Construction and aggregation of preference relations based on fuzzy partial orders

In group decision-making problems it is common to elicit preferences from human experts in the form of pairwise preference relations. When this is extended to a fuzzy setting, entries in the pairwise preference matrix are interpreted to denote strength of preference, however once logical properties such as consistency and transitivity are enforced, the resulting preference relation requires almost as much information as providing raw scores or a complete order over the alternatives. Here we instead interpret fuzzy degrees of preference to only apply where the preference over two alternatives is genuinely fuzzy and then suggest an aggregation procedure that minimizes a generalized Kemeny distance to the nearest complete or partial order. By focusing on the fuzzy partial order, the method is less affected by differences in the natural scale over which an expert expresses their preference, and can also limit the influence of extreme scores.

Aggregation and consensus for preference relations based on fuzzy partial orders

We propose a framework for eliciting and aggregating pairwise preference relations based on the assumption of an underlying fuzzy partial order. We also propose some linear programming optimization methods for ensuring consistency either as part of the aggregation phase or as a pre- or post-processing task. We contend that this framework of pairwise-preference relations, based on the Kemeny distance, can be less sensitive to extreme or biased opinions and is also less complex to elicit from experts. We provide some examples and outline their relevant properties and associated concepts.

A new fuzzy ranking method using fuzzy preference relations

In this paper, a new fuzzy ranking method for both type-1 and interval type-2 fuzzy sets (FSs) using fuzzy preference relations is proposed. The use of fuzzy preference relations to rank FSs with vertices has been introduced, and successfully implemented to undertake fuzzy multiple criteria hierarchical group decision-making problems. The proposed fuzzy ranking method is an extension of the results published in [1], and it is able to rank FSs with and without vertices. Besides that, it is important for a fuzzy ranking method to satisfy six reasonable fuzzy ordering properties as discussed in [6]-[8]. As a result, the capability of the proposed fuzzy ranking method in fulfilling these properties is analyzed and discussed. Issues related to time complexity of the proposed method are also examined.

Finding optimal alternatives based on efficient comparative preference inference

Choosing the right or the best option is often a demanding and challenging task for the user (e.g., a customer in an online retailer) when there are many available alternatives. In fact, the user rarely knows which offering will provide the highest value. To reduce the complexity of the choice process, automated recommender systems generate personalized recommendations. These recommendations take into account the preferences collected from the user in an explicit (e.g., letting users express their opinion about items) or implicit (e.g., studying some behavioral features) way. Such systems are widespread; research indicates that they increase the customers' satisfaction and lead to higher sales. Preference handling is one of the core issues in the design of every recommender system. This kind of system often aims at guiding users in a personalized way to interesting or useful options in a large space of possible options. Therefore, it is important for them to catch and model the user's preferences as accurately as possible. In this thesis, we develop a comparative preference-based user model to represent the user's preferences in conversational recommender systems. This type of user model allows the recommender system to capture several preference nuances from the user's feedback. We show that, when applied to conversational recommender systems, the comparative preference-based model is able to guide the user towards the best option while the system is interacting with her. We empirically test and validate the suitability and the practical computational aspects of the comparative preference-based user model and the related preference relations by comparing them to a sum of weights-based user model and the related preference relations. Product configuration, scheduling a meeting and the construction of autonomous agents are among several artificial intelligence tasks that involve a process of constrained optimization, that is, optimization of behavior or options subject to given constraints with regards to a set of preferences. When solving a constrained optimization problem, pruning techniques, such as the branch and bound technique, point at directing the search towards the best assignments, thus allowing the bounding functions to prune more branches in the search tree. Several constrained optimization problems may exhibit dominance relations. These dominance relations can be particularly useful in constrained optimization problems as they can instigate new ways (rules) of pruning non optimal solutions. Such pruning methods can achieve dramatic reductions in the search space while looking for optimal solutions. A number of constrained optimization problems can model the user's preferences using the comparative preferences. In this thesis, we develop a set of pruning rules used in the branch and bound technique to efficiently solve this kind of optimization problem. More specifically, we show how to generate newly defined pruning rules from a dominance algorithm that refers to a set of comparative preferences. These rules include pruning approaches (and combinations of them) which can drastically prune the search space. They mainly reduce the number of (expensive) pairwise comparisons performed during the search while guiding constrained optimization algorithms to find optimal solutions. Our experimental results show that the pruning rules that we have developed and their different combinations have varying impact on the performance of the branch and bound technique.

Preference relation-based markov random fields

A preference relation-based Top-N recommendation approach, PrefMRF, is proposed to capture both the second-order and the higher-order interactions among users and items. Traditionally Top-N recommendation was achieved by predicting the item ratings fi rst, and then inferring the item rankings, based on the assumption of availability of explicit feed-backs such as ratings, and the assumption that optimizing the ratings is equivalent to optimizing the item rankings. Nevertheless, both assumptions are not always true in real world applications. The proposed PrefMRF approach drops these assumptions by explicitly exploiting the preference relations, a more practical user feedback. Comparing to related work, the proposed PrefMRF approach has the unique property of modeling both the second-order and the higher-order interactions among users and items. To the best of our knowledge, this is the first time both types of interactions have been captured in preference relation-based method. Experiment results on public datasets demonstrate that both types of interactions have been properly captured, and signifi cantly improved Top-N recommendation performance has been achieved.

Preference relation-based Markov Random Fields for recommender systems

A preference relation-based Top-N recommendation approach is proposed to capture both second-order and higher-order interactions among users and items. Traditionally Top-N recommendation was achieved by predicting the item ratings first, and then inferring the item rankings, based on the assumption of availability of explicit feedback such as ratings, and the assumption that optimizing the ratings is equivalent to optimizing the item rankings. Nevertheless, both assumptions are not always true in real world applications. The proposed approach drops these assumptions by exploiting preference relations, a more practical user feedback. Furthermore, the proposed approach enjoys the representational power of Markov Random Fields thus side information such as item and user attributes can be easily incorporated. Comparing to related work, the proposed approach has the unique property of modeling both second-order and higher-order interactions among users and items. To the best of our knowledge, this is the first time both types of interactions have been captured in preference-relation based methods. Experimental results on public datasets demonstrate that both types of interactions have been properly captured, and significantly improved Top-N recommendation performance has been achieved.

PLA folding algorithm from compatibility relations

The letter reports an algorithm for the folding of programmable logic arrays. The algorithm is valid for both column and row folding, although it has been presented considering only the simple column folding. The pairwise compatibility relations among all the pairs of the columns of the PLA are plotted in a matrix called the compatibility matrix of the PLA. A foldable compatibility matrix (FCM), a new concept defined in the letter, is then derived from the compatibility matrix. Once an FCM is obtained, the ordered pairs of fold-able columns and the reordering of the rows are readily determined

Quasi-Transitive and Suzumura Consistent Relations

We examine properties of binary relations that complement quasi-transitivity and Suzumura consistency in the sense that they, together with the original axiom(s), are equivalent to transitivity. In general, the conjunction of quasi-transitivity and Suzumura consistency is strictly weaker than transitivity but in the case of collective choice rules that satisfy further properties, the conjunction of quasi- transitivity and Suzumura consistency implies transitivity of the social relation. We prove this observation by characterizing the Pareto rule as the only collective choice rule such that collective preference relations are quasi-transitive and Suzumura consistent but not necessarily complete.

Foldable compatibility matrix for the folding of programmable logic arrays

This paper describes a switching theoretic algorithm for the folding of programmable logic arrays (PLA). The algorithm is valid for both column and row folding, although it has been presented considering only the simple column folding. The pairwise compatibility relations among all the pairs of the columns of the PLA are mapped into a square matrix, called the compatibility matrix of the PLA. A foldable compatibility matrix (FCM), a new concept introduced by the author, is then derived from the compatibility matrix. A new theorem called the folding theorem is then proved. The theorem states that the existence of an m by 2m FCM is both necessary and sufficient to fold 2m columns of the n column PLA (2m ≤ n). Once an FCM is obtained, the ordered pairs of foldable columns and the re-ordering of the rows are readily determined.

Structured Learning for Non-Smooth Ranking losses

Learning to rank from relevance judgment is an active research area. Itemwise score regression, pairwise preference satisfaction, and listwise structured learning are the major techniques in use. Listwise structured learning has been applied recently to optimize important non-decomposable ranking criteria like AUC (area under ROC curve) and MAP(mean average precision). We propose new, almost-lineartime algorithms to optimize for two other criteria widely used to evaluate search systems: MRR (mean reciprocal rank) and NDCG (normalized discounted cumulative gain)in the max-margin structured learning framework. We also demonstrate that, for different ranking criteria, one may need to use different feature maps. Search applications should not be optimized in favor of a single criterion, because they need to cater to a variety of queries. E.g., MRR is best for navigational queries, while NDCG is best for informational queries. A key contribution of this paper is to fold multiple ranking loss functions into a multi-criteria max-margin optimization.The result is a single, robust ranking model that is close to the best accuracy of learners trained on individual criteria. In fact, experiments over the popular LETOR and TREC data sets show that, contrary to conventional wisdom, a test criterion is often not best served by training with the same individual criterion.

A multi-attribute ranking solutions confirmation procedure

Ranking problems arise from the knowledge of several binary relations defined on a set of alternatives, which we intend to rank. In a previous work, the authors introduced a tool to confirm the solutions of multi-attribute ranking problems as linear extensions of a weighted sum of preference relations. An extension of this technique allows the recognition of critical preference pairs of alternatives, which are often caused by inconsistencies. Herein, a confirmation procedure is introduced and applied to confirm the results obtained by a multi-attribute decision methodology on a tender for the supply of buses to the Porto Public Transport Operator. © 2005 Springer Science + Business Media, Inc.

A Characterization of Consistent Collective Choice Rules

We characterize a class of collective choice rules such that collective preference relations are consistent. Consistency is a weakening of transitivity and a strengthening of acyclicity requiring that there be no cycles with at least one strict preference. The properties used in our characterization are unrestricted domain, strong Pareto, anonymity and neutrality. If there are at most as many individuals as there are alternatives, the axioms provide an alternative characterization of the Pareto rule. If there are more individuals than alternatives, however, further rules become available.

Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model

When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and Mart´ınez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale.We find that the key of computational techniques
based on linguistic 2-tuples is to set suitable numerical scale with
the purpose of making transformations between linguistic 2-tuples
and numerical values. By defining the concept of the transitive
calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.