Transitive matrices, strict preference and intensity operators

Let X be a set of alternatives and a_{ij} a positive number expressing how much the alternative x_{i} is preferred to the alternative x_{j}. Under suitable hypothesis of no indifference and transitivity over the pairwise comparison matrix A= (a_{ij}), the alternatives can be ordered as a chain . Then a coherent priority vector is a vector giving a weighted ranking agreeing with the obtained chain and an intensity vector is a coherent priority vector encoding information about the intensities of the preferences. In the paper we look for operators F that, acting on the row vectors translate the matrix A in an intensity vector

A general unified framework for pairwise comparison matrices in multicriterial methods

In a Multicriteria Decision Making context, a pairwise comparison matrix $A=(a_{ij})$ is a helpful tool to determine the weighted ranking on a set $X$ of alternatives or criteria. The entry $a_{ij}$ of the matrix can assume different meanings: $a_{ij}$ can be a preference ratio (multiplicative case) or a preference difference (additive case) or $a_{ij}$ belongs to $[0,1]$ and measures the distance from the indifference that is expressed by 0.5 (fuzzy case). For the multiplicative case, a consistency index for the matrix $A$ has been provided by T.L. Saaty in terms of maximum eigenvalue. We consider pairwise comparison matrices over an abelian linearly ordered group and, in this way, we provide a general framework including the mentioned cases. By introducing a more general notion of metric, we provide a consistency index that has a natural meaning and it is easy to compute in the additive and multiplicative cases; in the other cases, it can be computed easily starting from a suitable additive or multiplicative matrix

A survey on pairwise comparison matrices over abelian linearly ordered groups

In this paper, we provide a survey of our results about the pairwise comparison matrices defined over abelian linearly ordered groups

Generalized Consistency and Representation of Preferences by Pairwise Comparisons

A crucial problem in a decision making process is the determination of a scale of relative importance for a set X of alternatives either with respect to a criterion C or an expert E. A widely used tool in Multicriteria Decision Making is the pairwise comparison matrix constituted by positive numbers expressing how much each alternative is preferred to each other. Under suitable hypothesis of no indifference and transitivity over the matrix the actual qualitative ranking on the set X is achievable. We focus on the properties weaker than the consistency and linked to theexistence of cardinal evaluation vectors

Matrici di valutazione ordinale e matrici intensità

Vengono considerate le proprietà di una matrice di confronti a coppie che permettono di determinare un ordinamento di stretta preferenza nell'insieme delle alternative e i vari livelli ai quali tale ordinamento può essere rappresentato. In particolare vengono indagate le matrici priorità le cui colonne sono vettori di valutazione ordinale e le matrici intensità le cui colonne sono vettori di valutazione cardinale ( dei rapporti

Independence and convergence in non additive settings

Some properties of convergence for Archimedean t-conorms areanalyzed and a definition of independence for events, evaluated by adecomposable measure, is introduced. This definition generalize theconcept of independence provided by Kruse and Qiang for thelambda - additive fuzzy measures. Then related properties ofBorel-Cantelli's type are stated

Coherent families of probabilities: properties and extensions

Let B be a Boolean Algebra and S a subset of B. An S-coherent family of probabilities is defined as a family associating to every element of S a probability defined over B, and verifying some coherence conditions. This notion allows to indicate an alternative way to explore conditional probabilities. Indeed, a coherent family of probabilities is naturally attached to every conditional probability. In the paper some conditions are provided, under which an S -coherent family of probabilities can be built up by a finite number of steps and connections between the notions of S- coherent family of probabilities and layers of events are given

Matrici di confronto a coppie: ordine di stretta preferenza e vettori intensità

Si indicano le proprita di una matrice di confronti a coppie che permettono di determinare un ordine di stretta preferenza nell'insieme delle alternative, si discutono quindi le condizioni sulla matrice che sono legate all'esistenza di vettori di valutazione dei rapporti, detti vettori intensit