26 research outputs found
On the coincidence of optimal completions for small pairwise comparison matrices with missing entries
Incomplete pairwise comparison matrices contain some missing judgements. A
natural approach to estimate these values is provided by minimising a
reasonable measure of inconsistency after unknown entries are replaced by
variables. Two widely used inconsistency indices for this purpose are Saaty's
inconsistency index and the geometric inconsistency index, which are closely
related to the eigenvector and the logarithmic least squares priority deriving
methods, respectively. The two measures are proven to imply the same optimal
filling for incomplete pairwise comparison matrices up to order four but not
necessarily for order at least five.Comment: 10 pages, 1 figur
Chapter Reducing inconsistency in AHP by combining Delphi and Nudge theory and network analysis of the judgements: an application to future scenarios
The Analytic Hierarchy Process (AHP) is a Multi-Criteria method in which a number of decision factors (typically criteria and alternatives) are compared pairwise by one or more experts, using the Saaty scale, with the goal of sorting the alternatives (Saaty, 1977; 1980). For group AHP the Delphi method can be used in parallel with the AHP (Di Zio and Maretti, 2014), and this allows the search for a consensus on each pairwise judgement. A big issue of the AHP regards the inconsistency of the pairwise comparison matrices and here we propose a new method to reduce the inconsistency. As a solution we exploit the Nudge theory (Thaler and Sunstein, 2008) and from the second round of the Delphi survey, we calculate and circulate a Nudge to “gentle push” the experts towards more consistent evaluations. Furthermore, we propose the representation of the AHP matrices through graphs. In a direct graph two nodes are linked with two direct and weighted edges (or one edge with the direction based on the weights), where the weights indicate the evaluation given by an expert or, for a group, the geometric mean of the judgements. This type of visualization facilitates the reading of the results and could also be used as real-time feedback in the Delphi process, by displaying on the edges also a measure of variability. An application is proposed, on the evaluation of four future scenarios on the regulation of genetic modification experiments, assessed by a panel of 27 experts according to different criteria (plausibility, consistency and simplicity). The application demonstrated that it is possible to: a) reduce the inconsistency; b) collect useful textual material which enrich the AHP itself; c) use the inconsistency index as a stopping criterion for the Delphi rounds; d) display the pairwise comparison matrices with graphs
How to avoid ordinal violations in incomplete pairwise comparisons
Assume that some ordinal preferences can be represented by a weakly connected
directed acyclic graph. The data are collected into an incomplete pairwise
comparison matrix, the missing entries are estimated, and the priorities are
derived from the optimally filled pairwise comparison matrix. Our paper studies
whether these weights are consistent with the partial order given by the
underlying graph. According to previous results from the literature, two
popular procedures, the incomplete eigenvector and the incomplete logarithmic
least squares methods fail to satisfy the required property. Here, it is shown
that the recently introduced lexicographically optimal completion combined with
any of these weighting methods avoids ordinal violation in the above setting.
This finding provides a powerful argument for using the lexicographically
optimal completion to determine the missing elements in an incomplete pairwise
comparison matrix.Comment: 11 pages, 2 figure
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
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
A Decision Support System for Improving the Inconsistency in AHP
This paper presents a DSS aimed at helping decision makers reduce and improve their inconsistency in eliciting their judgements when using the analytic hierarchy process (AHP). The DSS is designed for revising the judgements of a pairwise comparison matrix when the row geometric mean (RGM) is used as the prioritisation procedure and the geometric consistency index (GCI) as the inconsistency measure. The procedure employed guarantees that both the judgements and the derived priority vector will be close to the initial values. The DSS allows different degrees of participation of the decision maker in the review/modification of the judgements: no participation (automatic mode); prior participation (semi-automatic mode); and ongoing participation (interactive mode). The DSS also includes options to incorporate other requirements of the decision maker, such as limiting the modified values to an interval or improving inconsistency by modifying the lowest number of judgements, among others
On a Maximum Eigenvalue of Third‑Order Pairwise Comparison Matrix in Analytic Hierarchy Process and Convergence of Newton’s Method
Nowadays, the analytic hierarchy process is an established method of multiple criteria decision making in the field of Operations Research. Pairwise comparison matrix plays a crucial role in the analytic hierarchy process. The principal (maximum magnitude) eigenvalue of the pairwise comparison matrix can be utilized for measuring the consistency of the decision maker\u27s judgment. The simple transformation of the maximum magnitude eigenvalue is known to be Saaty\u27s consistency index. In this short note, we shed light on the characteristic polynomial of a pairwise comparison matrix of third order. We will show that the only real-number root of the characteristic equation is the maximum magnitude eigenvalue of the third-order pairwise comparison matrix. The unique real-number root appears in the area where it is greater than 3, which is equal to the order of the matrix. By applying usual Newton\u27s method to the characteristic polynomial of the third-order pairwise comparison matrix, we see that the sequence generated from the initial value of 3 always converges to the maximum magnitude eigenvalue