in the assessment.

3.1. Extended Similarity Measures Method

In the process of real-life MCGDM problems, each DM may have unique characteristics and differences with regard to knowledge background, professional perspective, and personality preferences, which signifies that the DMs have varying degrees of influence on the overall ranking and should be assigned different weights. To address this concern, the similarity measures method is extended in this paper to determine the weights of DMs and the definitions are clarified as follows:

Definition 4.

Let$a_j\hspace{0.33em}\hspace{0.33em}(j=1,2,\dots n)$be a set of single valued neutrosophic numbers (SVNN), then the SNWA and SNWG operators can be defined as [40]:

$$\begin{array}{l}SNWA(a_1,a_2,\dots ,a_n)=\\ 〈1-{\displaystyle \prod _{j=1}^n{(1-T_j)}^{w_j},1-{\displaystyle \prod _{j=1}^n{(1-I_j)}^{wj},{\displaystyle \prod _{j=1}^n{(1-F_j)}^{w_j}}}}〉,\end{array}$$

(14)

$$SNWG(a_1,a_2,\dots ,a_n)=〈{\displaystyle \prod _{j=1}^{n}T{j}^{w_j},{\displaystyle \prod _{j=1}^{n}I{j}^{w_j},{\displaystyle \prod _{j=1}^{n}F{j}^{w_j}}}}〉,$$

(15)

where $w={(w_1,w_2,\dots ,w_n)}^T$ is the weight vector of $a_j\hspace{0.33em}\hspace{0.33em}(j=1,2,\dots n)$, ${\displaystyle \sum _{j=1}^n{w}_j=1}$.

Definition 5.

Let $X^k={(a_{ij}^k)}_{m\times n},k\in [1,t],i\in [1,m],j\in [1,n]$represent the set of alternatives according to the criteria and let  $〈(T_{11}^k,I_{11}^k,F_{11}^k),(T_{11}^{k^{\prime }},I_{11}^{k^{\prime }},F_{11}^{k^{\prime }})〉$ be a basic form of triangular neutrosophic number to express the decision information of the first DM towards the alternative  $A_1$ . The decision matrix is denoted by $X^k={(a_{ij}^k)}_{m\times n}$, which can be yielded by the following operator under triangular neutrosophic environment:

$$a_{ij}^k=(a_{ij}^1,a_{ij}^2,\dots ,a_{ij}^t)=\left(\begin{array}{l}〈1-{\displaystyle \prod _{j=1}^n{(1-T_{ij}^k)}^{w_j},1-{\displaystyle \prod _{j=1}^n{(1-I_{ij}^k)}^{w_j},1-{\displaystyle \prod _{j=1}^n{(1-F_{ij}^k)}^{w_j}}}}〉,\\ 〈{\displaystyle \prod _{j=1}^n{(T_{ij}^{k^{\prime }})}^{w_j},{\displaystyle \prod _{j=1}^n{(I_{ij}^{k^{\prime }})}^{w_j},{\displaystyle \prod _{j=1}^n{(F_{ij}^{k^{\prime }})}^{w_j}}}}〉\end{array}\right).$$

(16)

For the average decision matrix, which is denoted by $X^{\ast }={(a_{ij}^{\ast })}_{m\times n}$ and the weight $w_j$ for each DM is $1/t$, that is $a_{ij}^{*}=\frac{1}{t}{\displaystyle \sum _{k=1}^t{a}_{ij}^k}$.

It is obvious that the closer the decision matrix $X^k$ is to the average decision matrix $X^{\ast }$, the better the value of the k-th DM weight and hence it occupies a more important role while making decisions, based on which, the ESM between the individual decision matrix $X^k$ and the ideal one $X^{\ast }$ can be defined as [55]:

$$sm(X^k,X^{\ast })=\frac{1}{mn}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n\frac{d(a_{ij}^k,{(a_{ij}^{\ast })}^C)}{d(a_{ij}^k,a_{ij}^{\ast })+d(a_{ij}^k,{(a_{ij}^{\ast })}^C)}}},$$

(17)

where

$$a_{ij}^{\ast }{}^C=\left(\begin{array}{l}〈1-{\displaystyle \prod _{j=1}^n{(1-F_{ij}^k)}^{w_j},{\displaystyle \prod _{j=1}^n{(1-I_{ij}^k)}^{w_j},1-{\displaystyle \prod _{j=1}^n{(1-T_{ij}^k)}^{w_j}}}}〉,\\ 〈{\displaystyle \prod _{j=1}^n{(F_{ij}^{k^{\prime }})}^{w_j},1-{\displaystyle \prod _{j=1}^n{(I_{ij}^{k^{\prime }})}^{w_j},{\displaystyle \prod _{j=1}^n{(T_{ij}^{k^{\prime }})}^{w_j}}}}〉\end{array}\right),$$

After calculations, the weight for each DM can be obtained using the following equation:

$$\lambda k=\frac{sm(X^k,X^{\ast })}{{\displaystyle {\sum }_{k=1}^{t}sm(X^k,X^{\ast })}},$$

(18)

where ${\lambda }_k\in [0,1],{\displaystyle {\sum }_{k=1}^t{\lambda }_k}=1,k\in [1,t]$, and we can obtain the weight vector of DMs $\lambda ={({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_t)}^T$ to aggregate all the individual decision matrix $X^k$ values into a collective decision-making matrix $X={(a_{ij})}_{m\times n}$.

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{C}_1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{C}_2\hspace{0.33em}\hspace{0.33em}\cdots \hspace{0.33em}\hspace{0.33em}{C}_n\\ X={\left[a_{ij}\right]}_{m\times n}=\begin{array}{c}{\mathrm{A}}_1\\ {\mathrm{A}}_2\\ ⋮\\ {\mathrm{A}}_m\end{array}\left[\begin{array}{cccc}{a}_{11}^{}& a_{12}^{}& \cdots & a_{1n}^{}\\ a_{21}^{}& a_{22}^{}& \cdots & a_{2n}^{}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}^{}& a_{m2}^{}& \cdots & a_{mn}^{}\end{array}\right],\end{array}$$

(19)

where $a_{ij}$ can be obtained as follows:

$$a_{ij}={\displaystyle \sum _{k=1}^t{\lambda }_k{X}_{}^k}=\left(\begin{array}{l}〈1-{\displaystyle \prod _{j=1}^n{(1-T_{ij}^k)}^{\lambda k},1-{\displaystyle \prod _{j=1}^n{(1-I_{ij}^k)}^{\lambda k},1-{\displaystyle \prod _{j=1}^n{(1-F_{ij}^k)}^{\lambda k}}}}〉,\\ 〈{\displaystyle \prod _{j=1}^n{(T_{ij}^{k^{\prime }})}^{\lambda k},{\displaystyle \prod _{j=1}^n{(I_{ij}^{k^{\prime }})}^{\lambda k},{\displaystyle \prod _{j=1}^n{(F_{ij}^{k^{\prime }})}^{\lambda k}}}}〉\end{array}\right).$$

3.2. Maximizing Deviation Method Optimization Model

Under this consideration, the larger the differences between the performance values of each alternative for a criterion, the more significant the criterion is and should be assigned a larger weight for selecting the optimal alternative [56]. The goal of this model is to estimate the optimal weight vector of each criterion through Euclidean distance measurement, thereby greatly reducing the inaccurate influence of the subjective evaluation of DMs under the interference of fuzzy and uncertain information. On the basis of the collective decision-making matrix obtained in Section 3.1, for the criteria $c_j\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(j\in [1,n])$, whose weights are denoted by $w=(w_1,w_2,\cdots ,w_n)$, the deviation of the alternatives $A_i(i\in [1,m])$ to all other alternatives between two triangular neutrosophic numbers, $a_{ij}$ and $a_{qj}$, can be clearly defined as follows:

$$D_{ij}={\displaystyle \sum _{q=1}^{m}d({\tilde{a}}_{ij},{\tilde{a}}_{qj})}={\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}},$$

(20)

where $d({\tilde{a}}_{ij},{\tilde{a}}_{qj})$ represents the Euclidean distance between two triangular neutrosophic values and was defined in Equation (4).

Let

$$D_j={\displaystyle \sum _{i=1}^m{D}_{ij}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^{m}d({\tilde{a}}_{ij},{\tilde{a}}_{qj})}}}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}}.$$

(21)

where $D_j$ represents the total deviation value of all alternatives to other alternatives for the criterion $c_j\hspace{0.33em}\hspace{0.33em}(j\in [1,n])$.

Next when the weight of criteria $w_j$ is taken into consideration.

Let

$$\begin{array}{l}D(w)={\displaystyle \sum _{j=1}^n{w}_j{D}_j=}\\ {\displaystyle \sum _{j=1}^n{w}_j}\left({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}}\right).\end{array}$$

(22)

where $D(w)$ represents the total deviation value of all alternatives to other alternatives for all the criteria.

Based on the above analysis regarding the optimal processing for the weights of criteria, a necessary maximizing selection step is needed, i.e., by constructing a nonlinear programming model as in following equation:

$$\max D(w)={\displaystyle \sum _{j=1}^{n}wj}\left({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}}\right),$$

(23)

$$s.t.\hspace{0.33em}\hspace{0.33em}{w}_j\ge 0,\hspace{0.17em}\hspace{0.33em}j=1,2,\dots ,n\hspace{0.17em},\hspace{0.33em}\hspace{0.33em}{\displaystyle \sum _{j=1}^n{w}_j^2}=1.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}$$

In order to solve this model, a Lagrange multipliers function is constructed and shown as follows:

$$L(w,\lambda )={\displaystyle \sum _{j=1}^n{w}_j\left({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}}\right)}+\frac{\lambda }{2}\left({\displaystyle \sum _{j=1}^n{w}_j^2-1}\right),$$

(24)

where $\lambda$ represents the Lagrange multiplier.

Based on the outcome of Equation (24), by differentiating it with respect to $w_j(j=1,2,\dots ,n)$ and $\lambda$, after setting these partial derivatives equal to zero, we can then obtain:

$$\begin{array}{l}\frac{\partial L}{{\partial }_{wj}}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}}+{\lambda }_{wj}=0\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\frac{\partial L}{\partial \lambda }=\frac{1}{2}({\displaystyle \sum _{j=1}^n{w_j}^2}-1)=0\end{array}$$

(25)

We can next determine the values of $w_j$ and $\lambda$ as follows:

$$w_j=\frac{-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}}}{\lambda },$$

(26)

$$\lambda =-\sqrt{{\displaystyle \sum _{j=1}^n{\left({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{q=1}^m\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}}\right)}^2}}$$

(27)

Therefore, we obtain $w_j$ by combining Equations (26) and (27).

$$w_j=\frac{{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{q=1}^m}\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2)\end{array}}{6}}}{\sqrt{{\displaystyle \sum _{j=1}^n}{\left({\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{q=1}^m}\sqrt{\frac{\begin{array}{l}({\left|T_{ij}^L-T_{qj}^L\right|}^2+{\left|T_{ij}^U-T_{qj}^U\right|}^2+{\left|I_{ij}^L-I_{qj}^L\right|}^2\\ +{\left|I_{ij}^U-I_{qj}^U\right|}^2+{\left|F_{ij}^L-F_{qj}^L\right|}^2+{\left|F_{ij}^U-F_{qj}^U\right|}^2\end{array}}{6}}\right)}^2}}$$

(28)

Finally, by normalizing $w_j\hspace{0.33em}\hspace{0.33em}(j=1,2,\dots ,n)$, making a summation of them into a unit, we can obtain $w_j^{*}=w_j/{\displaystyle \sum _{j=1}^n{w}_j}$, where $w_j^{*}$ can be considered as the optimal weight vector of criterion $j$. Therefore, the objective weights of all criteria are $w={(w_1,w_2,\dots ,w_n)}^T.$

3.3. Proposed Framework

This research constructs an integrated framework considering imprecise and vague information to address the problem of SSCF risk evaluation and financing enterprise selection. Five MCGDM techniques and methods are combined in this paper to improve the accuracy of decision-making results. The evaluation process is mainly divided into four phases: (1) describe the research problem and provide information for decision-making, which includes identifying the sets of DMS, criteria, alternatives, and obtaining evaluations of them from DMs; (2) transform the evaluation matrix into the P-NRN based on the calculations of NRN and the aggregation of the plithogenic set; (3) weight calculation by the ESM and MDMOM methods; and (4) evaluating alternative ranking and selecting by extended NRN-based COPRAS. The advantage and importance of the P-NRN lies not only in providing more accurate aggregation results while highly considering uncertainty, but also in comprehensively analyzing the upper and lower limits that contained vague information to better cope with increasingly complex environments. In addition, the application of weight-determination techniques can also objectively measure the amount of decision-making information and individual differences in the characteristics of DMs. Finally, an NRN-based COPRAS is proposed to identify the risk priority ranking of the enterprises applying for financing by measuring the relative importance and utility function of the criteria from alternatives. The specific steps of the proposed framework are illustrated in Figure 1 and its details will be described below:

Step 1: Assume that $A_i=\left\{A_1,A_2,\cdot \cdot \cdot A_m\right\}$ ($i\ge 1$) is a discrete set of $m$ alternatives, which are evaluated to select the optimal one based on a set of $C_j=\left\{c_1,c_2,\cdot \cdot \cdot c_n\right\}$ ($j\ge 1$) criteria. Suppose $w=(w_1,w_2,\cdots ,w_n)$ is the vector of criteria weights, where $0\le w_j\le 1,{\displaystyle {\sum }_{j=1}^n{w}_j}=1$. A group of DMs with experience are indicated by $D=\left\{D{M}_1,D{M}_2,\cdot \cdot \cdot D{M}_t\right\}$ ($t\ge 1$) where $t$ denotes the number of DMs who select the criteria and defines the evaluation elements of the initial decision-making matrix and the weight vector of them $\lambda =\left\{{\lambda }_1,{\lambda }_2,\cdot \cdot \cdot {\lambda }_t\right\}$ expressed accordingly, such that for each DM, $0\le {\lambda }_k\le 1,{\displaystyle {\sum }_{k=1}^t{\lambda }_k}=1$. A linguistic scale of evaluation is identified by DMs in the form of triangular neutrosophic numbers shown in

Table 1. Linguistic scale of evaluation.

Significance Linguistic Scale	Triangular Neutrosophic Scale
Very Weakly Significant (VWS)	((0.1,0.3,0.35),0.1,0.2,0.15)
Weakly Significant (WS)	((0.15,0.25,0.1),0.6,0.2,0.3)
Partially Significant (PS)	((0.4,0.35,0.5),0.6,0.1,0.2)
Equal Significant (ES)	((0.65,0.6,0.7),0.8,0.1,0.1)
Strong Significant (SS)	((0.7,0.65,0.8),0.9,0.2,0.1)
Very Strongly Significant (VSS)	((0.9,0.85,0.9),0.8,0.2,0.2)
Absolutely Significant (AS)	((0.95,0.9,0.95),0.9,0.1,0.1)

. Therefore, vector $x_{ij}$ can be applied to represent the value of element with i-th alternative by j-th criterion. Through summarization, the evaluation matrix of alternatives based on criteria is constructed.

Step 2: Transform the evaluation matrix values of alternatives toward criteria by DMs into NRNs to better express imprecise information according to Equations (4)–(11). The upper and lower limits of the evaluation are defined and we can obtain $A_i=(NRN(x_{i1}),NRN(x_{i2}),\dots ,$$NRN(x_{in}))$, where $NRN(x_{ij})=[((\underset{¯}{NRN}(x_{ij}^T),\underset{¯}{NRN}(x_{ij}^I),\underset{¯}{NRN}(x_{ij}^F)),\underset{¯}{NRN}(x_{ij}^{T^{\ast }}),\underset{¯}{NRN}(x{i}_{ij}^{I^{\ast }}),\underset{¯}{NRN}(x{i}_{ij}^{F^{\ast }})),$$((\overline{NRN}(x_{ij}^T),\overline{NRN}(x_{ij}^I),\overline{NRN}(x_{ij}^F)),\overline{NRN}(x_{ij}^{T^{\ast }}),\overline{NRN}(x{i}_{ij}^{I^{\ast }}),\overline{NRN}(x_{ij}^{F^{\ast }}))]$. In that way, we have an NRN decision-making matrix $X^D$ as follows:

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\begin{array}{llll}C1& \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}C2& \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\cdots & \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Cn\end{array}\\ X^D=\begin{array}{c}{\mathrm{A}}_1\\ {\mathrm{A}}_2\\ ⋮\\ {\mathrm{A}}_m\end{array}\left[\begin{array}{cccc}NRN(x_{11})& NRN(x_{12})& \cdots & NRN(x_{1n})\\ NRN(x_{21})& NRN(x_{22})& \cdots & NRN(x_{2n})\\ ⋮& ⋮& \ddots & ⋮\\ NRN(x_{m1})& NRN(x_{m2})& \cdots & NRN(x_{mn})\end{array}\right].\end{array}$$

(29)

Then the contradiction degree and appurtenance degree are defined to extend NRN to plithogenic form using Equations (1)–(3). After that, a de-neutrosophic calculation is processed to transform the P-NRN matrix into a crisp value RN matrix $X^{D^{\prime }}={[RN(x_{i{j}^{\prime }})]}_{m\times n}$ as in Equation (30).

$$\mathrm{S}\left(A\right)=\frac{1}{8}\left(T_A^{}(x)+I_A^{}(x)+F_A^{}(x)\right)\times \left(2+T_A^{\ast }(x)-T_A^{\ast }(x)-T_A^{\ast }(x)\right).$$

(30)

Step 3: Calculate the normalized decision-making matrix $Y^D={[RN(y_{ij})]}_{m\times n}$ using Equation (31) on the basis of the de-neutrosophic matrix $X^{D^{\prime }}$.

$$RN(y_{ij})=[y_{ij}^L,y_{ij}^U]=\left\{\begin{array}{l}\left[\frac{X_{i{j}^{\prime }}^L-\min X_{i{j}^{\prime }}^L}{\max X_{i{j}^{\prime }}^L-\min X_{i{j}^{\prime }}^L},\frac{X_{i{j}^{\prime }}^U-\min X_{i{j}^{\prime }}^U}{\max X_{i{j}^{\prime }}^U-\min X_{i{j}^{\prime }}^U}\right],\hspace{0.33em}\hspace{0.33em}{X}_{i{j}^{\prime }}^L\hspace{0.33em}is\hspace{0.33em}a\hspace{0.33em}benefit\hspace{0.33em}criteria.\\ \left[\frac{\max X_{i{j}^{\prime }}^L-X_{i{j}^{\prime }}^L}{\max X_{i{j}^{\prime }}^L-\min X_{i{j}^{\prime }}^L},\frac{\max X_{i{j}^{\prime }}^U-X_{i{j}^{\prime }}^U}{\max X_{i{j}^{\prime }}^U-\min X_{i{j}^{\prime }}^U}\right],\hspace{0.33em}\hspace{0.33em}{X}_{i{j}^{\prime }}^L\hspace{0.33em}is\hspace{0.33em}a\hspace{0.33em}\cos t\hspace{0.33em}criteria.\end{array}\right.$$

(31)

Step 4: Calculate the weight ${\lambda }_k$ of each DM based on the ESM method shown in Section 3.3 where $0\le {\lambda }_k\le 1,{\displaystyle {\sum }_{k=1}^t{\lambda }_k}=1$, and then obtain the comprehensive decision-making matrix.

Step 5: On the basis of the comprehensive decision matrix, the weights of criteria are calculated according to Equations (20)–(28) and we can determine $w={(w_1,w_2,\dots ,w_n)}^T$.

Step 6: In this step, the weighted normalized matrix $V$ is constructed in which the element value of normalized matrix $Y^D$ is multiplied by weights of criteria $RN(w_j)$

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\begin{array}{llll}{C}_1& \hspace{0.33em}\hspace{0.33em}{C}_2& \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\cdots & \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{C}_n\end{array}\\ V=\begin{array}{c}{\mathrm{A}}_1\\ {\mathrm{A}}_2\\ ⋮\\ {\mathrm{A}}_m\end{array}\left[\begin{array}{cccc}RN(v_{11})& RN(v_{12})& \cdots & RN(v_{1n})\\ RN(v_{21})& RN(v_{22})& \cdots & RN(v_{2n})\\ ⋮& ⋮& \ddots & ⋮\\ RN(v_{m1})& RN(v_{m2})& \cdots & RN(v_{mn})\end{array}\right],\end{array}$$

(32)

where $RN(v_{ij})=RN(y_{ij})\ast RN(w_j)$, $RN(y_{ij})$ are the elements of the normalized decision-making matrix $Y^D$ and $RN(w_j)=[w_j^L,w_j^U]$ are the weight coefficients of the criteria.

Step 7: The matrix $V$ is summed and summarized by columns according to the category (benefit or cost) to which the decision-making criterion belongs and we can obtain $RN(S_i^{+})$ and $RN(S_i^{-})$, respectively. For the alternative to be evaluated, both a higher level of $RN(S_i^{+})$ and a lower value of $RN(S_i^{-})$ can indicate a better achievement level. In this way, Equation (33) is applied for benefit criteria and Equation (34) for cost criteria.

$$RN(S_i^{+})={\displaystyle \sum _{j=1}^{r}RN(v_{ij})}={\displaystyle \sum _{j=1}^{r}RN(y_{ij})\ast RN(w_j)},$$

(33)

$$RN(S_i^{-})={\displaystyle \sum _{j=r+1}^{n}RN(v_{ij})}={\displaystyle \sum _{j=r+1}^{n}RN(y_{ij})\ast RN(w_j)},$$

(34)

where $r$ represents the number of benefit criteria that must be maximized, and the rest of criteria from $r+1$ to $n$ prefer lower values.

Step 8: Calculate the relative significance and utility degree of alternatives using Equations (35) and (36), respectively.

$$RN(Q_i)=RN(S_i^{+})+\frac{RN(S_{\min }^{-}){\displaystyle {\sum }_{i=1}^{m}RN(S_i^{-})}}{RN(S_i^{-}){\displaystyle {\sum }_{i=1}^m\left(\frac{RN(S_{\min }^{-})}{RN(S_i^{-})}\right)}},$$

(35)

$$U_i=\frac{RN(Q_i)}{RN{(Q_i)}^{\max }}\times 100\%,$$

(36)

where $RN{(Q_i)}^{\max }$ represents the maximum value of utility degree function. The ranking and selection of alternatives are based on the values of $U_i$. Better alternatives have a higher $U_i$, that is, the alternative with $RN{(Q_i)}^{\max }$ can be considered as the optimal one.

4. Numerical Application

In this section, we consider a practical case study concerning SSCF risk evaluation and financing enterprise selection in the medical industry to illustrate the effectiveness and applicability of the proposed novel model. Furthermore, the comparative analysis is also conducted to demonstrate the superiority of the proposed approaches.

The pervasive nature of the medical industry makes it relevant to everyone’s daily life. With the continuous upgrading of production equipment in response to the sustainable demand for “energy conservation and emission reduction” [57], and the information uncertainty [58] brought about by the increasingly fierce external competition for orders, many medical device companies are facing the problem of insufficient cash flow, and some of them are even forced to stop production, which seriously endangers the normal operation of the entire supply chain. Therefore, a direct strategy of reducing the production capital gap of related companies is for financial institutions to establish a rigorous and efficient decision-making approach to provide financing services for medical enterprises on the premise of risk assessment. According to the obtained risk evaluation results, it is also of great significance to formulate corresponding risk prevention and control measures for better selection of financing objects and development of medical services.

As a large state-owned commercial bank in China, Z actively carried out various financing models at the beginning of the rise of SSCF, and to a certain extent solved the problem of financing difficulties for enterprises within the scope of business radiation. Now Z intends to choose a medical device production company as the financing object of the cooperation on the basis of risk evaluation.

A committee of five DMs with extensive industry experience was formed to conduct the evaluation, denoted as $D=\left\{D{M}_1,D{M}_2,D{M}_3,D{M}_4\right\}$, in which the weight of each DM was determined by their differences in characteristics, and occupy different proportions as $\lambda =\left\{{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right\}$. After a primary election, four companies remained as the alternatives for further evaluation, which are denoted as $A=\{A_1,A_2,A_3,A_4\}$. In view of the DMs’ opinions and the business characteristics of the applicant enterprises, eleven risk indexes are identified as the evaluation criteria, which are composed of credit status (C1), profitability (C2), loan amount and frequency (C3), employee rights and interests (C4), community and government responsibility fulfillment (C5), environmental protection (C6), resource utilization (C7), organizational structure (C8), sustainable finance factors (C9), level of relevance and cooperation (C10), and information and control capability (C11). As summarized in

Table 2. Evaluation criteria for selecting the financing enterprise.

Name of Criterion	Objective
Credit status (C1)	max
Profitability (C2)	max
Loan amount and frequency (C3)	min
Employee rights and interests (C4)	max
Community and government responsibility fulfillment (C5)	max
Environmental protection (C6)	max
Resource utilization (C7)	max
Organizational structure (C8)	max
Sustainable finance factors (C9)	max
Level of relevance and cooperation (C10)	max
Information and control capability (C11)	max

, the criterion of loan amount and frequency is classified as the cost criteria (lower numerical value is more preferable), while the other ten are categorized as the benefit criteria (the-bigger-the-better). The weight vectors $w=(w_1,w_2,\cdots ,w_{11})$ of these criteria are completely unknown. The selection of financing enterprise can be modeled as a hierarchical structure, as shown in Figure 2. Based on the proposed approach discussed in Section 3, the considered problem is solved by the following steps:

Step 1: The DMs construct the decision evaluation matrix with respect to the five alternatives and eleven criteria according to the triangular neutrosophic linguistic scale shown in Table 1. During the evaluation process, each DM is asked to give their independent judgment towards the criteria and alternatives, the results are depicted in

Table 3. Evaluation matrix of the criteria-based alternatives.

Alternatives	DM	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11
A1	DM1	SS	SS	VSS	SS	ES	PS	ES	SS	SS	VSS	VSS
	DM2	VSS	SS	SS	ES	VSS	ES	ES	VSS	ES	AS	SS
	DM3	SS	VSS	ES	ES	SS	PS	PS	ES	ES	SS	ES
	DM4	AS	SS	ES	PS	SS	ES	SS	SS	VSS	AS	VSS
	DM5	VSS	ES	ES	ES	SS	PS	ES	VSS	SS	SS	SS
A2	DM1	SS	ES	ES	ES	ES	ES	SS	SS	ES	ES	VSS
	DM2	ES	PS	PS	ES	SS	SS	VSS	PS	PS	SS	SS
	DM3	SS	ES	ES	WS	ES	SS	SS	ES	PS	PS	SS
	DM4	SS	ES	WS	PS	SS	ES	ES	ES	SS	SS	SS
	DM5	ES	SS	PS	PS	ES	ES	SS	PS	ES	ES	VSS
A3	DM1	VSS	PS	AS	SS	PS	PS	SS	ES	ES	PS	VSS
	DM2	SS	PS	SS	VSS	WS	ES	PS	SS	SS	SS	VSS
	DM3	SS	ES	SS	ES	WS	ES	ES	SS	PS	PS	SS
	DM4	VSS	SS	ES	SS	PS	PS	ES	VSS	SS	SS	VSS
	DM5	VSS	ES	VSS	VSS	PS	PS	SS	SS	ES	ES	SS
A4	DM1	ES	SS	ES	ES	ES	SS	VSS	PS	VSS	SS	SS
	DM2	SS	VSS	SS	VSS	SS	ES	AS	SS	VSS	VSS	ES
	DM3	SS	SS	ES	ES	PS	SS	SS	VSS	SS	ES	PS
	DM4	VSS	VSS	ES	SS	ES	SS	SS	ES	SS	SS	SS
	DM5	SS	SS	SS	SS	ES	SS	VSS	SS	VSS	SS	SS

.

Step 2: According to the aggregation procedure in Section 3.3, we can firstly transform all the triangular neutrosophic values in Table 3 into NRN decision-making matrix $X^D$ using Equations (4)–(11). Here, we take the element $x_{11}$ = (SS, VSS, SS, AS, VSS) by five DM’s evaluation of C1 under A1 as an example:

${\mathrm{x}}_{11}^1={\mathrm{x}}_{11}^3=\mathrm{SS}=\left(\left(0.7,0.65,0.8\right),0.9,0.2,0.1\right);{\mathrm{x}}_{11}^2={\mathrm{x}}_{11}^5=\left(0.9,0.85,0.9\right),0.8,0.2,0.2);$ $x_{11}^4=\mathrm{AS}=\left(\left(0.95,0.9,0.95\right),0.9,0.1,0.1\right)$.

$\underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{\mathrm{T}})=\frac{1}{2}\times \left(0.7+0.7\right)=0.7$; $\underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{\mathrm{I}})=\frac{1}{2}\times \left(0.65+0.65\right)=0.65; \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{\mathrm{F}})=\frac{1}{2}\times \left(0.8+0.8\right)=0.8; \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{{\mathrm{T}}^{\prime }})=\frac{1}{5}\times \left(0.9+0.8+0.9+0.9+0.8\right)=0.86; \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{{\mathrm{I}}^{\prime }})=\frac{1}{5}\times \left(0.2+0.2+0.2+0.1+0.2\right)=0.18; \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{{\mathrm{F}}^{\prime }})=\frac{1}{3}\times \left(0.1+0.1+0.1\right)=0.1.$

$\overline{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{\mathrm{T}})=\frac{1}{5}\times \left(0.7+0.9+0.7+0.95+0.9\right)=0.83$; $\overline{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{\mathrm{I}})=\frac{1}{5}\times \left(0.65+0.85+0.65+0.9+0.85\right)=0.78; \overline{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{\mathrm{F}})=\frac{1}{5}\times \left(0.8+0.9+0.8+0.95+0.9\right)=0.87; \overline{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{{\mathrm{T}}^{\prime }})=\frac{1}{3}\times \left(0.9+0.9+0.9\right)=0.9; \overline{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{{\mathrm{I}}^{\prime }})=\frac{1}{4}\times \left(0.2+0.2+0.2+0.2\right)=0.2; \overline{\mathrm{NRN}}({\mathrm{x}}_{\mathrm{ss}}^{{\mathrm{F}}^{\prime }})=\frac{1}{5}\times \left(0.1+0.2+0.1+0.1+0.2\right)=0.14$.

Therefore, $\underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{11}^1)$= [((0.7,0.65,0.8),0.86,0.18,0.1), ((0.83,0.78,0.87),0.9,0.2,0.14)]. Similarly, the NRN evaluation matrix of C1 under A1 for the other four DMs is obtained as:

$$\begin{gathered} \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{11}^2)=\left[\right((0.8,0.75,0.85),0.8,0.18,0.14), ((0.92,0.87,0.92),0.86,0.2,0.2\left)\right]. \\ \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{11}^3)=\left[\right((0.7,0.65,0.8),0.86,0.18,0.1), ((0.83,0.78,0.87),0.9,0.2,0.14\left)\right]. \\ \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{11}^4)=\left[\right((0.83,0.78,0.87),0.86,0.1,0.1), ((0.95,0.9,0.95),0.9,0.18,0.14\left)\right]. \\ \underset{¯}{\mathrm{NRN}}({\mathrm{x}}_{11}^5)=\left[\right((0.8,0.75,0.85),0.8,0.18,0.14), ((0.92,0.87,0.92),0.86,0.2,0.2\left)\right]. \end{gathered}$$

Next, the extended NRN group evaluation matrix of C1 under A1 can be obtained based on the result of $\mathrm{NRN}\left(x_{11}\right)$ using Equations (1)–(3), and P-NRN $\left(x_{11}\right)$= [((0.26,0.72,1),0.41,0.16,0.46), ((0.55,0.84,1),0.54,0.2,0.6)]. Similarly, the P-NRN evaluation matrix of all criteria under alternatives is obtained, which is shown in

Table 4. Aggregation results of P-NRNs matrix.

	C1	C2	C3
A1	[((0.26,0.72,1),0.41,0.16,0.46), ((0.55,0.84,1),0.54,0.2,0.6)]	[((0.27,0.64,0.96),0.51,0.16,0.34), ((0.39,0.73,0.98),0.63,0.2,0.42)]	[((0.35,0.62,0.91),0.55,0.12,0.26), ((0.46,0.72,0.94),0.6,0.16,0.33)]
A2	[((0.13,0.62,1),0.41,0.14,0.41), ((0.16,0.64,1),0.54,0.18,0.41)]	[((0.16,0.51,0.94),0.33,0.1,0.34), ((0.25,0.6,0.97),0.52,0.14,0.41)]	[((0.01,0.35,0.66),0.3,0.1,0.35), ((0.23,0.52,0.86),0.42,0.13,0.51)]
A3	[((0.27,0.72,1),0.36,0.2,0.52), ((0.49,0.82,1),0.48,0.2,0.64)]	[((0.12,0.43,0.92),0.26,0.1,0.36), ((0.22,0.58,0.96),0.48,0.14,0.56)]	[((0.4,0.66,0.94),0.59,0.14,0.27), ((0.65,0.81,0.98),0.7,0.18,0.35)]
A4	[((0.15,0.64,1),0.41,0.16,0.42), ((0.28,0.73,1),0.54,0.2,0.52)]	[((0.33,0.68,0.98),0.52,0.2,0.37), ((0.49,0.78,0.98),0.63,0.2,0.56)]	[((0.35,0.61,0.92),0.58,0.12,0.26), ((0.38,0.63,0.94),0.67,0.16,0.26)]
	C4	…	C11
A1	[((0.32,0.51,0.8),0.51,0.1,0.21), ((0.44,0.61,0.87),0.67,0.14,0.27)]	…	[((0.96,0.66,0.4),0.98,0.16,0.012), ((0.98,0.78,0.58),0.99,0.2,0.018)]
A2	[((0.15,0.35,0.57),0.4,0.1,0.33), ((0.32,0.52,0.79),0.5,0.14,0.42)]	…	[((0.97,0.68,0.48),0.98,0.2,0.015), ((0.99,0.78,0.58),0.96,0.2,0.025)]
A3	[((0.52,0.66,0.91),0.65,0.16,0.24), ((0.7,0.78,0.95),0.73,0.2,0.32)]	…	[((0.96,0.72,0.51),0.98,0.2,0.012), ((0.98,0.82,0.63),0.99,0.2,0.02)]
A4	[((0.46,0.62,0.88),0.63,0.14,0.21), ((0.59,0.73,0.93),0.7,0.18,0.25)]	…	[((0.93,0.51,0.24),0.98,0.14,0.011), ((0.95,0.63,0.41),0.99,0.18,0.014)]

. A visual representation of this example is shown in Figure 3. Then, the P-NRN matrix is further transferred into a crisp value RN matrix $X^{D^{\prime }}$ using Equation (30).

Step 3: After converting the evaluation matrix to crisp values, the normalized decision-making matrix $Y^D$ of alternatives with respect to risk criteria can be calculated based on the nature of the criteria, as expressed in Equation (31), the result of which is presented in

Table 5. Normalized evaluation matrix of risk criteria.

	C1	C2	C3	C4	…	C11
A1	[1, 1]	[0.9, 0.99]	[0.13, 0.33]	[0.57, 0.52]	…	[0.68, 0.81]
A2	[0, 0]	[0.33, 0.42]	[1, 1]	[0, 0]	…	[0.85, 0.8]
A3	[0.05, 0.42]	[0, 0]	[0, 0]	[1, 1]	…	[1, 1]
A4	[0.02, 0.22]	[1, 1]	[0.1, 0.36]	[0.92, 0.85]	…	[0, 0]

.

Step 4. After calculating the normalized evaluation matrix of criteria in step 3, the weights of DMs are further derived by the ESM method, which was described in detail in Section 3.1, hence we can acquire the differential different weights shown in

Table 6. The weights of DMs.

	DM1	DM2	DM3	DM4	DM5
${\lambda }_k$	0.196	0.177	0.192	0.196	0.238

and the comprehensive decision matrix of all DMs’ evaluations using Equations (14)–(19).

Step 5: In order to evaluate the weights of the 11 criteria, MDMOM is applied. We can obtain the weight results according to Equations (20)–(28), which are presented in

Table 7. Weight vectors of criteria.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11
Weights	0.06	0.08	0.11	0.11	0.14	0.09	0.08	0.07	0.09	0.10	0.06

. The result shows that community and government responsibility fulfillment (C5) with weight 0.143, employee rights and interests (C4) with weight 0.113, and loan amount and frequency (C3) with weight 0.112 are the three most important risk criteria, occupying a larger proportion in the process of decision-making.

Steps 6–8: After the determination of criteria weights, the weighted normalized matrix $V$ can be calculated using Equation (32). According to Equations (33) and (34), values $RN(S_i^{+})$ for benefit criteria and values $RN(S_i^{-})$ for cost criteria can be obtained, as shown in the second and third columns of

Table 8. Values of risk criteria functions of alternatives and their ranking.

Alternative	$\mathit{R}\mathit{N} \left({\mathit{S}}_{\mathit{i}}^{+}\right)$	$\mathit{R}\mathit{N} \left({\mathit{S}}_{\mathit{i}}^{-}\right)$	$\mathit{R}\mathit{N} \left({\mathit{Q}}_{\mathit{i}}\right)$	$\mathit{R}\mathit{N} \left({\mathit{U}}_{\mathit{i}}\right)$	Rank
A1	[0.62, 0.63]	[0.01, 0.04]	[0.67, 0.71]	[93.0, 96.0]	2
A2	[0.36, 0.32]	[0.11, 0.11]	[0.36, 0.35]	[51.0, 47.0]	3
A3	[0.27, 0.29]	[0, 0]	[0.23, 0.26]	[32.0, 35.0]	4
A4	[0.65, 0.66]	[0.01, 0.04]	[0.72, 0.74]	[100.0, 100.0]	1

. In order to accurately determine the ranking order of enterprises, the values of $RN\left(Q_i\right)$ and $U_i$ were calculated as expressed in Table 8 using Equations (35) and (36), respectively. As the results show, alternative four lies at the top of the ranking, revealing that the fourth company is the optimal choice for financing, evaluated by the DMs according to the risk criteria, while alternative three is the least favored one.

The result suggests that: (1) The development level of enterprise SSCF is not only affected by internal economic factors, but also closely related to external social and environmental sustainability. The comprehensive performance under risk factors also determines whether an enterprise can successfully obtain financing. (2) Among the eleven defined risk criteria and the four alternative financing companies, we found that the importance of criterion: community and government responsibility fulfillment (C5), is most valued by financial institution DMs, while the fourth alternative (A4) has the highest degree of utility (most worthy of financing). (3) The proposed P-NRN-based model can help DMs in solving the dilemma of selecting the optimal medical enterprise for financing and formulating corresponding risk prevention measures under uncertainty to which the DMs may be exposed to during the decision-making process.

5. Comparison Analysis and Discussion

In this part, we apply some classical and representative MCGDM methods with uncertain vague information to analyze the risk evaluation and financing enterprise selection problem described above, and to compare the correlations and differences of ranking results obtained by different methods to demonstrate the effectiveness and superiority of the P-NRN-weight-determination techniques and P-NRN-COPRAS methods developed here. Three decision-making phases constitute the main structure of the proposed method, which are evaluation information integration, weight calculation, and ranking result acquisition. Since the proposed model integrates the advantages of NRN and plithogenic set theory in flexibly handing the uncertainties of DMs’ judgements, five other methods including neutrosophic weighted geometric average (NWGA) operator [59], BWM method [33], extended TOPSIS method [60], P-NRN-based-MABAC method [61], and VIKOR method [62] are selected and employed to perform the comparison of the same application example, of which the results are shown in

Table 9. Ranking results of different methods.

	NWGA Operator in [21]	$\mathit{R}\mathit{N}\left({\mathit{U}}_{\mathit{i}}\right)$	BWM Method	$\mathit{R}\mathit{N}\left({\mathit{U}}_{\mathit{i}}\right)$	Extended TOPSIS Method	$\mathit{R}\mathit{N}\left({\mathit{U}}_{\mathit{i}}\right)$
A1	2	[96.0, 97.0]	1	[100.0, 100.0]	2	[92.0, 95.0]
A2	3	[52.0, 48.0]	3	[44.0, 41.0]	3	[51.0, 48.0]
A3	4	[32.0, 35.0]	4	[33.0, 34.0]	4	[31.0, 35.0]
A4	1	[100.0, 100.0]	2	[80.0, 75.0]	1	[100.0, 100.0]
	P-NRN-Based MABAC	Preference Value	VIKOR Method	${\mathit{Q}}_{\mathit{i}}$	The Proposed Model	$\mathit{R}\mathit{N}\left({\mathit{U}}_{\mathit{i}}\right)$
A1	2	[0.17, 0.18]	4	0.85	2	[93.0, 96.0]
A2	3	[0.04, −0.01]	1	0.00	3	[51.0, 47.0]
A3	4	[−0.2, −0.2]	3	0.62	4	[32.0, 35.0]
A4	1	[0.22, 0.24]	2	0.55	1	[100.0, 100.0]

. It is worth noting that in the comparison we only change one method at a time, while the remaining methods are consistent with the model proposed in this paper.

From the results summarized in Table 9, it can be observed that rankings and optimal alternative obtained by different methods are generally consistent but slightly different individually. With the exception of the BWM and VIKOR methods, all methods show rankings consistent with the results of the proposed model, that is, $A_4\succ A_1\succ A_2\succ A_3$. Some inconsistent results are yielded among the listed methods, which can be explained and analyzed by the following reasons.

According to the first phase, comparison with evaluation information aggregation method, the first method listed in Table 9 applies a different aggregation operator, namely, the NWGA operator, to aggregate the evaluation values of different DMs in the form of NRNs and build the group assessment matrixes for further calculation. Likewise, the same weight determination method of risk criteria and DMs as in this paper are adopted in the NWGA-based model. Meanwhile, the COPRAS method is introduced to determine the relative significance and utility degree of the candidate companies. As can be seen from Table 9, the ranking results obtained by using the NWGA operator are exactly the same as those of the model proposed in this paper, with only slight differences in the upper and lower bounds of utility values. Although the two methods have their own advantages, the operator-based method requires a lot of complicated calculations and can thus easily cause loss of decision-making information. On the contrary, the plithogenic aggregating method based on the contradiction degree between the criteria proposed in this paper can not only preserve almost all the evaluation information of DMs, but can also improve the accuracy of aggregation with high consideration of uncertainty.

Next, we conduct a second-phase comparison, that is, the comparison of weighting methods, including the BWM method for determining criteria weights and the extended TOPSIS method for calculating the weights of DMs. (1) BWM is based on the best criterion and the worst criterion subjectively defined by DMs, and the weight for each criterion is obtained by comparing with other criteria [62]. The results obtained from Table 9 show that the ordering of the first two alternatives change while the rest remain the same. The difference is caused by a number of reasons, first, the characteristic of the BWM method is that the weighting is completely based on the subjective preference of DMs, and the different sizes of the set ideal weights reflect the different operation strategies used in criteria weighting. Therefore, different attitude preferences of DMs may affect the definition of these initial parameters, which in turn influences the value of the criterion weight, ultimately effecting the upper and lower bounds of utility value. It is still a tough challenge for DMs to make reasonable subjective judgments at the early stage of decision-making, and weight parameters need to be evaluated manually, with certain limitations in terms of flexibility and objectivity. By contrast, the MDMOM is more helpful to characterize the fuzzy and uncertain information in the DM’s judgment, which greatly enhances the objectivity and reduces the interference of unnecessary subjectivity during the determination process of criteria weights. (2) According to the results presented in Table 9, the rankings of alternatives obtained using the extended TOPSIS-based model showed the same decision, that $A_4$ is the optimal cooperative enterprise for the financing problem, consistent with the results obtained by the ESM method-based model applied in this paper. Although both methods have the ability to take into account the differential characteristics of DMs, the extended TOPSIS does not involve the common interaction features and similarities between DMs, but simply summarizes them as the distance from the maximum and minimum ideal solutions, which may lead to information bias in complex situations. Hence, the ranking orders of SSCF enterprises determined by the developed ESM-based model are more precise and comprehensive than those given by the traditional correlation methods when considering uncertainty evaluation information and the correlations among them.

When it comes to comparison of alternative ranking methods, different methods exhibit different ranking results, as shown in Table 9. The MABAC method, which prioritizes alternatives by calculating the distance between them and the border approximation area (BAA), is consistent with the results obtained in this paper. Different from the utility degree calculation of COPRAS, MABAC divides the decision-making matrix into upper and lower approximation regions, which also conforms to the definition of the upper and lower bounds of neutrosophic RNs. Therefore, the presented model can effectively address MCGDM problems with P-NRN information. In contrast, the proposed ranking method in this paper focuses more on the relative importance relationship among the alternatives, avoiding the precise evaluation of the classification, thus making the decision-making much easier. Finally, the most obvious ranking difference appears in the comparison with the VIKOR method, where $A_2$ is the most optimal and $A_1$ is the least favored. The following analysis can explain this consistent result, first, it should be noted that in the VIKOR method the P-NRN is not introduced for decision-making information aggregation, but only the general form of fuzzy number, ignoring the accuracy that can be improved by using the upper and lower bound features of the RN to summarize information in complex environments, which is likely to cause unnecessary information loss. In addition, the VIKOR method emphasizes maximizing group utility and minimizing personal regret to obtain a compromise solution that satisfies the corresponding constraints. Under this circumstance, the compromise solution may be one, two, or more, which may produce a reverse order and increase the burden of decision-making. The proposed method places more emphasis on the correlation characteristics of the local individual criteria and alternatives, rather than all ambiguous information as a whole, therefore it is more straightforward and more practical in real-life application. However, in contrast to Fang et al. [63], the proposed method does not take into account risk attitudes or aversion preferences among DMs, which is one of the aspects that needs to be improved in the future.

From the above comparative analysis, each method has its advantages, disadvantages, and applicable objects, none of which can always perform better than the others in any situation. Introducing the idea of P-NRN that combine the plithogenic aggregation operator and NRN in the process of MCGDM can comprehensively and specifically analyze, calculate, and evaluate the decision-making information from uncertain and vague environments. Therefore, the multi-dimension and multi-angle decision-making model constructed in this paper can obtain a more scientific information integration process, more reasonable weight information, and more feasible ranking results.

6. Conclusions

Due to the continuous improvement of people’s awareness of social, economic, and environmental harmony, the selection of financing enterprises aiming at sustainability has become a research hotspot of MCGDM. However, many factors exacerbate the impact of uncertainty and ambiguity inherent in decision-making, which may increase the burden on DMs and produce unrealistic evaluation outcomes. Inspired by classical RN and the plithogenic operator in group information aggregation, a P-NRN is presented in this paper to deal with the uncertainty and subjectivity in complex decisions by fusing plithogenic sets and RNs. An integrated MCGDM model is also introduced based on the application of P-NRNs. First, the construction process of P-NRNs is proposed, and some operational properties and aggregation operators about the plithogenic set and NRN are discussed. Next, the ESM method is applied to determine the weights of DMs, while the relative importance of each risk criterion is objectively determined by MDMOM, which avoids completely subjective judgments of DMs. Finally, we extend the COPRAS method to acquire the ranking results of the enterprises based on the obtained comprehensive evaluation value and weight information. The proposed model is validated through a case study of financing enterprise selection in the medical industry. Comparisons with representative MCGDM techniques also demonstrate the effectiveness and superiority of the proposed model.

The outstanding feature of this model is that it can make full use of objective information in the decision process and eliminate the negative influence brought about by uncertainty and subjectivity. The main advantages of this paper can be summarized as follows:

(1)

The application of P-NRN to express evaluation information can not only make up for the defects of traditional RNs that only use upper and lower approximate limit values to measure the diversity judgements of DMs, but also eliminate the subjectivity limitation brought about by using plithogenic aggregation alone.

(2)

Considering the weights of DMs and unknown risk criteria, we use the ESM and MDMOM methods to solve the weight information in the plithogenic environment.

(3)

We construct a COPRAS-based MCGDM model of the P-NRN environment, which enhances the persuasiveness of the decision-making results.

Despite the many superiorities in dealing with uncertain information, the proposed quantitative and objective decision-making model still has some limitations. Firstly, this paper only evaluates the initial information of financing alternatives from a quantitative perspective, but does not consider the possible probabilistic characteristics of criteria from a qualitative perspective. Secondly, the weights of risk criteria in the proposed model are calculated by an objective weighting method, without considering the comprehensive determination method combining subjectivity and objectivity. Thirdly, it is not enough to evaluate risk criteria numerically and ignore the decision-making attitudes of DMs toward risks. While in the application COPRAS, we did not take into account the risk-averse attitude of DMs. Therefore, in future research, we will focus on the combination of more complex language term sets with quantitative information to express uncertainty in a more comprehensive way [64]. Moreover, it is of great significance to develop a hybrid weighting method that considers both the subjectivity and objectivity of criteria. We also intend to apply other decision methods with unique advantages to the P-NRN-based MCGDM model to solve various real-life problems.