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The evaluation problem with three-parameter interval grey number (T-PIGN) widely exists in real world. To select effective evaluation indicators of the problem, this paper puts forward evaluation index system selection principle of T-PIGN based on distance entropy model, and gives out evaluation index system selection judgment criterion of T-PIGN. Furthermore, for the redundancy of evaluation index system with T-PIGN, a selection method of evaluation index system with T-PIGN is proposed. Finally, the applicability of the proposed method is verified by concrete examples.

The comprehensiveness and proper simplification of evaluation index system is an important and key step in multiple attributes evaluation problem. Although, there are many influencing factors on the evaluation effectiveness of evaluation objects, the evaluation index is not the more the better. The key problem of evaluation is whether the selected index is proper and reasonable. The omission of important index and the overlap of index information will make the evaluation result distorted, and too many evaluation indexes will increase the unnecessary workload and difficulty of the quantitatively calculation. Therefore, scientifically establishing the evaluation index system is an important part of the evaluation problem. There are a lot of achievements in this aspect [

In reality, evaluation index with three-parameter interval grey number (T-PIGN) exists widely, and there are many scholars studying this problem. Li et al. proposed a risky evaluation approach based on prospect theory to solve the multi-criteria evaluation problem with T-PIGN [

Definition 1:

Define the operation of T-PIGN, which is similar to the operation properties of the interval grey number.

Let

be the T-PIGN, we define:

Information entropy is an important concept in information theory, applied to measure the disorder degree of system. For a specific system, if the system is very random, chaotic and without order, the information entropy of the system will be large. Conversely, if a system is determinate, and obeys some order, the information entropy of the system will be small. Shannon proposed the information entropy equation [

Be similar to the operation of information entropy, the T-PIGN distance entropy can be defined [

Definition 2:

be the distance entropy of

Theorem: The closer

Proof:

Let

Thus, we can obtain that:

The derivationof

Let

Because

Similarly, when

Therefore, when

In the same way, the theorem that the farther

Meanwhile, the properties of grey distance entropy can be got:

1) It has the nonnegative, that is

2) It has the extremum, that is

3) It has the symmetry, that is

Proof:

Therefore, we can obtain that

Let

In the multi-attribute evaluation problems, if the difference of index value of the same index in all schemes is small, the impact of the index on the evaluation distinguishing degree is small. Conversely, it shows that the impact of the index on the evaluation distinguishing degree is great. Therefore, considering from this angle, the index that has greater difference degree should be retained. By the definition and theorem of the T-PIGN distance entropy, we know that

And let

Because

If the grey distance entropy of the index

Obtain the sum vector of grey distance entropy among the schemes under each index by calculating T-PIGN distance entropy, let sum vector

In the new evaluation sample matrix, because the grey distance entropy sum among the schemes under each index is constant, calculate the variance of surplus index grey distance entropy directly, and it is denoted as

Compare the relative error between

greater

Evaluation and selection of cadres is a multi-factors evaluation problem. A unit made 6 assessment index in the cadre assessment and selection: ideology and morality (

First, establish the evaluation matrix X.

Calculate the grey distance entropy of 5 candidates by using Equation (1), and the result is shown in

Based on

Thus,

Because the sum of grey distance entropy of each index is constant, the distance entropy is

0.6917 | 0.6925 | 0.6930 | 0.6927 | |||

0.6931 | 0.6926 | 0.6931 | 0.6931 | |||

0.6931 | 0.6931 | 0.6926 | 0.6931 | |||

0.6930 | 0.6931 | 0.6925 | 0.6930 | |||

0.6927 | 0.6930 | 0.6931 | 0.6931 | |||

0.6929 | 0.6928 | 0.6930 | 0.6918 | |||

0.6917 | 0.6930 | 0.6923 | 0.6928 | |||

0.6931 | 0.6929 | 0.6931 | 0.6931 | |||

0.6931 | 0.6931 | 0.6928 | 0.6931 | |||

0.6930 | 0.6931 | 0.6930 | 0.6931 | |||

0.6927 | 0.6923 | 0.6926 | 0.6929 | |||

0.6929 | 0.6931 | 0.6931 | 0.6927 | |||

0.6925 | 0.6930 | 0.6929 | 0.6931 | |||

0.6926 | 0.6929 | 0.6925 | 0.6926 | |||

0.6931 | 0.6931 | 0.6925 | 0.6931 | |||

0.6931 | 0.6931 | 0.6928 | 0.6931 | |||

0.6930 | 0.6923 | 0.6931 | 0.6929 | |||

0.6928 | 0.6931 | 0.6931 | 0.6927 | |||

0.6930 | 0.6923 | 0.6929 | 0.6930 | |||

0.6931 | 0.6931 | 0.6925 | 0.6931 | |||

0.6926 | 0.6928 | 0.6925 | 0.6925 | |||

0.6925 | 0.6930 | 0.6928 | 0.6929 | |||

0.6931 | 0.6926 | 0.6931 | 0.6931 | |||

0.6930 | 0.6931 | 0.6931 | 0.6925 | |||

0.6927 | 0.6928 | 0.6931 | 0.6930 | |||

0.6931 | 0.6931 | 0.6926 | 0.6931 | |||

0.6931 | 0.6931 | 0.6931 | 0.6925 | |||

0.6930 | 0.6931 | 0.6931 | 0.6929 | |||

0.6931 | 0.6929 | 0.6929 | 0.6931 | |||

0.6918 | 0.6927 | 0.6927 | 0.6925 |

According to

According to

Original evaluation index system | Optimal evaluation index system | ||||
---|---|---|---|---|---|

Relative closeness | Ranking | Relative closeness | Ranking | ||

0.5848 | 2 | 0.5671 | 2 | ||

0.6975 | 1 | 0.7089 | 1 | ||

0.4871 | 3 | 0.5117 | 3 | ||

0.4528 | 4 | 0.4254 | 4 | ||

0.4087 | 5 | 0.3797 | 5 | ||

The best candidate | |||||

nished. The optimal evaluation index system is

Sort the 5 candidates in the original evaluation index system and the optimal evaluation index system through the index system selection, and then get the sorting of candidates by calculating the relative closeness. The result is shown in

According to

This paper establishes T-PIGN distance entropy model that can be applied to the selection of evaluation index system with T-PIGN. The selection degree is related to the set value

In order to further verify the effectiveness and applicability of the index system selection method, this paper applies this method in [

The authors are grateful to anonymous referees for their helpful and constructive comments on this paper. This work was supported by the Soft-science Foundation of Henan Province (142400410727) and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (094100510013).

FanlinMeng,NaWang,BingjunLi, (2015) Selection Method of Evaluation Indicators with Three-Parameter Interval Grey Number. Open Journal of Applied Sciences,05,833-840. doi: 10.4236/ojapps.2015.512080