Multi-criteria semantic dominance: A linguistic decision aiding technique based on incomplete preference information

A linguistic decision aiding technique for multi-criteria decision is presented. We define a relation between alternatives as multi-criteria semantic dominance (MCSD). It adopts the similar ideal of the stochastic dominance by utilizing the partial information of the decision maker’s preference, which is only ordinal or partially cardinal. The MCSD rules based on three typical types of semanteme functions are introduced and proven. By using these rules, all the alternatives under consideration are divided into two mutually exclusive sets called efficient set and inefficient set. The decision maker who has such a semanteme function will never choose the alternative from the corresponding inefficient set as the optimal one. In such a way, when we analyze the linguistic decision information, the inherent fuzziness of preference can be handled and several controversial operations of the linguistic terms can be avoided. An example is also provided to illustrate the procedure of the proposed method.     

Extensions of the prudence principle to exploit a valued outranking relation

In many outranking methods, the alternatives are first compared pairwise to build a valued outranking relation, which is then exploited to derive a recommendation for the decision maker. This paper proposes extensions of Arrow and Raynaud’s prudence principle to exploit a valued outranking relation, even in cases where the outranking values have an ordinal interpretation. The idea is to build a ranking that maximizes the weakest support for its implicit pairwise comparisons. We present a family of exploitation models providing three types of solutions, depending on the decision maker’s request: a linear order, a weak order, or a partial preorder. An algorithm is provided which solves the exploitation models. Relations between these models and some of their properties are also discussed.     

DIS-CARD: a new method of multiple criteria sorting to classes with desired cardinality

In this paper, we present a new preference disaggregation method for multiple criteria sorting problems, called DIS-CARD. Real-life experience indicates the need of considering decision making situations in which a decision maker (DM) specifies a desired number of alternatives to be assigned to single classes or to unions of some classes. These situations require special methods for multiple criteria sorting subject to desired cardinalities of classes. DIS-CARD deals with such a problem, using the ordinal regression approach to construct a model of DM’s preferences from preference information provided in terms of exemplary assignments of some reference alternatives, together with the above desired cardinalities. We develop a mathematical model for incorporating such preference information via mixed integer linear programming (MILP). Then, we adapt the MILP model to two types of preference models: an additive value function and an outranking relation. Illustrative example is solved to illustrate the methodology.     

Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral

Within the multicriteria aggregation–disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM’s preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM’s preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modelled by the fuzzy measures, parameterizing our approach.     

Pricing and hedging in the presence of extraneous risks

Given an underlying complete financial market, we study contingent claims whose payoffs may depend on the occurrence of nonmarket events. We first investigate the almost-sure hedging of such claims. In particular, we obtain new representations of the hedging prices and provide necessary and sufficient conditions for a claim to be marketed. The analysis of various examples then leads us to investigate alternative pricing rules. We choose to embed the pricing problem into the agent’s portfolio decision and study reservation prices. We establish the existence and consistency of this pricing rule in a semimartingale model. We characterize the nonlinear dependence of the reservation price with respect to both the agent’s initial capital and the size of her position. The fair price arises as a limiting case.     

On the convergence of reference point methods in multiobjective programming

The purpose of this paper is to present some results about the convergence of interactive reference point methods in multiobjective programming. In particular, we describe how dual information may guide the decision maker in his choice of the successive reference points.In the literature different convergence models have been proposed. The analyst may induce convergence by selecting appropriate rules of the communication. Or he may rely on the learning process of the decision maker to induce some kind of ‘psychological’ convergence. In neither case are the activities of the decision maker precisely described. Consequently, the quality of the final decision cannot be established, and the question of convergence remains an unsolved issue.We describe different ways in which the decision maker may select his successive reference points, and we discuss the convergence of the resulting reference point procedures. Also, we comment on the relevance of these different assumptions about the decision maker's behavior. The procedures are illustrated by a small numerical example.     

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.     

ELECTRE: Robust ordinal regression for outranking methods

We present a new method, called ELECTRE^GKMS, which employs robust ordinal regression to construct a set of outranking models compatible with preference information. The preference information supplied by the decision maker (DM) is composed of pairwise comparisons stating the truth or falsity of the outranking relation for some real or fictitious reference alternatives. Moreover, the DM specifies some ranges of variation of comparison thresholds on considered pseudo-criteria. Using robust ordinal regression, the method builds a set of values of concordance indices, concordance thresholds, indifference, preference, and veto thresholds, for which all specified pairwise comparisons can be restored. Such sets are called compatible outranking models. Using these models, two outranking relations are defined, necessary and possible. Whether for an ordered pair of alternatives there is necessary or possible outranking depends on the truth of outranking relation for all or at least one compatible model, respectively. Distinguishing the most certain recommendation worked out by the necessary outranking, and a possible recommendation worked out by the possible outranking, ELECTRE^GKMS answers questions of robustness concern. The method is intended to be used interactively with incremental specification of pairwise comparisons, possibly with decreasing confidence levels. In this way, the necessary and possible outranking relations can be, respectively, enriched or impoverished with the growth of the number of pairwise comparisons. Furthermore, the method is able to identify troublesome pieces of preference information which are responsible for incompatibility. The necessary and possible outranking relations are to be exploited as usual outranking relations to work out recommendation in choice or ranking problems. The introduced approach is illustrated by a didactic example showing how ELECTRE^GKMS can support real-world decision problems.     

Classifier fusion in the Dempster-Shafer framework using optimized t-norm based combination rules

When combining classifiers in the Dempster-Shafer framework, Dempster’s rule is generally used. However, this rule assumes the classifiers to be independent. This paper investigates the use of other operators for combining non independent classifiers, including the cautious rule and, more generally, t-norm based rules with behavior ranging between Dempster’s rule and the cautious rule. Two strategies are investigated for learning an optimal combination scheme, based on a parameterized family of t-norms. The first one learns a single rule by minimizing an error criterion. The second strategy is a two-step procedure, in which groups of classifiers with similar outputs are first identified using a clustering algorithm. Then, within- and between-cluster rules are determined by minimizing an error criterion. Experiments with various synthetic and real data sets demonstrate the effectiveness of both the single rule and two-step strategies. Overall, optimizing a single t-norm based rule yields better results than using a fixed rule, including Dempster’s rule, and the two-step strategy brings further improvements.     

Self-consistency of decision rules for group decision making

The author treats, in this paper, a group of decision makers, where each of them already has preference on a given set of alternatives but the group as a whole does not have a decision rule to make their group decision, yet. Then, the author examines which decision rules are appropriate. As a criterion of “appropriateness” the author proposes the concepts of self-consistency and universal self-consistency of decision rules. Examining the existence of universally self-consistent decision rules in two cases: (1) decision situations with three decision makers and two alternatives, and (2) those with three decision makers and three alternatives, the author has found that all decision rules are universally self-consistent in the case (1), whereas all universally self-consistent decision rules have one and just one vetoer in the essential cases in (2). The result in the case (2) implies incompatibility of universal self-consistency with symmetry. An example of applications of the concept of self-consistency to a bankruptcy problem is also provided in this paper, where compatibility of self-consistency with symmetry in a particular decision situation is shown.     

Dual mechanism for an online retailer

While posted price and auction have typically been seen as alternatives to each other, we observe Web stores selling a product at a posted price and simultaneously running auctions for the identical product, a phenomenon that has not been studied fully. In this article, we study a dual mechanism, where an online retailer combines the two conventional mechanisms (posted price and auction) for multiple units of a product. We demonstrate that the dual mechanism can be used to achieve market segmentation when customers discount the expected utility of auctions. We characterize the customer’s decision rule and formulate a retailer’s profit function under the dual mechanism. Finally, we compare the performance of three selling mechanisms (posted price, auction, and dual) through computational experiments.     

An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.     

Implausible alternatives in eliciting multi-attribute value functions

Many methods to elicit preference models in multi-attribute decision making rely on evaluations of a set of sample alternatives by decision makers. Using orthogonal design methods to create this set of alternatives might require respondents to evaluate unrealistic alternatives. In this paper, we perform an empirical study to analyze whether the presence of such implausible alternatives has an effect on the quality of utility elicitation. Using a new approach to measure consistency, we find that implausible alternatives in fact, have a positive effect on consistency of intra-attribute preference information and consistency with dominance, but do not affect inter-attribute preference information.     

Reply to the note on article “The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees”

In a very recent note by Gao and Ni [B. Gao, M.F. Ni, A note on article “The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees”, European Journal of Operational Research, in press, doi:10.1016/j.ejor.2007.10.0381], they argued that Yen’s combination rule [J. Yen, Generalizing the Dempster–Shafer theory to fuzzy sets, IEEE Transactions on Systems, Man and Cybernetics 20 (1990) 559–570], which normalizes the combination of multiple pieces of evidence at the end of the combination process, was incorrect. If this were the case, the nonlinear programming models we proposed in [Y.M. Wang, J.B. Yang, D.L. Xu, K.S. Chin, The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees, European Journal of Operational Research 175 (2006) 35–66] would also be incorrect. In this reply to Gao and Ni, we re-examine their numerical illustrations and reconsider their analysis of Yen’s combination rule. We conclude that Yen’s combination rule is correct and our nonlinear programming models are valid.     

Cubature rule associated with a discrete blending sum of quadratic spline quasi-interpolants

A new cubature rule for a parallelepiped domain is defined by integrating a discrete blending sum of C¹ quadratic spline quasi-interpolants in one and two variables. We give the weights and the nodes of this cubature rule and we study the associated error estimates for smooth functions. We compare our method with cubature rules based on the tensor products of spline quadratures and classical composite Simpson’s rules.     

Intelligent data analysis and model interpretation with spectral analysis fuzzy symbolic modeling

This paper proposes fuzzy symbolic modeling as a framework for intelligent data analysis and model interpretation in classification and regression problems. The fuzzy symbolic modeling approach is based on the eigenstructure analysis of the data similarity matrix to define the number of fuzzy rules in the model. Each fuzzy rule is associated with a symbol and is defined by a Gaussian membership function. The prototypes for the rules are computed by a clustering algorithm, and the model output parameters are computed as the solutions of a bounded quadratic optimization problem. In classification problems, the rules’ parameters are interpreted as the rules’ confidence. In regression problems, the rules’ parameters are used to derive rules’ confidences for classes that represent ranges of output variable values. The resulting model is evaluated based on a set of benchmark datasets for classification and regression problems. Nonparametric statistical tests were performed on the benchmark results, showing that the proposed approach produces compact fuzzy models with accuracy comparable to models produced by the standard modeling approaches. The resulting model is also exploited from the interpretability point of view, showing how the rule weights provide additional information to help in data and model understanding, such that it can be used as a decision support tool for the prediction of new data.     

Ranking multicriteria alternatives: The method ZAPROS III

The new version of the method for the construction of partial order on the set of multicriteria alternatives is presented. This method belongs to the family of verbal decision analysis (VDA) methods and gives a more efficient means of problem solution. The method is based on psychologically valid operations for information elicitation from a decision maker: comparisons of two distances between the evaluations on the ordinal scales of two criteria. The information received from a decision maker is used for the construction of a binary relation between a pair of alternatives which yields preference, indifference and incomparability relations. The method allows construction of a partial order on the set of given alternatives as well as on the set of all possible alternatives. The illustrative example is given.