It has a wide attention about the methods for determining OWA operator weights. At the beginning of this dissertation, we provide a briefly overview of the main approaches for obtaining the OWA weights with a predefined degree of orness. Along this line, we next make an important generalization of these approaches as a special case of the well-known and more general problem of calculation of the probability distribution in the presence of uncertainty. All these existed methods for dealing these kinds of problems are quite complex. In order to simplify the process of computation, we introduce Yager’s entropy based on Minkowski metric. By analyzing its desirable properties and utilizing this measure of entropy, a linear programming (LP) model for the problem of OWA weight calculation with a predefined degree of orness has been built and can be calculated much easier. Then, this result is further extended to the more realistic case of only having partial information on the range of OWA weights except a predefined degree of orness. In the end, two numerical examples are provided to illustrate the application of the proposed approach.
The process of information aggregation has a great affect on the development of intelligent systems. Yager (1988) introduced a new information aggregation technique based on the ordered weighted averaging (OWA) operator. One key issue in the theory of the OWA operator is to determine its associated weights (Amin, 2006, Amin, 2007, Ahn, 2006, Filev and Yager, 1998, Nettleton and Torra, 2001 and Wang et al., 2007). O’Hagan (1988) proposed a maximum entropy approach, which involved a constrained nonlinear optimization problem with a predefined degree of orness as its constraint and the entropy as the objective function. Then, Fullér and Majlender (2001) transformed the maximum entropy model into a polynomial equation which can be solved analytically. Fullér and Majlender (2003) suggested a minimum variance approach to obtain the minimal variability OWA operator weights. Majlender (2005) proposed an approach for obtaining OWA operator weights based on maximal Rényi entropy for a given level of orness and pointed out that the maximum entropy approach and the minimum variance approach are its special cases, respectively. Liu and Chen (2004) proposed the PMEOWA operator and Liu (2006) proposed the MSEOWA operator. Few scholars study the nature of these models and the relationship between them. Moreover, there are two main shortcomings with in all the above approaches: (1) Generally, it is quite complex to acquire the solution of a constrained nonlinear optimization problem or a high-order nonlinear algebraic equation (Wang, 2005). (2) All of them are completely based on the assumption of given orness level. As a fact, it may be difficult for the decision maker (DM) to determine his/her orness in some circumstances. For instance, Xu and Da (2003) established an approach by considering the situation where partial weight information is available partially. The DM may also have other type of weight information except a predefined degree of orness (Kim and Ahn, 1999 and Park and Kim, 1997).
In order to simplify the complicated computation, we introduce Yager’s entropy based on Minkowski metric (Yager, 1995). By analyzing the desirable properties with this measure of entropy, we propose a novel approach to determine the weights of the OWA operator. It is a wide general method which can be specialized into many famous cases, such as the minimax model (Yager, 1993), the MSEOWA operator (Liu, 2006) and the minimum variance approach (Fullér & Majlender, 2003). We then extend it to a linear objective programming (LP) model with a predefined degree of orness. Further, we consider the (LP) model for the more realistic case, which miss a priori information for the desired orness and only has partial weight information as constraint.
The dissertation is organized as follows. Section 2 gives a brief overview and makes an important generalization of the main approaches for OWA weights with a priori information for the desired orness. Section 3 proposes a LP model for OWA weights based on Yager’s entropy. In Section 4, two numerical examples are provided. Section 5 concludes the paper.
In this dissertation, we make a important generalization of the existed approaches with a predefi- ned degree of orness for determining OWA operator weights. A linear objective programming (LP) model with maximal Yager’s entropy is proposed, which is simpler than the existing approaches. But, it relies 100% on subjective information regarding the orness level or the ranges of the OWA weights completely. So, the approach, which regards both the subjective information and the objective information in the data being aggregated, deserves to research in next step work.
