Intuitionistic fuzzy ordered weighted distance operator
Highlights
► A new intuitionistic fuzzy distance measure by using the OWD measure. ► The intuitionistic fuzzy ordered weighted distance (IFOWD) operator. ► Main properties and different families of the IFOWD operator. ► A new group decision-making method.
Introduction
The ordered weighted averaging (OWA) operator [55] is a very well-known aggregation operator that provides a parameterized family of aggregation operators that includes the maximum, the minimum, and the average, as special cases. The prominent characteristic of the OWA operator is the reordering step. Since its appearance, the OWA operator has been used in a wide range of applications such as [1], [2], [5], [7], [10], [15], [16], [20], [21], [22], [23], [24], [25], [26], [27], [28], [33], [37], [39], [40], [56], [57], [58], [59], [60].
An interesting extension of the OWA is the one that uses distance measures in the OWA operator. Motivated by the idea of the OWA operator, Xu and Chen [49] introduced the ordered weighted distance (OWD) measure, and gave some methods to determine its weights. The prominent characteristic of the OWD measure is that it can relieve (or intensify) the influence of unduly large or unduly small deviations on the aggregation results by assigning them low (or high) weights. This desirable characteristic makes the OWD measure very suitable to be used in many actual fields such as group decision-making, medical diagnosis, data mining, and pattern recognition. The OWD also generalizes a variety of well-known distance measures and aggregation operators, including the normalized Hamming distance, the normalized Euclidean distance, the OWA operator, etc. Yagexr [59] generalized Xu and Chen’s distance measures and provided a variety of ordered weighted averaging norms, based on which he proposed several similarity measures. Combining the OWA operator with Hamming distance, Merigó and Gil-Lafuente [27] introduced a new decision-making technique called the ordered weighted averaging distance (OWAD) operator. The main advantage of this operator is that we can take into account the attitudinal character of the decision-maker. Therefore, we are able to provide decision maker with an approach to the optimal choice according to his or her interests. Another advantage of the OWAD operator is that it provides a parameterized family of distance aggregation operators that ranges from the minimum to the maximum distance. Thus, they can provide a wide range of situations depending on the particular attitude taken by the decision maker in the specific problem considered. Moreover, with the OWAD, it is possible to establish an ideal, though unrealistic, alternative in order to compare it with available options in the decision-making problem. As such, the optimal choice is the alternative closest to the ideal one. Going a step further, Merigó and Gil-Lafuente [28] analyzed the use of the OWAD operator in the selection of human resources in sport management. For further research on the use of other different types of distance measures in the OWA operator, see, for example [14], [21], [22], [23], [24], [25], [52].
Usually, when using the OWD measure and the OWAD operator, it is assumed that the available information is clearly known and can be assessed with exact numbers. However, in the real-life world, due to the increasing complexity of the socioeconomic environment and the lack of knowledge or data about the problem domain, exact numbers are sometimes unavailable. Thus, the input arguments may be vague or fuzzy in nature. Atanassov [3] defined the notion of an intuitionistic fuzzy set (IFS), whose basic elements are intuitionistic fuzzy numbers (IFNs) [40], [46], [54], each of which is composed of a membership degree and a nonmembership degree. In many practical situations, particularly in the process of group decision making under uncertainty, the experts may come from different research areas and thus have different backgrounds and levels of knowledge, skills, experience, and personality. The experts may not have enough expertise or possess a sufficient level of knowledge to precisely express their preferences over the objects, and then, they usually have some uncertainty in providing their preferences, which makes the results of cognitive performance exhibit the characteristics of affirmation, negation, and hesitation. In such cases, the data or preferences given by the experts may be appropriately expressed in IFNs. For example, in multi-criteria decision-making problems, such as personnel evaluations, medical diagnosis, project investment analysis, etc., each IFN provided by the expert can be used to express both the degree that an alternative should satisfy a criterion and the degree that the alternative should not satisfy the criterion. The IFN is highly useful in depicting uncertainty and vagueness of an object, and thus can be used as a powerful tool to express data information under various different fuzzy environments which has attracted great attentions [36], [37], [41], [44], [48], [53], [54], [61]. Therefore, it is necessary to extend the above ordered weighted measures to accommodate the intuitionistic fuzzy situation, which is also the focus of this paper.
For this purpose, we shall develop an intuitionistic fuzzy ordered weighted distance (IFOWD) operator. This operator is very effective for the treatment of the data in the form of IFNs. The main advantage of the IFOWD operator is that it can alleviate the influence of unduly large (or small) deviations on the aggregation results by assigning them low (or high) weights. Moreover, it provides a robust formulation that includes a wide range of particular cases, such as the intuitionistic fuzzy max distance, the intuitionistic fuzzy min distance, the intuitionistic fuzzy normalized Hamming distance (IFNHD), the intuitionistic fuzzy normalized Euclidean distance (IFNED), the intuitionistic fuzzy normalized geometric distance (IFNGD), the intuitionistic fuzzy weighted Hamming distance (IFWHD), the intuitionistic fuzzy weighted Euclidean distance (IFWED), the intuitionistic fuzzy weighted geometric distance (IFWGD), the intuitionistic fuzzy ordered weighted Hamming distance (IFOWHD), the intuitionistic fuzzy ordered weighted Euclidean distance (IFOWED), the intuitionistic fuzzy ordered weighted geometric distance (IFOWGD) and the generalized intuitionistic fuzzy ordered weighted averaging (GIFOWA) operator [62]. Thus, the decision maker is able to consider a wide range of scenarios and select the one that is in accordance with his interests.
This paper is structured as follows. In Section 2, we review some aggregation operators and the ordered weighted distance measures. In Section 3, we develop the IFOWD operator, and study some its various properties. In Section 4, we analyze different types of IFOWD operators. Section 5 briefly describes the decision making process based on developed operators and we give a numerical example in Section 6. Section 7 summarizes the main conclusions of the paper.
Section snippets
Preliminaries
In this Section we briefly review some the OWA operator, the ordered weighted averaging distance (OWAD) and ordered weighted distance (OWD) measure.
Intuitionistic fuzzy sets
Intuitionistic fuzzy set (IFS) introduced by Atanassov [3] is an extension of the classical fuzzy set, which is a suitable way to deal with vagueness. Since its appearance, the IFS has been widely applied to the decision-making problems [4], [6], [8], [18], [29], [30], [31], [32], [34], [36], [37], [38], [41], [42], [43], [44], [45], [46], [47], [48], [51], [54], [61], [62]. It can be defined as follows. Definition 4 Let a set X = {x1, x2, … , xn} be fixed, an IFS A in X is given as following:
Families of the IFOWD operators
By using a different manifestation in the weighting vector W and parameter λ, we are able to obtain a wide range of particular types of the IFOWD operator. The selection of the particular case (or other cases found in the IFOWD) depends on the particular interest of the decision maker in the specific problem considered.
Multiple attribute group decision-making with the IFOWD operator
In this paper, we consider a decision-making application in the selection of investments under uncertainty. Let A = {A1, A2, … , Am} be a discrete set of alternatives, and C = {C1, C2, … , Cn} be the set of attributes (or characteristics). Let E = {e1, e2, … , et} be the set of decision makers (whose weighting vector is ). Each decision maker provides his own payoff matrix . Moreover, according to their objectives, the decision-makers establish a collective ideal investment
Illustrative example
In the following, we are going to develop a numerical example of the new approach. We analyze the results obtained by using different types of IFOWD operators and we see that depending on the aggregation operator used, the decision may be different. Note also that the IFOWD operator may be applied in similar problems as the OWD measure and the OWAD operator.
Assume a decision-maker wants to invest money in a company. After analyzing the market, he considers six possible alternatives:
- (1)
Invest in a
Conclusions
In this paper, we have suggested an intuitionistic fuzzy ordered weighted distance (IFOWD) operator, which is very useful to deal with the decision information represented in intuitionistic fuzzy numbers under uncertain situations. The main advantage of the IFOWD operator is that it can alleviate the influence of unduly large (or small) deviations on the aggregation results by assigning them low (or high) weights. Moreover, it provides a parameterized family of aggregation operators and
Acnowledgments
The author is very grateful to the editor and the anonymous referees for their insightful and constructive comments and suggestions, which have been very helpful in improving the paper. This research was supported by Zhejiang Province Natural Science Foundation (Grant No. Y6110777).
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