Ordered median problem with demand distribution weights

The ordered median function unifies and generalizes most common objective functions used in location theory. It is based on the ordered weighted averaging (OWA) operator with the preference weights allocated to the ordered distances. Demand weights are used in location problems to express the client demand for a service thus defining the location decision output as distances distributed according to measures defined by the demand weights. Typical ordered median model allows weighting of several clients only by straightforward rescaling of the distance values. However, the OWA aggregation of distances enables us to introduce demand weights by rescaling accordingly clients measure within the distribution of distances. It is equivalent to the so-called weighted OWA (WOWA) aggregation of distances covering as special cases both the weighted median solution concept defined with the demand weights (in the case of equal all the preference weights), as well as the ordered median solution concept defined with the preference weights (in the case of equal all the demand weights). This paper studies basic models and properties of the weighted ordered median problem (WOMP) taking into account the demand weights following the WOWA aggregation rules. Linear programming formulations were introduced for optimization of the WOWA objective with monotonic preference weights thus representing the equitable preferences in the WOMP. We show MILP models for general WOWA optimization.     

Interpreting membership functions: A constructive approach

In this paper, we study the semantics of fuzzy sets. We show that fuzzy sets can be interpreted as the aggregation of a set of observations. We formalize this interpretation by means of the OWA and the WOWA operators. The introduction of the WOWA operator allows the user to weigh each observation.     

Globally optimal solutions of max–min systems

A variety of problems in computer science, operations research, control theory, etc., can be modeled as non-linear and non-differentiable max–min systems. This paper introduces the global optimization into such systems. The criteria for the existence and uniqueness of the globally optimal solutions are established using the high matrix, optimal max-only projection set and k ^s -control vector of max–min functions. It is also shown that the global optimization can be accomplished through the partial max-only projection representation with algebraic and combinatorial features. The methods are constructive and lead to an algorithm of finding all globally optimal solutions.     

Deriving weights in multiple-criteria decision making with support vector machines

A key problem in Multiple-Criteria Decision Making is how to measure the importance of the different criteria when just a partial preference relation among actions is given. In this note we address the problem of constructing a linear score function (and thus how to associate weights of importance to the criteria) when a binary relation comparing actions and partial information (relative importance) on the criteria are given. It is shown that these tasks can be done viaSupport Vector Machines, an increasingly popular Data Mining technique, which reduces the search of the weights to the resolution of (a series of) nonlinear convex optimization problems with linear constraints. An interactive method is then presented and illustrated by solving a multiple-objective 0–1 knapsack problem. Extensions to the case in which data are imprecise (given by intervals) or intransitivities in strict preferences exist are outlined.     

Bottleneck combinatorial optimization problems with uncertain costs and the OWA criterion

In this paper a class of bottleneck combinatorial optimization problems with uncertain costs is discussed. The uncertainty is modeled by specifying a discrete scenario set containing a finite number of cost vectors, called scenarios. In order to choose a solution the Ordered Weighted Averaging aggregation operator (OWA for short) is applied. The OWA operator generalizes traditional criteria in decision making under uncertainty such as the maximum, minimum, average, median, or Hurwicz criterion. New complexity and approximation results in this area are provided. These results are general and remain valid for many problems, in particular for a wide class of network problems.     

Using the WOWA operator in robust discrete optimization problems

In this paper a class of discrete optimization problems with uncertain costs is discussed. The uncertainty is modeled by introducing a scenario set containing a finite number of cost scenarios. A probability distribution over the set of scenarios is available. In order to choose a solution the weighted OWA criterion (WOWA) is applied. This criterion allows decision makers to take into account both probabilities for scenarios and the degree of pessimism/optimism. In this paper the complexity of the considered class of discrete optimization problems is described and some exact and approximation algorithms for solving it are proposed. Applications to the selection and the assignment problems, together with results of computational tests are shown.     

Prioritized OWA aggregation

We indicate that the problem of prioritized criteria arises in situations in which there exists a relationship between the criteria so that lack of satisfaction by the higher priority criteria cannot be readily compensated for by satisfaction by lower priority criteria. Typical of this situation is the relationship between safety and cost. We consider the problem of criteria aggregation in this environment. Central to our approach is the use of importance weights to enforce this prioritization imperative. We apply our use of priority based importance weights to the case where the scope of the criteria aggregation is an OWA type aggregation.     

WRAO and OWA learning using Levenberg–Marquardt and genetic algorithms

The generalized Weighted Relevance Aggregation Operator (WRAO) is a non-additive aggregation function. The Ordered Weighted Aggregation Operator (OWA) (or its generalized form: Generalized Ordered Weighted Aggregation Operator (GOWA)) is more restricted with the additivity constraint in its weights. In addition, it has an extra weights reordering step making it hard to learn automatically from data. Our intension here is to compare the efficiency (or effectiveness) of learning these two types of aggregation functions from empirical data. We employed two methods to learn WRAO and GOWA: Levenberg–Marquardt (LM) and a Genetic Algorithm (GA) based method. We use UCI (University of California Irvine) benchmark data to compare the aggregation performance of non-additive WRAO and additive GOWA. We found that the non-constrained aggregation function WRAO was learnt well automatically and produced consistent results, while GOWA was learnt less well and quite inconsistently.     

On goal programming formulations of the reference point method

This paper discusses connections between the multi-criteria techniques of goal programming (GP) and the reference point method (RPM). Both approaches use a certain target point in the criterion (outcome) space as a key element to model decision maker preferences. Therefore, RPM can be expressed, similarly to GP, in the modelling framework of deviational variables. The paper gives a systematic survey and analysis of the lexicographic GP models of RPM. The corresponding preference models are formalised and analysed with respect to target values interpretations as well as the Pareto efficiency of their solutions. The properties of equity among the individual achievements of solutions are also analysed with respect to the Rawlsian principle of justice.     

A classification of the weighting schemes in reference point procedures for multiobjective programming

The reference point-based methods form one of the most widely used class of interactive procedures for multiobjective programming problems. The achievement scalarizing functions used to determine the solutions at each iteration usually include weights. In this paper, we have analysed nine weighting schemes from the preferential point of view, that is, examining their performance in terms of which reference values are given more importance and why. As a result, we have carried out a systematic classification of the schemes attending to their preferential meaning. This way, we distinguish pure normalizing schemes from others where the weights have a preferential interpretation. This preferential behaviour can be either designed (thus, predetermined) by the method, or decided by the decision maker. Besides, several figures have been used to illustrate the way each scheme works. This paper enables the potential users to choose the most appropriate scheme for each case.     

Aggregation of ordinal data using ordered weighted averaging operator weights

In multi-criteria decision-making problems, ordinal data themselves provide a convenient instrument for articulating preferences but they impose some difficulty on the aggregation process since ambiguity prevails in the preference structure inherent in the ordinal data. One of the key concerns in the aggregation of ordinal data is to differentiate among the rank positions by reflecting decision-maker??s preferences. Since individual attitude is fairly different, it is presumable that each ranking position has different importance. In other words, the quantification schemes among the rank positions could vary depending on the individual preference structure. We find that, among others, the ordered weighted averaging (OWA) operator can help to take this concept into effect on several reasons. First, the OWA operator provides a means to take into account a discriminating factor by introducing the measure of attitudinal character. Second, it can produce appropriate ranking weights corresponding to each rank position by solving a mathematical program subject to the constraint of attitudinal character. To better understand the attitudinal character playing a role as a discriminating factor, we develop centered ranking weights from ordinal weak relations among the ranking positions and then investigate their properties to relate them with the OWA operator weights having the maximum entropy. Finally, we present a method for generating the OWA operator weights via rank-based weighting functions.     

An interactive method for multiple criteria group decision analysis based on interval type-2 fuzzy sets and its application to medical decision making

The theory of interval type-2 fuzzy sets provides an intuitive and computationally feasible way of addressing uncertain and ambiguous information in decision-making fields. The aim of this paper is to develop an interactive method for handling multiple criteria group decision-making problems, in which information about criterion weights is incompletely (imprecisely or partially) known and the criterion values are expressed as interval type-2 trapezoidal fuzzy numbers. With respect to the relative importance of multiple decision-makers and group consensus of fuzzy opinions, a hybrid averaging approach combining weighted averages and ordered weighted averages was employed to construct the collective decision matrix. An integrated programming model was then established based on the concept of signed distance-based closeness coefficients to determine the importance weights of criteria and the priority ranking of alternatives. Subsequently, an interactive procedure was proposed to modify the model according to the decision-makers’ feedback on the degree of satisfaction toward undesirable solution results for the sake of gradually improving the integrated model. The feasibility and applicability of the proposed methods are illustrated with a medical decision-making problem of patient-centered medicine concerning basilar artery occlusion. A comparative analysis with other approaches was performed to validate the effectiveness of the proposed methodology.     

基于四参数区间数的多准则决策方法

针对准则权重和准则值均为四参数区间数的多准则决策问题,提出了一种基于四参数区间集结算子的多准则决策方法。首先,定义了四参数区间数以及加权的连续四参数区间数有序加权平均 (WCFP-OWA) 算子和加权的连续四参数区间数有序加权几何平均 (WCFP-OWG) 算子,并利用连续四参数区间数有序加权平均 (CFP-OWA) 算子对四参数区间数准则权重进行处理,然后利用WCFP-OWA算子和WCFP-OWG算子对四参数区间数准则值进行集成得到综合值以确定方案排序。最后将该方法应用于勘探公司的选择,验证了该方法的合理性和有效性。     

Aggregation of fuzzy preference relations to multicriteria decision making

Weighted aggregation of fuzzy preference relations on the set of alternatives by several criteria in decision-making problems is considered. Pairwise comparisons with respect to importance of the criteria are given in fuzzy preference relation as well. The aggregation procedure uses the composition between each two relations of the alternatives. The membership function of the newly constructed fuzzy preference relation includes t-norms and t-conorms to take into account the relation between the criteria importance. Properties of the composition and new relation, giving a possibility to make a consistent choice or to rank the alternatives, are proved. An illustrative numerical study and comparative examples are presented.     

Using TODIM to evaluate green supply chain practices under uncertainty

This study approaches the problem of exploring the importance and performance levels in green supply chain practices (GSCP) under conditions of uncertainty. This contribution creates a mechanism that assists in the process of analyzing and selecting the alternatives aligned with the proposed criteria on both the qualitative and quantitative scales. This analysis uses the acronym from the interactive and multi-criteria decision-making method (known as TODIM in Portuguese), a discrete multi-criteria method based on prospect theory. However, the criteria weights and alternatives are described as linguistic preferences and involve quantitative data. The qualitative preferences transformed into crisp values and the quantitative data converts into comparable scale. The results indicate the discrepancies between the importance and performance levels of the GSCP. This study identified set of principal criteria that can influence the recommendations for strategic direction in the most systematic and wide-ranging manner.     

Learning Weights in the Generalized OWA Operators

This paper discusses identification of parameters of generalized ordered weighted averaging (GOWA) operators from empirical data. Similarly to ordinary OWA operators, GOWA are characterized by a vector of weights, as well as the power to which the arguments are raised. We develop optimization techniques which allow one to fit such operators to the observed data. We also generalize these methods for functional defined GOWA and generalized Choquet integral based aggregation operators.     

Parameterized OWA operator weights: An extreme point approach

Since Yager first presented the ordered weighted averaging (OWA) operator to aggregate multiple input arguments, it has received much attention from the fields of decision science and computer science. A critical issue when selecting an OWA operator is the determination of the associated weights. For this reason, numerous weight generating methods, including rogramming-based approaches, have appeared in the literature. In this paper, we develop a general method for obtaining OWA operator weights via an extreme point approach. The extreme points are represented by the intersection of an attitudinal character constraint and a fundamental ordered weight simplex. The extreme points are completely identified using the proposed formula, and the OWA operator weights can then be expressed by a convex combination of the identified extreme points. With those identified extreme points, some new OWA operator weights can be generated by a centroid or a user-directed method, which reflects the decision-maker’s incomplete preferences. This line of reasoning is further extended to encompass situations in which the attitudinal character is specified in the form of interval with an aim to relieve the burden of specifying the precise attitudinal character, thus obtaining less-specific expressions that render human judgments readily available. All extreme points corresponding to the uncertain attitudinal character are also obtained by a proposed formula and then used to prioritize the multitude of alternatives. Meanwhile, two dominance rules are effectively used for prioritization of alternatives.     

A Universality Result for the Smallest Eigenvalues of Certain Sample Covariance Matrices

After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy–Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Péché.     

Preference programming for robust portfolio modeling and project selection

In decision analysis, difficulties of obtaining complete information about model parameters make it advisable to seek robust solutions that perform reasonably well across the full range of feasible parameter values. In this paper, we develop the Robust Portfolio Modeling (RPM) methodology which extends Preference Programming methods into portfolio problems where a subset of project proposals are funded in view of multiple evaluation criteria. We also develop an algorithm for computing all non-dominated portfolios, subject to incomplete information about criterion weights and project-specific performance levels. Based on these portfolios, we propose a project-level index to convey (i) which projects are robust choices (in the sense that they would be recommended even if further information were to be obtained) and (ii) how continued activities in preference elicitation should be focused. The RPM methodology is illustrated with an application using real data on road pavement projects.     

On the Parameterized OWA Operators for Fuzzy MCDM Based on Vague Set Theory

The family of Ordered Weighted Averaging (OWA) operators, as introduced by Yager, appears to be very useful in multi-criteria decision-making (MCDM). In this paper, we extend a family of parameterized OWA operators to fuzzy MCDM based on vague set theory, where the characteristics of the alternatives are presented by vague sets. These families are specified by a few parameters to aggregate vague values instead of the intersection and union operators proposed by Chen. The proposed method provides a “soft” and expansive way to help the decision maker to make his decisions. Examples are shown to illustrate the procedure of the proposed method at the end of this paper.