Proposals for the Set of Pairwise Comparisons

Szádoczki, Zsombor; Bozóki, Sándor; Tekile, Hailemariam Abebe

doi:10.13033/isahp.y2020.054

Cited by 4 publications

(6 citation statements)

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“…Presently, finding an analytical expression for this function is problematic. However, we can see that it directly depends on the diameters of spanning tree graphs, used during priority aggregation (as shown in Szádoczki et al, 2020). In the context of our problem, graph diameter is the length of longest path between its two vertices (nodes).…”

Section: Motivationmentioning

confidence: 99%

“…Finally, ideas, suggested in Szádoczki et al (2020) and , outlined a new direction of combinatorial method development, based on such concepts from graph theory as graph diameter and regularity.…”

Section: Literature Reviewmentioning

confidence: 99%

“…As the method is based on combinatorial enumeration of spanning trees, its computational complexity may become a burden, especially under larger n. For example, in (Szádoczki et al, 2020) the largest dimensionality analyzed is 24 n = . Complete enumeration of all spanning trees would take considerable time or even lie beyond the computational power of any machine.…”

Section: Experimental Research: Motivational Examplesmentioning

confidence: 99%

“…Smaller-diameter spanning tree graphs accumulate smaller expert errors. Another critical component of stability and credibility of pair-wise comparisons, addressed in (Szádoczki et al, 2020) (besides graph diameter), is graph regularity. A graph is considered regular if degrees of all its vertices are the same, or quasi-regular, if the degree of exactly one vertex exceeds the degrees of all other vertices by 1 exactly.…”

Section: Application Of Graph (Quasi)-regularity Concept To Enumeration Of Spanning Treesmentioning

confidence: 99%

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An update on combinatorial method for aggregation of expert judgments in AHP

Kadenko

Tsyganok

Szádoczki

et al. 2021

Prod.

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This article represents an extended version of the respective paper (Kadenko & Tsyganok, 2020), presented at the International Symposium on the Analytic Hierarchy Process (ISAHP-2020). The key additions to the original conference paper include: elaborated description of several versions of combinatorial method, motivational examples, and results of the authors' most recent research on the subject.The best way to evaluate alternative decision options according to intangible criteria, that cannot be numerically described, is to compare them among themselves. This suggestion, made and empirically confirmed by Tom Saaty and his followers (Saaty, 1980(Saaty, , 1996, provided the basis for a whole family of AHP/ANP expert evaluation methods. These methods are widely used by decision-makers around the world. AHP/ANP application domains cover almost all spheres of human activity (as we can see from proceedings of multiple editions of

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Section: Motivationmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Experimental Research: Motivational Examplesmentioning

confidence: 99%

Section: Application Of Graph (Quasi)-regularity Concept To Enumeration Of Spanning Treesmentioning

confidence: 99%

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An update on combinatorial method for aggregation of expert judgments in AHP

Kadenko

Tsyganok

Szádoczki

et al. 2021

Prod.

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“…The results obtained by any weight calculation methods for IPCMs is strongly dependent on the number of known comparisons, namely the number of edges of the representing graph (e), and the arrangements of these known elements. Several properties have been examined in connection with the positioning of the known items, among which (some sense of) regularity of comparisons seems to be an especially important one (Szádoczki et al, 2020;Wang and Takahashi, 1998;Kulakowski et al, 2019), which can be also described by the representing graph.…”

Section: Basic Concepts: Pcms and Their Graph Representationmentioning

confidence: 99%

Optimal filling in sequences for incomplete pairwise comparison matrices

Bozóki¹,

Szádoczki²

2022

Preprint

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Pairwise comparisons form the basis of preference modelling and decision making. It is important from both theoretical and practical points of view to determine the number of questions and their arrangements to ask from the decision maker. We focus on incomplete pairwise comparison matrices, and provide the optimal filling in patterns, which result in the closest (LLSM) weight vectors on average to the complete case for at most six alternatives and for all possible number of comparisons, when the underlying representing graph is connected. These results are obtained by extensive numerical simulations with large sample sizes. Many of the optimal filling structures resulted in optimal filling in sequences, one optimal case can be reached by adding a comparison to a previous one, which are presented on GRAPH of graphs for the different number of alternatives. The star graph is revealed to be optimal among the cases with the same cardinality (spanning trees), while the optimal graphs are always close to bipartite ones. Regular graphs also correspond to optimal cases for the examined parameters. Regularity is important for all optimal graphs, as the degrees of the different vertices are always as close to each other as possible. Besides applying the optimal filling structure in given decision making problems, practitioners can utilize the optimal filling sequences in the cases, when the decision maker can abandon the problem at any period of the process. The effect of an additional comparison is also examined for the optimal cases, which can serve as a guide to determine the minimal sufficient number of comparisons in a given problem.

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Incomplete pairwise comparison matrices based on graphs with average degree approximately 3

et al. 2022

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A crucial, both from theoretical and practical points of view, problem in preference modelling is the number of questions to ask from the decision maker. We focus on incomplete pairwise comparison matrices based on graphs whose average degree is approximately 3 (or a bit more), i.e., each item is compared to three others in average. In the range of matrix sizes we considered, $$n=5,6,7,8,9,10$$ n = 5 , 6 , 7 , 8 , 9 , 10 , this requires from 1.4n to 1.8n edges, resulting in completion ratios between 33% ($$n=10$$ n = 10 ) and 80% ($$n=5$$ n = 5 ). We analyze several types of union of two spanning trees (three of them building on additional ordinal information on the ranking), 2-edge-connected random graphs and 3-(quasi-)regular graphs with minimal diameter (the length of the maximal shortest path between any two vertices). The weight vectors are calculated from the natural extensions, to the incomplete case, of the two most popular weighting methods, the eigenvector method and the logarithmic least squares. These weight vectors are compared to the ones calculated from the complete matrix, and their distances (Euclidean, Chebyshev and Manhattan), rank correlations (Kendall and Spearman) and similarity (Garuti, cosine and dice indices) are computed in order to have cardinal, ordinal and proximity views during the comparisons. Surprisingly enough, only the union of two star graphs centered at the best and the second best items perform well among the graphs using additional ordinal information on the ranking. The union of two edge-disjoint spanning trees is almost always the best among the analyzed graphs.

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Proposals for the Set of Pairwise Comparisons

Cited by 4 publications

References 3 publications

An update on combinatorial method for aggregation of expert judgments in AHP

An update on combinatorial method for aggregation of expert judgments in AHP

Optimal filling in sequences for incomplete pairwise comparison matrices

Incomplete pairwise comparison matrices based on graphs with average degree approximately 3

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