This paper presents an approach for weighting indices in the comprehensive evaluation. In accordance with the principle that the entire difference of various evaluation objects is to be maximally differentiated, an adjusted weighting coefficient is introduced. Based on the idea of maximizing the difference between the adjusted evaluation scores of each evaluation object and their mean, an objective programming model is established with more obvious differentiation between evaluation scores and the combined weight coefficient determined, thereby avoiding contradictory and less distinguishable evaluation results of single weighting methods. The proposed model is demonstrated using 2,044 observations. The empirical results show that the combined weighting method has the least misjudgment probability, as well as the least error probability, when compared with four single weighting methods, namely, G1, G2, variation coefficient, and deviation methods.
In the process of comprehensive evaluation, a complete set of evaluation indices must be determined, with the corresponding weight of each evaluation index. The index weight reflects the relative importance of each index, that is, the status and function of each factor in the evaluation and decisionmaking process. Hence, the determination of index weight relates to the reliability and validity of ranking results of the project. For example, an enterprise credit risk evaluation is related to its financial factors, nonfinancial factors, and macro factors. In terms of large enterprises, financial condition can reflect the repayment ability and willingness to reimburse well; for this reason, there must be substantial weights for the financial factors in a large enterprise credit risk evaluation. When the financial system of small enterprises is not standard and sound, financial information cannot fully reflect their operating condition, and nonfinancial and macro factors have greater influence. As a result, if we cannot allocate the weights of financial, nonfinancial, and macro factors, the irrational phenomenon that small enterprises with poor credit might have a high ranking can result in losses to the bank. Therefore, weight determination methods have been a major focus in comprehensive evaluation research.
Weighting methods for comprehensive evaluation can be divided into three categories: subjective, objective, and combined. A subjective weighting method is a means whereby decision makers obtain index weights on the basis of their own experience and the emphasis that they subjectively place on each index. For this reason, the index weight obtained using subjective weighting depends on experts’ knowledge, experience, and personal preferences, with no consideration for the characteristics and regularities of actual sample data [
Though an objective weighting method provides weights according to the actual sample data, it is easily subject to the influence of sample data. Moreover, different objective weighting methods tend to yield different results. Therefore, many scholars have proposed combined weighting methods that consider the advantages and disadvantages of both subjective and objective weighting. A combined weighting model has the advantage of integrating the benefits from both subjective and objective weighting models and mitigating limitations of single weighting models [
Although existing research has made great progress, there are still drawbacks. First, there is usually randomness in the method selection for the determination of objective and subjective weights. Meanwhile, the method for the coefficient of objective and subjective weights is not appropriate. Second, a reasonable test is lacking to verify the rationality of a combined weighting method.
This paper proposes a combined weighting method based on difference maximization. Different evaluation scores for the same object are obtained using different weighting methods. According to the principle that the entire difference of different evaluation objects must be maximally differentiated, an adjusted weighting coefficient is introduced. Based on the idea of maximizing the difference between the adjusted evaluation scores of each evaluation object and their mean, an objective programming model is established with more obvious differentiation between evaluation scores guaranteed and the combined weight coefficient determined. At the same time, with 2,044 small private businesses from a Chinese governmentowned commercial bank as empirical samples, a higher differentiation accuracy of the combined weighting method compared with four kinds of single weighting methods, namely, G1, G2, variation coefficient, and deviation methods, is ensured by using misjudgment probability (MP) and error probability (EP) to test evaluation results from the use of different weighting methods.
The rest of the paper is structured as follows. Section
The purpose of index data standardization is to transform the index data into
There are four types of indexes: positive, negative, interval, and qualitative. Positive indices are those for which greater values are better, such as “
Let
Equation (
The two equations have similar meanings. Equation (
Let
For all qualitative indices, we establish a grading standard suitable for small private businesses. Note that the data of qualitative indices fall in
G1 reflects the importance of indices by their sequential order. After the order is determined, rational weights are assigned based on comparing the adjacent indices. Therefore, the relative importance of any two adjacent indices is certain, assuring that the importance corresponds to the weight. The steps of G1 are as follows.
Experts specify the sequence order of indices.
Define
Based on the rational ratio
Equation (
Note that
The symbolic meanings in (
Equation (
Equation (
Similarly, iteration continues until
In (
As a result of (
The essential role of G2 is to reflect index importance by index order; a more important index corresponds to a greater weight. We first specify index order by expert experience and find the least important index labeling
Equation (
Let
Then,
Equations (
Let
Different single weighting models lead to varying evaluation results. Standardization is necessary to establish the comparability of weighting models. Let
Let
The
Based on the principle of reflecting the entire difference of different evaluation objects most fully, the variance of the evaluation scores adjusted by (
Matrix
The economic significance of (
According to [
The accuracy of the evaluation results for each weighting model is tested using the receiver operating characteristic (ROC) curve. ROC requires two indices, MP (misjudgment probability), the probability of rejecting a good sample as a bad sample by misjudgment, and EP (error probability), the probability of accepting a bad sample as a good sample by misjudgment. In the case of measuring a credit evaluation model, there are default samples and nondefault samples in the sample set; hence each weighting model can discriminate a sample as either default or nondefault. Greater MP and EP indicate that the model has less discrimination accuracy, while lower MP and EP indicate that the model has greater discrimination accuracy.
Based on available indices from a Chinese governmentowned commercial bank, this paper selects 21 indices of small private businesses, including six feature layers, “
Index system for small private businesses.
(1) Number  (2) Feature layer  (3) Indices  (4) Type of index 

1 


Qualitative 
2 

Interval  
3 

Qualitative  
4 

Qualitative  
5 

Positive  
6 

Negative  


7 


Qualitative 
8 

Positive  
9 

Qualitative  
10 

Qualitative  


11 


Positive 
12 

Negative  


13 


Positive 
14 

Positive  
15 

Positive  


16 


Positive 
17 

Positive  
18 

Qualitative  


19 


Positive 
20 

Positive  
21 

Interval 
Data are collected from a Chinese governmentowned commercial bank that deals with 2,044 small private businesses from 30 provinces [
According to the type in Column 4 of Table
The scoring criteria of qualitative indices.
(1) Number  (2) Feature layer  (3) Indices  (4) Options number  (5) Options  (6) Scoring 

1 


1  Undergraduate and above  1 
2  2  Junior college  0.8  
3  3  High school and technical secondary school  0.6  
4  4  Junior high school  0.4  
5  5  Primary school  0.2  
6  6  Other  0  


⋮  ⋮  ⋮  ⋮  ⋮  ⋮ 


39 


1  Eight years or more  1 
40  2  Five years or more and less than eight years  0.75  
41  3  Two years or more and less than five years  0.5  
42  4  Less than two years  0.25  
43  5  Missing Data  0 
Through interviews with many customer managers from a Chinese governmentowned commercial bank, the order rational ratio
Weights of different types of weighting method.
(1) Number  (2) Feature layer  (3) Indices  (4) Order rational ratio 
(5) G1 weight 
(6) Importance degree ratio 
(7) G2 weight 
(8) Variation coefficient 
(9) Variation coefficient weight 
(10) Deviation weight 
(11) Combined weight 

1 


—  0.0537  1.2  0.0432  0.1967  0.0106  0.0224  0.0270 
2 

0.9  0.0596  1.3  0.0468  0.6725  0.0364  0.0283  0.0392  
3 

0.8  0.0745  1.5  0.0540  0.3947  0.0213  0.0507  0.0462  
4 

1.1  0.0677  1.6  0.0576  0.1578  0.0085  0.0333  0.0347  
5 

1.2  0.0565  1.4  0.0504  2.3502  0.1270  0.0324  0.0701  
6 

1.1  0.0513  1.3  0.0468  0.6758  0.0365  0.0481  0.0448  


7 


1.2  0.0428  1.3  0.0468  0.2293  0.0124  0.0264  0.0273 
8 

0.8  0.0535  1.5  0.0540  3.2307  0.1746  0.0093  0.0769  
9 

1.1  0.0486  1.4  0.0504  1.2054  0.0652  0.0062  0.0387  
10 

1.1  0.0442  1.2  0.0432  0.6646  0.0359  0.0408  0.0401  


11 


1.1  0.0402  1.3  0.0468  0.1349  0.0073  0.0156  0.0211 
12 

1.0  0.0402  1.3  0.0468  0.4407  0.0238  0.0302  0.0319  


13 


0.8  0.0502  1.6  0.0576  0.3874  0.0209  0.0758  0.0508 
14 

1.1  0.0456  1.6  0.0576  4.1970  0.2269  0.0525  0.1087  
15 

1.1  0.0415  1.4  0.0504  0.4961  0.0268  0.0688  0.0476  


16 


0.9  0.0461  1.3  0.0468  0.4517  0.0244  0.0799  0.0515 
17 

1.1  0.0419  1.2  0.0432  0.7739  0.0418  0.1127  0.0680  
18 

1.1  0.0381  1.2  0.0432  0.3333  0.0180  0.0685  0.0432  


19 


1.1  0.0346  1.1  0.0396  0.2148  0.0118  0.0437  0.0310 
20 

1.0  0.0346  —  0.0352  0.9733  0.0526  0.0873  0.0598  
21 

1.0  0.0346  1.1  0.0396  0.3194  0.0173  0.0671  0.0414 
Through interviews with many customer managers from a Chinese governmentowned commercial bank, we can determine the least important index of 21 indices “
The mean
The deviation weights, shown in Table
The evaluation scores of the four kinds of single weighting methods are obtained by substituting the G1 weight
The combined weight
Results of the five kinds of weighting method are shown in Figure
Results of five kinds of weighting method.
To test the accuracy of the five kinds of weighting method, we used the remaining 1,022 small private businesses as the test sample. There are 174 default samples and 848 nondefault samples. The results of MP and EP can be obtained using the ROC curve. The accuracy of weighting results can be estimated by comparing the MP and EP of the different kinds of weighting method. In this way, we can evaluate the advantages and disadvantages of the different weighting methods as well. The results of MP and EP based on the different kinds of weighting method are shown in Figure
MP and EP of the different kinds of weighting models.
Figure
In a comprehensive evaluation, the index weight reflects the relative importance of each index. In other words, it reflects the status and function of each factor in the evaluation and decisionmaking process. Hence, the determination of index weight relates to the reliability and validity of ranking results of the project. It is for this reason that weight determination methods have been a major focus in comprehensive evaluation research. To this end, many mathematical models have been explored as decision support methods.
This paper proposed a combined weighting method based on difference maximization. Different evaluation scores for the same object are obtained using different weighting methods. An adjusted weighting coefficient was introduced in accordance with the principle that the entire difference of different evaluation objects is to be maximally differentiated. An objective programming model was established with more obvious differentiation between evaluation scores guaranteed and the combined weight coefficient determined. Our model is based on the idea of maximizing the difference between the adjusted evaluation scores of each evaluation object and their mean. The proposed model is demonstrated using 2,044 observations. The empirical results show that the combined weighting method has the smallest misjudgment probability, as well as the smallest error probability, when compared with four kinds of single weighting methods, G1, G2, variation coefficient, and deviation methods.
The research contribution in theory was as follows. An adjusted weighting coefficient was introduced in accordance with the principle of reflecting the entire difference of different evaluation objects to the maximum. An objective programming model was established to calculate the combined weight coefficients of the indices, based on the idea of maximizing the difference between the adjusted evaluation scores of each evaluation object and their mean. Therefore, our model avoids the contradictory and less distinguishable evaluation results that are typical in single weighting methods.
The authors declare that there is no conflict of interests regarding the publication of the paper.
This research is supported by the National Natural Science Foundation of China (nos. 71171031, 7147027, and 71503199), Banking Information Technology Risk Management Project of China Banking Regulatory Commission (CBRC) (no. 20124005), Science and Technology Research Project of Ministry of Education of China (no. 201110), the Bank of Dalian as credit rating and loan pricing systems for Small business (no. 201201), and Credit Risks Evaluation and Loan Pricing For Petty Loan Funded for the Head Office of Post Savings Bank of China (no. 200907). The authors thank the organizations mentioned above.