For a multipleattribute group decisionmaking problem with interval intuitionistic fuzzy sets, a method based on extended TODIM is proposed. First, the concepts of interval intuitionistic fuzzy set and its algorithms are defined, and then the entropy method to determine the weights is put forward. Then, based on the Hamming distance and the Euclidean distance of the interval intuitionistic fuzzy set, both of which have been defined, function mapping is given for the attribute. Finally, to solve multipleattribute group decisionmaking problems using interval intuitionistic fuzzy sets, a method based on extended TODIM is put forward, and a case that deals with the site selection of airport terminals is given to prove the method.
Zadeh [
Multipleattribute decision making (MADM) belongs to multicriteria decision making (MCDM), which has characteristics of discrete types and limited alternatives. The process of decision making is to gather the opinions of all the decision makers for several alternatives. MADM means that there is not only one attribute, and how to integrate the attributes of various alternatives is very important. Therefore, many researchers have devoted themselves to the study of MADM, and rich achievements have been obtained. Specific methods have been used to solve the problems of multipleattribute decision making, such as the method of choice [
Park et al. [
However, the methods used to solve the problems that concerned MADM with IIFS have been extended. Some methods involving MADM with IIFS can be understood as follows. Hu and Xu [
Gomes and Lima [
The traditional TODIM method mainly handles real numbers where the weights are known and only one decision maker appears in the MADM problems. The IIFS is used to express the opinions of every expert for every alternative, because its description is more accurate than other mathematical linguistics. In this paper, multipleattribute group decision making is studied by using the extended TODIM method, which is represented by IIFSs. The TODIM method aims to optimize the ranking of alternatives. In addition, the entropy method for the determination of weights is put forward. We can derive the expert weights according to the score function, and the attribute weights can be derived by the idea of maximum entropy.
The structure of this paper is arranged as follows. In Section
Because the concept of IIFS is derived from intuitionistic fuzzy sets, we first define IFS. On the basis of that, IIFS will be defined. The basic algorithms of the IIFS are given, as well as a description of its operators.
Let
In this definition,
This expression should satisfy the following conditions:
Let
In this definition,
This expression should satisfy the following conditions:
Given the following assumptions, let
Then we have these basic algorithms:
The interval intuitionistic fuzzy weighted averaging (IIFWA) operator is a kind of aggregating operator, which evolved from the intuitionistic fuzzy weighted averaging (IFWA) operator. This operator has been proven, and its specific form is
Alternatives will be evaluated according to the distance of IIFS. Based on the Hamming distance and the Euclidean distance of IFS, we can extend to the normalized Hamming distance and the normalized Euclidean distance of IIFS. Its specific form is as follows.
Let
When
For a multipleattribute group decisionmaking problem, we suppose that there is an alternative set
The IIFWA operator is adopted to aggregate the expert matrix. In this case, we suppose that the weight of expert
The concept of entropy is derived from thermodynamics. C. E. Shannon has recently applied it to information theory. The basic principle is that entropy can be used as a measure of the useful information that the data provides. The entropy weight method has been widely used in the decisionmaking process. In this paper, the entropy coefficient method is applied to determine the weights of experts and attributes in group decision making using IIFS.
Let
Formula (
Using the properties of entropy, we find that if the degree of disorder in a system is high, the entropy value will correspondingly be larger. In group decision making, if experts have similar opinions about different alternatives, the entropy value will be larger. That is, the greater the entropy value of expert
Let
Formula (
the relative importance of evaluation attributes and alternatives is independent;
if there is
Formula (
According to the theory of entropy, when the entropy of attribute
Based on the above discussion, which included the concept of IIFSs, as well as its basic algorithms and the gathering method of the interval intuitionistic fuzzy matrix, a new method of processing multiattribute group decision making is given, which is called the extended TODIM method. The traditional TODIM method measures the comprehensive degree of an alternative better than others and is extended in this paper to handle IIFS. Assuming the conditions that (a) weights of the experts and attributes are unknown and (b) there is more than one decision maker, we outline the steps of how to use the TODIM method in the next segment.
There are
Calculate the expert weights. The weight measure
Aggregate all of the assessment matrices
Calculate the attribute weights. On the basis of Step 3, the weight measure
According to the Hamming distance (Definition 5.1) and the Euclidean distance (Definition 5.2) of IIFSs, the distance between every element in the aggregating matrix and the interval intuitionistic fuzzy number
The interval intuitionistic fuzzy number
When matrix
In expression (
According to expression (
Rank the values of function
An airfield plans to construct different airport terminals in surrounding cities. After preliminary screening, several cities remain for further analysis and research. First, a comprehensive evaluation index system should be established, starting from the following three aspects: number of potential customers, degree of traffic connections, and the existing competitive capacity.
The number of potential customers can refer not only to the volume of airline passenger transportation and the rate of growth in recent years, but also to the local economic conditions, such as local GDP, industrial patterns, and other enterprises. The degree of traffic connections mainly takes other modes of transport into account, considers whether the others are convenient and fast and have a high coverage rate throughout the city, and also considers the convenience degree of each traffic connection from the city to the surrounding areas. The existing competitive capacity mainly means the competitive pressures coming from highspeed rail and airport terminals of other airfields, the degree of local government supporting the air transportation industry, and the advantages of relevant policies.
Suppose there are five alternative cities
Specific steps are as follows.
Determination of basic data: the three matrices below represent the three experts’ evaluation of the five cities for every attribute:
Calculating the values of expert weight: according to formulas (
Finally, the expert weights can be worked out and the results are






0.850  0.575  0.350 

1.625  1.475  0.850 

0.900  0.850  1.275 

0.950  1.425  0.100 

1.100  0.925  0.850 


Total  5.425  5.250  3.425 
The above three matrices of the five cities can apply Definition


 


0.448  0.561  0.161  0.225  0.453  0.554  0.209  0.274  0.347  0.448  0.439  0.532 

0.664  0.739  0.139  0.229  0.387  0.451  0.125  0.227  0.613  0.682  0.225  0.294 

0.459  0.531  0.125  0.176  0.748  0.853  0.000  0.130  0.370  0.435  0.209  0.310 

0.284  0.380  0.321  0.417  0.613  0.683  0.161  0.213  0.591  0.411  0.321  0.401 

0.858  1.000  0.222  0.321  0.212  0.312  0.321  0.374  0.588  0.674  0.200  0.255 
Calculate the values of the attribute weights. On the basis of Step 3, the weight measure
Finally, the attribute weights can be worked out and the results are as follows:






0.778  0.812  0.912 

0.732  0.760  0.798 

0.751  0.622  0.812 

0.842  0.753  0.880 

0.651  0.796  0.783 


Total  3.754  3.743  4.185 
Calculate the distance between the IIFSs in matrix
Calculating by the Hamming distance, the results are as follows:
Calculating by the Euclidean distance, the results are as follows:
The degree by which a given alternative is superior to all of the others can be calculated using the extended TODIM method. First, work out the values of function
Calculating by the Hamming distance, the results are
According to expression (
The values of function
No matter what kinds of distance formulas were adapted in this process, such as the Hamming distance formula or the Euclidean distance formula, the final results of the sorting are consistent. And it is
A method of multipleattribute group decision making based on TODIM is proposed to evaluate the information of the IIFSs. The differences between this paper and other studies that discuss the TODIM method of group decision making in the case of unknown weights [
First, we show that expert weights can be obtained using the original expert matrices. Second, we show how the matrices of IIFSs including different expert opinions can be integrated. Then, we discuss how the attribute weights can be worked out on top of the aggregated expert matrix. Finally, using the Hamming distance and the Euclidean distance of the IIFSs, we accomplish the transformation of the matrix. The traditional TODIM method is improved so that it can be used to handle the IIFS information, allowing for a general ranking of the alternatives. Finally, a case concerning the site selection of airport terminals is described to show that the steps of the above method are operable and easy.
However, according to the Hamming distance and the Euclidean distance of the IIFSs, the transformation of the matrix is simple, and there must be a more accurate mapping function that can be used to express the relationship. Another deficiency of this paper is the simplicity of the method used to determine weights. Some studies have proposed other methods to solve the unknown weights problems, which may be applied to the extended TODIM method for group decision making with the IIFSs.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors also would like to express sincere appreciation to the anonymous reviewers and Editors for their very helpful comments that improved the paper. This paper is supported by the National Social Science Fund of China (no. 12BJY112) and the Fundamental Research Funds for the Central Universities of China (no. ZXH2010B002).