OWA aggregation over a continuous interval argument with applications to decision making

We briefly describe the ordered weighted average (OWA) operator. We discuss its role in decision making under uncertainty. We provide an extension of the OWA operator to the case in which our argument is a continuous valued interval rather than a finite set of values. We look at some examples of this type of aggregation. We show how it can be used in some tasks that arise in decision making. We consider the extension of the continuous interval argument OWA operator to the more general case in which the argument values have importance weights. We use this to introduce the idea of an attitudinal-based expected value associated with a continuous random variable.     

OWA aggregation of intuitionistic fuzzy sets

We recall the concept of an intuitionistic fuzzy subset (IFS). Fundamental to an IFS is the fact that it is defined using two values, a degree of membership and degree of non-membership. The ordered weighted averaging (OWA) operator is introduced and several of its features are described. Particularly notable is the idea of the dual of an OWA operator. We next discuss the aggregation of a collection of IFS using a prescribed OWA operator. It is shown that while the aggregation of the degrees of membership is performed using the prescribed OWA operator, the aggregation of the degrees of non-membership requires use of the dual of the prescribed OWA operator. The Choquet integral aggregation operator is introduced and applied to the aggregation of IFSs. Here again the concept of the dual is needed to perform the aggregation of the degrees of non-membership. We also discuss the aggregation of IFSs using the Sugeno integral. Fundamental to this work is our realisation of the importance of the concept of the dual operators in dealing with the aggregation of IFS.     

Quantifier guided aggregation using OWA operators

We consider multicriteria aggregation problems where, rather than requiring all the criteria be satisfied, we need only satisfy some portion of the criteria. The proportion of the critera required is specified in terms of a linguistic quantifier such as most . We use a fuzzy set representation of these linguistic quantifiers to obtain decision functions in the form of OWA aggregations. A methodology is suggested for including importances associated with the individual criteria. A procedure for determining the measure of “orness” directly from the quantifier is suggested. We introduce an extension of the OWA operators which involves the use of triangular norms. © 1996 John Wiley & Sons, Inc.     

A class of aggregation functions encompassing two-dimensional OWA operators

In this paper we prove that, under suitable conditions, Atanassov’s K_α operators, which act on intervals, provide the same numerical results as OWA operators of dimension two. On one hand, this allows us to recover OWA operators from K_α operators. On the other hand, by analyzing the properties of Atanassov’s operators, we can generalize them. In this way, we introduce a class of aggregation functions - the generalized Atanassov operators - that, in particular, include two-dimensional OWA operators. We investigate under which conditions these generalized Atanassov operators satisfy some properties usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc. We also show that if we apply these aggregation functions to interval-valued fuzzy sets, we obtain an ordered family of fuzzy sets.     

The uncertain OWA aggregation with weighting functions having a constant level of orness

Since the ordered weighted averaging (OWA) operator was introduced by Yager [IEEE Trans Syst Man Cybern 1988;18:183–190], numerous aggregation operators have been presented in academic journals. Apart from a setting where exact numerical assessments on weights and input arguments can be obtained, the issue of generalizing the OWA to take into account uncertainties in weights and/or input arguments has been considered. Recently, Xu and Da [Int J Intell Syst 2002;17:569–575] proposed an uncertain OWA operator in which input arguments are given in the form of interval numbers. The interval numbers within the interval sometimes do not have the same meaning for the decision maker as is implied by the use of interval ranges. Thus, we present a way of prioritizing interval numbers, taking into account the strength of preference based on the probabilistic measure. Further, rank‐based weighting functions having constant values of orness irrespective of the number of objectives aggregated are presented and a final rank ordering of courses of action is performed by the use of those weighing functions. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 469–483, 2006.     

A priori identification of preferred alternatives of OWA operators by relational analysis of arguments

We present a reverse decision-aiding method that is distinct from previously reported ordered weighted averaging (OWA) aggregation methods. The proposed method is implemented in two phases. In the first phase, potentially best alternative, defined as one having any weighting vector that enables it to be at least as good as the others, is identified. In the second phase, the maximum and the minimum attitudinal characters for such an alternative are computed as the highest and the lowest values which it can attain under the weights-set identified in the first phase. These two phases are governed only by the relational analysis of input arguments, without soliciting the decision-maker to supply a specific attitudinal character. The proposed method can be applied to cases when it is difficult to obtain a precise attitudinal character and when, even if a precise one is obtained, the OWA operator weights are different, depending on the weights generating methods adopted. Further, if uncertain attitudinal character in the form of the interval number is available, its projection into the results of the proposed method yields less alternatives of consideration, in some cases, a single best alternative. The proposed method also allows for a priori identification of alternatives prone to change at a particular range of attitudinal character.     

基于扩展OWA算子的数据信息聚合方法研究

在数据信息聚合的过程中通常会用到有序加权平均聚合算子,然而有序加权平均聚合算子只是考虑了数据信息所处聚合位置的重要度,却很少考虑数据本身的重要度。针对这种缺点和不足,提出了一种扩展的有序加权几何平均聚合算子,证明了该扩展聚合算子的一些基本性质定理;从理论上分析了该扩展聚合算子的科学性和合理性;通过一个算例的对比分析,证实了该扩展的聚合算子在数据信息聚合时更能真实地反映实际情况。     

Using trapezoids for representing granular objects: Applications to learning and OWA aggregation

We discuss the role and benefits of using trapezoidal representations of granular information. We focus on the use of level sets as a tool for implementing many operations on trapezoidal sets. We point out the simplification that the linearity of the trapezoid brings by requiring us to perform operations on only two level sets. We investigate the classic learning algorithm in the case when our observations are granule objects represented as trapezoidal fuzzy sets. An important issue that arises is the adverse effect that very uncertain observations have on the quality of our estimates. We suggest an approach to addressing this problem using the specificity of the observations to control its effect. We next consider the OWA aggregation of information represented as trapezoids. An important problem that arises here is the ordering of the trapezoidal fuzzy sets needed for the OWA aggregation. We consider three approaches to accomplish this ordering based on the location, specificity and fuzziness of the trapezoids. From these three different approaches three fundamental methods of ordering are developed. One based on the mean of the 0.5 level sets, another based on the length of the 0.5 level sets and a third based on the difference in lengths of the core and support level sets. Throughout this work particular emphasis is placed on the simplicity of working with trapezoids while still retaining a rich representational capability.     

An OWA operator with fuzzy ranks

The ordered weighted averaging (OWA) operator of Yager was introduced to provide a method for aggregating several inputs which lies between the max and min operators. The fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged according to their integer ranks. In this paper, we generalize the OWA operator to include the case of real-number or fuzzy ranks. © 1998 John Wiley & Sons, Inc.13: 69–81, 1998     

Probabilities in the OWA operator

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.     

Data aggregation for scheduling of resource-intensive computations under uncertainty

A control process for resource-intensive computations in heterogeneous specialized environment is considered. For each task, a concept of computational job required to execute it is introduced. Two interconnected problems—of capability and performance—are studied. A parametric approach to scheduling is proposed that allows for balanced resource allocation while meeting user requirements under uncertainty.     

Generalizing predicates with string arguments

The least general generalization (LGG) of strings may cause an over-generalization in the generalization process of the clauses of predicates with string arguments. We propose a specific generalization (SG) for strings to reduce over-generalization. SGs of strings are used in the generalization of a set of strings representing the arguments of a set of positive examples of a predicate with string arguments. In order to create a SG of two strings, first, a unique match sequence between these strings is found. A unique match sequence of two strings consists of similarities and differences to represent similar parts and differing parts between those strings. The differences in the unique match sequence are replaced to create a SG of those strings. In the generalization process, a coverage algorithm based on SGs of strings or learning heuristics based on match sequences are used. Ilyas Cicekli received a Ph.D. in computer science from Syracuse University in 1991. He is currently a professor of the Department of Computer Engineering at Bilkent University. From 2001 till 2003, he was a visiting faculty at University of Central Florida. His current research interests include example-based machine translation, machine learning, natural language processing, and inductive logic programming. Nihan Kesim Cicekli is an Associate Professor of the Department of Computer Engineering at the Middle East Technical University (METU). She graduated in computer engineering at the Middle East Technical University in 1986. She received the MS degree in computer engineering at Bilkent University in 1988; and the PhD degree in computer science at Imperial College in 1993. She was a visiting faculty at University of Central Florida from 2001 till 2003. Her current research interests include multimedia databases, semantic web, web services, data mining, and machine learning.     

Single machine scheduling problems with uncertain parameters and the OWA criterion

In this paper a class of single machine scheduling problems is discussed. It is assumed that job parameters, such as processing times, due dates, or weights are uncertain and their values are specified in the form of a discrete scenario set. The ordered weighted averaging (OWA) aggregation operator is used to choose an optimal schedule. The OWA operator generalizes traditional criteria used in decision making under uncertainty, such as the maximum, average, median, or Hurwicz criterion. It also allows us to extend the robust approach to scheduling by taking into account various attitudes of decision makers towards a risk. In this paper, a general framework for solving single machine scheduling problems with the OWA criterion is proposed and some positive and negative computational results for two basic single machine scheduling problems are provided.     

FORTRAN routines with optional arguments

In designing a general purpose subroutine package to solve a class of problems, one often has to write subroutines with a large number of arguments. Although these arguments are required to cover a range of possibilities, many of these arguments have some commonly occurring values. The user is thus burdened with supplying a long list of arguments and making sure that the number and types match. An alternate solution is to write such subroutines in assembly language so that they could have a variable number of arguments. This approach is expensive and eliminates a large class of program designers who do not and do not want to know assembler language. This paper describes a facility which enables these program designers to write their routines completely in a higher level language (FORTRAN) and yet enjoy the ‘luxury’ of having a variable number of arguments in the calling sequence.     

S‐H OWA Operators with Moment Measure

Step‐like or Hurwicz‐like ordered weighted averaging (OWA) (S‐H OWA) operators connect two fundamental OWA operators, step OWA operators and Hurwicz OWA operators. S‐H OWA operators also generalize them and some other well‐know OWA operators such as median and centered OWA operators. Generally, there are two types of determination methods for S‐H OWA operators: One is from the motivation of some existed mathematical results; the other is by a set of “nonstrict” definitions and often via some intermediate elements. For the second type, in this study we define two sets of strict definitions for Hurwitz/step degree, which are more effective and necessary for theoretical studies and practical usages. Both sets of definitions are useful in different situations. In addition, they are based on the same concept moment of OWA operators proposed in this study, and therefore they become identical in limit forms. However, the Hurwicz/step degree (HD/SD) puts more concerns on its numerical measure and physical meaning, whereas the relative Hurwicz/step degree (rHD/rSD), still being accurate numerically, sometimes is more reasonable intuitively and has larger potential in further studies and practical applications.     

On the dispersion measure of OWA operators

We describe some basic features of the OWA operator. We turn to the problem of determining the weights associated with this operator and particularly the maximal dispersion (entropy) approach. We consider the possibility of using minimization of dispersion. After discussing concerns with both maximization and minimization of dispersion we investigate the possibility of finding an optimal solution intermediate to these extremes. We next consider alternative measures of dispersion. We introduce a fundamental requirement for a measure of dispersion called the Preference for Equal Division. A number of general classes of dispersion measures are provided notable among these are those based on t-norm and t-conorm operators.     

Sensitivity analysis of the OWA operator.

The successful design and application of the ordered weighted averaging (OWA) method as a decision-making tool depend on the efficient computation of its order weights. The most popular methods for determining the order weights are the fuzzy linguistic quantifiers approach and the minimal variability method, which give different behavior patterns for the OWA. These two methods will be first analyzed in detail by using sensitivity analysis on the outputs of the OWA with respect to the optimism degree of the decision maker, and then the two methods will be compared. The fuzzy linguistic quantifiers approach gives more information about the behavior of the OWA outputs in comparison to the minimal variability method. However, in using the minimal variability method, the OWA has a linear behavior with respect to the optimism degree, and, therefore, it has better computation efficiency. Since maximizing the combined goodness measure and minimizing its sensitivity to optimism degree are conflicting objectives, a new composite measure of goodness will be defined to have more reliability in obtaining optimal solutions. The theoretical results will be illustrated in a water resources management problem.     

Parameterized additive neat OWA operators with different orness levels

The article proposes an extension of the BADD OWA operator—ANOWA (additive neat OWA) operator—and defines its orness measure. Some properties of the weighting function associated with orness level are analyzed. Then two special classes of ANOWA operator with maximum entropy and minimum variance are proposed, and the orness of the BADD OWA operator is discussed. For a given orness level, these ANOWA operators can be uniquely determined. Their aggregation values for any aggregation elements set always monotonically increase with their orness levels. Therefore they can be used as a parameterized aggregation method with orness as its control parameter and to represent the decision maker's preference. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1045–1072, 2006.