A Method for Multiple Attribute Decision Making Based on the Fusion of Multisource Information

We propose a new method for the multiple attribute decision making problem. In this problem, the decision making information assembles multiple source data. Two main advantages of this proposed approach are that (i) it provides a data fusion technique, which can efficiently deal with the multisource decision making information; (ii) it can produce the degree of credibility of the entire decision making. The proposed method performs very well especially for the scenario that there exists conflict among the multiple source information. Finally, a traffic engineering example is given to illustrate the effect of our method.


Introduction
In the decision-making theory, many methods and their applications have been extensively studied.Recently, multiple attribute decision making (MADM) problems [1,2], whose decision making information comes from multiple source data, receive more and more attention.Among these problems, the MADM problems which have the subjective and the objective information [1][2][3][4][5] at the same time, and the multiple attribute group decision making (MAGDM) [6][7][8][9] problems are the two hot topics in this research field.
The key to the two kinds of problems is to fuse various pieces of information [10].For example, the following literature is to solve the first kind of problems.The literature [3] has proposed an optimization model to deal with the MADM problems with preference information on alternatives, which were given by decision maker in a fuzzy relation.With respect to the MADM problems with intuitionists fuzzy information, the literature [4] has proposed an optimization model based on the maximum deviation method.By this model, we can derive a simple and an exact formula for determining the completely unknown attribute weights.The literature [5] has proposed a linguistic weighted arithmetic averaging operator to solve the MADM problems, where there is linguistic preference information and the preference values take the form of linguistic variables and so forth.
In the respect of MAGDM problems, the literature [6] has researched the MAGDM problem with different formats of preference information on attributes; the literature [7] has researched the 2-tuple linguistic MAGDM problems with incomplete weight information and established an optimization model based on the maximizing deviation method; the literature [8] has presented a new approach to the MAGDM problems, where cooperation degree and reliability degree are proposed for aggregating the vague experts' opinions; the literature [9] has developed a compromise ratio methodology for fuzzy MAGDM problems and so forth.
Through these literatures, we could find that most of the solutions have used some subjective attitudes or information [10,11], which were not provided by the problem itself.This is seriously out of line with the social needs.In order to overcome this defect, this paper presents two methods for the above two kinds of problems.The proposed methods are based on strong calculation and combined with the optimization theory [12] or the variation coefficient method [13].
The highlights of this new method could be summarized into two points.The first, it can efficiently deal with the

The MADM Problems under the Condition of Information
Conflict.We will introduce this problem as follows.Let  = { 1 ,  2 , . . .,   } be a discrete set of  feasible alternatives, let  = { 1 ,  2 , . . .,   } be a finite set of attributes, and let   =   (  ) ( = 1, 2, . . ., ;  = 1, 2, . . ., ) be the values of the alternative   under the attribute   .In this paper, we only consider the situation that   is given in real numbers.The decision matrix of attribute set  with regard to the set  is expressed by the matrix For convenience, we suppose that the decision matrix  has been normalized and denote  = {1, 2, . . ., },  = {1, 2, . . ., }.For specific details of standardization, please see the literature [3,14].
Based on the above conditions, the problem is to select and rank the alternatives.In this paper, we mainly consider the situation where there are serious conflicts between the subjective information and the objective information [15].
Here, the same as the above subsection, we suppose that the decision matrix  has been normalized, and we only consider the situation that   is given in real numbers.
The problem is to solve the MAGDM problem with the above conditions.

Main Tool of Our Research
The common character of the two problems is that they all involve the operation of interval numbers.In addition, we must point out that the situation we have no weight information equals to the situation where the weight is a variable located in the interval [0, 1].In the following, we would give a new method for operating the interval numbers.
The new method originates from the basic of strong calculation by modern computer.
Without loss of generality, we take calculating the distance between ([ Step 2. Take any element  from  and take any element  from ; according to the formula we could calculate the distance.By doing so, we could get ( * ) 2 distances.Then, we calculate the average value of these distances and denote the average value as .
Step Obviously, the main advantage of this method is that the calculation procedure is in an objective, consistent way, and Abstract and Applied Analysis 3 there is no subjective information involved in the calculation procedure.

Decision Methods
At the beginning of this section, we would introduce a method called the simple additive weighting method [1].Now, we consider a problem in hypothetical situation, where we have known the weight vector  = ( 1 ,  2 , . . .,   )  and the attribute values.In this situation, we could get the comprehensive attribute value   ( = 1, 2, . . ., ) by Obviously, the bigger   () leads to the more excellent   .Therefore, we could accomplish the process of getting the best alternative and ranking all of the alternatives by ( 5), and we could see that the determination of the weight vector  = ( 1 ,  2 , . . .,   )  is the core of the MADM problem in general conditions.

Decision Method 1.
In this paper, we only consider the situation where there is significant difference among these weight vectors; that is, the Kendall consistence [13] of the above weight vectors is imperfect.
The following, we present a new decision making method for solving the problems of subsection 2.1.The characteristic of our method is that it could provide the credibility of the decision maker for the subjective information as well as the entire decisional results.Specific decision steps are as follows.
Step 1. Solve the single objective programming model and denote the result of model ( 6) as  max  ( ∈ ), and  max  is the ideal value of the comprehensive attribute value of   ( ∈ ).
Solve the single objective programming model and denote the result of model (7) as  min  ( ∈ ), and  min  is the negative ideal value of the comprehensive attribute value of   ( ∈ ).
Step 4. If the optimal solution  * is consistent with the subjective decision information , we consider it as the final optimal solution of the entire decision making process, and consider as the degree of believing for the subjective information.In (10), the value √( − 1)/, which is obtained by optimization theory is the max Euclidean distance between ((1/), (1/), . . ., (1/))  and any possible weight vector  * .The value  1 reflects the similarity scale of  * and ((1/), (1/), . . ., (1/))  .From the aspect of set-valued statistics, the bigger the  1 is, the more support would be got from the data of the objective information.
Because there is coordination between the subjective and objective information, and they all support the optimal alternative  * , we set the credibility of the entire decision as 1.
Step 5.If the optimal solution  * is inconsistent with the subjective decision information , we would believe that the subjective information has got no support from the objective information.Here, we correct the value  1 and set  1 as zero.Define and define x as the alternative corresponding to the index λ .Obviously, the alternative x could represent the subjective information to some extent.
Step 6.We use the parameter  2 to represent the credibility of the entire decisional results.In this step, we assume that the MADM problems have the alternative set of { * , x} only.
Because the weight information is unknown, we consider the weight vector as random element in weight space and the random element follows a uniform distribution.
Step 7. By using the main tool of our research, which has been introduced in Section 3, we compare the advantages of the alternative  * with the alternative x and calculate the credibility of them.By (5), every element of  would support one optimal alternative.Based on this, each alternative would be supported by a region of hypercube , and the ranking of all alternatives could be solved by comparing the regions of hypercube .The result of this regions comparison could be got by the technique of numerical simulation [16].It's worth mentioning that if the sum of the weight vector is not one, by normalization, it is equivalent to a weight vector with the sum one.

Decision Method 2. Now, we present a new decision mak-
ing method for solving the problem of subsection 2.2.In this paper, we only consider the situation where there is significant difference among these weight vectors  1 ,  2 , . . .,   ; that is, the Kendall consistence [13] of the above weight vectors is imperfect.
For convenience, we denote the expert weight vector of set  as Obviously, the relative attribute weights of the set  could be got by ) . ( The result of the formula [13] is one interval number column vector; we denote it as In the following, we would present the new decision making method.
Step 1. Denote the weight vectors which are provided by experts   and   as The distance between   and   would be got by the main tool of our research, which has been introduced in Section 3. The computation follows the formula [5].
Step Step 6.Take any element  from W * , take any  from , and denote According to comparing each   ( ∈ ), any element  from W * will support one alternative.Thus, the set W * would be divided into  subsets.We denote the element number of these subsets as  1 ,  2 , . . .,   and denote the support degree for each   ( ∈ ) as   = (  / * ).
Step 7. Increase the value  * gradually and repeat the above steps.When the value   ( ∈ ) holds steady to two digits after the decimal point, the procedure is ended and the final result   ( ∈ ) is seen as the support degree for each   ( ∈ ).
Step 8.By comparing each   ( ∈ ), we could sort and select the optimal alternatives.

An Application Example
In this section, we would present an example to illustrate our proposed methodology.Because the first method is relatively complex, and the two methods are similar, we only give an example to verify the first method.This example is coming from the traffic engineering.In this example,  = { 1 ,  2 , . . .,  5 } is the set of alternatives,  = { 1 ,  2 ,  3 ,  4 } is the set of attributes, and all attributes are of benefit types.Consider  = ( 1 ,  2 ,  3 ,  4 )  as the weight vector of all attributes.Here, we have no information about .The standardized decision matrix is given as The evaluation vector, which is given towards the set { 1 ,  2 , . . .,  5 } and determined by the decision maker, is  = (0.45, 0.34, 0.27, 0.25, 0.25) . ( Now we seek to rank these alternatives and find the most desirable one. Firstly, according to the method proposed in this paper, we build one optimal decision model as follows: min  * = ( * 1 − 0.45) Afterwards, by using the simple additive weighting method [1], the vector of comprehensive attribute values of each alternative could be obtained and it is (0.7734, 0.7786, 0.7476, 0.4571, 0.6081)  . (26) Obviously we can conclude that  2 would be the optimal alternative and it is inconsistent with the subjective decision information .Thus, we set the degree of believing for the subjective information as 0.
Then, we use the method of hypercube segmentation [16] as a tool to compare the advantages of alternative  1 and alternative  2 .Results show that  2 comes to be the best alternative, and its credibility is 98.7654%.
In a word, there are conflicts between the subjective and objective decision-making information in this case.According to our method,  2 is the best alternative, with the reliability 0 for the subjective information and the reliability 98.7654% for the entire decision.

Conclusions
Firstly, from the above example, it could be found that the uncertainty of the multisource decision making information has been studied and fused, and the process of the given method is objective, with no subjective factors.This is the highlight of our new method.
Secondly, though our two methods are all based on the new algorithms of interval numbers, they also have the diversity.The first method combines with the optimization theory, and the second method combines with the principle that the minority is subordinate to the majority.Thirdly, from the example it can be seen that it is easy and convenient to use the two new methods, and the numerical example illustrates that our proposed method can deal with the multisource decision-making information well.
So, the proposed method may have a higher availability and a better application prospect.