Intuitionistic fuzzy-valued Choquet integral and its application in multicriteria decision making
Introduction
As an aggregation function, the Choquet integral [9] with respect to fuzzy measures [43] has performed successfully in multicriteria decision making (MCDM) [13], [14], [16], [18], [32], [38]. The main characteristic of this aggregation function is that it is able to flexibly describe the relative importance of decision criteria as well as their interactions [16], [32]. Some well-known aggregation functions, such as the weighted arithmetic mean (WAM), the ordered weighted averaging (OWA) [60], and the weighted minimum and maximum, are special cases of the Choquet integral [14], [20], [21]. There are many works on the Choquet integral of single-valued functions [38], [39], [40], [42], [48], [50]. Jang et al. [23] introduced the Choquet integral of set-valued functions and studied their mathematical properties. Zhang et al. [66] performed a further investigation on the set-valued Choquet integral. Yang et al. [61] extended the Choquet integral with a measurable interval valued integrand and proposed the fuzzification of Choquet integral in which the integrand is fuzzy-valued and the integration result is a fuzzy number. Wang et al. [51] studied the real-valued Choquet integral of a fuzzy-valued integrand.
In fuzzy set theory [62], [64], [65], the membership of an element to a fuzzy set is a single value between 0 and 1, and the degree of non-membership is just equal to 1 minus the degree of membership. As an extension of the fuzzy set, the intuitionistic fuzzy set (IFS) [1], [2] is characterized by two functions expressing the degree of membership and the degree of non-membership respectively. The only constraint is that the sum of the two degrees must not exceed 1. In many decision making problems, most of the information the decision maker presents is usually imprecise or uncertain due to time pressure, lack of data, or the decision maker’s limited attention and information processing capabilities [46], [58]. In such situation, the IFS, as compared with the fuzzy set, becomes a more suitable tool for describing the imprecise or uncertain decision information and dealing with the hesitation and uncertainty in decision making. Many researchers have shown great interest in the IFS theory as well as its application in decision making [3], [8], [11], [22], [31], [44], [55], [56], [58], [59]. Atanassov [1] defined some basic operations and relations over IFSs. De et al. [16] proposed some new operations on IFSs, such as scalar multiplication and exponentiation. Xu [55] introduced the score function and accuracy function-based method to compare intuitionistic fuzzy values and developed the intuitionistic fuzzy weighted averaging operator, the ordered weighted averaging operator and the hybrid aggregation operator to aggregate intuitionistic fuzzy information. Xu and Yager [58] proposed some types of geometric aggregation operators of the intuitionistic fuzzy values and presented an approach of intuitionistic fuzzy multi-attribute decision making by using the intuitionistic fuzzy hybrid geometric operator.
It is of interest to combine the Choquet integral and the IFS theory for MCDM under intuitionistic fuzzy environment, because, by doing this, we cannot only deals with the imprecise and uncertain decision information but also efficiently take into account the various interactions among the decision criteria.
The intuitionistic fuzzy-valued Choquet integral, the combination of the Choquet integral and the IFS theory, can also act as an aggregation tool employed in MCDM as well as other multicriteria analysis field. Some works on this topic have been performed recently, see [45], [57]. In this paper, we focus on the aggregation properties and characteristics of the intuitionistic fuzzy-valued Choquet integral (IFCI) and the intuitionistic fuzzy-valued conjugate Choquet integral (IFCCI).
The paper is organized as follows. After the introduction, we introduce some notions about fuzzy measures, the Choquet integral and IFSs, and discuss the operational properties and the order rations on intuitionistic fuzzy values in Section 2. In Section 3, we investigate the aggregation properties and characteristics of the IFCI and the IFCCI. In Section 4, the intuitionistic fuzzy valued Choquet integrals-based MCDM method is presented and analyzed. Section 5 provides an application example to illustrate the MCDM method. Finally, we conclude and discuss future work in Section 6.
Throughout this paper, X = {x1, x2, … , xn} is a finite non-empty set and is the power set of X.
Section snippets
Fuzzy measure and Choquet integral
Definition 1 A fuzzy measure on X is a set function such that μ(∅) = 0, μ(X) = 1; A,B ⊆ X, A ⊆ B implies μ(A) ⩽ μ(B).[15], [17], [25], [42], [43]
One can see that a fuzzy measure is a normal monotone set function which vanishes at the empty set.
Furthermore, a fuzzy measure on X is said to be
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additive if μ(A ∪ B) = μ(A) + μ(B) for all disjoint subsets A,B ⊆ X,
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subadditive if μ(A ∪ B) ⩽ μ(A) + μ(B) for all disjoint subsets A,B ⊆ X,
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superadditive if μ(A ∪ B) ⩾ μ(A) + μ(B) for all disjoint subsets A,B ⊆ X,
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cardinality-based if, for any A ⊆ X, μ(A) depends only on the
Intuitionistic fuzzy-valued Choquet integral
In this section, we will generalize the original Choquet integral shown in Eq. (2) by introducing the intuitionistic fuzzy-valued Choquet integral (IFCI) and the intuitionistic fuzzy-valued conjugate Choquet integral (IFCCI).1 Definition 8 Let be an intuitionistic fuzzy-valued function on X, μ be a fuzzy measure on X, the intuitionistic fuzzy-valued
Multicriteria decision-making method based on intuitionistic fuzzy-valued Choquet integral
This section presents a new method for MCDM, in which the partial evaluations of the alternatives are given by IFVs and the interaction among the criteria are allowed.
Let Y = {y1, y2, … , ym} be an alternative set of m alternatives, the decision maker will choice the best one(s) from Y according to a criterion set X = {x1, x2, … , xn}. In the intuitionistic fuzzy MCMD [3], [31], [44], [55], [59], the partial evaluation of each alternative on each criterion is an IFV. The partial evaluations of an
An illustrative example
This section presents an application of the proposed method to assess the software development risks. Risk management has played an important role in software project management [5]. Risk assessments have been typically employed in the development process to identify and evaluate the risk inherent in software projects [6], [7], [34], [41].
The software development risk assessment problem involves more than one criterion (risk factor), and criteria often conflict with each other. Almost all of
Conclusions and remarks
In this paper, we combine the Choquet integral and the IFS theory to propose two intuitionistic fuzzy aggregation functions, the IFCI and the IFCCI, and investigate their aggregation properties and characteristics in the intuitionistic fuzzy MCDM framework.
The aggregation properties of the IFCI and the IFCCI greatly depend on the give total order relation on IFVs. The order relation based on the score and accuracy functions, ⩽sh, is not an operation-invariant total order. Accordingly, based on
Acknowledgements
We are grateful to the Editor-in-Chief, Professor Witold Pedrycz, and the anonymous referees for their valuable comments and suggestions that helped us to improve the earlier versions of this paper. The authors also acknowledge Professor Jianzhen Zhang, Hongxia Sun, Shengju Sang, and Fanyong Meng, for their help in improving the linguistic quality of the paper. The work was supported by the National Natural Science Foundation of China (No. 71201110, 70771010), Humanities and Social science
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