The aim of this paper is to develop a method for ranking trapezoidal intuitionistic fuzzy numbers (TrIFNs) in the process of decision making in the intuitionistic fuzzy environment. Firstly, the concept of TrIFNs is introduced. Arithmetic operations and cut sets over TrIFNs are investigated. Then, the values and ambiguities of the membership degree and the nonmembership degree for TrIFNs are defined as well as the value-index and ambiguity-index. Finally, a value and ambiguity-based ranking method is developed and applied to solve multiattribute decision making problems in which the ratings of alternatives on attributes are expressed using TrIFNs. A numerical example is examined to demonstrate the implementation process and applicability of the method proposed in this paper. Furthermore, comparison analysis of the proposed method is conducted to show its advantages over other similar methods.
Multiattribute decision making (MADM) is an important research field of decision science, operational research, and management science. MADM is the process of identifying the problem, constructing the preferences, evaluating the alternatives, and determining the best alternatives. The classical decision making methods assume that accurate data is available to determine the best alternatives among the available options. However, in practice, due to the inherent uncertainty and impression of the available data, it is often impossible to obtain accurate information. Therefore, decision making under fuzzy environment problem is an interesting research topic having received more and more attention from researchers during the last several years.
The fuzzy set [
For decision making using the IF sets, it is required to rank the IFNs. So far, several methods have been developed for ranking the IFNs. Mitchell [
In this paper, TrIFNs are introduced as a special type of IFNs, which have appealing interpretations and can be easily specified and implemented by the decision maker. The concept of the TrIFNs and ranking method as well as applications are discussed in depth.
This paper is organized as follows. In Section
A TrIFN
TrIFN.
If
In a similar way to arithmetic operations of IFNs, the arithmetic operations of TrINFs can be defined as follows.
Let
From Definition
According to the cut sets of the IF set defined in [
An (
A
Using (
A
Using (
In this section, the value and ambiguity of TrIFNs are defined as follows.
Let
The function
According to (
Let
It is easy to see that
According to (
Likewise, according to (
Based on the above value and ambiguity of a TrIFN, a new ranking method of TrIFNs is proposed in this subsection. A value-index and an ambiguity-index for
Let
It is easily seen that
Let
According to Definition
Furthermore, from
Let
According to Definition
Let
If
If
If
If
If
Wang and Kerre [
Let
It is derived from (
Similarly, it follows that
Let
According to Theorem
Similarly, it follows that
In this section, we will apply the above ranking method of TrIFNs to solve MADM problems in which the ratings of alternatives on attributes are expressed using TrIFNs. Sometimes such MADM problems are called as MADM problems with TrIFNs for short. Suppose that there exists an alternative set
Due to the fact that different attributes may have different importance, assume that the relative weight of the attribute
Identify the evaluation attitudes and alternatives.
Pool the decision maker’s opinion to get the ratings of alternatives on alternatives on attributes, that is, the TrIFN decision matrix
Normalize the TrIFNs decision matrix. In order to eliminate the effect of different physical dimensions on the final decision making results, the normalized TrIFNs decision matrix can be calculated using the following formulae:
Calculate the weighted comprehensive values of alternatives. Using Definition
Rank all alternatives. The ranking order of the alternatives
Let us suppose there is an investment company, which wants to invest a sum of money in best option. There is a panel with four possible to invert the money:
TrIFNs decision matrix.
|
|
| |
---|---|---|---|
|
(0.26, 0.36, 0.46, 0.56; | (0.34, 0.44, 0.54, 0.64; | (0.12, 0.22, 0.32, 0.42; |
0.16, 0.36, 0.46, 0.66) | 0.24, 0.44, 0.54, 0.74) | 0.04, 0.22, 0.32, 0.50) | |
|
(0.50, 0.60, 0.70, 0.80; | (0.30, 0.55, 0.70, 0.80; | (0.34, 0.44, 0.54, 0.54; |
0.42, 0.60, 0.70, 0.88) | 0.22, 0.55, 0.70, 0.88) | 0.34, 0.44, 0.54, 0.74) | |
|
(0.55, 0.55, 0.68, 0.68; | (0.54, 0.64, 0.74, 0.84; | (0.36, 0.46, 0.56, 0.56; |
0.28, 0.55, 0.68, 0.78) | 0.46, 0.54, 0.74, 0.92) | 0.16, 0.46, 0.56, 0.66) | |
|
(0.66, 0.76, 0.86, 0.96; | (0.55, 0.63, 0.78, 0.86; | (0.18, 0.28, 0.38, 0.48; |
0.64, 0.76, 0.86, 0.98) | 0.55, 0.63, 0.78, 0.92) | 0.08, 0.28, 0.38, 0.58) |
Using (
According to (
Using (
It is easy to know that
To further illustrate the superiority of the decision method proposed in this paper, we apply some of the other methods to rank the TrIFNs
Ranking of TrIFNS using different methods.
Ranking methods |
|
|
|
|
Ranking results |
---|---|---|---|---|---|
Nan et al. [ |
0.7425 | 1.1775 | 1.0425 | 1.22 |
|
Nehi [ |
0.39 | 0.598 | 0.54 | 0.622 |
|
Wang and Nie [ |
0.37125 | 0.58875 | 0.52125 | 0.61 |
|
From Table
In this paper, we have studied two characteristics of a TrIFN, that is, the value and ambiguity, which are used to define the value index and ambiguity index of the TrIFN. Then, a ranking method is developed for the ordering of TrIFNs and applied to solve MADM problems with TrIFNs. Due to the fact that a TrIFN is a generalization of a trapezoidal fuzzy number, the other existing methods of ranking fuzzy numbers may be extended to TrIFNs. More effective ranking methods of TrIFNs will be investigated in the near future.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the Key Program of National Natural Science Foundation of China (no. 71231003), the National Natural Science Foundation of China (nos. 71171055, 71101033, and 71001015), the Program for New Century Excellent Talents in University (the Ministry of Education of China, NCET-10-0020), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20113514110009) as well as Science and Technology Innovation Team Cultivation Plan of Colleges and Universities in Fujian Province.