769 research outputs found
Parametric shortest-path algorithms via tropical geometry
We study parameterized versions of classical algorithms for computing
shortest-path trees. This is most easily expressed in terms of tropical
geometry. Applications include shortest paths in traffic networks with variable
link travel times.Comment: 24 pages and 8 figure
A lexicographically optimal completion for pairwise comparison matrices with missing entries
Estimating missing judgements is a key component in many multi-criteria
decision making techniques, especially in the Analytic Hierarchy Process.
Inspired by the Koczkodaj inconsistency index and a widely used solution
concept of cooperative game theory called the nucleolus, the current study
proposes a new algorithm for this purpose. In particular, the missing values
are substituted by variables, and the inconsistency of the most inconsistent
triad is reduced first, followed by the inconsistency of the second most
inconsistent triad, and so on. The necessary and sufficient condition for the
uniqueness of the suggested lexicographically optimal completion is proved to
be a simple graph-theoretic notion: the undirected graph associated with the
pairwise comparisons, where the edges represent the known elements, should be
connected. Crucially, our method does not depend on an arbitrarily chosen
measure of inconsistency as there exists essentially one reasonable triad
inconsistency index.Comment: 17 pages, 2 figure
The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices
Complete and incomplete additive/multiplicative pairwise comparison matrices
are applied in preference modelling, multi-attribute decision making and
ranking. The equivalence of two well known methods is proved in this paper. The
arithmetic (geometric) mean of weight vectors, calculated from all spanning
trees, is proved to be optimal to the (logarithmic) least squares problem, not
only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S.
(2017): The mathematical equivalence of the "spanning tree" and row geometric
mean preference vectors and its implications for preference analysis, European
Journal of Operational Research 257(1) 197-208, but for incomplete matrices as
well. Unlike the complete case, where an explicit formula, namely the row
arithmetic/geometric mean of matrix elements, exists for the (logarithmic)
least squares problem, the incomplete case requires a completely different and
new proof. Finally, Kirchhoff's laws for the calculation of potentials in
electric circuits is connected to our results.Comment: 21 pages, 6 figure
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