A credit rating model of microfinance based on fuzzy ... - IOS Press

3095

Journal of Intelligent & Fuzzy Systems 31 (2016) 3095–3102 DOI:10.3233/JIFS-169195 IOS Press

A credit rating model of microfinance based on fuzzy cluster analysis and fuzzy pattern recognition: Empirical evidence from Chinese 2,157 small private businesses Baofeng Shia,∗ , Nan Chenb and Jing Wanga a College b UNSW

of Economics & Management, Northwest A & F University, Yangling, Shaanxi, China Business School, The University of New South Wales, Sydney, New South Wales, Australia

Abstract. Small private businesses provide employment for citizens. Their revenue and profit also contribute to GDP. Therefore, they are an important part of economic development in China. However, the key factor that can impede the development of small private businesses is financial problems. In order to solve this problem, we set up a credit rating model to analyze the credit status of small private businesses. The contributions of the paper are threefold. First, this paper introduces a novel technique that divides the customers’ credit ratings by using a fuzzy cluster analysis, as well as distinguishes the customer’s credit level by utilizing a fuzzy pattern recognition approach, which is helpful to evaluate and predict the customer’s credit level. Second, the proposed model predicts the credit rating of a new loan customer by utilizing the lattice degree of nearness between the center vector of each credit rating and the data vector of a new loan applicant. This seems to offer a new insight into the credit rating of customers. Third, by utilizing the microfinance data of 2,157 Chinese small private businesses, the empirical results indicate that our research is not only significant for assessing the credit status in China’s small private businesses, but also serves as a useful tool for worldwide customers’ credit ratings. Keywords: Credit rating, fuzzy cluster, fuzzy pattern recognition, microfinance, small private business

1. Introduction Small private businesses have developed rapidly in recent years and they play an important role in the economic development in China. According to the relevant, statistical data in China, there were 52.85 million individual businesses and 18.02 million private businesses by the end of September in 2015, which provide more than a third of employment in China. More than a quarter of urban citizens are working in private businesses [18]. However, due to ∗ Corresponding author. Baofeng Shi, College of Economics & Management, Northwest A & F University, Yangling, Shaanxi, China. Tel.: +86 15388615466; Fax: +86 029 87081209; E-mail: fengbei [email protected]; [email protected].

the small size, lack of mortgage and a short period of small private businesses [3], it is difficult to find the classical approach to describe the credit status of small private businesses. Therefore, the financing status and approach to obtaining loans for small private businesses are more serious, and it is urgent to solve the issue of credit rating for small private businesses. The main proposed references about credit rating included the establishment of credit scoring models and the division of credit ratings. Early studies included the development of the five-variable Z-Score credit rating model and the Zeta rating model [7, 8]. Hwang et al. proposed a credit risk prediction method based on an ordered, semiparametric Probit approach [14]. Blancoa et al. revealed that the

1064-1246/16/$35.00 © 2016 – IOS Press and the authors. All rights reserved

3096

B. Shi et al. / A credit rating model of microfinance based on fuzzy cluster analysis

neural network credit rating model outperformed the LDA, QDA and LR models [1]. Harris established the support vector machine credit-scoring model [20]. In order to assess the credit risk of retailers, Baradaran and Keshavarz proposed a credit scoring model by combining the integration of the system dynamics model and the fuzzy inference system (FIS) [23]. Sohn et al. developed a fuzzy credit scoring model to handle the fuzzy input and output data in the credit risk evaluation [19]. Based on the established credit rating index system, Wu et al. proposed an enterprise credit rating model that combines the fuzzy cluster and decision tree approach [10]. In order to extract important credit risk evaluation information, the fuzzy logic and neural network approaches were developed [13, 16]. Research on how to issue loans requires the division of customers’ credit ratings. When using the customers’ default distance, DD, belonging to different intervals, the loan customers had been divided into three levels [6]. Florez-Lopez calculated the probability default (PD) by using the econometrics technique and artificial intelligence method. Then, the loan customers were divided into five ratings by using five-rating virtual variables [15]. Since the loan customers’ credit scoring belonged to different score intervals, the China Construction Bank divided small loan enterprises into five levels [21]. On the basis of customer numbers of all credit ratings following a normal distribution, Shi et al. divided loan customers into nine ratings [5]. Although the existing studies have made great progress in dealing with credit rating issues, there are still some drawbacks. First, the credit rating system of microfinance for small private businesses needs further study. Second, few studies focus on the establishment of the credit rating model by combining the methods of the fuzzy cluster analysis with the fuzzy pattern recognition. This paper advances in threefold. Firstly, this is the first study that proposes a credit rating model to analyze credit risk of the small private businesses by using the approaches of the fuzzy cluster and the fuzzy pattern recognition. Secondly, an empirical study on the microfinance data of 2,157 small private businesses is carried out in order to verify the accuracy of the proposed model. Thirdly, our study is not only significant for assessing the credit status in China’s small private businesses, but also serves as a useful tool for customers’ credit ratings in the world. This paper proceeds as follows. In the next section, we present the establishment of a credit rating model.

Section 3 introduces the data and empirical analysis. Section 4 describes our results.

2. Methodology of the study In this section, we will introduce the process of creating a credit rating model for microfinance that is based on the fuzzy cluster and fuzzy pattern recognition. A step-by-step instruction is provided. 2.1. Establishment of the credit rating index system Through investigating the literature and combining the available indices from a Chinese national commercial bank [22], the first criteria layer of credit risk assessment is comprised of six feature layers. They are ‘X1 Basic information’, ‘X2 Guarantee and joint guarantee’, ‘X3 Capacity of repayment’, ‘X4 Capacity of profitability’, ‘X5 Capacity of operation’, and ‘X6 Macro environment’. Then, a mass-election credit rating index system is created that is composed of 64 indices. The process of screening the credit rating indices is as follows. First, we removed six unavailable indices, and the other 58 indices are left. Second, by utilizing the logistic regression model and the Wald test, we selected the 24 indices that can effectively distinguish default customers from non-default customers. Third, by using the Pearson correlation analysis, we deleted five indices of large correlation from the whole mass-election index system. In summary, we selected 19 credit rating indices, as shown in Table 1. The empirical process of establishing a credit rating index system based on 2157 small private businesses can be found in reference [5]. 2.2. Construction of the credit rating model based on fuzzy clustering method (1) Data standardization of evaluation indices The indices of credit rating can be divided into two categories: the quantitative indices and the qualitative indices. The quantitative indices include interval indices, positive indices and negative indices. This paper uses the Max-Min normalization technique to transform the positive indices and the negative indices [4]. LetX = xij denote the standard score set; let V = vij denote the original data set; let m stand for the number of indices; let n stand for the number of small private businesses. The standardization

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

X6

X5

X4

X2 X3

(c) Indices

X1,1 Marital status X1,2 Gender X1,3 Age X1,4 Living status X1,5 Industry type X1,6 Household spending X1,7 Loan purpose Guarantee and joint guarantee X2,1 Guarantor power Capacity of repayment X3,1 Asset-liability ratio X3,2 The ratio of reimbursement to net income Capacity of profitability X4,1 Return on net assets X4,2 Net profit margin X4,3 Monthly average tax Capacity of operation X5,1 Total assets turnover X5,2 Planting duration X5,3 Number of Employees Macro environment X6,1 Net per capita disposable income X6,2 Engel coefficient X6,3 Industry cycle index

X1 Basic information

(a) No. (b) Feature layers Qualitative Qualitative Interval Qualitative Qualitative Negative Qualitative Qualitative Negative Negative Positive Positive Positive Positive Qualitative Qualitative Positive Negative Positive

(d) Index type 1 1 37 1 2 3000 3 2000 0.195 35.28 0.066 0.054 0 1.953 4 7 14367.5 42.60 131.625

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

1 1 31 1 1 2000 3 2500 0.000 64.84 0.009 0.181 100 0.097 2 0 20551.7 37.75 96.85

1 1 1.000 1 0.75 0.970 0.50 0.081 0.930 0.688 0.041 0.055 0.000 0.234 0.75 0.083 0.144 0.618 1.000

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

1 1 1.000 1 1.00 0.981 0.50 0.101 1.000 0.426 0.005 0.189 0.002 0.011 0.25 0.000 0.510 0.324 0.000

Original data of indices vij Standardized data of indices xij (1) An S.M. . . . (2157) Zhu W.H. (2158) An S.M. . . . (4314) Zhu W.H.

Table 1 The credit rating index system of microfinance for small private business

B. Shi et al. / A credit rating model of microfinance based on fuzzy cluster analysis 3097

3098


formulas of positive indices and negative indices are shown as Formula (1) and Formula (2), respectively. xij =

vij − min (vij ) 1≤j≤n

xij =

(1)

max (vij ) − min (vij )

C = (cij )m×n = R1 ◦ R2 ∈ P(X × Z)

1≤j≤n

1≤j≤n

max (vij ) − vij

1≤j≤n

(2)

max (vij ) − min (vij )

1≤j≤n

1≤j≤n

Let q1 and q2 denote the left boundary and the right boundary of the ideal interval, respectively. The standard scores of the interval indices can be obtained by utilizing Formula (3). ⎧ q1 − vij 1− , ⎪ ⎪ max(q1 − min (vij ), max (vij ) − q2 ) ⎪ ⎪ 1≤j≤n 1≤j≤n ⎪ ⎪ ⎪ ⎪ ⎨ vij − q2 xij =

1−

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

max(q1 − min (vij ), max (vij ) − q2 ) 1≤j≤n

1

,

vij < q1 (a)

vij > q2 (b)

q1 ≤ vij ≤ q2 (c) (3)

By using the approach of reference [5], the standard scoring of the qualitative indices can be obtained. (2) Construction of the fuzzy similar matrix among small private businesses After the standardization of the data, we can construct a fuzzy similar matrix by the means of a vector-angular cosine to calculate the similar coefficient sij between the customer xi and the customer xj . Then, the fuzzy similar matrix R among customers can be calculated by Formula (4) [11]. ⎞ ⎛ s11 · · · s1n ⎟ ⎜ . . . . . ... ⎟ R=⎜ (4) ⎠ ⎝ . sn1 · · · snn n×n

The similar coefficient sij can be obtained by using Formula (5): m xik xjk sij = k=1 (5) m m 2 2 k=1 xik k=1 xjk (3) Determination of credit ratings of small private businesses based on fuzzy clustering ① Calculation of the composition matrix C

(6)

where cij = ∨sk=1 (aik ∧ bkj ), aik ∧ bkj = min (aik , bkj ) and aik ∨ bkj = max (aik , bkj ). ② Calculation of the transitive closure t (R) Let R denote a fuzzy similar matrix; we can calculate R, R2 , R4 , ..., R2i by utilizing Formula (6). If Rk ◦ Rk = Rk (k = 1, 2, 4, ..., 2i), the matrix Rk is called a transitive matrix. Then, the transitive closure t (R) = Rk = R∗ can be obtained and R∗ is defined as the fuzzy equivalence matrix. Ifλ denote the intercept, we can change the value of λ from 1 to 0 to get the dynamic clustering results of the credit rating for small private business. ③ Division of credit ratings

1≤j≤n

,

Let X = {x1 , x2 , ..., xm }, Y = {y1 , y2 , ..., ym } and Z = {z1 , z2 , ..., zm } be a finite set; if R1 = (aik )m×s ∈ P (X × Y ) , R2 = (bki )s×m ∈ P (Y × Z) , the composition matrix C of R1 and R2 is given by Formula (6) [24]:

Let the discourse universe V = {v1 , v2 , ..., vn } be the sample space, and every small private business vj has m indices: vj = vj1 , vj2 , ..., vjm ;then, we can get the original data matrix V = vij m×n . Let v¯ k = nj=1 vkj n denote the average of the original indices data vjk of the j-th customer. Then, the center vector of the total number of customers can be defined as v¯ = (¯v1 , v¯ 2 , ..., v¯ m ). Corresponding to λ, the classification number is r, the total number of customers in the j-th classifica(j) (j) (j) tion is nj , v1 , v2 , ..., vnj are the customers in the j-th classification, and the cluster center of the j-th (j) (j) classification is v¯ (j) = (¯v1 , v¯ 2 , ..., v¯ m (j) ). F-statistics is created by Formula (7) [2]: r (j) vk , v¯ )]2 j=1 nj [1 − cos(¯ (r − 1) F = r nj (j) (j) 2 vi , v¯ )] j=1 i=1 [1 − cos(¯ (n − r) (7) where m (j) ¯ k v¯ k k=1 v cos(¯v(j) , v¯ ) = (8) (j) 2 m m 2 ¯k vk ) k=1 (¯ k=1 v F-statistics obeys F distribution, whose degrees of freedom are r − 1 and n − r. If F > Fα (r − 1, n − r) , we can deduce that the difference among classifications are remarkable and the classification approach is reasonable.


Change the value of λ from 1 to 0, and we can get different classification schemes. Corresponding to λ0 , we can get a specific classification scheme. If the value of F for the specific classification scheme can pass F-test, we can know that the result of customers’ credit ratings is reasonable and we define the λ0 as the best classification threshold value. ④ Sortation of the p credit ratings Let wi stand for the variation coefficient weight [4]; we have: n 1 ¯ i )2 j=1 (xij − x n wi = (9) x¯ i Let disclose universe V = {customers}, all the customers can be divided into p ratings. All the p ratings’ center vectors can be calculated and they are shown as Formula (10): A1 = (a11 , a12 , ..., a1m ) A2 = (a21 , a22 , ..., a2m ) ... AP = ap1 , ap2 , .., apm

(10)

The p vectors {A1 , A2 , ..., AP } form a standard model base on disclose universe V . Then, we can calculate the credit score of each rating by Formula (11) and rank them in a descending order. m Sk = wAk = wi aik (K = 1, 2, ..., p) i=1 (11) 2.3. Prediction of the credit rating of a small private business based on fuzzy pattern recognition (1) Calculation of lattice degree of nearness σ (A, B) If A and B are fuzzy sets on the disclose universe V , the lattice degree of nearness between A and B is defined by Formula (12) [12, 17]: σ(A, B) =

1 [A ◦ B + (1 − A ⊗ B)] 2

(12)

where A ⊗ B = ∧ (A(x) ∨ B(x)). x∈V

(2) Fuzzy pattern recognition of a new loan applicant

3099

There are m indices to describe a new loan applicant’s credit status, which is shown as Formula (13): B = (b1 , b2 , ..., bm )

(13)

Then, we can calculate the lattice degree of nearness σk = σ (Ak , B) between Ak and B (k =1, 2, ..., p). If σk0 = max (σ1 , σ2 , ..., σp is the largest value in all the values of the lattice degree of nearness, the new customer belongs to the k-th rating. 3. Empirical study 3.1. Sample and data source For our case, the 2,157 small private businesses each have fifty-eight conditional indices (indicators or characteristics). The conditional indices include six categories of information: ‘X1 Basic information’, ‘X2 Guarantee and joint guarantee’, ‘X3 Capacity of repayment’, ‘X4 Capacity of profitability’, ‘X5 Capacity of operation’, and ‘X6 Macro environment’. The established credit rating index system and the original data is shown in the Columns a to d and Columns 1 to 2,157 of Table 1. 3.2. Establishment of the credit rating model (1) Data standardization According to the index type in Column d of Table 1, using the data standardization approaches as mentioned in subsection 2.1, the standardized scoring of quantitative indices can be calculated. In terms of the scoring standard of qualitative indices in reference [5], the standard scores of qualitative indices are obtained. (2) Construction of the fuzzy similar matrix R We can obtain the fuzzy similar matrix R by taking the standardized data xij into Formula (4) and Formula (5). ⎛ 1.0000 0.9202 0.9236 ... 0.9011 ⎞ ⎜ 0.9202 ⎜ R = ⎜ 0.9236 ⎜ ⎝ ...

1.0000

0.9458 ... 0.9247

0.9458

1.0000 ... 0.8764

...

0.9011 0.9247

...

...

...

0.8764 ... 1.0000

⎟ ⎟ ⎟ ⎟ ⎠ 2157×2157

(3) Division of the credit ratings of 2,157 small private businesses After calculating R, R2 , R4 , ..., R2i by using Matlab R2010a, we can find that R8 ◦ R8 = R8 ,

3100


which means that R8 is the transitive matrix of the process of the credit rating for 2,157 small private businesses. Therefore, the transitive closure t (R) = R8 = R∗ can be obtained and R8 = R∗ is the fuzzy equivalence matrix. R8 = R∗

⎛ 1.0000 ⎜ 0.9854 ⎜ ⎜ ⎝ ...

0.9854 0.9889 1.0000

= ⎜ 0.9889 0.9854 ...

0.9736 0.9736

... 0.9736⎞

0.9854 ... 0.9736⎟

⎟ ⎟ ⎠

1.0000 ... 0.9736 ⎟ ...

...

...

0.9736 ... 1.0000

2157×2157

In changing the value of λ from 1 to 0 by using Matlab R2010a, we can know that there are 324 credit rating results, which are shown as Fig. 1. Then, we calculate the F values for each classification scheme by utilizing Formula (7) and Formula (8). By comparing the 324 calculated F values with Fα (r − 1, n − r), we found that F146 = 32.6411 > Fα (r − 1, n − r) = F0.05 (292, 1865) = 1.1526 (at this point λ=0.9854). Therefore, the 146th credit rating result is reasonable. The best classification threshold value λ0 equals 0.9854. Meanwhile, the 2,157 small private businesses are divided into nine ratings. The center vectors for the nine ratings are shown in Table 2. (4) Sortation of the nine ratings According to Formula (9), the variation coefficient weight of the 19 indices can be obtained: w=(0.019, 0.006, 0.011, 0.037, 0.026, 0.008, 0.008, 0.138, 0.007, 0.018, 0.155, 0.092, 0.137, 0.085, 0.029, 0.099, 0.030, 0.043, 0.052). As is known from the indices’ weights, the ranking results of the weights in descending order are: X4 = 0.384 > X5 = 0.213 > X2 = 0.138 > X6 = 0.125 > X1 = 0.115 > X3 = 0.025. In other words, we can

Table 2 Center vectors for each classification (λ0 = 0.9854) (1) No. 1 2 3 4 5 6 7 8 9

(2) Vectors Ai

(3) Center vectors

A1 A2 A3 A4 A5 A6 A7 A8 A9

(0.640, 0.857,..., 0.538, 0.298)1 × 19 (0.750, 0.957,..., 1.000, 0.275)1 × 19 (0.453, 0.956,..., 0.536, 0.256)1 × 19 (0.480, 0.965,..., 0.519, 0.230)1 × 19 (1.000, 0.956,..., 0.000, 0.227)1 × 19 (0.750, 0.961,..., 1.000, 0.227)1 × 19 (0.750, 0.959,..., 1.000, 0.210)1 × 19 (1.000, 0.961,..., 0.000, 0.203)1 × 19 (1.000, 0.964,..., 0.000, 0.181)1 × 19

deduce that the order of importance of the six criterion layers is X4 Capacity of profitability > X5 Capacity of operation > X2 Guarantee and joint guarantee > X6 Macro environment > X1 Basic information > X3 Capacity of repayment. When substituting the center vectors’ data from Table 2 and the variation coefficient weight w into Formula (11), the credit scores Sk of the nine ratings can be obtained. Then, the credit ratings AAA, AA, A,..., C, that correspond to the credit scores Sk from high to low order, can be obtained, respectively, as shown in Table 3. The number of small private businesses in each credit rating and the small private business’s order number are given in Table 4. By using the data on the third Column of Table 4, the distribution of small private businesses in each credit rating is shown in Fig. 2. 3.3. Credit rating of a new loan applicant We assume that the 19 indices data of a new loan applicant vnew =(1.000, 1.000, 1.000, 0.000, 1.000, 0.959, 0.500, 0.121, 0.977, 0.919, 0.008, 0.014, 0.091, 0.085, 0.250, 0.083, 0.510, 0.324, 0.503). When substituting the nine center vectors data from the third Column of Table 2 and the

Fig. 1. The dynamic clustering results of 2,157 small private businesses.


3101

Table 3 Credit score and ratings in each credit rating (1) No. (2) Vectors Ai 1 2 3 4 5 6 7 8 9

A1 A2 A3 A4 A5 A6 A7 A8 A9

(3) Credit scores Sk

(4) Credit ratings

0.2556 0.2273 0.2749 0.2270 0.1808 0.2098 0.2300 0.2029 0.2975

A BB AA B C CCC BBB CC AAA

Table 4 Small private businesses in each credit rating (1) No. 1 2 3 4 5 6 7 8 9

(2) Credit rating

(3) The number of customers

AAA AA A BBB BB B CCC CC C

328 335 784 25 359 146 42 18 120

(4) Customers’ order number 1727, 555, . . . , 1940 1, 226, . . . , 1463 2, 1476, . . . , 1972 15, 492, . . . , 511 10, 97, . . . , 2015 88, 1283, . . . , 686 530, 1587, . . . , 1765 118, 396, . . . , 1165 17, 307, . . . , 355

data vnew into Formula (12) and Formula (13), the lattice degree of nearness σk = σ (Ak , vnew ) between Ak and vnew (k = 1, 2, ..., 9) can be calculated. They are σ1 = σ (A1 , vnew )=0.967, σ2 = σ (A2 , vnew )=0.980, σ3 = σ (A3 , vnew )=0.958, σ4 = σ (A4 , vnew )=0.997, σ5 = σ (A5 , vnew ) =0.999, σ6 = σ (A6 , vnew )=0.975, σ7 = σ (A7 , vnew )=0.952, σ8 = σ (A8 , vnew )=0.981, σ9 = σ (A9 , vnew )=0.911. Among the nine the lattice degrees of nearness σ 0 = max (σ1 , σ2 , ..., σ9 ) = σ5 = 0.999. Therefore, according to the fifth Row of Table 3, the new small private business vnew belongs to the credit rating C. 4. Conclusion It is well known that the methods of fuzzy clustering and fuzzy neural networks have been successfully applied to credit rating for many years [1, 9, 10, 13, 16, 19, 23]. However, the disadvantage of the models used in previous studies is that it is difficult to predict the credit rating of a new loan customer. In order to make up the gap and help small private businesses get credit funds and help financial institutions select quality customers in the complex environment, this paper introduces a novel technique that divides the customers’ credit ratings by using

Fig. 2. Distribution of customers in each credit rating.

the fuzzy cluster analysis. It also distinguishes the customer’s credit level by utilizing the fuzzy pattern recognition approach, which is helpful to evaluate and predict the credit level. The developed model is verified by the data of 2,157 small private businesses. The empirical analysis results are as follows. (1) The proposed method can accurately divide the credit ratings of microfinance for small private businesses, and can also find out the quality of small private businesses from all of the small private businesses who apply for loans, which can decrease the bank’s loan loss. (2) In the evaluation of credit risk of microfinance for small private businesses, the order of importance of the six criterion layers is as follows: X4 Capacity of profitability > X5 Capacity of operation > X2 Guarantee and joint guarantee > X6 Macro environment > X1 Basic information > X3 Capacity of repayment. The contributions of the article are as follows. First, this is the first study that proposes a credit rating model to analyze the credit status of small private businesses by combining the approaches of the fuzzy cluster analysis and the fuzzy pattern recognition. Second, the proposed model predicts the credit rating of a new loan customer by utilizing the lattice degree of nearness σ (Ak , B) between the center vector of each credit rating Ak and the data vector of a new loan applicantB, which seems to offer new insight into the credit rating of customers. Although we have introduced a novel credit rating approach, there is some room for further study. In this empirical analysis, we utilized the data of the 2,157 small private businesses. Relevant conclusions are derived from the exploratory analysis results, which may not be generalized enough for all small private business loans in China. There are also concerns about more relevant small private business data. In addition, customers’ loan price can be taken into consideration based on the proposed model.

3102


Acknowledgments The research was supported by National Natural Science Foundation of China (Nos. 71503199, 71471027 and 71373207), Project Funded by China Postdoctoral Science Foundation (Nos. 2015M572608 and 2016T90957), Basic Business Project of Humanities Social Sciences for Central University (No. 2015RWYB09), Shaanxi Province Postdoctoral Science Foundation Funded Project, Natural Science Basic Research Project in Shaanxi Province (No. 2016JQ7005), China Ministry of Education Social Sciences and Humanities Research Youth Fund Project (No. 16YJC630102). References [1]

[2] [3]

[4]

[5]

[6]

[7]

[8]

[9]

A. Blancoa, R. Pino-Mej´ıasb, J. Larac and S. Rayoc, Credit scoring models for the microfinance industry using neural networks: Evidence from Peru, Expert Systems with Applications 40(1) (2013), 356–364. A.C. Rencher and G.B. Schaalje Linear Models in Statistics, John Wiley & Sons, Hoboken, NJ, USA 2008. B.F. Shi and G.T. Chi, A credit risk evaluation index screening model of petty loans for small private business and its application, Advances in information Sciences and Service Sciences 5(7) (2013), 1116–1124. B.F. Shi, H.F. Yang, J. Wang and J.X. Zhao, City green economy evaluation: Empirical evidence from 15 sub-provincial cities in China, Sustainability 8(6) (2016), 1–39. B.F. Shi, J. Wang, J.Y. Qi and Y.J. Cheng, A novel imbalanced data classification approach based on Logistic regression and Fisher discriminant, Mathematical Problems in Engineering 2015 (2015), 1–12. C.C. Yeh, F.Y. Lin and C.Y. Hsu, A hybrid KMV model, random forests and rough set theory approach for credit rating, Knowledge-Based Systems 33 (2012), 166–172. E.I. Altman, Financial ratios, discrimination analysis and the prediction of corporate bankruptcy, Journal of Finance 23(4) (1968), 589–609. E.I. Altman, R.G. Haldeman and P. Narayanan, ZETATM analysis: A new model to identify bankruptcy risk of corporations, Journal of Banking and Finance 1(1) (1977), 29–54. H.M. Moisés, J.T. Rodr´ıguez and J. Montero, Credit rating using fuzzy algorithms, Actas de la XVI Conferencia de la Asociación Española para la Inteligencia Artificial, CAEPIA’15, Albacete, Spain, 2015, pp. 539-548

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

H.X. Wu, X.Q. Liu and Y.H. Liu, Enterprise credit rating model based on fuzzy clustering and decision tree, Proceedings of the 2010 Third International Symposium on Information Science and Engineering (ISISE ’10), IEEE Computer Society, Washington, DC., USA (2010), 105–108. J.C. Xu, T.H. Xu, L. Sun and J.Y. Ren, An efficient gene selection technique based on fuzzy C-means and neighborhood rough set, Applied Mathematics & Information Sciences 8(6) (2014), 3101–3110. J.M. Keller and P. Gader, et al., Advances in fuzzy integration for patter recognition, Fuzzy Sets and Systems 65(2-3) (1994), 273–283. P. Hájek, Credit rating analysis using adaptive fuzzy rulebased systems: An industry-specific approach, Central European Journal of Operations Research 20(3) (2012), 421–434. R.C. Hwang, H.M. Chung and C.K. Chu, Predicting issuer credit ratings using a semiparametric method, Journal of Empirical Finance 17(1) (2010), 120–137. R. Florez-Lopez, Modelling of insurers’ rating determinants: An application of machine learning techniques and statistical models, European Journal of Operational Research 183(3) (2007), 1488–1512. R.H. Abiyev, Credit rating using type-2 fuzzy neural networks, Mathematical Problems in Engineering 2014 (2014), 1–8. S. Mitra and S.K. Pal, Fuzzy sets in pattern recognition and machine intelligence, Fuzzy Sets and Systems 156 (2005), 381–386. State Administration for Industry & Commerce of the People’s Republic of China, The Report on the relationship between private businesses and employment in China, 2015, http://www.saic.gov.cn/zwgk/tjzl/ zxtjzl/xxzx/201510/t20151030 163438.html. S.Y. Sohn, D.H. Kim and J.H. Yoon, Technology credit scoring model with fuzzy logistic regression, Applied Soft Computing 43 (2016), 150–158. T. Harris, Quantitative credit risk assessment using support vector machines: Broad versus narrow default definitions, Expert Systems with Applications 40(11) (2013), 4404–4413. The China Construction Bank, The credit rating method of small enterprises of China Construction Bank, China Construction Bank (2007), 1–8. The Postal Savings Bank of China (PSBC), Credit risks decision and evaluation system of microcredit for merchants for postal savings Bank of China, The Postal Savings Bank of China, the second edition, 2011. V. Baradaran and M. Keshavarz, An integrated approach of system dynamics simulation and fuzzy inference system for retailers’ credit scoring, Economic Research-Ekonomska Istraˇzivanja 28(1) (2015), 959–980. W.N. Wang and Y.J. Zhang, On fuzzy cluster validity indices, Fuzzy Sets and Systems 158 (2007), 2095–2117.