Cui Jie,Liu Sifeng,Xie Naiming

(1.College of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing,211106,P.R.China;2.Faculty of Economics and Management,Huaiyin Institute of Technology,Huaian,223001,P.R.China)

INTRODUCTION

Multi-attribute decision making is generally implemented in economy, engineering,management and some other fields for information fusion of a number of attributes and their weigh ts from different programs.Multi-attribute decision making is delivered through some forms of operators to get the value of comprehensive evaluation of each program,sort the evaluation values and then select the best solution.If the available factor information,such as attribute weights,attribute values,is completely unknown or partially unknown,a multi-attribute decision making problem is an uncertain one.Because of the disturbing factors present within and outside all systems,and the limited capability of human′s understanding of the real world,the information obtained in the process of researching systems usually has some uncertainties[1-3]. Therefore,the in-depth studies on the uncertain multiattribute decision making has a broad prospect.At present,domestic and foreign researches on this issue are fruitful.Li,et al proposed a grey multi-attribute decision making method, and applied it to the problem of selecting vendors[4].Xiao,et al proposed a decision model of optimal gray clustering[5].Luo,et al constructed a grey multi-criteria risk decision making[6].Cui,et al proposed a new method for solving the weight of multi-attribute decision making[7]. From the perspective of risk-type indicator,Rao,et al and Luo,et al presented the gray correlation matrix and the grey multi-attribute group decision making,respectively[8-9].Chen,et al proposed a new method of grey fuzzy multi-attribute decision making based on complementary judgment matrix[10]. Gong gave the information on alternatives preference with partly unknown attribute weights and presented the attribute values in the forms of triangular fuzzy numbers[11].Since the attribute values were the interval-valued trapezoidal intuitionistic fuzzy numbers and the weights of attributes were intervals, a multi-attribute decision making method was proposed based on fractional programming[12]. Aiming at hybrid multiattribute decision-making under the risk of interval probability with weight unknown,a decision approach was presented based on entropy weight and projection theory[13]. Zhang,et al investigated the multi-attribute group decision making problems with integrating conditions of attribute weight information and unknown attribute weights into the framework of intervalvalued intuitionistic fuzzy set[14].Wei investigated the dynamic hybrid multi-attribute decision making problems. In his study, the decision information, provided by decision makers at different periods,was expressed in real numbers,interval numbers or linguistic labels(linguistic labels was described by triangular fuzzy numbers), respectively[15]. Current researches focused on grey decision-making problems with a finite number of programs. Grey multi-stage decision making research is still rare.From the perspective of grey system theory,almost all of the systems are grey in reality.The identification of system information by human beings will gradually transform from grey to white state,and gradually approach its true face.It is significant to study the grey multi-stage decision model to improve the science and rationality of grey decision-making. Therefore,a grey multi-stage decision model is constructed and an implementation algorithm is developed. A numerical analysis is given to demonstrate the applicability of the method.

1 PRELIMINARY KNOWLEDGE

In the multi-attribute decision making problem with weights completely unknown and attribute values as interval grey numbers,to facilitate the description,the subscript set is denoted as I={1,2,… ,n},J={1,2,… ,m},the time set T={1,2,… ,t},i∈ I,j∈ J,k∈ T.The alternative set is X={x1,x2,…,xn}.U={u1,u2,…,um}is theset of attributes of each alternative.The information of attribute weights is completely unknown.For the alternative xi,its attribute ujis calculated and obtained with the attribute value aijAll of the attribute values form the following decision matrix A

Assume that W={w1,w2,…,wm}is the set of all attribute values,wj∈W,and it can be expressed as a vector

Each alternative′s synthetical score at the k th time point is denoted as

The weight vector of each alternative′s synthetical score at the k th time point is denoted as

All alternatives′scores in the t th point form the evaluation matrix of scores

The comprehensive score of the i th alternative is denoted as

The problem of how to select the best scheme or rank the attribute weights is completely unknown in stages and relies on experts′evaluation on the attribute values.

2 ALGORITHM FOR GREY MULTI-STAGE DECISION MAKING MODEL

is the possibility of a(? )＞ b(? ).

The calculation steps are listed as follows to solve the uncertain multi-attribute decisionmaking problem with attribute weights completely unknown and attribute values as interval grey numbers.

Step 1 Standardize decision matrix.

Let

If ujis an efficiency-type index,then

If uj is a cost-typeindex,then

After standardizing decision matrix A by this method, the following decision matrix B is abtained

Step 2 Calculate the attribute weights according to the methods in Ref.[7].

Step 3 Compute the evaluating score matrix G*,composed of the evaluation scores of all alternatives,evaluated at t points.

Step 4 Calculate the weights vector W*,composed of evaluation scores of all alternatives,evaluated at t points.

Step 5 Calculate the final scoreof the i th alternative.

Step 6 Rank the priorities of all alternatives.The bigger the value ofis,the priorer the alternative xi i

3 NUMERICAL ANALYSIS

A numerical analysis is conducted to demonstrate the applicability of the algorithm.

An investment organization plans to invest four enterprises:E1,E1,E3,E4.Now it selects the ratio of investment net output, the rate of investment profit,the rateof internal return,and the level of environmental pollution of these companies to evaluate the decision.

A1:the ratio of investment net output,A2:the rate of investment profit,A3: the rate of internal return,A4: the level of environmental pollution.

This investment organization exerts the first survey of the above indicators of the four companies,and obtains the data decision matrix X(1)

where A1,A2,A3 are efficient indicators,A4 is a cost typeindicator.

B(1)is a standardization decision matrix of grey numbers

Assume that ten experts give th e following score of the four indicators,respectively

According to the new weight calculating method proposed in Ref.[7](see Appendix),the following results are obtained

wA1=0.35,wA2=0.25,wA3=0.25,wA4=0.15

To further understand the formation of the above indicators A1,A2,A3,A4 of the four companies,the investment institution delivers another two surveys and obtains decision-making matrixes X(2),X(3)

According to the standardization method of the above indicators, after the treatment of decision-making matrixes X(2)is performed,X(3)is developed and the standardization decision matrixes B(2),B(3)are obtained B(2)=

According to Step 3 of the algorithm,the three-time point score matrix of the four enterprises is obtained

The weight of each point in time is as follows W*=[w*(1) w*(2) w*(3)]T=

The final overall score of the four enterprises are

The following possibility degrees are calculated

P(S1＞ S2)=0.27,P(S1＞ S3)=0.87

P(S1＞ S4)=0.16,P(S2＞ S3)=0.98

P(S2＞ S4)=0.34,P(S3＞ S4)=0

Therefore,the rank of the four enterprises is E4?0.66E2?0.73E1?0.87E3.

The calculation results show the score of enterprise E4 is the highest,therefore,it has the highest value of investment. Investment institution should invest enterprise E4.

4 CONCLUSION

As an important embranchment of modern decision science,grey decision making has a very broad application prospect.Based on the existing research results, a grey multi-stage decision model is established.The study results can rich the theory of grey system decision making model and expand its applicative range

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