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Abstract

With the development of the internet economy, e-commerce has rapidly risen, and a large number of small and micro e-commerce enterprises have emerged. However, these enterprises have low financial information transparency, small scale, and high development uncertainty. Therefore, combining the characteristics of the internet economy, it is of great significance to dynamically evaluate credit risk. This not only helps to enhance the quality and rationality of credit risk evaluation results, but also helps to improve financing efficiency and reduce financing risks. The credit evaluation for small and micro enterprises is a multiple-attribute group decision-making (MAGDM). Recently, the TODIM (an acronym in Portuguese of interactive and multicriteria decision making) and TOPSIS method has been inaugurated to cope with MAGDM issues. The 2-tuple linguistic neutrosophic sets (2TLNSs) are inaugurated as an effective tool for characterizing uncertain information during the credit evaluation for small and micro enterprises. In this paper, the 2-tuple linguistic neutrosophic TODIM-TOPSIS (2TLNN-TODIM-TOPSIS) method is inaugurated to solve the MAGDM under 2TLNSs. Finally, a numerical case study for credit evaluation for small and micro enterprises is inaugurated to confirm the proposed method. The prime contribution of this paper are outlined: (1) The information entropy based on score function and accuracy function are built on the 2TLNSs to obtain weight information; (2) an integrated the 2-tuple linguistic neutrosophic TODIM-TOPSIS (2TLNN-TODIM-TOPSIS) method is established to cope with MAGDM; (3) An illustrative example for credit evaluation for small and micro enterprises has accomplished to illustrate the 2TLNN-TODIM-TOPSIS; (4) some comparative analysis are employed to verify the 2TLNN-TODIM-TOPSIS method.

Keywords

Multiple-attribute group decision-making (MAGDM)2-tuple linguistic neutrosophic sets (2TLNSs)TODIM TOPSIS credit evaluation for small and micro enterprises

1. Introduction

In the process of economic integration, the operational status of small and micro enterprises relies more on the external economic environment, and uncertainty has become the norm for their operations [1, 2]. Therefore, there is an increasing demand for credit information from small and micro enterprises among trading parties. The main theme of current development is that many macroeconomic control policies such as energy conservation and emission reduction, structural adjustment, etc. pose serious challenges to the development of small and micro enterprises [3, 4]. The development of small and micro enterprises, especially the traditional manufacturing industry, faces prominent problems such as market fluctuations, sharp increase in operating costs, and difficulties in sustained operation [5, 6, 7]. Therefore, small and micro enterprises have the characteristics of fast establishment, rapid transformation, and fast bankruptcy, and uncertainty has become the norm in the operation of small and micro enterprises. In the operation process of small and micro enterprises, they have a strong dependence on the environment they are located in [8, 9]. Once the environment changes, small and micro enterprises will be the first to be affected. The characteristics of the environment have become the background characteristics of small and micro enterprise operation, mainly manifested in the following aspects: firstly, the geographical characteristics of small and micro enterprises are obvious. In the actual operation process, the business projects and methods of small and micro enterprises are constrained by their location, and enterprises located in the same region are in the same business content. There is a strong similarity in business models [10, 11, 12, 13]. Secondly, the characteristics of the relationship are obvious, as compared to large enterprises, small and micro enterprises have narrower commercial relationships and are only concentrated in basic business units such as customers and management departments. Thirdly, there is a strong dependence on the economic environment, weak capital management strength, low risk resistance, and strong uncertainty in the operating environment. Fourthly, similar financial backgrounds refer specifically to the similar financial difficulties faced by small and micro enterprises. Small and micro enterprises have a small and frequent demand for funds, and have high requirements for effective service of funds [14, 15]. However, on the contrary, small and micro enterprises are unable to obtain sufficient funds, posing a high risk to their survival. Compared with large enterprises, although small and micro enterprises face many unfavorable factors such as small scale, weak capital strength, narrow business relationships, and weak risk resistance, they also have unique advantages in their business processes: high flexibility of capital, relatively concentrated business content, flexible organizational management, and more sensitive responses to market conditions and factors affecting business conditions [16, 17, 18]. The flexibility of organizational management endows enterprises with corresponding flexibility in their operations. The characteristic of survival in the gaps makes small and micro enterprises grasp market opportunities more advanced. Once business decisions are correct, they will adjust their business direction in a timely manner. Faced with extremely strong uncertainty and their own weak position in business operations, small and micro enterprises face enormous challenges and competitive pressure in competition [19, 20, 21]. Especially during times of economic downturn, stakeholders of small and micro enterprises hold a wait-and-see attitude. For example, banks observe the operational status and future development of small and micro enterprises on the one hand, and wait for national financial regulation policies on the other hand. Local governments are also watching the contribution of small and micro enterprises to the local community. In this situation, enterprises with good operating conditions and those that contribute greatly to the local government will receive support from the local government, enterprises with good financial credit will receive financial support from banks, and supplier customers will choose their trading partners based on the credit of the enterprise. Therefore, credit has become one of the competitiveness of enterprises [22, 23, 24]. There are a large number of small and micro enterprises in China with a large financing gap, and financial institutions such as banks will inevitably receive a large number of credit applications from these enterprises [25, 26]. In order to maximize the mutual benefit and win-win situation between financial institutions and enterprises, it is crucial to scientifically and reasonably evaluate the credit of small and micro enterprises [27, 28, 29, 30].

As decision problems and backgrounds become increasingly complex, it is difficult for a single expert to directly provide decision results, so more and more experts need to join in to solve the problem together [31, 32, 33, 34, 35]. In this case, multiple-attribute group decision-making (MAGDM) methods have emerged. MAGDM can gather expert evaluation information through different decision-making methods or aggregation operators, and then use various scoring functions and sorting methods to obtain the final decision result [36, 37, 38, 39, 40]. Among them, the background of MAGDM methods can be based on the qualitative and quantitative environment already provided, or it can be based on the fuzzy background environment, which provides experts with more evaluation environment choices [41, 42, 43, 44]. In today’s real life, most of the problems people encounter have a certain degree of fuzziness, that is, people cannot directly provide accurate evaluation information to make decisions [45, 46, 47, 48]. This provides research space for MAGDM problems in fuzzy backgrounds, and also attracts more and more scholars to study this [49, 50, 51]. The?credit evaluation for small and micro enterprises is MAGDM. Recently, the TODIM [52, 53] and TOPSIS [54] method was used to tackle the MAGDM. The 2TLNSs [55] are employed as an effective tool for depicting uncertain information during the credit evaluation for small and micro enterprises. Until now, no or few algorithms have been studied on information entropy and TODIM-TOPSIS under 2TLNSs. Therefore, an integrated the 2-tuple linguistic neutrosophic TODIM-TOPSIS (2TLNN-TODIM-TOPSIS) method is established to manage MAGDM. An illustrative example for credit evaluation for small and micro enterprises is inaugurated to validate the proposed method. The prime motivation and objectives of this paper are outlined: (1) The information entropy based on score function and accuracy function are built on the 2TLNSs to obtain weight information; (2) an integrated the 2-tuple linguistic neutrosophic TODIM-TOPSIS (2TLNN-TODIM-TOPSIS) method is established to cope with MAGDM; (3) An illustrative example for credit evaluation for small and micro enterprises has accomplished to illustrate the 2TLNN-TODIM-TOPSIS; (4) some comparative analysis are employed to verify the 2TLNN-TODIM-TOPSIS method.

The structure of this paper is inaugurated below. In Section 2, the 2TLNSs is introduced. In Section 3, 2TLNN-TODIM-TOPSIS method is inaugurated under 2TLNSs with entropy. Section 4 inaugurate an illustrative example for credit evaluation for small and micro enterprises and some comparative analysis. Some remarks are inaugurated in Section 5.

2. Preliminaries

Wang et al. [55] inaugurated the 2TLNSs.

Definition 1 [28, 29]. Let $ws_{1},ws_{2},\ldots,ws_{W}$ be linguistic variables. Any label $ws_{i}$ shows a possible linguistic variable, and $w s$ is inaugurated:

$\displaystyle ws=\left\{{\begin{array}[]{l}ws_{0}=\textit{extremely poor},ws_{% 1}=\textit{very poor},ws_{2}=\textit{poor},ws_{3}=\textit{medium},\\ ws_{4}=\textit{good},ws_{5}=\textit{very good},ws_{6}=\textit{extremely good}.% \\ \end{array}}\right\}$

Let $\Theta$ be a fix set, the SVNSs $w\eta$ is [6, 7]:

$\displaystyle w\eta=\{{{({w,\phi_{\eta}(w),\varphi_{\eta}(w),\gamma_{\eta}(w)}% )}|w\in\Theta}\}$ (1)

where $\phi_{\eta}(w),\varphi_{\eta}(w),\gamma_{\eta}(w)\in[{0,1}]$ represent truth-membership (TM), indeterminacy-membership (IM) and falsity-membership (FM), $0\leqslant\phi_{\eta}(w)+\varphi_{\eta}(w)+\gamma_{\eta}(w)\leqslant 3$ .

Definition 2 [55]. Let $w\delta_{j}({j=1,2,\ldots,k})$ be a 2TLSs. If $w\delta=\langle{({ws_{t},w\xi}),({ws_{i},w\psi}),({ws_{f},w\zeta})}\rangle$ is inaugurated for $ws_{t},ws_{i},ws_{f}\in ws,w\xi,w\psi,w\zeta\in[{0,{0.5})}$ , where $({ws_{t},w\xi}),({ws_{i},w\psi}),({ws_{f},w\zeta})$ depict the TM, IM and the FM through 2TLSs, then the 2TLNSs are designed:

$\displaystyle w\delta_{j}=\langle{({ws_{t_{j}},w\xi_{j}}),({ws_{i_{j}},w\psi_{% j}}),({ws_{f_{j}},w\zeta_{j}})}\rangle$ (2)

where $0\leqslant\Delta^{-1}({ws_{t_{j}},w\xi_{j}})\leqslant W,0\leqslant\Delta^{-1}(% {ws_{i_{j}},w\psi_{j}})\leqslant W,0\leqslant\Delta^{-1}({ws_{f_{j}},w\zeta_{j% }})\leqslant W$ , and $0\leqslant\Delta^{-1}({ws_{t_{j}},w\xi_{j}})+\Delta^{-1}({ws_{i_{j}},w\psi_{j}% })+\Delta^{-1}({ws_{f_{j}},w\zeta_{j}})\leqslant 3W$ .

Definition 3 [27]. Let $w\delta=\langle{({ws_{t},w\xi}),({ws_{i},w\psi}),({ws_{f},w\zeta})}\rangle$ . The score function $SF({w\delta})$ and accuracy function $AF({w\delta})$ are designed:

$\displaystyle SF({w\delta})=\frac{({2W+\Delta^{-1}({ws_{t},w\xi})-\Delta^{-1}(% {ws_{i},w\psi})-\Delta^{-1}({ws_{f},w\zeta})})}{3W},SF({w\delta})\in[{0,1}]$ (3) $\displaystyle AF(w\delta)=\frac{\Delta^{-1}({ws_{t},w\xi})-\Delta^{-1}({ws_{f}% ,w\zeta})+W}{2W},AF(w\delta)\in[{0,1}].$ (4)

Definition 4 [27]. Let $w\delta_{1}=\langle{({ws_{t_{1}},w\xi_{1}}),({ws_{i_{1}},w\psi_{1}}),({ws_{f_{% 1}},w\zeta_{1}})}\rangle$ and $w\delta_{2}=\langle({ws_{t_{2}},w\xi_{2}}),(ws_{i_{2}},\linebreak w\psi_{2}),(% {ws_{f_{2}},w\zeta_{2}})\rangle$ , then

$\displaystyle(1)\text{if }SF({w\delta_{1}})\prec SF({w\delta_{2}}),w\delta_{1}% \prec w\delta_{2};$ $\displaystyle(2)\text{if }SF({w\delta_{1}})=SF({w\delta_{2}}),AH({w\delta_{1}}% )\prec AH({w\delta_{2}}),w\delta_{1}\prec w\delta_{2};$ $\displaystyle(3)\text{if }SF({w\delta_{1}})=SF({w\delta_{2}}),AH({w\delta_{1}}% )=AH({w\delta_{2}}),w\delta_{1}=w\delta_{2};$

Definition 5 [27]. Let $w\delta_{1}=\langle{({ws_{t_{1}},w\xi_{1}}),({ws_{i_{1}},w\psi_{1}}),({ws_{f_{% 1}},w\zeta_{1}})}\rangle$ , $w\delta_{2}=\langle({ws_{t_{2}},w\xi_{2}}),({ws_{i_{2}},w\psi_{2}}),\linebreak% ({ws_{f_{2}},w\zeta_{2}})\rangle$ and $w\delta=\langle{({ws_{t},c\xi}),({ws_{i},w\psi}),({ws_{f},w\zeta})}\rangle$ be three 2TLNNs, then

$\displaystyle(1)w\delta_{1}\oplus w\delta_{2}=\left\{{\begin{array}[]{l}\Delta% \left({W\left({\frac{\Delta^{-1}({ws_{t_{1}},w\xi_{1}})}{W}+\frac{\Delta^{-1}(% {ws_{t_{2}},w\xi_{2}})}{W}-\frac{\Delta^{-1}({ws_{t_{1}},w\xi_{1}})}{W}\cdot% \frac{\Delta^{-1}({ws_{t_{2}},w\xi_{2}})}{W}}\right)}\right),\\ \\ \Delta\left({W\left({\frac{\Delta^{-1}({ws_{i_{1}},w\psi_{1}})}{W}\cdot\frac{% \Delta^{-1}({ws_{i_{2}},w\psi_{2}})}{W}}\right)}\right),\\ \\ \Delta\left({W\left({\frac{\Delta^{-1}({ws_{f_{1}},w\zeta_{1}})}{W}\cdot\frac{% \Delta^{-1}({ws_{f_{2}},w\zeta_{2}})}{W}}\right)}\right)\\ \end{array}}\right\};$ $\displaystyle(2)w\delta_{1}\otimes w\delta_{2}=\left\{{\begin{array}[]{l}% \Delta\left({W\left({\frac{\Delta^{-1}({ws_{t_{1}},w\xi_{1}})}{W}\cdot\frac{% \Delta^{-1}({ws_{t_{2}},w\xi_{2}})}{W}}\right)}\right),\\ \\ \Delta\left({W\left({\frac{\Delta^{-1}({ws_{i_{1}},w\psi_{1}})}{W}+\frac{% \Delta^{-1}({ws_{i_{2}},w\psi_{2}})}{W}-\frac{\Delta^{-1}({ws_{i_{1}},w\psi_{1% }})}{W}\cdot\frac{\Delta^{-1}({ws_{i_{2}},w\psi_{2}})}{W}}\right)}\right),\\ \\ \Delta\left({W\left({\frac{\Delta^{-1}({ws_{f_{1}},w\zeta_{1}})}{W}+\frac{% \Delta^{-1}({ws_{f_{2}},w\zeta_{2}})}{W}-\frac{\Delta^{-1}({ws_{f_{1}},w\zeta_% {1}})}{W}\cdot\frac{\Delta^{-1}({ws_{f_{2}},w\zeta_{2}})}{W}}\right)}\right)\\ \end{array}}\right\};$ $\displaystyle(3)\alpha w\delta=\left\{{\begin{array}[]{l}\Delta\left({W\left({% 1-\left({1-\frac{\Delta^{-1}({ws_{t},w\xi})}{W}}\right)^{\alpha}}\right)}% \right),\\ \\ \Delta\left({W\left({\frac{\Delta^{-1}({ws_{i},w\psi})}{W}}\right)^{\alpha}}% \right),\Delta\left({W\left({\frac{\Delta^{-1}({ws_{f},w\zeta})}{W}}\right)^{% \alpha}}\right)\\ \end{array}}\right\},\alpha>0;$ $\displaystyle(4)w\delta^{\alpha}=\left\{{\begin{array}[]{l}\Delta\left({W\left% ({\frac{\Delta^{-1}({ws_{t},w\xi})}{W}}\right)^{\alpha}}\right),\Delta\left({W% \left({1-\left({1-\frac{\Delta^{-1}({ws_{i},w\psi})}{W}}\right)^{\alpha}}% \right)}\right),\\ \\ \Delta\left({W\left({1-\left({1-\frac{\Delta^{-1}({ws_{f},w\zeta})}{W}}\right)% ^{\alpha}}\right)}\right)\\ \end{array}}\right\},\alpha>0.$

Definition 6 [56]. Let $w\delta_{1}=\langle{({ws_{t_{1}},w\xi_{1}}),({ws_{i_{1}},w\psi_{1}}),({ws_{f_{% 1}},w\zeta_{1}})}\rangle$ and $w\delta_{2}=\langle({ws_{t_{2}},w\xi_{2}}),(ws_{i_{2}},\linebreak w\psi_{2}),(% {ws_{f_{2}},w\zeta_{2}})\rangle$ , then the Hamming distance is inaugurated:

$\displaystyle HD({w\delta_{1},w\delta_{2}})=\frac{1}{3}\left({\begin{array}[]{% l}\left|{\frac{\Delta^{-1}({ws_{t_{1}},w\xi_{1}})-\Delta^{-1}({ws_{t_{2}},w\xi% _{2}})}{W}}\right|\\ \\ {}+\left|{\frac{\Delta^{-1}({ws_{i_{1}},w\psi_{1}})-\Delta^{-1}({ws_{i_{2}},w% \psi_{2}})}{W}}\right|\\ \\ {}+\left|{\frac{\Delta^{-1}({ws_{f_{1}},w\zeta_{1}})-\Delta^{-1}({ws_{f_{2}},w% \zeta_{2}})}{W}}\right|\\ \end{array}}\right)$ (5)

The 2TLNNWA is inaugurated as follow:

Definition 7 [55]. Let $w\delta_{j}=\{{({ws_{t_{j}},w\alpha_{j}}),({ws_{i_{j}},w\beta_{j}}),({ws_{f_{j% }},w\chi_{j}})}\}$ , the 2TLNNWA is presented:

$\displaystyle 2\text{TLNNWA}({w\delta_{1},w\delta_{2},\ldots,w\delta_{n}})$ $\displaystyle=w_{1}w\delta_{1}\oplus w_{2}w\delta_{2}\ldots\oplus w_{n}w\delta% _{n}=\mathop{\oplus}\limits_{j=1}^{n}w_{j}w\delta_{j}$ (6) $\displaystyle=\left({\begin{array}[]{l}\Delta\left({W\left({1-\prod\limits_{j=% 1}^{n}{\left({1-\frac{\Delta^{-1}({ws_{t_{j}},w\alpha_{j}})}{W}}\right)^{w_{j}% }}}\right)}\right),\\ \\ \Delta\left({W\prod\limits_{j=1}^{n}{\left({\frac{\Delta^{-1}({ws_{i_{j}},w% \beta_{j}})}{W}}\right)^{w_{j}}}}\right),\\ \\ \Delta\left({W\prod\limits_{j=1}^{n}{\left({\frac{\Delta^{-1}({ws_{f_{j}},w% \chi_{j}})}{W}}\right)^{w_{j}}}}\right).\\ \end{array}}\right)$

3. 2TLNN-TODIM-TOPSIS method for MAGDM with entropy weight

3.1 2TLNN-MAGDM issues description

In this section, 2TLNN-TODIM-TOPSIS method is built for MAGDM. Let $WA=\{WA_{1},WA_{2},\ldots,\linebreak WA_{m}\}$ be alternatives, and the attributes set $WG=\{{WG_{1},WG_{2},\ldots,WG_{n}}\}$ with weight $w c$ , where $ww_{j}\in[{0,1}]$ , $\sum\limits_{j=1}^{n}{ww_{j}}=1$ and a set of invited experts $WE=\{{WE_{1},WE_{2},\ldots,WE_{q}}\}$ , let expert’s weight be $\{{dw_{1},dw_{2},\ldots,dw_{t}}\}$ .

Then, 2TLNN-TODIM-TOPSIS method is inaugurated for MAGDM. The calculating steps are depicted:

Step 1. Build the 2TLNN-matrix $WR=[{WR_{ij}^{t}}]_{m\times n}=\{({ws_{t_{ij}}^{t},w\alpha_{ij}^{t}}),({ws_{i_% {ij}}^{t},w\beta_{ij}^{t}}),(ws_{f_{ij}}^{t},\linebreak w\chi_{ij}^{t})\}_{m% \times n}$ and calculate the average matrix $WR=[{WR_{ij}}]_{m\times n}$ :

$\displaystyle\begin{array}[]{l}\qquad\qquad\qquad\qquad\qquad\qquad{\begin{% array}[]{*{20}c}{\,WG_{1}}&{\,\,\,\,WG_{2}}&\,\,\ldots&{\,\,WG_{n}}\\ \end{array}}\\ R=[{WR_{ij}^{t}}]_{m\times n}={\begin{array}[]{*{20}c}{WA_{1}}\\ {WA_{2}}\\ \vdots\\ {WA_{m}}\\ \end{array}}\left[{{\begin{array}[]{*{20}c}{WR_{11}^{t}}&{WR_{12}^{t}}&\ldots&% {WR_{1n}^{t}}\\ {WR_{21}^{t}}&{WR_{22}^{t}}&\ldots&{WR_{2n}^{t}}\\ \vdots&\vdots&\vdots&\vdots\\ {WR_{m1}^{t}}&{WR_{m2}^{t}}&\ldots&{WR_{mn}^{t}}\\ \end{array}}}\right]\end{array}$ (7) $\displaystyle\begin{array}[]{l}\quad\qquad\qquad\qquad\qquad\qquad\qquad{% \begin{array}[]{*{20}c}{\,WG_{1}}&{\,\,\,\,WG_{2}}&\,\,\ldots&{\,\,WG_{n}}\\ \end{array}}\\ WR=[{WR_{ij}}]_{m\times n}={\begin{array}[]{*{20}c}{WA_{1}}\\ {WA_{2}}\\ \vdots\\ {WA_{m}}\\ \end{array}}\left[{{\begin{array}[]{*{20}c}{WR_{11}}&{WR_{12}}&\ldots&{WR_{1n}% }\\ {WR_{21}}&{WR_{22}}&\ldots&{WR_{2n}}\\ \vdots&\vdots&\vdots&\vdots\\ {WR_{m1}}&{WR_{m2}}&\ldots&{WR_{mn}}\\ \end{array}}}\right]\end{array}$ (8)

Based on 2TLNNWA, the $WR=[{WR_{ij}}]_{m\times n}=\{{({ws_{t_{ij}},w\alpha_{ij}}),({ws_{i_{ij}},w% \beta_{ij}}),({ws_{f_{ij}},w\chi_{ij}})}\}_{m\times n}$ is:

$\displaystyle WR_{ij}=dw_{1}WR_{ij}^{1}\oplus dw_{2}WR_{ij}^{2}\oplus\ldots% \oplus dw_{t}WR_{ij}^{t}$ $\displaystyle=\left\{{\begin{array}[]{l}\Delta\left({W\left({1-\prod\limits_{d% =1}^{t}{\left({1-\frac{\Delta^{-1}({ws_{t_{ij}},w\alpha_{ij}})^{t}}{W}}\right)% ^{dw_{t}}}}\right)}\right),\\ \Delta\left({W\prod\limits_{d=1}^{t}{\left({\frac{\Delta^{-1}({ws_{i_{ij}},w% \beta_{ij}})^{t}}{W}}\right)^{dw_{t}}}}\right),\\ \Delta\left({W\prod\limits_{d=1}^{t}{\left({\frac{\Delta^{-1}({ws_{f_{ij}},w% \chi_{ij}})^{t}}{W}}\right)^{dw_{t}}}}\right)\\ \end{array}}\right\}$ (9)

Step 2. Normalize the $WR=[{WR_{ij}}]_{m\times n}$ into $\textit{NWR}=[{\textit{NWR}_{ij}}]_{m\times n}$ .

For benefit attributes:

$\displaystyle\textit{NWR}_{ij}=WR_{ij}=\{{({ws_{t_{ij}},w\alpha_{ij}}),({ws_{i% _{ij}},w\beta_{ij}}),({ws_{f_{ij}},w\chi_{ij}})}\}$ (10)

For cost attributes:

$\displaystyle\textit{NWR}_{ij}=\{{({ws_{t_{ij}},w\alpha_{ij}}),({ws_{i_{ij}},w% \beta_{ij}}),({ws_{f_{ij}},w\chi_{ij}})}\}$ (11) $\displaystyle=\{{\Delta({W-\Delta^{-1}({ws_{t_{ij}},w\alpha_{ij}})}),\Delta({W% -\Delta^{-1}({ws_{i_{ij}},w\beta_{ij}})}),\Delta({W-\Delta^{-1}({ws_{f_{ij}},w% \chi_{ij}})})}\}$

3.2 Inaugurate the attributes weight by information entropy

Entropy [57] is a conventional theory to derive weight. Firstly, the normalized 2TLNN-matrix $WW_{ij}$ is obtained:

$\displaystyle WW_{ij}=\frac{\left({\begin{array}[]{l}SF\{{({ws_{t_{ij}},w% \alpha_{ij}}),({ws_{i_{ij}},w\beta_{ij}}),({ws_{f_{ij}},w\chi_{ij}})}\}\\ {}+AF\{{({ws_{t_{ij}},w\alpha_{ij}}),({ws_{i_{ij}},w\beta_{ij}}),({ws_{f_{ij}}% ,w\chi_{ij}})}\}\\ \end{array}}\right)}{\sum\limits_{i=1}^{m}{\left({\begin{array}[]{l}SF\{{({ws_% {t_{ij}},w\alpha_{ij}}),({ws_{i_{ij}},w\beta_{ij}}),({ws_{f_{ij}},w\chi_{ij}})% }\}\\ {}+AF\{{({ws_{t_{ij}},w\alpha_{ij}}),({ws_{i_{ij}},w\beta_{ij}}),({ws_{f_{ij}}% ,w\chi_{ij}})}\}\\ \end{array}}\right)}},$ (12)

Then, the 2TLNN Shannon entropy $2\textit{TLNNSE}=({2\textit{TLNNSE}_{1},2\textit{TLNNSE}_{2},\ldots,2\textit{% TLNNSE}_{n}})$ is inaugurated:

$\displaystyle 2\textit{TLNNSE}_{j}=-\frac{1}{\ln m}\sum\limits_{i=1}^{m}{WW_{% ij}\ln WW_{ij}}$ (13)

and $WW_{ij}\ln WW_{ij}=0$ if $WW_{ij}=0$ .

Then, the weights $ww=({ww_{1},ww_{2},\ldots,ww_{n}})$ is inaugurated:

$\displaystyle ww_{j}=\frac{1-2\textit{TLNNSE}_{j}}{\sum\limits_{j=1}^{n}{({1-2% \textit{TLNNSE}_{j}})}},j=1,2,\ldots,n.$ (14)

3.3 2TLNN-TODIM-TOPSIS method for MAGDM

Then, the 2TLNN-TODIM-TOPSIS method is inaugurated to cope with MAGDM.

(1) Inaugurate relative weight of $WG_{j}$ as:

$\displaystyle\textit{rww}_{j}={ww_{j}}/{ww_{j}^{*}},$ (15)

$ww_{j}^{*}$ is the maximum weight of the attributes, and $0\leqslant ww_{j}\leqslant 1$ .

(2) The dominance degree (DD) $DD_{j}({WA_{i},WA_{t}})$ of $WA_{i}$ over $WA_{t}$ for $WG_{j}$ is inaugurated.

$\displaystyle DD_{j}({WA_{i},WA_{t}})=\left\{{{\begin{array}[]{ll}{\sqrt{\frac% {\textit{rww}_{j}\times HD({\textit{NWR}_{ij},\textit{NWR}_{tj}})}{\sum% \nolimits_{j=1}^{n}{\textit{rww}_{j}}}}}&{\text{if }SF({\textit{NWR}_{ij}})>SF% ({\textit{NWR}_{tj}})}\\ 0&{\text{if }SF({\textit{NWR}_{ij}})=SF({\textit{NWR}_{tj}})}\\ {-\frac{1}{\theta}\sqrt{\frac{\sum\nolimits_{j=1}^{n}{\textit{rww}_{j}\times HD% ({\textit{NWR}_{ij},\textit{NWR}_{tj}})}}{\textit{rww}_{j}}}}&{\text{if }SF({\textit{NWR}_{ij}})<SF({\textit{NWR}_{tj}})}\\ \end{array}}}\right.$ (16)

where the parameter $\theta$ means the risk factor for the gain and loss.

The DD matrix $DD_{j}({WA_{i}})({j=1,2,\ldots,n})$ is inaugurated:

$\displaystyle DD_{j}({WA_{i}})=[{DD_{j}({WA_{i},WA_{t}})}]_{m\times m}$ $\displaystyle\qquad\qquad\qquad\qquad{\begin{array}[]{*{20}c}{WA_{1}}&{\;\;\;% \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;WA_{2}}&{\;\;\;\;\;\;\;\;\;\ldots}&{\;\;\;% \;\;\;\;\;\;\;\;WA_{m}}\\ \end{array}}$ $\displaystyle={\begin{array}[]{*{20}c}{WA_{1}}\\ {WA_{2}}\\ \vdots\\ {WA_{m}}\\ \end{array}}\left[{{\begin{array}[]{*{20}c}0&{DD_{j}({WA_{1},WA_{2}})}&\ldots&% {DD_{j}({WA_{1},WA_{m}})}\\ {DD_{j}({WA_{2},WA_{1}})}&0&\ldots&{DD_{j}({WA_{2},WA_{m}})}\\ \vdots&\vdots&\ldots&\vdots\\ {DD_{j}({WA_{m},WA_{1}})}&{DD_{j}({WA_{m},WA_{2}})}&\ldots&0\\ \end{array}}}\right]$

(3) Inaugurate the overall DD of $WA_{i}$ over other alternatives for $WG_{j}$ :

$\displaystyle DD_{j}({WA_{i}})=\sum\limits_{t=1}^{m}{DD_{j}({WA_{i},WA_{t}})}$ (17)

The overall DD is inaugurated:

$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\@setsize{% \scriptsize}{9.5pt}{\viiipt}{\@viiipt}\begin{array}[]{*{20}c}{WG_{1}}&{\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,WG_{2}}&{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ldots}&{\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,WG_{n}}\\ \end{array}}$ $\displaystyle DD=({DD_{ij}})_{m\times n}={\begin{array}[]{*{20}c}{WA_{1}}\\ WA_{2}\\ \vdots\\ {WA_{m}}\\ \end{array}}\left[{{\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}\begin{% array}[]{cccc}{\sum\limits_{t=1}^{m}{DD_{1}({WA_{1},WA_{t}})}}&{\sum\limits_{t% =1}^{m}{DD_{2}({WA_{1},WA_{t}})}}&\ldots&{\sum\limits_{t=1}^{m}{DD_{n}({WA_{1}% ,WA_{t}})}}\\ {\sum\limits_{t=1}^{m}{DD_{1}({WA_{2},WA_{t}})}}&{\sum\limits_{t=1}^{m}{DD_{2}% ({WA_{2},WA_{t}})}}&\ldots&{\sum\limits_{t=1}^{m}{DD_{n}({WA_{2},WA_{t}})}}\\ \vdots&\vdots&\vdots&\vdots\\ {\sum\limits_{t=1}^{m}{DD_{1}({WA_{m},WA_{t}})}}&{\sum\limits_{t=1}^{m}{DD_{2}% ({WA_{m},WA_{t}})}}&\ldots&{\sum\limits_{t=1}^{m}{DD_{n}({WA_{m},WA_{t}})}}\\ \end{array}}}\right]$

(4) Produce the 2TLNN positive ideal solution (2TLNNPIS) and 2TLNN negative ideal solution (2TLNNNIS):

$\displaystyle 2\textit{TLNNPIS}=({2\textit{TLNNPIS}_{1},2\textit{TLNNPIS}_{1},% \ldots,2\textit{TLNNPIS}_{n}})$ (18) $\displaystyle 2\textit{TLNNNIS}=({2\textit{TLNNNIS}_{1},2\textit{TLNNNIS}_{1},% \ldots,2\textit{TLNNNIS}_{n}})$ (19) $\displaystyle 2\textit{TLNNPIS}_{j}=\mathop{\max}\limits_{j=1}^{n}DD_{ij},2% \textit{TLNNNIS}_{j}=\mathop{\min}\limits_{j=1}^{n}DD_{ij}$ (20)

(5) Produce the 2TLNN Euclidean distance and the 2TLNN closeness coefficient (2TLNNCC) to the 2TLNNPIS. The alternative has the maximum 2TLNNCC which is the most desirable alternative.

$\displaystyle 2\textit{TLNNED}({WA_{i},2\textit{TLNNPIS}})=\sqrt{\sum\limits_{% j=1}^{n}{({DD_{ij}-2\textit{TLNNPIS}_{j}})^{2}}}$ (21) $\displaystyle 2\textit{TLNNED}({WA_{i},2\textit{TLNNNIS}})=\sqrt{\sum\limits_{% j=1}^{n}{({DD_{ij}-2\textit{TLNNNIS}_{j}})^{2}}}$ (22) $\displaystyle 2\textit{TLNNCC}({WA_{i},2\textit{TLNNPIS}})$ (23) $\displaystyle=\frac{2\textit{TLNNED}({WA_{i},2\textit{TLNNNIS}})}{2\textit{% TLNNED}({WA_{i},2\textit{TLNNNIS}})+2\textit{TLNNED}({WA_{i},2\textit{TLNNPIS}% })}$

4. An empirical example and comparative analysis

4.1 An empirical example

For over 20 years, corporate credit has been a hot topic of concern for various banks, and corporate credit is the main basis for banks to issue loans. Therefore, the construction of a credit system has always been a major task for banks. The “Regulations on the Administration of Credit Reporting Industry”, which came into effect on March 15, 2013, limited the main credit reporting agencies to various banks. However, the collection, organization, storage, processing, and provision of credit product information to information users for enterprises are not limited to banks. Credit rating agencies also engage in similar credit product supply businesses. At present, the demand for credit products is mainly concentrated in banks and the government. However, in an uncertain environment, due to information asymmetry, the demand for the credit status of small and micro enterprises is increasing and showing a diversified trend. The demand entities include not only credit institutions, but also small credit enterprises, suppliers of enterprises, customers, and other entities. However, there are significant differences in evaluation methods, content, and format of credit reports when banks and rating agencies provide credit products. Therefore, it is necessary to establish reasonable credit evaluation indicators and establish a comprehensive credit rating system to provide comparable and universal credit information for credit demanders, in order to meet the needs of all parties and achieve the goal of saving social costs. The credit evaluation for small and micro enterprises is MAGDM issue. Therefore, the credit evaluation for small and micro enterprises is presented to demonstrate the approach developed in this paper. There is a panel with five potential small and micro enterprises $WA_{i}({i=1,2,3,4,5})$ to choose. The experts select four attributes to assess the five potential small and micro enterprises [1]:

⟀ WG₁ is the financial credit of an enterprise. The financial credit of an enterprise is the core of enterprise credit evaluation. The construction of financial strength is composed of past business accumulation, future financial planning, and current fund operation status, and is a comprehensive credit evaluation indicator. This indicator is evaluated based on debt paying ability, and is analyzed and evaluated through a series of financial indicators. However, when using this indicator for evaluation, the profit centered indicator system should be changed and a cash flow centered indicator system should be established. Firstly, attach importance to the analysis of the cash inflow ability of enterprise operating activities, which reflects the core “hematopoietic” function of the enterprise. In a normally operating enterprise, the cash inflow from its operations should be obtained through operating activities to ensure the circulation and turnover of funds. For the cash inflows from asset disposal or asset realization, they are more reflected in investment activities. For the cash flows brought about by the company’s investment income, it can strengthen the company’s hematopoietic capacity. However, if the cash flows brought about by the disposal of long-term assets, a more in-depth analysis of the reasons for the disposal should be conducted. For the financing ability of a company, to some extent, it reflects its “blood transfusion” ability. In the expansion stage of the company, this cash flow is undoubtedly the financial support for the development of the company, but in other development stages, the guarantee of the company’s operating activities cash flow for the company’s funds should be considered. Secondly, pay attention to the proportion of enterprise research and development investment. The R&D capability of an enterprise belongs to its soft power, and the proportion of R&D investment to a certain extent reflects the future operational and financial strength of the enterprise.

⟁ WG₂ is the development credit of an enterprise. The development credit of an enterprise is a credit indicator that measures its future development status and is influenced by the following three aspects: firstly, the management, strategy, and competitive strength of the enterprise are mainly reflected in its hard power and soft power. The hard power of the enterprise is determined by its basic information, specifically referring to its human, financial, and other resources. The existence of tangible and tangible elements such as equipment, capital strength, and economic scale of an enterprise is an objective condition for its development. Soft power is centered around the system, technology, competitiveness, and other contents of the enterprise, and is realized as the human resource team, the charisma and planning power of decision-makers, the execution power of employees, and various systems of the enterprise. Secondly, the ability of a company to resist risks refers to its ability to maintain normal business operations under uncertain conditions, and can also refer to its ability to respond to risks. In uncertain environments, the risks faced by enterprises are intensifying and showing a diversified trend. The risk resistance ability of enterprises is related to their future development ability and level. In the credit evaluation process, the evaluation of future development status must consider the enterprise’s risk resistance ability. The level of a company’s ability to resist risks depends on its ability to raise funds, innovate technology, expand production and market, manage flexibility, be aware of risks, and control risks. Thirdly, the innovation ability of a company, as its technological innovation becomes the source of its competitive strength and the fundamental foundation for its survival and development, is related to its future development and profitability. Therefore, the evaluation of a company’s technological innovation capability has become a factor that affects its credit level. The evaluation of an enterprise’s innovation ability should be based on multiple perspectives, including the composition of the R&D team, the quality of the R&D personnel, the investment of R&D funds, the promotion and application of research results, and the scientific research cooperation between the enterprise and external research institutions (such as universities).

⟂ WG₃ is willingness credit. The measurement of a company’s willingness and strength is mainly based on the willingness and quality of the company to actively take on social responsibility in its past business processes, as well as the integrity of small and micro enterprise leaders and decision-makers. Willingness credit belongs to a higher level of credit in the process of credit evaluation, belonging to the moral level, and the overall connotation of credit is also a moral category. The two are to some extent consistent, and the past willingness credit of enterprises will be reflected in their past credit records. For example, in the past business process, whether the enterprise has paid taxes in a timely manner, whether it has evaded taxes, and whether it has defaulted on employee wages; Whether there is collusion behavior in bidding activities, whether there are illegal business activities, whether there have been fines from environmental protection departments, whether there have been malicious payments owed to customers, whether there have been falsified false financial information to obtain credit reports, and whether there are serious financial problems, reflecting the willingness of the enterprise to assume responsibility in the past business process and leading to corresponding credit records in the corresponding period. The strength of a company’s willingness depends on the character of its leaders, the abilities and qualities of its leaders and decision-makers, the charisma of its leaders, and the execution ability of its employees. The willingness and credit of a company can enhance its soft power, attract external attention and recognition, enhance its credibility, and directly enhance its comprehensive competitiveness.

⟃ WG₄ is quality credit. In the process of building a credit system, quality credit is the foundation of the entire credit system. In the operation process of small and micro enterprises, due to the lack of support from credit funds, the funds obtained through the operation process are difficult to maintain the demand for funds for independent innovation and expansion of the enterprise. Due to the drive of interests, short-sighted behavior of enterprises, inadequate supervision and management systems, and inadequate punishment, 90% of quality hazards come from small and micro enterprises or small and medium-sized suppliers of large enterprises. Therefore, the quality credit evaluation of small and micro enterprises is a necessary factor to consider, but in the traditional credit evaluation process, the quality of enterprise products or services is not included in the credit evaluation indicator system.

The five possible small and micro enterprises $WA_{i}({i=1,2,3,4,5})$ are to be evaluated with linguistic scales (see Table 1) with the four criteria by three experts $WE_{t}({t=1,2,3})$ (Suppose expert’s weight is (0.35, 0.35, 0.30).

Table 1
Linguistic scale and 2TLNNs

Linguistic terms	2TLNNs
Exceedingly Terrible-WET	$\{(ws_{0},0),(ws_{5},0),(ws_{6},0)\}$
Very Terrible-WVT	$\{(ws_{1},0),(ws_{4},0),(ws_{5},0)\}$
Terrible-WT	$\{(ws_{2},0),(ws_{3},0),(ws_{4},0)\}$
Medium-WM	$\{(ws_{3},0),(ws_{3},0),(ws_{3},0)\}$
Well-WW	$\{(ws_{4},0),(ws_{3},0),(ws_{2},0)\}$
Very Well-WVW	$\{(ws_{5},0),(ws_{2},0),(ws_{1},0)\}$
Exceedingly Well-WEW	$\{(ws_{6},0),(ws_{1},0),(ws_{0},0)\}$

The 2TLNN-TODIM-TOPSIS method is inaugurated to solve the credit evaluation for small and micro enterprises.

Step 1. Inaugurate the 2TLNN matrix $WR=[{WR_{ij}^{t}}]_{5\times 4}$ (see Tables 2–4).

Table 2

Evaluation information by $WE_{1}$

	WG₁ (benefit)	WG₂ (cost)	WG₃ (benefit)	WG₄ (benefit)
WA₁	WW	WM	WVT	WVW
WA₂	WM	WT	WVW	WW
WA₃	WW	WVW	WVT	WM
WA₄	WVT	WVW	WM	WVT
WA₅	WVW	WVT	WM	WT

Table 3

Evaluation information by $WE_{2}$

	WG₁ (benefit)	WG₂ (cost)	WG₃ (benefit)	WG₄ (benefit)
WA₁	WVW	WW	WM	WT
WA₂	WVW	WT	WVT	WM
WA₃	WM	WVW	WT	WW
WA₄	WVT	WM	WVW	WW
WA₅	WVT	WVW	WM	WW

Table 4

Evaluation information by $WE_{3}$

	WG₁ (benefit)	WG₂ (cost)	WG₃ (benefit)	WG₄ (benefit)
WA₁	WVW	WW	WM	WT
WA₂	WVT	WT	WW	WVW
WA₃	WT	WM	WM	WW
WA₄	WT	WM	WW	WVW
WA₅	WM	WW	WVT	WVT

Then according to 2TLNNWA, the $WR=[{WR_{ij}}]_{5\times 4}$ is inaugurated (see Table 5).

Step 2. Normalize the $WR=[{WR_{ij}}]_{5\times 4}$ into $\textit{NWR}=[{\textit{NWR}_{ij}}]_{5\times 4}$ (see Table 6).

Step 3. Inaugurate the weight: $ww_{1}=0.1005,ww_{2}=0.3026,ww_{3}=0.3506,ww_{4}=0.2463$ .

Step 4. Inaugurate the relative weight: $\textit{rww}=\{0.2867,0.8631,1.0000,0.7025\}$ .

Step 5. Inaugurate the $DD=({DD_{ij}})_{5\times 4}$ (see Table 7).

Step 7. Obtain the 2TLNNPIS and 2TLNNNIS (see Table 8).

Step 8. Calculate the $({DD_{ij}-2\textit{TLNNPIS}_{j}})^{2}$ and $({DD_{ij}-2\textit{TLNNNIS}_{j}})^{2}$ (see Tables 9 and 10).

Step 17. Calculate the $2\textit{TLNNED}({WA_{i},2\textit{TLNNPIS}})$ , $2\textit{TLNNED}({WA_{i},2\textit{TLNNNIS}})$ and $2\textit{TLNNCC}\linebreak({WA_{i},2\textit{TLNNPIS}})$ (see Tables 11 and 12).

Thus, the best small and micro enterprise is $WA_{1}$ .

4.2 Comparative analysis

Then, the 2TLNN-TODIM-TOPSIS method is compared with 2TLNNWA operator and 2TLNNWG operator [55], 2-tuple linguistic neutrosophic weighted Hamy mean (2TLNWHM) operator [58], 2-tuple linguistic neutrosophic weighted dual Hamy mean (2TLNWDHM) operator [58], 2TLNN-GRA method [59], 2TLNN-CODAS method [60], and 2TLNN-TODIM method [56]. The comparative decision results are shown in Table 13.

Table 5
The $WR=[{WR_{ij}}]_{5\times 4}$

	WG₁	WG₂
WA₁	$\{{({ws_{5},-0.38}),({ws_{2},0.32}),({ws_{1},0.34})}\}$	$\{{({ws_{3},-0.42}),({ws_{3},0.26}),({ws_{3},0.43})}\}$
WA₂	$\{{({ws_{4},0.14}),({ws_{3},-0.38}),({ws_{2},-0.16})}\}$	$\{{({ws_{4},0.42}),({ws_{3},-0.49}),({ws_{2},-0.06})}\}$
WA₃	$\{{({ws_{2},-0.26}),({ws_{3},0.23}),({ws_{4},0.28})}\}$	$\{{({ws_{3},-0.42}),({ws_{3},0.21}),({ws_{3},0.43})}\}$
WA₄	$\{{({ws_{4},0.03}),({ws_{3},0.12}),({ws_{2},0.16})}\}$	$\{{({ws_{5},-0.26}),({ws_{2},0.23}),({ws_{1},0.39})}\}$
WA₅	$\{{({ws_{4},-0.19}),({ws_{3},-0.26}),({ws_{2},0.34})}\}$	$\{{({ws_{3},-0.27}),({ws_{3},0.37}),({ws_{3},0.23})}\}$
	WG₄	WG₃
WA₁	$\{{({ws_{3},0.32}),({ws_{3},0.26}),({ws_{3},-0.35})}\}$	$\{{({ws_{4},-0.15}),({ws_{3},-0.32}),({ws_{2},0.16})}\}$
WA₂	$\{{({ws_{3},0.05}),({ws_{3},0.18}),({ws_{3},-0.02})}\}$	$\{{({ws_{4},0.32}),({ws_{2},0.45}),({ws_{2},-0.36})}\}$
WA₃	$\{{({ws_{4},-0.43}),({ws_{3},-0.23}),({ws_{2},0.45})}\}$	$\{{({ws_{3},-0.27}),({ws_{3},0.29}),({ws_{3},0.12})}\}$
WA₄	$\{{({ws_{4},-0.13}),({ws_{3},-0.43}),({ws_{2},0.15})}\}$	$\{{({ws_{2},-0.27}),({ws_{3},0.32}),({ws_{4},0.09})}\}$
WA₅	$\{{({ws_{4},-0.16}),({ws_{3},-0.26}),({ws_{2},0.19})}\}$	$\{{({ws_{1},0.39}),({ws_{4},-0.34}),({ws_{5},-0.17})}\}$

Table 6

The $\textit{NWR}=[{\textit{NWR}_{ij}}]_{5\times 4}$

	WG₁	WG₂
WA₁	$\{{({ws_{5},-0.38}),({ws_{2},0.32}),({ws_{1},0.34})}\}$	$\{{({ws_{3},-0.42}),({ws_{3},0.26}),({ws_{3},0.43})}\}$
WA₂	$\{{({ws_{4},0.14}),({ws_{3},-0.38}),({ws_{2},-0.16})}\}$	$\{{({ws_{4},0.42}),({ws_{3},-0.49}),({ws_{2},-0.06})}\}$
WA₃	$\{{({ws_{2},-0.26}),({ws_{3},0.23}),({ws_{4},0.28})}\}$	$\{{({ws_{3},-0.42}),({ws_{3},0.21}),({ws_{3},0.43})}\}$
WA₄	$\{{({ws_{4},0.03}),({ws_{3},0.12}),({ws_{2},0.16})}\}$	$\{{({ws_{5},-0.26}),({ws_{2},0.23}),({ws_{1},0.39})}\}$
WA₅	$\{{({ws_{4},-0.19}),({ws_{3},-0.26}),({ws_{2},0.34})}\}$	$\{{({ws_{3},-0.27}),({ws_{3},0.37}),({ws_{3},0.23})}\}$
	WG₄	WG₃
WA₁	$\{{({ws_{3},0.32}),({ws_{3},0.26}),({ws_{3},-0.35})}\}$	$\{{({ws_{4},-0.15}),({ws_{3},-0.32}),({ws_{2},0.16})}\}$
WA₂	$\{{({ws_{3},0.05}),({ws_{3},0.18}),({ws_{3},-0.02})}\}$	$\{{({ws_{4},0.32}),({ws_{2},0.45}),({ws_{2},-0.36})}\}$
WA₃	$\{{({ws_{4},-0.43}),({ws_{3},-0.23}),({ws_{2},0.45})}\}$	$\{{({ws_{3},-0.27}),({ws_{3},0.29}),({ws_{3},0.12})}\}$
WA₄	$\{{({ws_{4},-0.13}),({ws_{3},-0.43}),({ws_{2},0.15})}\}$	$\{{({ws_{2},-0.27}),({ws_{3},0.32}),({ws_{4},0.09})}\}$
WA₅	$\{{({ws_{4},-0.16}),({ws_{3},-0.26}),({ws_{2},0.19})}\}$	$\{{({ws_{1},0.39}),({ws_{4},-0.34}),({ws_{5},-0.17})}\}$

Table 7

The $DD=({DD_{ij}})_{5\times 4}$

	WG₁	WG₂	WG₃	WG₄
WA₁	$-$ 0.2112	0.9536	0.3092	$-$ 0.3643
WA₂	0.2029	0.3192	$-$ 1.7283	0.2791
WA₃	$-$ 0.9466	$-$ 0.7968	0.7723	$-$ 0.7844
WA₄	0.6422	$-$ 0.4474	$-$ 0.4604	0.6223
WA₅	$-$ 0.8590	$-$ 1.2653	$-$ 0.5488	$-$ 0.0383

Table 8

The 2TLNNPIS and 2TLNNNIS

	WG₁	WG₂	WG₃	WG₄
2TLNNPIS	0.6422	0.9536	0.7723	0.6223
2TLNNNIS	$-$ 0.9466	$-$ 1.2653	$-$ 1.7283	$-$ 0.7844

Table 9

The $({DD_{ij}-2\textit{TLNNPIS}_{j}})^{2}$

	FG₁	FG₂	FG₃	FG₄
WA₁	0.7283	0.0000	0.2145	0.9734
WA₂	0.1929	0.4025	6.2533	0.1178
WA₃	2.5242	3.0639	0.0000	1.9788
WA₄	0.0000	1.9627	1.5196	0.0000
WA₅	2.2536	4.9236	1.7453	0.4363

Table 10

The $({DD_{ij}-2\textit{TLNNNIS}_{j}})^{2}$

	FG₁	FG₂	FG₃	FG₄
WA₁	0.5408	4.9236	4.1516	0.1765
WA₂	1.3215	2.5106	0.0000	1.1310
WA₃	0.0000	0.2195	6.2533	0.0000
WA₄	2.5242	0.6690	1.6077	1.9788
WA₅	0.0077	0.0000	1.3914	0.5567

Table 11

The $2\textit{TLNNED}({WA_{i},2\textit{TLNNPIS}})$ , $2\textit{TLNNED}({WA_{i},2\textit{TLNNNIS}})$

Alternative	$2\textit{TLNNED}({WA_{i},2\textit{TLNNPIS}})$	$2\textit{TLNNED}({WA_{i},2\textit{TLNNNIS}})$
WA₁	1.3843	3.1293
WA₂	2.6394	2.2278
WA₃	2.7508	2.5442
WA₄	1.8661	2.6038
WA₅	3.0592	1.3985

Table 12

The $2\textit{TLNNCC}({WA_{i},2\textit{TLNNPIS}})$

Alternative	$2\textit{TLNNCC}({WA_{i},2\textit{TLNNPIS}})$	Order
WA₁	0.6933	1
WA₂	0.4577	4
WA₃	0.4805	3
WA₄	0.5825	2
WA₅	0.3137	5

Table 13

Order of the different methods

	Order
2TLNNWA operator [55]	$WA_{1}>WA_{4}>WA_{3}>WA_{2}>WA_{5}$
2TLNNWG operator [55]	$WA_{1}>WA_{4}>WA_{2}>WA_{3}>WA_{5}$
2TLNWHM operator [58]	$WA_{1}>WA_{4}>WA_{3}>WA_{2}>WA_{5}$
2TLNWDHM operator [58]	$WA_{1}>WA_{4}>WA_{3}>WA_{2}>WA_{5}$
2TLNN-GRA method [59]	$WA_{1}>WA_{4}>WA_{3}>WA_{2}>WA_{5}$
2TLNN-CODAS method [60]	$WA_{1}>WA_{4}>WA_{3}>WA_{2}>WA_{5}$
2TLNN-TODIM method [56]	$WA_{1}>WA_{4}>WA_{2}>WA_{3}>WA_{5}$

From the above comparative analysis, it could be known that the order of these models is slightly different, but all these decision models have the same optimal small and micro enterprise and worst small and micro enterprise. This verifies the 2TLNN-TODIM-TOPSIS method is reasonable and effective. Moreover, the proposed 2TLNN-TODIM-TOPSIS method in this paper supplied a more reliable and precise decision analysis than other decision methods of decision-making. The main advantages of this paper are that the proposed 2TLNN-TODIM-TOPSIS method could consider the bounded rationality of the decision makers and solve MAGDM problems with conflicting or non-commensurable attributes.

5. Conclusion

Financial indicators such as profitability and solvency in the credit evaluation system of small and micro enterprises have been seriously affected by the COVID-19 epidemic. The government’s various policy tools to support small and micro enterprises will play a positive role in their survival and development, helping them overcome difficulties, resume production, and improve profitability. At the same time, the top-level national policy design also provides basic guidance and guidance for commercial banks to apply the credit evaluation system for small and micro enterprises under the epidemic and implement credit policies. The?credit evaluation for small and micro enterprises is classical MAGDM. Recently, the TODIM-TOPSIS method was inaugurated to solve MAGDM. The 2TLNSs are inaugurated as a tool for depicting uncertain information during the credit evaluation for small and micro enterprises. In this paper, the 2TLNN-TODIM-TOPSIS model is inaugurated to deal with the MAGDM under 2TLNSs. In the end, a numerical case study for credit evaluation for small and micro enterprises is inaugurated to show the 2TLNN-TODIM-TOPSIS model. The prime contribution of this paper are outlined: (1) The information entropy based on score function and accuracy function are built on the2TLNSs to obtain weight information; (2) an integrated the 2-tuple linguistic neutrosophic TODIM-TOPSIS (2TLNN-TODIM-TOPSIS) method is established to cope with MAGDM; (3) An illustrative example for credit evaluation for small and micro enterprises has accomplished to illustrate the 2TLNN-TODIM-TOPSIS; (4) some comparative analysis are employed to verify the 2TLNN-TODIM-TOPSIS method.

Based on the research conclusions of this article, the following suggestions are proposed: (1) Continuously improve the credit rating system for small and medium-sized enterprises. Continuously building and improving the credit evaluation system for small and medium-sized enterprises, formulating scientific and unified credit evaluation standards for small and medium-sized enterprises, achieving credit data interconnection, providing support for financial institutions to conduct risk assessments of small and medium-sized enterprises, and effectively addressing their financing difficulties; (2) Improve the quality of data collection and improve artificial intelligence technology. Continuously improve the comprehensiveness and effectiveness of enterprise data collection, and be cautious of missing and incorrect important indicator data. At the same time, continuously delving into machine learning model methods to improve the accuracy of credit risk assessment models; (3) Financial institutions should strengthen research and analysis on the management of small and medium-sized enterprises. Financial institutions should not only pay attention to the basic operating and financial conditions of enterprises, but also strengthen their attention to the management of enterprises, focusing on the collection and analysis of data on the size, gender, age, shareholding ratio, and meeting frequency of the management of enterprises; (4) Small and medium-sized enterprises should strengthen their attention to their innovation capabilities and focus on controlling the relationship between enterprise innovation and credit risk. Enterprises should control their investment in innovation and stimulate business vitality while ensuring the normal operation of their production and business activities and invest cautiously in high-risk innovative businesses.

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Enhanced group decision-making through an intelligent algorithmic approach for multiple-attribute credit evaluation with 2-tuple linguistic neutrosophic sets

Abstract

Keywords

1. Introduction

2. Preliminaries

3.1 2TLNN-MAGDM issues description

4.1 An empirical example

Table 1 Linguistic scale and 2TLNNs

Table 5 The W ⁢ R = [ W ⁢ R i ⁢ j ] 5 × 4

References

Table 1
Linguistic scale and 2TLNNs

Table 5
The $WR=[{WR_{ij}}]_{5\times 4}$