Left and right compatibility of strict orders with fuzzy tolerance and fuzzy equivalence relations

The notion of extensionality of a fuzzy relation w.r.t. a fuzzy equivalence was first introduced by Hohle and Blanchard. Belohlavek introduced a similar definition of compatibility of a fuzzy relation w.r.t. a fuzzy equality. In [14] we generalized this notion to left compatibility, right compatibility and compatibility of arbitrary fuzzy relations and we characterized them in terms of left and right traces introduced by Fodor. In this note, we will again investigate these notions, but this time we focus on the compatibility of strict orders with fuzzy tolerance and fuzzy equivalence relations

Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C ψ: [0, 1]2 → [0, 1], C ψ(x, y) = ψ{C[ψ−1(x), ψ−1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails

A novel algorithm for fusing preference orderings by rank-ordered agents

Yager proposed an algorithm to combine multi-agent preference orderings of several alternatives into a single consensus fused ordering, when the agents’ importance is expressed through a rank-ordering and not a set of weights. This algorithm is simple and automatable but has some limitations which reduce its range of application, e.g., (i) preference orderings should not include incomparabilities between alternative and/or omissions of some of them, and (ii) the fused ordering may sometimes not reflect the majority of the multi-agent preference orderings. The aim of this article is to present an enhanced version of the Yager’s algorithm, which overcomes the above limitations. Some practical examples support the description of the new algorithm

Checking the consistency of the solution in ordinal semi-democratic decision making problems

An interesting decision-making problem is that of aggregating multi-agent preference orderings into a consensus ordering, in the case the agents’ importance is expressed in the form of a rank-ordering. Due to the specificity of the problem, the scientific literature encompasses a relatively small number of aggregation techniques. For the aggregation to be effective, it is important that the consensus ordering well reflects the input data, i.e., the agents’ preference orderings and importance rank-ordering. The aim of this paper is introducing a new quantitative tool – represented by the so-called p indicators – which allows to check the degree of consistency between consensus ordering and input data, from several perspectives. This tool is independent from the aggregation technique in use and applicable to a wide variety of practical contexts, e.g., problems in which preference orderings include omissions and/or incomparabilities between some alternatives. Also, the p indicators are simple, intuitive and practical for comparing the results obtained from different techniques. The description is supported by various application examples

Indicators of Inequality and Poverty

This essay aims at a broad, main-stream account of the literature on inequality and poverty measurement in the space of income and, additionally, deals with measures of disparity and deprivation in the more expanded domain of capabilities and functionings. In addition to an introductory and a concluding part, the paper has four sections. The first of these, on measurement of income inequality, deals with preliminary concepts and definitions; a visual representation of inequality (the Lorenz curve); real-valued indices of inequality; properties of inequality indices; some specific inequality measures; and the relationship between Lorenz, welfare, and inequality orderings. The second section, on poverty, deals with the identification and aggregation exercises; properties of poverty indices; some specific poverty measures; the problem of plurality and unambiguous rankings; poverty measures and anti-poverty policy; and other issues in the measurement of poverty. The third section considers aspects of both congruence and conflict in the relationship amongst poverty, inequality, and welfare. The final substantive section advances the rationale for a more comprehensive assessment of human wellbeing than is afforded by the income perspective, it briefly reviews measurement concerns relating to generalized indices of deprivation and disparity, and it discusses the data and policy implications of the more expansive view of well-being adopted in the section.inequality, disparity, poverty, deprivation, measurement, income, capability, functioning, well-being

A paired-comparison approach for fusing preference orderings from rank-ordered agents

The problem of aggregating multi-agent preference orderings has received considerable attention in many fields of research, such as multi-criteria decision aiding and social choice theory; nevertheless, the case in which the agents’ importance is expressed in the form of a rank-ordering, instead of a set of weights, has not been much debated. The aim of this article is to present a novel algorithm – denominated as ‘‘Ordered Paired-Comparisons Algorithm’’ (OPCA), which addresses this decision-making problem in a relatively simple and practical way. The OPCA is organized into three main phases: (i) turning multi- agent preference orderings into sets of paired comparisons, (ii) synthesizing the paired-comparison sets, and (iii) constructing a fused (or consensus) ordering. Particularly interesting is phase two, which introduces a new aggregation process based on a priority sequence, obtained from the agents’ importance rank-ordering. A detailed description of the new algorithm is supported by practical examples

Algorithms to Detect and Rectify Multiplicative and Ordinal Inconsistencies of Fuzzy Preference Relations

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Consistency, multiplicative and ordinal, of fuzzy preference relations (FPRs) is investigated. The geometric consistency index (GCI) approximated thresholds are extended to measure the degree of consistency for an FPR. For inconsistent FPRs, two algorithms are devised (1) to find the multiplicative inconsistent elements, and (2) to detect the ordinal inconsistent elements. An integrated algorithm is proposed to improve simultaneously the ordinal and multiplicative consistencies. Some examples, comparative analysis, and simulation experiments are provided to demonstrate the effectiveness of the proposed methods

Typicality, graded membership, and vagueness

This paper addresses theoretical problems arising from the vagueness of language terms, and intuitions of the vagueness of the concepts to which they refer. It is argued that the central intuitions of prototype theory are sufficient to account for both typicality phenomena and psychological intuitions about degrees of membership in vaguely defined classes. The first section explains the importance of the relation between degrees of membership and typicality (or goodness of example) in conceptual categorization. The second and third section address arguments advanced by Osherson and Smith (1997), and Kamp and Partee (1995), that the two notions of degree of membership and typicality must relate to fundamentally different aspects of conceptual representations. A version of prototype theory—the Threshold Model—is proposed to counter these arguments and three possible solutions to the problems of logical selfcontradiction and tautology for vague categorizations are outlined. In the final section graded membership is related to the social construction of conceptual boundaries maintained through language use