1,931 research outputs found
Decision support model for the selection of asphalt wearing courses in highly trafficked roads
The suitable choice of the materials forming the wearing course of highly trafficked roads is a delicate task because of their direct interaction with vehicles. Furthermore, modern roads must be planned according to sustainable development goals, which is complex because some of these might be in conflict. Under this premise, this paper develops a multi-criteria decision support model based on the analytic hierarchy process and the technique for order of preference by similarity to ideal solution to facilitate the selection of wearing courses in European countries. Variables were modelled using either fuzzy logic or Monte Carlo methods, depending on their nature. The views of a panel of experts on the problem were collected and processed using the generalized reduced gradient algorithm and a distance-based aggregation approach. The results showed a clear preponderance by stone mastic asphalt over the remaining alternatives in different scenarios evaluated through sensitivity analysis. The research leading to these results was framed in the European FP7 Project DURABROADS (No. 605404).The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under Grant Agreement No. 605404
On optimal completions of incomplete pairwise comparison matrices
An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper
On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices
The aim of the paper is to obtain some theoretical and numerical properties of Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices (PRM). In the case of 3 × 3 PRM, a differentiable one-to-one correspondence is given between Saaty’s inconsistency ratio and Koczkodaj’s inconsistency index based on the elements of PRM. In order to make a comparison of Saaty’s and Koczkodaj’s inconsistencies for 4 × 4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n × n PRM is formulated, the elements aij (i < j) of which were randomly chosen from the ratio scale ... ...
with equal probability 1/(2M − 1) and a ji is defined as 1/a ij . By statistical analysis, the empirical distributions of the maximal eigenvalues of the PRM depending on the dimension number are obtained. As the dimension number increases, the shape of distributions gets similar to that of the normal ones. Finally, the inconsistency of asymmetry is dealt with, showing a different type of inconsistency
Right-left asymmetry of the eigenvector method: A simulation study
The eigenvalue method, suggested by the developer of the extensively used
Analytic Hierarchy Process methodology, exhibits right-left asymmetry: the
priorities derived from the right eigenvector do not necessarily coincide with
the priorities derived from the reciprocal left eigenvector. This paper offers
a comprehensive numerical experiment to compare the two eigenvector-based
weighting procedures and their reasonable alternative of the row geometric mean
with respect to four measures. The underlying pairwise comparison matrices are
constructed randomly with different dimensions and levels of inconsistency. The
disagreement between the two eigenvectors turns out to be not always a
monotonic function of these important characteristics of the matrix. The
ranking contradictions can affect alternatives with relatively distant
priorities. The row geometric mean is found to be almost at the midpoint
between the right and inverse left eigenvectors, making it a straightforward
compromise between them.Comment: 19 pages, 6 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
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
Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process
We propose to extend the aggregation scheme of Saaty’s AHP, from the stan- dard weighted averaging to the more general Choquet integration. In our model, a measure of inconsistency between criteria is derived from the main pairwise comparison matrix and it is used to construct a non-additive capacity, whose associated Choquet integral reduces to the standard weighted mean in the con- sistency case. In the general inconsistency case, however, the new aggregation scheme based on Choquet integration tends to attenuate (resp. emphasize) the priority values of the criteria with higher (resp. lower) average inconsistency with the remaining criteria.Aggregation Functions, Multiple Criteria Analysis, AHP, Inconsintency, non-additive measures, Choquet integral, and Shapley values.
Inconsistency evaluation in pairwise comparison using norm-based distances
AbstractThis paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation
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