معرف های کمی فازی در تجزیه و تحلیل حساسیت از عملگر OWA

The efficient use of the Ordered Weighted Averaging (OWA) operator in decision making problems depends on the choice of the order weights. Using fuzzy quantifiers is one of the most popular methods to obtain them. In this study, a new method will be introduced for determining the order weights from the quantifiers, which is especially useful in the case of unimodal quantifiers. The new method is generic and has better computational efficiency in comparison to the previously applied methods. In addition, a new measure for sensitivity analysis on the outputs of OWA operator will be introduced. The theoretical results will be illustrated by a Ph.D. student selection problem discussed earlier in the literature.

The model of Ordered Weighted Averaging (OWA) operator has been in focus of research in the last two decades. OWA as an aggregation operator was initiated by Yager (1988) and has been applied in many fields including Multi Criteria Decision Making (MCDM). An n -dimensional OWA operator is a mapping F : I n ↦ I defined as: equation(1) View the MathML sourceFa1,a2,…,an=∑j=1nwjbj=w1b1+w2b2+⋯+wnbn, Turn MathJax on where b j is the j th largest element in the set of inputs {a 1, a 2, …, a n}, n being the number of the inputs and {w 1, w 2, …, w n} are the order weights. It is usually assumed that w j ⩾ 0 for all j , and View the MathML source∑j=1nwj=1. Vector View the MathML sourcew̲=(w1,w2,…..,wn) is called the order weights vector. F is the combined goodness measure of a decision alternative if the inputs are its evaluations with respect to n criteria. Any alternative with the highest F value will be considered the most preferred decision. Note that the components of the input vector have been ordered before multiplying them by the order weights. The OWA operator encompasses several operators, since it can implement different aggregation rules by changing the order weights. Indeed the OWA category of operators allows easy adjustment of the ANDness and ORness degrees embedded in the aggregation. This scheme is improved newly by the Unified AND–OR operator introduced by Khan and Engelbrecht (2007). The order weights of OWA depend on the optimism degree (also known as the ORness degree) of the decision maker (DM). The greater the weights at the beginning of the weight vector, the higher the optimism degree (risk acceptance). Yager (1988) has defined the optimism degree, θ, as: equation(2) View the MathML sourceθ=1n-1∑j=1n(n-j)wj. Turn MathJax on The well-known methods to obtain the OWA weights are listed in Table 1. The main conceptual difference among these methods is the way how they reflect the preferences of the DM (e.g. to be risk prone or risk averse). Table 1. Major methods to obtain the OWA weights Method Approach Reference Fuzzy linguistic quantifiers Using the fuzzy linguistic quantifiers to characterize the aggregation inputs Yager (1988) Maximum entropy Maximizing the entropy measure of Shannon (1948) for the order weights for a given ORness degree O’Hagan (1988) S-OWA Defining two specific equations for OR-like and AND-like OWA operators Yager (1993) Neat OWA Using the BADD (BAsic Defuzzification Distribution transformation) OWA operator in which the weights depend on the inputs and the results are neat OWA Yager, 1993 and Yager and Filev, 1994 Learning method Obtaining the weights by minimizing the distance of outputs of OWA operator from the real data Filev and Yager (1998) Exponential OWA Defining two specific graphs to obtain the weights for optimistic and pessimistic OWA operators Filev and Yager (1998) Minimal variability Minimizing the variance of the weights for a given ORness degree Fullér and Majlender (2003) Minimax disparity Minimizing the maximum difference between any two adjacent weights for a given ORness degree Wang and Parkan, 2005 and Amin and Emrouznejad, 2006 Least squares deviation and χ2 models Producing as equally important OWA operator weights as possible for a given ORness degree Wang et al. (2007) Gaussian method Obtaining the weights by the Normal distribution Xu, 2005 and Yager, 2007 Table options Some of the methods listed in Table 1 are already compared in the literature. Zarghami, Ardakanian, and Szidarovszky (2007) showed the different behavior of fuzzy linguistic quantifiers and the minimal variability method. Wang, Luo, and Liu (2007) have compared the weights obtained by the maximum entropy, minimal variability, minimax disparity, least-squares deviation and χ2 models by a numerical example. Liu (2007) showed the equivalence of the solutions of the minimax disparity approach and the minimal variability method. This paper discusses the method of fuzzy quantifiers and then will apply it to the sensitivity analysis of the OWA operator. Due to the drawbacks of the existing methods, we will introduce a new measure to obtain the order weights of any type of quantifiers. Sensitivity analysis is an important tool to gain deeper insight into the behavior of the mathematical models and their solutions. A comprehensive review of the sensitivity analysis for MCDM models can be found in the literature (e.g. Triantaphyllou & Sanchez, 1997). Torra (2001) analyzed the sensitivity of the OWA operator concerning the weights of the criteria and the evaluations of the alternatives. Wang and Lin (2003) analyzed the sensitivity of the OWA operator with respect to the quantifiers in a flight simulator development project. Ben-Arieh (2005) analyzed the effect of various weight vectors on the outputs of four selected weighted aggregation operators. In our case however the weight vectors are obtained as the order weights of the OWA operator. This paper will mainly focus on the methods discussed by Ben-Arieh (2005) with the major difference that he did not discuss how to obtain the order weights in the case of unimodal quantifiers but here an efficient method will be introduced for their computation as well. The paper is organized as follows: Section 2 describes fuzzy quantifiers and our new method for obtaining the order weights. Section 3 introduces the sensitivity analysis model for the OWA outputs concerning the fuzzy quantifiers. The theoretical results will be illustrated in Section 4 by a case study known already from the literature, which the results show the advantages of the new method.

This paper introduced a generic method, called ABS, to obtain the order weights of the OWA operator. This new method can be applied for every type of fuzzy quantifiers. This study examined its application for RIM, RDM and RUM quantifiers. The ABS method has better efficiency without the disadvantages of the other methods. The sensitivity of the alternatives’ rankings is analyzed by using a method known already from the literature. The results demonstrated that any change in the base quantifier might have structural effect on the results. Therefore a new sensitivity measure was introduced, which is independent from the base quantifier, so it can be used without asking the base quantifier from the DM. The new measure was illustrated and applied successfully in a case study from the literature.