engleski

A Graph approach to MCDM

The Potential Method (PM) is a ranking procedure which uses a \emph{; ; ; preference graph}; ; ; as input. The nodes of the the preference graph are the alternatives in consideration. Two nodes $u, v$ are adjacent if they are compared, in a pairwise comparison, and the arc is oriented towards the more preferred node. The intensity of that preference, on some scale, is the value of the \emph{; ; ; preference flow}; ; ; $\F$ defined on the set of arcs. $\F$ is \emph{; ; ; consistent}; ; ; if there is no component of the flow in a cycle-space of the graph, or equivalently, if the flow is an element of column space of an incidence matrix $A$ of the graph. In that case there exists a real function $X$ (\emph{; ; ; potential}; ; ; ) defined in the set of nodes such that $AX=\F$. If $\F$ is consistent then, $X$ is a measurable value function in the sense of von Neumann and Morgenstern. For inconsistent flow, the \emph{; ; ; measure of inconsistency}; ; ; is defined as the angle between the flow and the column space of incidence matrix. In that case the equation $AX=\F$ has no solution and we calculate the best approximation of $\F$ by the column space of the incidence matrix and calculate its potential. An advantage of the graph approach, with respect to some other methods (AHP for example) is that the graph need not to be complete i.e. the decision maker is not forced to make all pairwise comparisons. In this article we shall point out the 'good' sides and the 'weakness' of the PM and give a comparative study of PM and some other methods of MCDM. Some specific examples in Data Envelopment Analysis are solved, as well as some examples of the aggregation of grades which were successfully considered only by fuzzy integral. A review of open problems is given at the end.

multi-criteria decision making; preference graph; potential method

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