An extended minimax absolute and relative disparity approach to obtain the OWA operator weights

1 Introduction

The ordered weighted averaging (OWA) operator defined by Yager [1] provides a general class of parametric aggregation operators that includes the min, max and average, and has shown to be useful for modeling many different kinds of aggregation problems. The OWA operator has been used in a wide range of applications such as neural networks [2, 3], fuzzy logic controllers [4, 5], decision making [6–10] and data mining [11, 12], etc. To apply the OWA operator, a very crucial issue is the determination of the weights of the operator.

A number of techniques have been suggested for generating the weights. O’Hagan [13] first determined the OWA operator weights and suggested a maximum entropy method. Fullér and Majlender [14] showed that the maximum entropy model could be transformed into a polynomial equation that can be solved analytically. Fullér and Majlender [15] also suggested a minimum variance method to obtain the minimal variability OWA operator weights.

Liu [16, 17] presented the minimal variability OWA operator generating method with the equidifferent OWA operator. Wang and Parkan [18] proposed a linear programming model with a minimax disparity approach to obtain the OWA operator. Majlender [19] extended the maximum entropy method to Rényi entropy and proposed a maximum Rényi entropy OWA operator. Wang et al. [20] proposed minimize the relative disparity method to determine the OWA operator weights.

As Wang and Parkan [18] mentioned to determine the OWA operator weights when the α= orness(W) constraint is taken into consideration, this models above mentioned could be interpreted as making all the weights as close to each other as possible for the given degree of orness. So they imposed the disparities between two adjacent weights are made as small as possible. Minimizing the difference between two adjacent weights is a very strong constraint in the current disparity approach that renders the weights less usable in practice [21]. Thus, the minimax disparity approach was also extended by Wang et al. [20], Amin and Emrouznejad [21, 22]. Amin [23] explored some properties of the extended minimax disparity model. This paper aims to relax this condition and improves the relative disparity method by minimax absolute and relative disparity of any distinct pairs of weights instead of adjacent weights.

The remainder of this paper is organized as follows. In Section 2, we give a review of the OWA operator disparity models for determining its weights. In Section 3, we present two new models for determining OWA weights. In Section 4, we present one numerical example to examine the two methods. In Section 5, we use search engine selection problem to illustrate the proposed method. The conclusions are discussed in Section 6.

2 The OWA operator disparity models

An ordered weighted aggregation (OWA) operator to aggregate information was introduced by Yager [1]. This operator has been investigated in many documents and used in an astonishingly wide range of applications.

Definition 1. An OWA operator of dimension n is a mapping, OWA : R ⁿ  → R, that has an associated n vector = (w₁,  w₂, …,  w _n ) such that w _i  ∈ [0, 1] and ∑ i = 1 n w i = 1. Furthermore, OWA w ( a 1 , a 2 , … , a n ) = ∑ j = 1 n w j b j(1) where b _j is the jth largest element of the collection of the aggregated objects a₁,  a₂, …,  a _n . Yager further introduced a characterizing measure called orness measure associated with the weighting vector w of an OWA operator, where the orness measure of the aggregation is defined as orness ( w ) = 1 n - 1 ∑ i = 1 n ( n - i ) w i(2)

Which lies in the unit interval [0,1] and characterizes the degree to which the aggregation is like an or operation.

One important issue in the OWA operator is to determine its associated weights. To determine OWA operator weights, O’Hagan [13] suggested a maximum entropy method (MEM), which requires the solution of the following constrained nonlinear optimization model (3).

Fullér and Majlender [15] proposed a minimum variance method (MVM), which minimizes the variance of OWA operator weights under a given level of orness. Their method requires the solution of the following mathematical programming model (4).

Max Disp ( W ) = - ∑ i = 1 n w i ln w i s . t . orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(3) and

Min D 2 ( W ) = 1 n ∑ i = 1 n ( w i - 1 n ) 2 s . t . orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(4)

Wang and Parkan [18] proposed a minimax disparity approach (MDA) for generating OWA operator weights. They presented the following model for minimizing the maximum disparity:

Min { Max i = 1 , … , n - 1 | w i - w i + 1 | } s . t . orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(5)

There is a common characteristic for above mentioned models. That is the OWA operator weights should be made as equally important as possible. So, based on such a characteristic, Wang and Luo [20] propose the chi-square model (CSM) for obtaining OWA operator weights.

Min J ( W ) = ∑ i = 1 n - 1 ( w i w i + 1 + w i + 1 w i - 2 ) s . t . orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(6)

Amin and Emrouznejad [21] propose an extended minimax disparity model (EMDM) by minimax disparity of any distinct pairs of weights instead of adjacent weights. That is:

Min { Max i = 1 , … , n - 1 j = i + 1 , … , n | w i - w j | } s . t . orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(7)

Emrouznejad and Amin [22] propose a new disparity model imposing less restriction of disparity between w _j and w _i as follows:

Min ∑ i = 1 n - 1 ∑ j = i + 1 n δ ij s . t . - δ ij ≤ w i - w j ≤ δ ij orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n δ ij ≥ 0 , i = 1 , … , n - 1 , j = i + 1 , … , n(8)

The following section provides two new disparity OWA weight determination models as an extension to the above models.

As Wang et al. [20] mentioned to determine the OWA operator weights, when the orness constraint is taken into consideration, models (5) and (8) could be understood as making all the weights as close to each other as possible under a given degree of orness. In this paper, two difference models between any distinct pairs of weights are made. We propose the follow model for minimizing the maximum disparity:

Min J 1 ( W ) = min { max i = 1 , … , n - 1 j = i + 1 , … , n | w i w j - 1 | } s . t . orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(9) and

Min J 2 ( W ) = min { max i = 1 , … , n - 1 j = i + 1 , … , n | w i w j - 1 | + max i = 1 , … , n - 1 j = i + 1 , … , n | w i - w j | } s . t . orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(10)

The two new models both produce as equally important OWA operator weights as possible for a given degree of orness. Note that model (9) is an extension of model (6), which minimizes the maximum disparity any two weights under a given level of orness. The model (10) is an extension of model (7), which takes into account the absolute deviation, relative deviation, and any distinct pairs of OWA operator weights. For convenience, we refer to model (9) as the extended relative difference model (ERDM) for determining the OWA operator weights and model (10) as the extended relative and absolute difference model (ERADM). The ERDM (9) and ERADM (10) can be reconfigured as the follow models:

Min ρ s . t . 1 - ρ ≤ w i w j ≤ 1 + ρ , i = 1 , … , n - 1 , j = i + 1 , … , n orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(11) and

Min { ρ + ξ } s . t . 1 - ρ ≤ w i w j ≤ 1 + ρ , i = 1 , … , n - 1 , j = i + 1 , … , n - ξ ≤ w i - w j ≤ ξ , i = 1 , … , n - 1 , j = i + 1 , … , n orness ( W ) = ∑ i = 1 n n - i n - 1 · w i = α , 0 ≤ α ≤ 1 ∑ i = 1 n w i = 1 , 0 ≤ w i ≤ 1 , i = 1 , ⋯ , n(12)

Models (11) and (12) are both nonlinear and can be solved by using LINGO or MATLAB software package. Note that model (11) and (12) are not applicable to α= 0 and α= 1, which are two extreme degrees of orness, but can be approached by using α= 0.0001 and α= 0.9999. We will use these two values to represent the two extreme cases in the next section.

In this section, we examine the two new models with a numerical example and verify their applicability in determining OWA operator weights. The weights determined by the two models are also compared with above mentioned models and the NOWA weights [24]. Suppose n = 5 and it is needed to determine the OWA operator weights satisfying different degrees of orness: α= 0.5, 0.6,  …  , 1, which are provided by the decision maker.

Tables 1 and 2 show the OWA operator weights determined by models (9) and (10), respectively, which are also depicted in Figs. 1 and 2. The models are solved by using MATLAB software package.

It is easy to see from Tables 1–4 that the OWA operator weights determined by different methods are slightly different from one another, but there are no significant differences among them. This shows the use of the two new models to determine OWA operator weights is not only feasible but alsoeffective.

To find the slight differences among the above mentioned methods, we consider the distributions of the OWA operator weights in Table 5. Take α= 0.7 for example. Table 5 shows the distribution of the OWA operator weights determined by the eight methods under the given degree of orness (α) = 0.7, from which it can be seen very clearly that the weights determined by the two new methods vary neither in the form of exponential nor in the form of arithmetical progression, but in a very general way. This is what we are expecting because there is no evidence to support the OWA operator weights to follow a very regular distribution, and must be adjacent weight minimum requirements [20].

5 Application of the proposed OWA weights in internet search engine

Keyhanipour et al. [25] used optimistic OWA (O-OWA) operator to measure the performance of the search engines by factors such as average click rate, total relevancy of the returned results and the variance of the clicked results.

Relevance judgments were done according to users’ preferences, e.g., overall value of the returned results. Relevant items were scored as 2, irrelevant ones as 0 and undecided items as 1. One sample from the judgment of a user which is extracted from Keyhanipour is shown in the following matrix.

In their method they used O-OWA operator as introduced in Yager [1] to combine the decisions of underlying search engines which have different degrees of importance in a certain category. Emrouznejad [26] also used most preferred OWA (MP-OWA) operator to rank the search engines.

To compare the results with other models, Table 7 shows the score is given to each search engine using ERDM and ERADM as developed in this paper. According to the models (9) and (10), for an orness level of 0.75, the highest score is assigned to the WebFusion-OWA search engine, and the lowest score is given to the LookSmart or Lycos search engine, which is consistent with the method of O-OWA and MP-OWA. Some disagreements exist between the models-for example search engine AltaVista is given a very high rank (3rd place) by models (9) and (10), and a very low rank (9th or 6th place) using O-OWA and MP-OWA. In fact, from the matrix of judgment of a user for a sample query results (Table 6), we can find search engines AltaVista and ProFusion have no significant difference. Therefore, the rank of our method is more reasonable.

6 Conclusions

In this paper, we have proposed two new models for determining OWA operator weights. The two new models prove to be practical and effective and can produce the OWA operator weights that are very close to those obtained by existing methods. But the weights determined by considering the absolute deviation and relative deviation of any distinct pairs of weights, thus the two new models do not follow a regular distribution and therefore make more sense.

In future research, we will further investigate the related properties of the absolute deviation and relative deviation model through an analytic solution. A detailed application study of different models may also be a topic of interest.