In the field of decisionmaking, for the multiple attribute decisionmaking problem with the partially unknown attribute weights, the evaluation information in the form of the intuitionistic fuzzy numbers, and the preference on alternatives, this paper proposes a comprehensive decision model based on the intuitionistic fuzzy cross entropy distance and the grey correlation analysis. The creative model can make up the deficiency that the traditional intuitionistic fuzzy distance measure is easy to cause the confusion of information and can improve the accuracy of distance measure; meanwhile, the grey correlation analysis method, suitable for the small sample and the poor information decisionmaking, is applied in the evaluation. This paper constructs a mathematical optimization model of maximizing the synthesis grey correlation coefficient between decisionmaking evaluation values and decisionmakers’ subjective preference values, calculates the attribute weights with the known partial weight information, and then sorts the alternatives by the grey correlation coefficient values. Taking venture capital firm as an example, through the calculation and the variable disturbance, we can see that the methodology used in this paper has good stability and rationality. This research makes the decisionmaking process more scientific and further improves the theory of intuitionistic fuzzy multiple attribute decisionmaking.
Professor Zadeh [
Besides, in the practice of decisionmaking, to get more comprehensive and accurate understanding of the alternatives and to achieve the utility maximization better, decisionmakers often try to obtain the preference information of the alternatives by all means. The preference information on alternatives usually refers to the tendency and the emotion hidden inside the decisionmakers. Rational decisionmakers select alternatives in accordance with a predetermined optimization principle. Although their preference information contains subjective elements, the decision criterion is still based on the maximization of the expected utility. Therefore, the decisionmaking problem with preference information on alternatives still belongs to the multiple attribute decisionmaking problem under the complete rational perspective.
In recent years, scholars have studied multiple attribute decisionmaking problems of different data types with preference on alternatives, and such problems have become a hotspot in research of multiple attribute decisionmaking problems. Fan et al. [
At the same time, the concept and the theory of the cross entropy in information entropy are applied to the fuzzy multiple attribute decisionmaking. Shang and Jiang [
From the existing results, the research method of the intuitionistic fuzzy multiple attribute decisionmaking based on preference on alternatives is not perfect. Firstly, most of the existing decision models based on the deviation are the distance optimization model, while the existing intuitionistic fuzzy number distance measure formula has some defects, which often cannot distinguish the data size or the information confusion; secondly, the research is not deep enough for the unknown or partially unknown attribute weights, which often overlooks the case of the uncertainty of attribute weights. From the perspective of improving the accuracy of decision results, the intuitionistic fuzzy multiple attribute decisionmaking method based on preference on alternatives still needs further indepth study.
In view of the above analysis, we propose a comprehensive decision model based on the intuitionistic fuzzy cross entropy distance and the grey correlation analysis to solve the problem of the intuitionistic fuzzy multiple attribute decisionmaking with preference on alternatives in this paper, which makes up the deficiency of causing the information confusion easily for the traditional intuitionistic fuzzy distance measure, improves the accuracy of the distance measure, solves the attribute weights by combining with the grey correlation analysis suitable for the small sample and the poor information decisionmaking, setting up the mathematical programming model with the maximum synthesis grey correlation coefficient between the evaluation value and the subjective preference value of the decisionmakers, and then sorts the alternatives according to the change of the grey identification coefficients to demonstrate the stability. At last, the validity of the model is proved by the example of a venture capital firm.
In this section, we introduce some basic knowledge and the necessary concepts related to the intuitionistic fuzzy set and the distance measure formula.
Suppose
If
Based on the geometric distance model, Xu [
Let
When
When
Assume a domain
The intuitionistic fuzzy cross entropy
The intuitionistic fuzzy cross entropy distance
If
(1) One has
According to Jansen’s inequality [
Because the logarithmic function above is strictly convex, then
(2) When
(3) As can be seen from the complementation of the intuitionistic fuzzy sets,
(4) Because
As can be seen from property (2), when the two intuitionistic fuzzy sets are exactly equal, the intuitionistic fuzzy cross entropy is the least; therefore, the cross entropy can be used to measure the difference between two intuitionistic fuzzy sets. The intuitionistic fuzzy cross entropy adds the meaning of the information entropy on the basis of the original intuitionistic fuzzy complete information. It can be used to measure the fuzzy degree and the uncertainty degree of the intuitionistic fuzzy sets. The greater the cross entropy of two intuitionistic fuzzy numbers, the further the distance [
For example, there are three intuitionistic fuzzy numbers
In this paper, the multiple attribute decisionmaking problems are assumed to have a certain subjective preference for the decisionmakers. Generally the problems can be abstracted as follows: the decisionmakers can give the attribute values in the form of the intuitionistic fuzzy numbers
Intuitionistic fuzzy decision matrix



 





















Suppose the decisionmakers have some preference on alternatives
In this paper, the intuitionistic fuzzy multiple attribute decisionmaking method with preference on alternatives draws lessons from the theory of the intuitionistic fuzzy cross entropy distance and the grey correlation analysis method to solve the optimum alternative selection problem in the case where the weights are partly unknown.
Determine the alternatives
Calculate the grey correlation coefficient between the subjective evaluation of each alternative based on the intuitionistic fuzzy cross entropy distance and the subjective preference of the decisionmakers. The formula is as follows:
The grey correlation coefficient here shows the approximation degree of the subjective evaluation to the subjective preference of each
In formula (
The actual meaning has been clear by last step; therefore, each attribute weight can be determined by constructing the mathematical programming model with the purpose of maximizing the grey correlation coefficient.
Let the attribute weight be
If the attribute weight
Since there is no preference relationship between the various alternatives, it is fair to compete, so
The attribute weights can be obtained by using the software MATLAB_R2014a. If the attribute weight part is known, then
Substitute the obtained attribute weight
Sort the alternatives by the synthesis grey correlation coefficient
Make the perturbation analysis according to the variation of the distinguishing coefficient
In order to prove the accuracy and the validity of the method in this chapter, we use the example in document [
The intuitionistic fuzzy decision matrix



 


(0.4, 0.5)  (0.5, 0.4)  (0.2, 0.7)  (0.3, 0.5) 

(0.7, 0.2)  (0.5, 0.4)  (0.2, 0.5)  (0.1, 0.7) 

(0.5, 0.3)  (0.3, 0.4)  (0.6, 0.2)  (0.4, 0.4) 

(0.6, 0.4)  (0.6, 0.3)  (0.6, 0.3)  (0.3, 0.6) 

(0.5, 0.5)  (0.4, 0.5)  (0.4, 0.4)  (0.5, 0.4) 
The subjective preference of the decisionmakers on alternatives
Specific calculation steps are as follows.
Determine the alternatives
Calculate the intuitionistic fuzzy cross entropy distances between the subjective evaluation and the subjective preference of each alternative and form the distance matrix
Construct the mathematical programming model with the goal of maximizing the grey correlation coefficient:
Substitute the obtained attribute weight
Sort the alternatives by the synthesis grey correlation coefficient
Make the perturbation analysis according to the variation of the distinguishing coefficient
The attribute weight values under different grey resolutions.










0.28  0.28  0.28  0.28  0.28  0.28  0.28 

0.20  0.25  0.25  0.25  0.25  0.25  0.25 

0.22  0.22  0.22  0.22  0.22  0.22  0.22 

0.30  0.25  0.25  0.25  0.25  0.25  0.25 
The decisionmaking results under different grey resolutions.










Rank 

Rank 

Rank 

Rank 

Rank 

Rank 

Rank  

0.70  3  0.86  3  0.91  3  0.93  3  0.95  3  0.96  3  0.96  3 

0.37  5  0.53  5  0.61  5  0.67  5  0.71  5  0.74  5  0.76  5 

0.71  2  0.89  2  0.93  2  0.95  2  0.96  2  0.97  2  0.97  2 

0.56  4  0.77  4  0.83  4  0.86  4  0.89  4  0.90  4  0.91  4 

0.86  1  0.95  1  0.97  1  0.98  1  0.98  1  0.99  1  0.99  1 
As can be seen from Table
However, the result of this paper is difficult compared to that of [
In this paper, a comprehensive decision model based on the intuitionistic fuzzy cross entropy distance and the grey correlation analysis is proposed for the multiple attribute decisionmaking problems with the attribute weights partially unknown, the evaluation information in the form of intuitionistic fuzzy numbers, and the preference information on alternatives. The model introduces the intuitionistic fuzzy cross entropy distance to substitute the traditional geometric distance, calculates the attribute weights by constructing the mathematical optimization model of maximizing the synthesis grey correlation coefficient between the decisionmaking evaluation values and decisionmakers’ subjective preference values and making use of the known partial weight information, then sorts the alternatives by the synthesis grey correlation coefficient values, and at last demonstrates the effectiveness of the proposed model through the comparison and the analysis of two calculation examples. In this paper, the model has strong pertinence, high accuracy, and simple calculation and has further perfected and enriched the intuitionistic fuzzy multiple attribute decisionmaking theory. In future research, we will focus on the decision model and method constructed in this paper. Comparing to the study of alternatives with the definite preference, this research field will be more complex and more innovative in the multiple attribute decisionmaking method.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was supported by the National Natural Science Foundation of China (71271070), Specialty of College Comprehensive Reform Pilot Project (ZG0429), and Specialty and Curriculum Integration Project of Guangxi High School (GXTSZY016).