Valued Preference Relations Research Articles

Dominance Lagrange Optimized Rule Generation for Decision Table Evaluation

Manipulating the volumes of data and recognizing significant information was the challenging task in organizing and managing with uncertain data. In data management, evidence theory and fuzzy measure evaluated the uncertain information for decision execution from an arranged decision table. Despite, improvement in the decision performance and validity, an optimized rule generation has to be studied. Through this paper, a Dominance Principle-based Reduct with Dominance Lagrange Optimized Rule Generation (DLO-RG) model is proposed for the analysis of obtained decision rule set. In DLO-RG, the degree of coherence to the dominance factor is defined. Then, the coherence degrees of all objects are combined, called combined valued preference relation dominance factor to generate the reducts of an input decision table. To the generated reducts, a Deterministic Global Optimization model using Lagrange is designed to optimize the rule generation. Tight lower bounds and upper bounds are generated using Lagrange formulation. The Deterministic Global Optimization model contributes optimum lower bounds, eliminating uncertainty while extracting rules and therefore improving rule extraction accuracy. Theoretical analysis conducted using the DLO-RG proves that the extracted reducts have lower redundancy, therefore reducing the rule generation time and are more diverse when compared with those obtained by existing evidence theory. Experimental results prove the rules framed by the DLO-RG are more significant when compared with that of rules from existing methods.

Incomplete interval valued fuzzy preference relations

An interval valued preference relation is a preference structure that is used to describe uncertainty in complex decision making problems. Retrieving complete information from experts is improbable in real life scenarios. Discarding incomplete information leads to loss of important data. In this paper, we introduce an upper bound condition to deal with incomplete interval valued fuzzy preference relations. With the help of this condition, missing preferences are estimated such that they are expressible. Moreover, the resultant complete relation is consistent. In case if an expert is unable to abide by the proposed upper bound condition, an algorithm is formulated to assist the expert in complying to the upper bound condition.

Valued preference-based instantiation of argumentation frameworks with varied strength defeats

A Dung-style argumentation framework aims at representing conflicts among elements called arguments. The basic ingredients of this framework is a set of arguments and a Boolean abstract (i.e., its origin is not known) binary defeat relation. Preference-based argumentation frameworks are instantiations of Dung's framework in which the defeat relation is derived from an attack relation and a preference relation over the arguments. Recently, Dung's framework has been extended in order to consider the strength of the defeat relation, i.e., to quantify the degree to which an argument defeats another argument. In this paper, we instantiate this extended framework by a preference-based argumentation framework with a valued preference relation. As particular cases, the latter can be derived from a weight function over the arguments or a Boolean preference relation. We show under some reasonable conditions that there are “less situations” in which a defense between arguments holds with a valued preference relation compared to a Boolean preference relation. Finally, we provide some conditions that the valued preference relation shall satisfy when it is derived from a weight function.

Algorithms for improving consistency or consensus of reciprocal [0,1]-valued preference relations

We investigate the consistency and consensus of reciprocal [0,1]-valued preference relations (also called fuzzy preference relations by many authors) based on the multiplicative consistency property, which is an important issue in fuzzy set theory. An algorithm is first developed to improve the consistency level of a reciprocal [0,1]-valued preference relation, and the corresponding algorithm for the incomplete reciprocal [0,1]-valued preference relation is also developed. We further propose the consensus improving algorithms for individual reciprocal [0,1]-valued preference relations or incomplete ones. The convergence and robustness of the algorithms are proven and some important conclusions are obtained. In addition, the proposed algorithms can improve the consistency or consensus of reciprocal [0,1]-valued preference relations with less interactions with the decision makers, which can save a lot of time and obtain the results quickly.

The role of fuzzy sets in decision sciences: Old techniques and new directions

We try to provide a tentative assessment of the role of fuzzy sets in decision analysis. We discuss membership functions, aggregation operations, linguistic variables, fuzzy intervals and the valued preference relations they induce. The importance of the notion of bipolarity and the potential of qualitative evaluation methods are also pointed out. We take a critical standpoint on the state-of-the-art, in order to highlight the actual achievements and question what is often considered debatable by decision scientists observing the fuzzy decision analysis literature.

Ranking of admissible alternatives in interval decision making

In the framework of interval decision making, the available information is vague and numerically imprecise, and decision situations are modelled by imprecise probabilities and utilities that are simply represented by suitable intervals and comparisons. Alternatives are therefore evaluated in terms of interval expected utilities, which are then used for expressing crisp preferences among these alternatives. In this work, we construct a valued preference relation expressing the degree to which an alternative is considered as better than another alternative, based on the overlap between these interval expected utilities. In particular, we study a chain of interval order relations associated with the proposed valued preference relation and introduce the notion of α-admissibility in terms of non-dominated alternatives induced by such relations. Furthermore, we consider a possible ranking of the admissible alternatives w.r.t. the corresponding degrees of preference/dominance. In addition, the decision maker is provided with the possibility to state a threshold, expressing his/her own ideas, understanding, views etc., based upon which an alternative can be regarded as better than another one. Thus the admissible alternatives can be defined as non-dominated alternatives w.r.t. the stated degree of preference.

Crisp dimension theory and valued preference relations

Representation of binary preference relations in a real space where each coordinate suggests the existence of underlying criteria is a standard and indeed suggestive approach. Classical dimension theory addresses this problem, showing that whenever crisp preferences define a partial order set, it can be represented in a real space, and then we can search for a minimal representation. Valued preference relation being a much more complex structure, there is an absolute need for meaningful representations, being manageable by decision-makers. In this paper, we continue analyzing the concept of a generalized dimension function of valued preference relations, i.e. a mapping assigning a generalized dimension value to every α-cut of any given valued preference relation, as introduced in a previous paper. We should of course be expecting deep computational problems within this generalized dimension context, since they are already present in crisp dimension theory. In this paper, we present some properties of such a generalized dimension function, pointing out that our approach allows alternative representations depending on some underlying rationality core the decision-maker may change.

Searching for the dimension of valued preference relations

The more information a preference structure gives, the more sophisticated representation techniques are necessary, so decision makers can have a global view of data and therefore a comprehensive understanding of the problem they are faced with. In this paper we propose to explore valued preference relations by means of a search for the number of underlying criteria allowing its representation in real space. A general representation theorem for arbitrary crisp binary relations is obtained, showing the difference in representation between incomparability––related to the intersection operator––and other inconsistencies––related to the union operator. A new concept of dimension is therefore proposed, taking into account inconsistencies in source of information. Such a result is then applied to each α-cut of valued preference relations.

Soft dimension theory

Classical dimension theory, when applied to preference modeling, is based upon the assumption that linear ordering is the only elemental notion for rationality. In fact, crisp preferences are in some way decomposed into basic criteria, each one being a linear order. In this paper, we propose that indeed dimension is relative to a previous idea of rationality, but such a rationality is not unique. In particular, we explore alternative approaches to dimension, based upon a more general representation and allowing different classes of orders for basic criteria. In this way, classical dimension theory is generalized. As a first consequence, we explore the existence of crisp preference representations not being based upon linear orders. As a second consequence, it is suggested that an analysis of valued preference relations can be developed in terms of the representations of all α-cuts.

Improved engineering design concept selection using fuzzy sets

Conceptual design is important in producing a successful product, because a poor design concept can rarely be compensated for at the later design stages. Pugh's concept selection method has been widely used in industries to help designers to select promising design concepts. However, experience shows that many designers feel uncomfortable with adopting the highest scoring result, because the Pugh's concept selection method does not provide enough information to allow designers to put faith in the obtained result. The objective of this paper is to extend the Pugh's concept selection method with fuzzy set theory to measure the quality of a chosen concept. The performance ratings of a design concept are represented with linguistic terms that can be further characterized with fuzzy sets. The proposed concept selection model produces a partial order of design concepts based on the valued preference relations between different design concepts. The best design concepts can be identified for continuous improvement or further development in the next design stage. In addition, three measures, including the non-domination index, the imbalance index, and the optimality index, are developed to help designers evaluate the quality of the candidate concepts and to determine possible ways for further improvement. An outpatient syringe example is used to illustrate the approach developed.

On Aggregation Models Leading to Valued Preference Relations

Abstract Various methods have been proposed in the literature for aggregating multi-dimensional preferences into a valued preference relation. Among these methods one can cite ELECTRE III and IV or PROMETHEE; the Condorcet voting method, in a different field of application, can also be viewed as pertaining to this framework. The above mentioned methods generally result in valued preference relations without special desirable properties such as transitivity or acyclicity. Our aim is to describe and characterize a general model where the overal valued preference is obtained through the aggregation of “differences of preferences” along the various viewpoints. The relevance of the model for understanding the principles on which some of the existing methods are built will be discussed.

A characterization of ‘min’ as a procedure for exploiting valued preference relations and related results

AbstractThis paper is concerned with procedures which transform valued preference relations on a set of alternatives into crisp relations. We present a simple characterization of a procedure that ranks alternatives in decreasing order of their minimal performance. This is done by means of three axioms that are shown to be independent. Among other results, we characterize in a very similar manner a procedure called ‘leximin’ and investigate two families of procedures whose intersection is the ‘min’ procedure.

Valued preference structures

An approach to the axiomatics of valued preference modelling is suggested. In the framework of this approach, general definitions of strict preference, indifference and incomparability relations associated with a valued preference relation are established via solving a system of functional equations. Any solution of this system satisfies most of the classical properties. Characterizations of important particular solutions are also proved.

Ranking methods for valued preference relations: A characterization of a method based on leaving and entering flows

In this paper we study a particular method that builds a partial ranking on the basis of a valued preference relation. This method, which is used in the MCDM method PROMETHEE I, is based on ‘leaving’ and ‘entering’ flows. We show that this method is characterized by system of three independent axioms.

Ranking methods based on valued preference relations: A characterization of the net flow method

This paper deals with the problem of ranking several alternatives on the basis of a valued preference relation. A system of three independent axioms is shown to characterize a ranking method based on ‘net flows’ which contains as particular cases the rules of Copeland and Borda and is used in one of the Promethee methods.

On strict preference relations

A general notion of a valued strict preference relation associated with a valued preference relation is introduced. The transitivity property of a valued strict preference relation is established.

Strict preference relations based on weak t-norms

In this paper we introduce and investigate weak t-norms and related concepts in order to give a general representation of valued strict preference relations associated with a given valued preference relation. All the well-known forms of valued strict preference relations are particular cases of our results. Moreover, when some types of ‘rationality’ assumptions are made, our unified approach seems to be the only possible one.