Elsevier

Expert Systems with Applications

Volume 39, Issue 13, 1 October 2012, Pages 11456-11467
Expert Systems with Applications

Probabilities in the OWA operator

https://doi.org/10.1016/j.eswa.2012.04.010Get rights and content

Abstract

We analyze the use of the probability in the ordered weighted average (OWA). We introduce the probabilistic OWA (POWA) operator. It is an aggregation operator that provides a parameterized family of aggregation operators between the minimum and the maximum that considers the degree of importance that the probability and the OWA operator have in the aggregation. We study some of its main properties and particular cases. We also study the construction of interval and fuzzy numbers with POWA operators. We study the applicability of the POWA operator and we see that it is very broad because all the previous studies that use the probability can be revised with this new approach. We develop an application in a group decision making problem regarding investment selection.

Highlights

► The probabilistic OWA operator. ► A new model that unifies the OWA operator with the probability. ► Construction of interval numbers with the POWA operator. ► An application in multi-person decision making.

Introduction

Decision making problems (Gil-Lafuente & Merigó, 2010) are very common in the literature. There are different ways and methods for solving the decision process (Liu, 2011, Merigó and Gil-Lafuente, 2010, Zavadskas and Turskis, 2011, Zhou and Chen, 2010). Usually, the method used for solving the decision problem depends on the available information. For example, when the decision maker has probabilistic information he will solve the problem calculating the expected value for each alternative. Thus, he will be in a problem of decision making under risk environment. However, in other problems the decision maker may not have probabilistic information. Therefore, he must use another approach for solving the problem such as the use of the ordered weighted averaging (OWA) operator (Yager, 1988) that reflects the attitudinal character (degree of optimism) of the decision maker. In this case, we are in a situation of decision making under uncertainty.

The OWA operator is a very useful technique for aggregating the information. It provides a parameterized family of aggregation operators between the maximum and the minimum. In decision making, it is very useful for representing the attitudinal character of the decision maker. Since its appearance, the OWA operator has been studied and applied in a wide range of problems (Beliakov et al., 2007, Yager et al., 2011). For example, Merigó and Gil-Lafuente (2009) extended the OWA by using induced and generalized aggregation operators forming the induced generalized OWA (IGOWA) operator. They extended the IGOWA operator by using the adequacy coefficient (Merigó, Gil-Lafuente, & Gil-Aluja, 2011). Chen and Zhou (2011) developed the induced generalized continuous OWA (IGCOWA) operator and Merigó and Casanovas (2011a) the induced OWA distance (IOWAD) operator. Several authors used the OWA operator in a fuzzy environment by using fuzzy numbers (Merigó & Casanovas, 2010a), harmonic means (Wei, 2011) and intuitionistic fuzzy sets (Xu and Xia, 2011, Zhao et al., 2010).

Yager, Engemann, and Filev (1995) presented a new model that tried to unify the concept of probability with the OWA operator. They introduced the immediate probabilities. This method formulates an aggregation operator that uses probabilities and OWA operators at the same time. Therefore, this approach permits to consider decision making problems under risk environment and under uncertainty at the same time. The concept of immediate probabilities has been further studied by Engemann, Filev, and Yager (1996) and Merigó (2010). It is very useful in some particular situations. However, further analysis (as those shown in the paper) show that it can only unify probabilities and OWAs giving a neutral degree of importance to each case. But the situation that we find in the real world is that sometimes the decision maker believes more on the probabilistic information or on the OWAs. Thus, in order to overcome this issue, the real unification must consider the degree of importance of these two concepts in the aggregation process.

The aim of this paper is to present a new formulation that unifies the probabilities and the OWA operators considering the degree of importance of each case in the aggregation process. We call this new approach as the probabilistic ordered weighted averaging (POWA) operator. The main advantage of this model is that it provides more complete information of the decision process because we can unify the probabilistic information and the attitudinal character according to the available information in the decision process. Thus, we can under or overestimate the probability according to a degree of orness (degree of optimism). We study some of its main properties and we see that the OWA operator and the usual probabilistic aggregation are particular cases of this approach. We study other particular cases of the POWA operator such as the Hurwicz-POWA, the olympic-POWA and the S-POWA.

We analyze the construction of interval and fuzzy numbers (FNs) by using POWA operators. We see that we can construct interval and FNs considering the probabilistic information available in the problem. We see the probability from an objective way although it is also possible to consider it from a subjective perspective. Therefore, we find the objective interval numbers (OIN) and the objective fuzzy numbers (OFN). Their main advantage is that they can consider objective bounds. That is, the maximum and the minimum subject to the probability. Thus, we can form more specific intervals that permits to obtain a more specific information of the problem.

We study the applicability of the POWA operator and we see that it is incredibly broad because all the previous studies that use the probability can be revised and extended by using this approach. We focus on a group decision making problem regarding the selection of investments. We develop a multi-person decision making process where there are several experts giving their opinion regarding the problem considered. We introduce the multi-person POWA (MP-POWA) operator. It is an aggregation operator that deals with the opinion of several experts in the analysis. It includes a wide range of aggregation operators including the multi-person arithmetic mean (MP–AM), the multi-person probabilistic aggregation (MP–PA) and the multi-person OWA (MP–OWA) operator. We see the usefulness of the POWA operator because we can use situations of decision making under risk and under uncertainty at the same time.

This paper is organized as follows. In Section 2 we briefly review the probabilistic aggregation operators, the OWA operator and the immediate probability. Section 3 presents the POWA operator and Section 4 different particular cases. Section 5 analyzes the use of POWA operators in the construction of interval and fuzzy numbers and Section 6 analyzes the applicability of the POWA operator. In Section 7 we present a group decision making problem with the POWA operator in investment selection and in Section 8 and illustrative example. In Section 9, we summarize the main conclusions of the paper.

Section snippets

Preliminaries

In this Section, we briefly review the probabilistic aggregations, the OWA operator and the immediate probabilities.

The probabilistic OWA operator

The probabilistic ordered weighted averaging (POWA) operator is an extension of the OWA operator for situations where we find probabilistic information. Therefore, it provides a parameterized family of aggregation operators between the minimum and maximum that includes the probability in the aggregation process. Its main advantage is that it can represent the degree of importance of the probability and the OWA in the aggregation. It can also be seen as a unification between decision making

Families of POWA operators

By using different expressions in the weighting vector of the POWA operator, we can obtain a wide range of particular cases. First of all we are going to consider the two main cases of the POWA operator that are found by analyzing the coefficient β. Basically, if β = 0, then, we get the probabilistic aggregation and if β = 1, the OWA operator.

By choosing a different manifestation of the weighting vector in the POWA operator, we are able to obtain different types of aggregation operators. For

Group decision making with POWA operators

In this following, we are going to consider a decision-making application in investment selection, using a multi-person analysis. A multi-person analysis provides a more complete representation of the problem because it is based on the opinion of several experts. Therefore, we can aggregate the opinion of different experts to obtain a representative view of the problem.

The procedure to select investments with the POWA operator in multi-person decision-making is described in this section. Note

Illustrative example

In this paper, we develop an application in a multi-person decision making regarding selection of investments. The main reason for using the POWA operator is that we are able to assess the decision making problem considering probabilities and the attitudinal character of the decision maker. Thus, we get a more complete representation of the decision problem.

In the following, we present a numerical example of the new approach in investment selection. We analyze a company that operates in Europe

Conclusions

We have presented an approach for group decision making with probabilities and OWA operators. We have developed a new aggregation operator that unifies the probability with the OWA operator. We have called it the POWA operator. Its main advantage is that it gives a more complete representation of the aggregation process because we are taking into account the probabilistic information and the attitudinal character (degree of orness) of the decision maker and considering the degree of importance

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