A novel MABAC approach for multi-attribute group decision-making with single-valued neutrosophic sets: An application in assessing microfinance group lending performance

Abstract

In the 40 years of global microcredit practice, loan technology has played a positive role in microcredit as one of the most important supporting elements. The development and evolution of microcredit institution lending technology is the result of comprehensive consideration of specific regional economic, social, cultural, and geographical factors. In the context of the diversified trend of microcredit technology, choosing loan technology reasonably, exploring flexible guarantee conditions, and innovating diversified loan technology combinations will become practical problems faced by microcredit institutions, and also the direction of theoretical research. The timely innovation of group loan technology in microcredit has practical value and theoretical significance for promoting the innovation of financial agricultural products in the implementation of China’s rural revitalization strategy, as well as bridging the theoretical controversy of microcredit loan technology. The performance evaluation of microfinance groups lending is a MAGDM issues. In this paper, the distances measures of single-valued neutrosophic sets (SVNSs) and maximizing deviation method (MDM) is used to obtain the attribute weight values. Based on the classical Multi-Attributive Border Approximation area Comparison (MABAC) method, the single-valued neutrosophic numbers MABAC (SVNN-MABAC) method is constructed for MAGDM under SVNSs. Finally, an example for performance evaluation of microfinance groups lending and some comparative decision analysis are constructed to verify the SVNN-MABAC model.

Keywords

MAGDM single-valued neutrosophic sets (SVNSs)MABAC model MDM microfinance groups lending

1. Introduction

In the practice of microcredit in China for over 20 years, group loan technology has always been the most important loan technology [1, 2]. Group loans have always been the most important loan technology in poverty alleviation projects of international institutions, microcredit practices of poverty alleviation societies, and government promoted microcredit poverty alleviation projects [3, 4]. However, during the implementation process, technical elements such as weekly meetings, sequential loans, and group funds were revised. In addition, among the small loans promoted by rural credit cooperatives, the loan technology arrangement of “letter of credit $+$ joint guarantee loan” has made significant revisions to the group loan technology of Grameen Bank. Since 2005, commercial small loan companies have been piloting more diversified loan technologies such as mortgage loans and guaranteed loans [5, 6]. Overall, the credit technology of microcredit institutions in China can be divided into two situations: one is the complete introduction and adaptive transformation of group loan technology from the first-generation model of Grameen Bank, and the other is that group loan has not been the main loan technology since the beginning [7, 8, 9]. The former case can be seen as follows: the “Poverty Alleviation Community” practice of the Agricultural Development Institute of the Chinese Academy of Social Sciences started in October 1993, the microfinance poverty alleviation practice of the Luonan County government started in 1993, the microfinance demonstration in Yilong County of Sichuan Province started in 1995, and the microfinance of rural cooperative financial institutions started in 1998. These credit projects have basically completely introduced group loan technology from the first-generation model of the Bangladesh Grameen Bank, and have adapted some practices during operation, such as abandoning the weekly meeting system and changing the “2-2-1” loan to a one-time unified loan. Facts have proven that group loan technology has played a positive role in maintaining credit discipline and ensuring repayment rates [10, 11, 12]. However, from another perspective, it can also be seen that for the success of a project, loan technology is only a supporting condition, and the social environment in the operation of credit projects also plays a crucial role [13, 14, 15]. The repeated practice of the Poverty Alleviation Society and the overall failure of the Luonan County government to help the poor through credit have verified that the macro financial system is a more important factor than the loan technology from one side. Only by maintaining the independent operation of credit projects and avoiding government intervention, can group loan technology play its due role. The latter situation begins with the practice of commercialized microcredit. In the pilot program of commercial small loan companies that began in 2005, there was a trend of diversification in loan technology. Most commercial microcredit companies mainly rely on mortgage loans, guaranteed loans, and pledged loans, but they are clearly more flexible [16, 17, 18]. Items that cannot be mortgaged by other financial institutions, such as houses, agricultural machinery, etc., can also become valuable collateral for small loan companies on a voluntary basis. Small loan companies can recognize guarantees that are not recognized by other financial institutions. The overall conclusion of theoretical analysis in the past 20 years is that group loans have created a horizontal supervision mechanism and alleviated Moral hazard in rural credit. However, the horizontal supervision mechanism also faces supervision costs and certain limitations, resulting in limited effective space for action. If we review and prospect the technological development process of microcredit, it is not difficult to find that the emergence of group loan technology and the influence of cultural, geographical and other factors play an important role in the replication process in various regions, leading to the revision of group loan technology [19, 20]. Various variations of group loan technology creatively maintain the most important technical elements while correcting some unsuitable technical elements, taking into account regional factors. Similarly, the importance attached to loan technology factors other than joint and several liability also indicates that there are new technological factors that affect the repayment rate of microcredit, which has impacted the monopoly position of group loan technology in the microcredit industry [21]. These practical activities confirm each other with relevant theoretical conclusions. At the same time, the development of microcredit over the past 30 years also indicates that the diversification trend of microcredit technology is becoming increasingly evident. The early monopoly position of group loans in microcredit has gradually been broken. In the context of commercialization and industrial development, group loan technology is also facing the impact of loan technologies such as progressive loans, flexible mortgage conditions, and financial collateral [17, 22]. Even in some institutions, they have abandoned group lending technology and completely shifted to other individual based lending technologies [13]. Therefore, under the condition of diversified technologies coexisting, the choice and trade-off of group loan technology by microcredit institutions are issues that need further explanation. For example, the impact of different loan technology choices on the selection of institutional customer groups, the interrelationship with social capital, the difficulty of financial regulatory enforcement, and the connection with institutional social goals still need to be further studied in the next step.

The multi-attribute decision-making (MADM) method is widely used in various fields such as project evaluation management and engineering technology analysis [23, 24, 25, 26, 27, 28, 29]. It is a decision-making method that comprehensively measures multiple attribute factors to obtain the optimal solution [30, 31]. With the rapid development of technology and the deepening understanding of social practice, people are facing increasingly complex decision-making problems [32, 33, 34, 35, 36]. With the rapid development of computer technology, the ways and types of information obtained by people about the same thing are also gradually increasing, making attribute information gradually showing diversified characteristics. In the current context, MADM problems with single attribute values are gradually not meeting practical needs. Traditional MADM problems mainly focus on studying deterministic decision-making problems. However, due to the limitations of human cognitive level and the disturbance of objective things, data information often exhibits uncertainty [37, 38, 39, 40, 41, 42]. Multi attribute group decision-making (MAGDM) refers to a decision maker composed of multiple decision experts who jointly express opinions on multiple attributes of a set of alternative solutions, thereby ranking the advantages and disadvantages of each alternative solution and selecting the best solution [43, 44, 45, 46, 47]. At present, the core issues of MAGDM methods mainly include the following two aspects: firstly, how decision information is aggregated, namely the information aggregation operator [48, 49, 50, 51, 52]; The other is how to meet group consensus in the decision-making process, that is, the consensus reaching process [53, 54, 55, 56, 57].

Pamucar and Cirovic [58] first constructed the MABAC in 2015. Su et al. [59] extended MABAC model based on prospect theory for MAGDM under probabilistic uncertain linguistic environment. Simic et al. [60] integrated CRITIC and MABAC based type-2 neutrosophic model to cope with public transportation pricing system selection. Mishra et al. [61] chose the sustainable supplier selection by using HF-DEA-FOCUM-MABAC technique. Jiang et al. [62] defined the Picture fuzzy MABAC method based on prospect theory for MAGDM for supplier selection. Ahmad et al. [63] defined the MABAC under non-linear diophantine fuzzy numbers for emergency decision support systems. Zhang et al. [64] defined the CPT-MABAC method for spherical fuzzy MAGDM and its application to green supplier selection. Rong et al. [65] assessed the MOOCs based on multigranular unbalanced hesitant fuzzy linguistic MABAC model. Estiri et al. [66] defined the multi-attribute framework for the selection of high-performance work systems based on the hybrid DEMATEL-MABAC method. Chen et al. [67] defined the MABAC and CRITIC method under the linguistic Z-numbers environment. Su et al. [59] defined the MABAC method based on prospect theory for MAGDM under probabilistic uncertain linguistic environment. Shen et al. [68] extended Z-MABAC model based on regret theory for regional circular economy development program selection with Z-information. However, it’s obvious that the existing decision studies about the modified MABAC method [58] based on the maximizing deviation method (MDM) model [69] under SVNNs with completely unknown weight information is not existing. Hence, it’s very crucial to investigate the modified MABAC method based on the maximizing deviation method (MDM) model under SVNNs with completely unknown weight information. The main purpose of this work is to construct the SVNN-MABAC model based on MDM model to cope with MAGDM with completely unknown weight information more effectively. In this paper, the distances measures of single-valued neutrosophic sets (SVNSs) and maximizing deviation method (MDM) is used to obtain the attribute weight values. Based on the classical Multi-Attributive Border Approximation Area Comparison (MABAC) method, the single-valued neutrosophic numbers MABAC (SVNN-MABAC) method is constructed for MAGDM under SVNSs. Finally, an example for performance evaluation of microfinance groups lending and some comparative decision analysis are constructed to verify the SVNN-MABAC model.

The reminder framework of this paper proceeds. The SVNSs is constructed in Sec. 2. The SVNN-MABAC method for MAGDM is constructed in Sec. 3. An example for performance evaluation of microfinance groups lending is constructed to verify the superiority in Sec. 4. the conclusion is fully constructed in Sec. 5.

2. Preliminaries

Wang et al. [70] constructed the given SVNSs.

Definition 1 [70]. The SVNSs $P C$ in $Y$ is constructed:

$\displaystyle PC=\{{({y,CT(y),CI(y),CF(y)})|{y\in Y}}\}$ (1)

with constructed truth-membership $CT(y)$ , constructed indeterminacy-membership $CI(y)$ and constructed falsity-membership $CF(y),CT(y),CI(y),CF(y)\in[{0,1}]$ , $0\leqslant CT(y)+CI(y)+CF(y)\leqslant 3$ .

The SVNN is expressed as $PC=({CT,CI,CF})$ , $CT,CI,CF\in[{0,1}]$ , $0\leqslant CT+CI+CF\leqslant 3$ .

Definition 2 [71]. Let $PC=({CT,CI,CF})$ be the SVNN, the score value is:

$\displaystyle SV({PC})=\frac{({2+CT-CI-CF})}{3},SV({PC})\in[{0,1}].$ (2)

Definition 3 [71]. Let $PC=({CT,CI,CF})$ be the SVNN, the accuracy value is:

$\displaystyle AV({PC})=CT-CF,HV({PC})\in[{-1,1}].$ (3)

Peng et al. [71] came up with order decision relation for SVNNs.

Definition 4 [71]. Let $PC=({CT,CI,CF})$ and $PD=({DT,DI,DF})$ be SVNNs, let $SV({PC})=\frac{({2+CT-CI-CF})}{3}$ and $SV({PD})=\frac{({2+DT-DI-DF})}{3}$ , and let $AV({PC})=CT-CF$ and $AV({PD})=DT-DF$ , if $SV({PC})<SV({PD})$ , $PC<PD$ ; if $SV({PC})=SV({PD})$ , then (1) if $AV({PC})=AV({PD})$ , $PC=PD$ ; (2) if $AV({PC})=AV({PD})$ , $PC<PD$ .

Definition 5 [72]. Let $PC=({CT,CI,CF})$ and $PD=({DT,DI,DF})$ be SVNNs, the basic operations are constructed:

$\displaystyle(1)\;PC\oplus PD=({CT+DT-CT\cdot DT,CI\cdot DI,CF\cdot DF});$ $\displaystyle(2)\;PC\otimes PD=({CT\cdot DT,CI+DI-CI\cdot DI,CF+DF-CF\cdot DF});$ $\displaystyle(3)\;\lambda PC=({1-({1-CT})^{\lambda},({CI})^{\lambda},({CF})^{% \lambda}}),\lambda>0;$ $\displaystyle(4)\;({PC})^{\lambda}=({({CT})^{\lambda},({CI})^{\lambda},1-({1-% CF})^{\lambda}}),\lambda>0.$

Definition 6 [73]. Let $PC=({CT,CI,CF})$ and $PD=({DT,DI,DF})$ , then the distance measure between $PC=({CT,CI,CF})$ and $PD=({DT,DI,DF})$ is constructed:

$\displaystyle DM({PC,PD})=\max\left\{{\frac{2+CT-CI-CF}{3},\frac{2+DT-DI-DF}{3% }}\right\}{}-\min\left\{{\frac{2+CT-CI-CF}{3},\frac{2+DT-DI-DF}{3}}\right\}$ (4)

3. The MDM-MABAC method for MAGDM with SVNNs

Let $CZ=\{{CZ_{1},CZ_{2},\ldots,CZ_{n}}\}$ be attributes, $wc=\{{wc_{1},wc_{2},\ldots,wc_{n}}\}$ be the weight of attributes $CZ_{j}$ . Let $CP=\{{CP_{1},CP_{2},\ldots,CP_{m}}\}$ be alternatives. And $CQ=({CQ_{ij}})_{m\times n}=(CT_{ij},CI_{ij},\linebreak CF_{ij})_{m\times n}$ is the SVNNmatrix. The MDM-MABAC method is constructed for MAGDM with SVNNs (See Fig. 1).

Figure 1.

MDM-MABAC method for MAGDM with SVNNs.

Step 1. Construct the SVNN matrix $CQ=({CQ_{ij}})_{m\times n}=({CT_{ij},CI_{ij},CF_{ij}})_{m\times n}$ .

$\displaystyle CQ=({CQ_{ij}})_{m\times n}=\left[{{\begin{array}[]{*{20}c}{CQ_{1% 1}}&{CQ_{12}}&\ldots&{CQ_{1n}}\\ {CQ_{21}}&{CQ_{22}}&\ldots&{CQ_{2n}}\\ \vdots&\vdots&\vdots&\vdots\\ {CQ_{m1}}&{CQ_{m2}}&\ldots&{CQ_{mn}}\\ \end{array}}}\right]$ (5)

Step 2. Normalize matrix $CQ=({CQ_{ij}})_{m\times n}=({CT_{ij},CI_{ij},CF_{ij}})_{m\times n}$ to $\textit{NCQ}=({\textit{NCQ}_{ij}})_{m\times n}=({\textit{NCT}_{ij},\textit{NCI% }_{ij},\textit{NCF}_{ij}})_{m\times n}$ .

$\displaystyle\textit{NCQ}_{ij}=({\textit{NCT}_{ij},\textit{NCI}_{ij},\textit{% NCF}_{ij}})=\left\{{{\begin{array}[]{l}({CT_{ij},CI_{ij},CF_{ij}}),CZ_{j}% \textit{ is a benefit criterion}\\ ({CF_{ij},CI_{ij},CT_{ij}}),CZ_{j}\textit{ is a cost criterion}\\ \end{array}}}\right.$ (6)

Step 3. Construct the weight information through the MDM model.

The maximizing deviation method (MDM) model [69] is a useful tool to construct the weight information. Until now, the MDM model has been employed in different decision domains [74, 75, 76, 77, 78, 79, 80, 81, 82]. Then, the MDM model is constructed to obtain the weight values under SVNNs.

(1) Construct the deviation of $CP_{i}$ from other alternatives is defined through $\textit{NCQ}=({\textit{NCQ}_{ij}})_{m\times n}=({\textit{NCT}_{ij},\textit{NCI% }_{ij},\textit{NCF}_{ij}})_{m\times n}$ .

$\displaystyle\textit{SVNNDM}_{ij}=\sum\limits_{t=1}^{m}{wc_{j}\cdot DM({% \textit{NCQ}_{ij},\textit{NCQ}_{tj}})}$ (7)

where

$\displaystyle DM({\textit{NCQ}_{ij},\textit{NCQ}_{tj}})=\max\left\{{\frac{2+% \textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3},\frac{2+\textit{NCT}% _{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}{}-\min\left\{{\frac{2+% \textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3},\frac{2+\textit{NCT}% _{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}.$

(2) Construct the weighted deviation.

$\displaystyle\textit{SVNNDM}_{j}({wc})=\sum\limits_{i=1}^{m}{\textit{SVNNDM}_{% ij}}({wc})=\sum\limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{wc_{j}}}\left({\begin{% array}[]{l}\max\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF% }_{ij}}{3},\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}% \right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right)$ (8)

(3) Construct the programming decision model.

$\displaystyle({M-1})$ (9) $\displaystyle\left\{{\begin{array}[]{l}\max DM({wc})=\sum\limits_{j=1}^{n}{% \sum\limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{wc_{j}}}\left({\begin{array}[]{l}% \max\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3},% \frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right)}\\ s.t.\,wz_{j}\geqslant 0,\sum\limits_{j=1}^{n}{wc_{j}^{2}=1}\\ \end{array}}\right.$

To cope with the constructed model, The Lagrange function is employed to cope with this model.

$\displaystyle L({wc,\xi})=\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{m}{\sum% \limits_{t=1}^{m}{wc_{j}}}\left({\begin{array}[]{l}\max\left\{{\frac{2+\textit% {NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3},\frac{2+\textit{NCT}_{tj}-% \textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right){}+\frac{\xi}{2}\left({\sum\limits_{j=1}^{n}{wc_{j}^{2}-1}}\right)$ (10)

where $\xi$ is the Lagrange decision multiplier, the partial derivatives are constructed.

$\displaystyle\left\{{\begin{array}[]{l}\frac{\partial L}{\partial wc_{j}}=\sum% \limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{\left({\begin{array}[]{l}\max\left\{{% \frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3},\frac{2+% \textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right)+\xi wc_{j}=0}}\\ \frac{\partial L}{\partial\xi}=\frac{1}{2}\left({\sum\limits_{j=1}^{n}{wc_{j}^% {2}-1}}\right)=0\\ \end{array}}\right.$ (11)

And the weight information is constructed.

$\displaystyle wc_{j}^{*}=\frac{\sum\limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{% \left({\begin{array}[]{l}\max\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{% ij}-\textit{NCF}_{ij}}{3},\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{% NCF}_{tj}}{3}}\right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right)}}}{\sqrt{\sum\limits_{j=1}^{n}{\left({\sum\limits_{i=1}^{m% }{\sum\limits_{t=1}^{m}{\left({\left({\begin{array}[]{l}\max\left\{{\frac{2+% \textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3},\frac{2+\textit{NCT}% _{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right)}\right)}}}\right)^{2}}}}$ (12)

Finally, the normalized weight values are constructed.

$\displaystyle wc_{j}=\frac{\sum\limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{\left({% \begin{array}[]{l}\max\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-% \textit{NCF}_{ij}}{3},\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}% _{tj}}{3}}\right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right)}}}{\sum\limits_{j=1}^{n}{\sum\limits_{i=1}^{m}{\sum\limits% _{t=1}^{m}{\left({\begin{array}[]{l}\max\left\{{\frac{2+\textit{NCT}_{ij}-% \textit{NCI}_{ij}-\textit{NCF}_{ij}}{3},\frac{2+\textit{NCT}_{tj}-\textit{NCI}% _{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ -\min\left\{{\frac{2+\textit{NCT}_{ij}-\textit{NCI}_{ij}-\textit{NCF}_{ij}}{3}% ,\frac{2+\textit{NCT}_{tj}-\textit{NCI}_{tj}-\textit{NCF}_{tj}}{3}}\right\}\\ \end{array}}\right)}}}}$ (13)

Step 4. Construct the weighted matrix $\textit{WCQ}=({\textit{WCQ}_{ij}})_{m\times n}$ :

$\displaystyle\textit{WCQ}_{ij}=({\textit{WCT}_{ij},\textit{WCI}_{ij},\textit{% WCF}_{ij}})=wc_{j}\cdot\textit{NCQ}_{ij}=({1-({1-\textit{NCT}_{ij}})^{wc_{j}},% ({\textit{NCI}_{ij}})^{wc_{j}},({\textit{NCF}_{ij}})^{wc_{j}}})$ (14)

Step 5. Construct the SVNN border approximation area (SVNNBAA).

$\displaystyle\textit{SVNNBAA}_{j}=({\textit{WCT}_{j}^{\textit{BAA}},\textit{% WCI}_{j}^{\textit{BAA}},\textit{WCF}_{j}^{\textit{BAA}}})=\left({\prod\limits_% {i=1}^{m}{({\textit{WCT}_{ij}})^{\frac{1}{m}}},\prod\limits_{i=1}^{m}{({% \textit{WCI}_{ij}})^{\frac{1}{m}}},1-\prod\limits_{i=1}^{m}{({1-\textit{WCF}_{% ij}})^{\frac{1}{m}}}}\right)$ (15)

Step 6. Construct the SVNN distance matrix (SVNNDM) from SVNNBAA.

$\displaystyle\textit{SVNNDM}_{ij}=\left\{{\begin{array}[]{l}({DM({\textit{WCQ}% _{ij},\textit{SVNNBAA}_{j}})}),\text{if }SV({\textit{WCQ}_{ij}})\geqslant S({% \textit{SVNNBAA}_{j}}),\\ -({DM({\textit{WCQ}_{ij},\textit{SVNNBAA}_{j}})}),\text{if }SV({\textit{WCQ}_{% ij}})<S({\textit{SVNNBAA}_{j}}).\\ \end{array}}\right.$ (16)

Step 7. Construct the SVNN order value (SVNNOV).

$\displaystyle\textit{SVNNOV}_{i}=\sum\limits_{j=1}^{n}{\textit{SVNNDM}_{ij}}$ (17)

Step 8. Sort the alternatives with $\textit{SVNNOV}_{i}$ . The higher value of $\textit{SVNNOV}_{i}$ is, the optimal choice will be.

4. The empirical example and comparative analysis

4.1 An empirical example

Muhammad Yunus once said: “Complaining that the poor cannot provide loan support without collateral is like complaining that people cannot fly to the sky without wings.” More than 100 years ago, the Wright brothers invented the airplane, which successfully made human beings soar. 70 years later, Yunus and the Grameen Bank under his guidance issued loans to the extremely poor in rural Bangladesh through group lending, successfully achieving extremely high repayment rates. Group lending is the most legendary lending technology in the field of microcredit. It involves the voluntary formation of credit groups by the poor, who are jointly and severally responsible for repayment, effectively solving the repayment problem of unsecured loans. Since the 1980s to 1990s, with the promotion of institutions such as the World Bank, the Grameen Bank’s group loan technology has been promoted in rural credit markets in Asia, Africa, and Latin America. Even in developed countries such as the United States and Canada, small and medium-sized enterprise credit has adopted group loan technology similar to that of Grameen Bank. Group lending is the most legendary lending technology in the field of microcredit. It involves the voluntary formation of credit groups by the poor, who are jointly and severally responsible for repayment, effectively solving the repayment problem of unsecured loans. Since the 1980s to 1990s, with the promotion of institutions such as the World Bank, the Grameen Bank’s group loan technology has been promoted in rural credit markets in Asia, Africa, and Latin America. Even in urban credit in developed countries such as the United States and Canada, small and medium-sized enterprise credit has adopted group loan technology similar to the Grameen Bank. Due to the unique nature of organizational forms, group loans have become synonymous with microcredit to a certain extent. People often first pay attention to the unique credit groups and joint liability of group loans. In theoretical research, the joint and several liability of group loans is also considered an important guarantee for low default rates of small credit institution customers. However, the development of microcredit practices has reversed this certain trend. In other regions, the adaptability of group lending technology has been increasingly questioned. Therefore, it is of great significance to track the development and evolution of the relevant theoretical literature’s understanding of the technical performance of group loans, sort out the relevant explanations of the technical mechanism of group loans, clear up the relevant misunderstandings about group loans, objectively analyze the adaptability of group loans, and select and develop appropriate microfinance technologies. The performance evaluation of microfinance groups lending is the MAGDM. In this paper, an empirical example for performance evaluation of microfinance groups lending was constructed based on the SVNN-MABAC model. There are five microfinance groups $CP_{i}({i=1,2,3,4,5})$ are to evaluated the performance of microfinance groups lending. In order to assess these microfinance groups fairly, the experts depict their assessment information with four constructed attributes depicted in Table 1.

Table 1
Four constructed attributes

Attributes	Attributes descriptions
CZ ${}_{1}$ (benefit)	Pay back ability for microfinance groups
CZ ${}_{2}$ (cost)	Loan amount for microfinance groups
CZ ${}_{3}$ (benefit)	Loan utilization rate for microfinance groups
CZ ${}_{4}$ (benefit)	Loan repayment ability for microfinance groups

Then, the MDMMABAC method is employed to cope with the performance evaluation of microfinance groups lending with SVNNs. The built MDMMABAC method involves the following steps:

Step 1. Construct the SVNN-matrix $CQ=({CQ_{ij}})_{5\times 4}$ (Table 2) according to five experts’ decision information with equal weight values.

Table 2

The $CQ=({CQ_{ij}})_{5\times 4}$

	CZ ${}_{1}$	CZ ${}_{2}$	CZ ${}_{3}$	CZ ${}_{4}$
PP ${}_{1}$	(0.52, 0.36, 0.58)	(0.38, 0.43, 0.54)	(0.58, 0.36, 0.25)	(0.48, 0.29, 0.35)
PP ${}_{2}$	(0.54, 0.17, 0.53)	(0.51, 0.64, 0.25)	(0.62, 0.21, 0.46)	(0.52, 0.41, 0.26)
PP ${}_{3}$	(0.46, 0.15, 0.37)	(0.42, 0.29, 0.53)	(0.34, 0.21, 0.52)	(0.51, 0.38, 0.32)
PP ${}_{4}$	(0.57, 0.51, 0.36)	(0.34, 0.42, 0.76)	(0.31, 0.43, 0.29)	(0.59, 0.46, 0.27)
PP ${}_{5}$	(0.63, 0.39, 0.42)	(0.34, 0.35, 0.47)	(0.37, 0.25, 0.37)	(0.39, 0.63, 0.54)

Step 2. Normalize $CQ=({CQ_{ij}})_{5\times 4}$ to $\textit{NCQ}=[{\textit{NCQ}_{ij}}]_{5\times 4}$ (See Table 3).

Table 3

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

	CZ ${}_{1}$	CZ ${}_{2}$	CZ ${}_{3}$	CZ ${}_{4}$
PP ${}_{1}$	(0.52, 0.36, 0.58)	(0.54, 0.43, 0.38)	(0.58, 0.36, 0.25)	(0.48, 0.29, 0.35)
PP ${}_{2}$	(0.54, 0.17, 0.53)	(0.25, 0.64, 0.51)	(0.62, 0.21, 0.46)	(0.52, 0.41, 0.26)
PP ${}_{3}$	(0.46, 0.15, 0.37)	(0.53, 0.29, 0.42)	(0.34, 0.21, 0.52)	(0.51, 0.38, 0.32)
PP ${}_{4}$	(0.57, 0.51, 0.36)	(0.76, 0.42, 0.34)	(0.31, 0.43, 0.29)	(0.59, 0.46, 0.27)
PP ${}_{5}$	(0.63, 0.39, 0.42)	(0.47, 0.35, 0.34)	(0.37, 0.25, 0.37)	(0.39, 0.63, 0.54)

Step 3. Construct the attribute weights in Table 4.

Table 4

The attributes weights

	CZ ${}_{1}$	CZ ${}_{2}$	CZ ${}_{3}$	CZ ${}_{4}$
$wc_{j}$	0.2425	0.1747	0.2816	0.3012

Step 4. Construct the weighted matrix $\textit{WCQ}=({\textit{WCQ}_{ij}})_{5\times 4}$ (Table 5).

Table 5

The $\textit{WCQ}=({\textit{WCQ}_{ij}})_{5\times 4}$

CZ ${}_{1}$	CZ ${}_{2}$
(0.4215, 0.3872, 0.4218)	(0.4309, 0.2217, 0.2064)
(0.2236, 0.4153, 0.3675)	(0.3702, 0.2904, 0.3907)
(0.2803, 0.1894, 0.2906)	(0.2674, 0.3775, 0.2095)
(0.2365, 0.2803, 0.2053)	(0.2906, 0.2074, 0.3976)
(0.2759, 0.3067, 0.4092)	(0.4532, 0.2906, 0.4084)
CZ ${}_{3}$	CZ ${}_{4}$
(0.2954, 0.3298, 0.2086)	(0.2097, 0.3075, 0.4093)
(0.3562, 0.2709, 0.3554)	(0.4218, 0.3623, 0.3421)
(0.2749, 0.3982, 0.4096)	(0.3027, 0.2538, 0.3083)
(0.3721, 0.2709, 0.2875)	(0.2904, 0.3986, 0.4095)
(0.2903, 0.4095, 0.3571)	(0.3409, 0.3994, 0.3026)

Step 5. Construct the SVNNBAA (Table 6).

Table 6

SVNNBAA

	SVNNBAA
CZ ${}_{1}$	(0.2178, 0.1914, 0.2097)
CZ ${}_{2}$	(0.2526, 0.2097, 0.1893)
CZ ${}_{3}$	(0.2216, 0.1942, 0.1795)
CZ ${}_{4}$	(0.2198, 0.1997, 0.1659)

Step 6. Construct the $\textit{SVNNDM}=({\textit{SVNNDM}_{ij}})_{5\times 4}$ (Table 7).

Table 7

The $\textit{SVNNDM}=({\textit{SVNNDM}_{ij}})_{5\times 4}$

	CZ ${}_{1}$	CZ ${}_{2}$	CZ ${}_{3}$	CZ ${}_{4}$
CP ${}_{1}$	$-$ 0.0849	0.0259	0.0406	$-$ 0.1037
CP ${}_{2}$	0.1340	$-$ 0.0700	0.1316	$-$ 0.1442
CP ${}_{3}$	$-$ 0.1070	$-$ 0.0468	$-$ 0.1094	$-$ 0.1179
CP ${}_{4}$	$-$ 0.0487	$-$ 0.0393	$-$ 0.0297	$-$ 0.1079
CP ${}_{5}$	$-$ 0.1245	$-$ 0.0658	$-$ 0.1439	$-$ 0.2182

Step 7. Construct the $\textit{SVNNOV}_{i}$ (Table 8).

Table 8

The SVNNOV values

Alternative	SVNNOV
CP ${}_{1}$	$-$ 0.1222
CP ${}_{2}$	0.0513
CP ${}_{3}$	$-$ 0.3811
CP ${}_{4}$	$-$ 0.2256
CP ${}_{5}$	$-$ 0.5524

Step 8. From the Table 8, the order is: $CP_{2}>CP_{1}>CP_{4}>CP_{3}>CP_{5}$ and $CP_{2}$ is the optimal microfinance group.

4.2 Compare analysis

The SVNN-MABAC is compared with SVNNWA and SVNNWG operators [71], weighted singlevalued neutrosophic copula power average (WSVNCPA) operator [83], weighted singlevalued neutrosophic copula power geometric (WSVNCPG) operator [83], SVNN-CODAS method [84], SVNN-EDAS method [85] and CPT-SVN-TODIM method [86]. The order results of different methods are constructed in Table 9.

Table 9
Results for different given methods

Methods	Order	The optimal choice	The worst choice
SVNNWA operator [71]	$CP_{2}>CP_{1}>CP_{4}>CP_{3}>CP_{5}$	$CP_{2}$	$CP_{5}$
SVNNWG operator [71]	$CP_{2}>CP_{1}>CP_{3}>CP_{4}>CP_{5}$	$CP_{2}$	$CP_{5}$
WSVNCPA operator [83]	$CP_{2}>CP_{1}>CP_{4}>CP_{3}>CP_{5}$	$CP_{2}$	$CP_{5}$
WSVNCPG operator [83]	$CP_{2}>CP_{1}>CP_{3}>CP_{4}>CP_{5}$	$CP_{2}$	$CP_{5}$
SVNN-CODAS method [84]	$CP_{2}>CP_{1}>CP_{4}>CP_{3}>CP_{5}$	$CP_{2}$	$CP_{5}$
SVNN-EDAS method [85]	$CP_{2}>CP_{1}>CP_{4}>CP_{3}>CP_{5}$	$CP_{2}$	$CP_{5}$
CPT-SVN-TODIM method [86]	$CP_{2}>CP_{1}>CP_{3}>CP_{4}>CP_{5}$	$CP_{2}$	$CP_{5}$
SVNN-MABAC method	$CP_{2}>CP_{1}>CP_{4}>CP_{3}>CP_{5}$	$CP_{2}$	$CP_{5}$

From Table 9, obviously, the order of these decision models is slightly different, however, the best microfinance group is $CP_{2}$ and the worst microfinance group is $CP_{5}$ for all the decision models. This verifies the SVNN-MABAC method is reasonable and effective.

5. Conclusion

Due to the unique nature of organizational forms, group loans have become synonymous with microcredit to a certain extent. People often first pay attention to the unique credit groups and joint liability of group loans. In theoretical research, the joint and several liability of group loans is also considered an important guarantee for low default rates of small credit institution customers. However, the development of microcredit practices has reversed this certain trend. In other regions, the adaptability of group lending technology has been increasingly questioned. Therefore, it is of great significance to track the development and evolution of the relevant theoretical literature’s understanding of the technical performance of group loans, sort out the relevant explanations of the technical mechanism of group loans, clear up the relevant misunderstandings about group loans, objectively analyze the adaptability of group loans, and select and develop appropriate microfinance technologies. The performance evaluation of microfinance groups lending is a MAGDM issues. In this paper, the distances measures of single-valued neutrosophic sets (SVNSs) and maximizing deviation method (MDM) method is employed to obtain the attribute weight values. Based on the classical MABAC method, the single-valued neutrosophic numbers MABAC (SVNN-MABAC) method is constructed for MAGDM under SVNSs. Finally, an example for performance evaluation of microfinance groups lending and some comparative decision analysis are constructed to verify the SVNN-MABAC model.

This article studies the decision methods under SVNNs. Although some research results have been obtained, however, there are still many shortcomings that we need to further research in the future: (1) There is little research on the methods of combination weighting and further improvement is needed. (2) The current research mainly focuses on the improvement of a single MABAC method, and research on combination decision methods still needs further improvement. (3) With the further development of novel fuzzy sets, and the research on some novel fuzzy set still needs to be further investigated, such as the probabilistic neutrosophic sets [87, 88, 89, 90] and Trapezoidal Neutrosophic sets [91, 92, 93, 94, 95].

Footnotes

Funding

The work was supported by the 2022 high-level talent research start-up funding project of Liuzhou Vocational and Technical College, titled “Research on Rural Small Group Loans Based on Interval Intuitive Fuzzy Multiple Attribute Decision Method” (No. 2022GCQD14).

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A novel MABAC approach for multi-attribute group decision-making with single-valued neutrosophic sets: An application in assessing microfinance group lending performance

Abstract

Keywords

1. Introduction

2. Preliminaries

4.1 An empirical example

Table 1 Four constructed attributes

Table 9 Results for different given methods

Footnotes

Funding

References

Table 1
Four constructed attributes

Table 9
Results for different given methods