Generalized ordered weighted power utility averaging operator and its applications to group decision-making
Abstract
This paper develops a new operator called the generalized ordered weighted power utility averaging (GOWPUA) operator, which first introduces the risk attitude of decision makers (DMs) in the aggregation process. We study its properties and families. To determine the GOWPUA operator weights, we put forward an orness measure of the GOWPUA operator and analyze its properties. Considering that different DMs may have different perspectives towards decision-making, which can be characterized by different degrees of orness, we construct a new nonlinear optimization model to determine the optimal weights which can aggregate all the individual sets of weights into an overall set of weights. Finally, based on the GOWPUA operator, a method for multiple attribute group decision-making (MAGDM) is developed.
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Introduction
Multiple attribute group decision-making (MAGDM) considers the problem of selecting alternatives that are associated with incommensurate and conflicting attributes by a cooperative group, known as the group decision-making [1]. To choose a desirable alternative, decision makers (DMs) often present their preference information which needs to be aggregated via some proper approaches. There are many methods for aggregating the information [2-17]. One of the most popular methods for aggregating decision formation is the ordered weighted averaging (OWA) operator developed by Yager [14]. It provides a general class of parametric aggregation operators and has been shown to be useful for modeling many different kinds of aggregation problems. Up to now, OWA operator has been used in a wide range of applications [7-10, 16, 18].
Motivated by the OWA operator, an extension to the OWA operator is the generalized OWA (GOWA) operator, which combined the OWA operator with the generalized mean operator [15]. It generalized a wide range of aggregation operators such as the OWA operator [14], the ordered weighted geometric averaging (OWGA) operator [19], the ordered weighted harmonic averaging (OWHA) operator [15], etc. Based on the optimization theory, Zhou and Chen [20] presented the generalized ordered weighted logarithm averaging (GOWLA) operator, which is an extension of the OWGA operator. Other extension of the OWA operator can be founded in literature [6, 21]. However, the above aggregation operators only focused on using the mean to eliminate the difference, and did not consider the DMs’ risk attitude in the aggregation process.
Another important issue of applying the OWA operator for MAGDM is how to determine the associated weights. Many researchers have focused on this issue, and developed some useful approaches to obtaining the OWA weights. For example, O’Hagan [22] suggested a maximum entropy approach for obtaining the OWA operator weights for a given level of orness. Fullér and Majlender [4] proposed an analytic approach for obtaining maximal entropy OWA operator weights for a given orness level. Wang and Parkan [23] proposed a minimax disparity approach for obtaining OWA operator weights for a given orness level. Majlender [24] developed a maximal Rényi entropy method for generating a parametric class of OWA operators and the maximal Rényi entropy OWA weights. Other extension approaches to determining the OWA operator weights can be founded in literature [6, 21, 25, 26]. The methods mentioned above assume that any individual weight vector is equal to the optimal weight vector and correspondingly, and there is only one degree of orness to characterize the DMs’ attitude towards decision-making. As a result, there is only one set of OWA operator weights to be generated. However, this is not consistent with the real situation. In fact, multiple DMs may join in decision-making process to reach a holistic opinion that reflects all the participants’ collective view, and different DMs may have different degrees of orness, which leads to the corresponding OWA operator weights may also be different. So it is necessary to introduce a new method to aggregate all the participants’ preference in MAGDM.
Conclusion
In this paper, we developed the generalized ordered weighted power utility averaging (GOWPUA) operator, which is an extension of the GOWLA operator. The main character of the GOWPUA operator is that it introduced the risk attitude of DMs in the aggregation process. We investigated some properties of the GOWPUA operator and proved that it was commutative, monotonic, bounded and idempotent. In addition, we discussed the families of the GOWPUA operator and found that they included a wide range of aggregation operators such as the OWLGA operator, OWGA operator, OWA operator, OWHA operator, etc. To determine the GOWPUA operator weights, we addressed an orness measure of the GOWPUA operator and analyzed its properties. We developed a new nonlinear optimization model to determine the optimal weight vector of the GOWPUA operator. The main character of the model is that it considered different DMs may have different degrees of orness, and can aggregate all the individual sets of weights into an overall set of weights. Furthermore, a new approach for MAGDM was given based on the GOWPUA operator. This approach is also applicable to different group decision-making problems effectively such as human resource management and financial management, etc.
In further research, it would be very interesting to extend our analysis to the case of more sophisticated situation such as introducing the behavior theory of DMs in the GOWPUA operator. Nevertheless, we leave that point to future research, since our methodology cannot be applied to that extended framework, which will result in more sophisticated calculation and which we cannot tackle here.