Livia D’Apuzzo

Investigating Properties of the ⊙-Consistency Index

We consider pairwise comparison matrices on a real divisible and continuous abelian linearly ordered group \(\mathcal{G}= (G, \odot, \leq)\), focusing on a proposed ⊙-consistency measure and its properties. We show that the proposed general ⊙-(in)consistency index satisfies some basic properties that can be considered naturally characterizing a consistency measure.

Ranking and weak consistency in the A.H.P. context

The actual ranking of a set of alternatives is obtainable in a simple way assuming that the matrixA of pairwise comparisons isr-transilive. We show that, in some cases of inconsistency, the weights assigned to the alternatives by means of some well-known methods, suggested by the A.H.P., do not agree with the ranking. Further we introduce a condition, theweak consistency ofA, that ensures the mentioned methods provide weights according with the ranking.RiassuntoNel lavoro si indica un modo per ottenere l’ordinamento effettivo di un insieme di alternative confrontate rispetto ad un criterioC, nelle ipotesi che sia assegnata una matriceA di confronti a coppie. Vengono confrontati con rale ordinamento gli ordinamenti che si deducono dai vettori delle valutazioni numeriche che sono determinate con il metodo dell’autovettore proposto dall’A.H.P. o da metodi che vengono usualmente adoperati nello stesso contesto. Si osserva che in caso di inconsistenza della matriceA i diversi ordinamenti possono non concordare e si dà un condizione per la matriceA, la debole consistenza, sotto la quale gli ordinamenti indicati dalle valutazioni numeriche concordano con l’ordinamento effettivo dell’insieme delle alternative.

Pairwise Comparison Matrices: Some Issue on Consistency and a New Consistency Index

In multicriteria decision making, the pairwise comparisons are an useful starting point for determining a ranking on a set X = {x 1,x 2,..., x n } of alternatives or criteria; the pairwise comparison between x i and x j is quantified in a number a ij expressing how much x i is preferred to x j and the quantitative preference relation is represented by means of the matrix A = (a ij ). In literature the number a ij can assume different meanings (for instance a ratio or a difference) and so several kind of pairwise comparison matrices are proposed. A condition of consistency for the matrix A = (a ij ) is also considered; this condition, if satisfied, allows to determine a weighted ranking that perfectly represents the expressed preferences. The shape of the consistency condition depends on the meaning of the number a ij . In order to unify the different approaches and remove some drawbacks, related for example to the fuzzy additive consistency, in a previous paper we have considered pairwise comparison matrices over an abelian linearly ordered group; in this context we have provided, for a pairwise comparison matrix, a general definition of consistency and a measure of closeness to consistency. With reference to the new general unifying context, in this paper we provide some issue on a consistent matrix and a new measure of consistency that is easier to compute; moreover we provide an algorithm to check the consistency of a pairwise comparison matrix and an algorithm to build consistent matrices.

Analysis of Qualitative and Quantitative Rankings in Multicriteria Decision Making

The decision procedures applied in MCDM (Multicriteria Decision Making) are the most suitable in coping with problems involved by social choices, which have to satisfy a high number of criteria. In such a framework an important role is played by the Analytic Hierarchy Process (A.H.P., for short), a procedure developed by T.L. Saaty at the end of the 70s [14], [15], [16], and widely used by governments and companies in fixing their strategies [10], [16], [19]. The A.H.P. shows how to use judgement and experience to analyze a complex decision problem by combining both qualitative and quantitative aspects in a single framework and generating a set of priorities for alternatives.

Pairwise Comparison Matrices over Abelian Linearly Ordered Groups: A Consistency Measure and Weights for the Alternatives

he pairwise comparison matrices play a basic role in multi-criteria decision making methods such as the Analytic Hierarchy Process (AHP).

A General Framework for Individual and Social Choices

Suitable algebraic structures for individual and social choices are proposed. Some relevant properties are illustrated.

INVESTIGATING CONDITIONS ENSURING RELIABILITY OF THE PRIORITY VECTORS

In this paper, we investigate conditions, weaker than consistency, that a pairwise comparison matrix has to satisfy in order to ensure that priority vectors proposed in literature are ordinal evaluation vectors for the actual ranking. In particular, we introduce a partial order on the rows of a pairwise comparison matrix; if it is a simple order, then the matrix is transitive, the actual ranking is easily established and priority vectors are ordinal evaluation vectors for the actual ranking. Keywords: pairwise comparison matrices, ordinal evaluation vectors, simple order