Filling in pattern designs for incomplete pairwise comparison matrices: (Quasi-)regular graphs with minimal diameter

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

doi:10.1016/j.omega.2021.102557

Cited by 17 publications

(9 citation statements)

References 34 publications

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“…We focus on a similar problem as the reduction of comparisons, however, we assume that the whole questionnaire has to be prepared before the decision making process and we cannot ask the decision makers to modify some answers, thus our approach is in some sense similar to Amenta et al (2021). There are only a few studies dealing with the question that which pattern of comparisons should we use in a given problem (Szádoczki et al, 2022), and to our knowledge, none of them regards additional ordinal information as well, thus in this paper we would like to eliminate this research gap. The comparison of different priority vectors has an important role in our study, which has been also used in the different optimizations of aggregation of group preferences (Duleba et al, 2021) and in several weight calculation problems (Kou and Lin, 2014).…”

Section: Literature Reviewmentioning

confidence: 99%

“…As k-regularity is not possible, when both the number of vertices (n) and the level of regularity (k) are odd, Szádoczki et al (2022) defined the graphs that are the closest to k-regularity in this case, as follows.…”

Section: Definition (Consistent Pcm)mentioning

confidence: 99%

“…Our simulations are in a sense similar to the ones in Szádoczki et al (2022), but (besides using other filling patterns and other metrics) we apply elementwise perturbations instead of their matrixwise solution, and the handling of scales is improved in our case.…”

Section: The Simulation Processmentioning

confidence: 99%

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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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Section: Literature Reviewmentioning

confidence: 99%

Section: Definition (Consistent Pcm)mentioning

confidence: 99%

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

et al. 2022

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“…When both the number of vertices (n) and the level of regularity (k) are odd, kregularity is not possible. However, the graphs that are the closest to k-regularity in this case are called k-quasi-regular graphs (Szádoczki et al, 2022).…”

Section: Basic Concepts: Pcms and Their Graph Representationmentioning

confidence: 99%

“…Incompleteness (the absence of some comparisons) occurs quite often in practical problems (Bozóki et al, 2016), as well as in theoretical questions (Fedrizzi and Giove, 2007;Bozóki et al, 2010;Csató and Rónyai, 2016;Kułakowski and Talaga, 2020). In connection with decision making problems, one major source of missing data is the lack of willingness or time of the decision maker, as completing all comparisons -especially in case of many different levels, criteria and alternatives -can be exhausting and lingering (Szádoczki et al, 2022;Fedrizzi and Giove, 2013).…”

Section: Introductionmentioning

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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Significance of the order of pair‐wise comparisons in Analytic Hierarchy Process: an experimental study

Andriichuk,

Kadenko,

Tsyganok

2024

Multi Criteria Decision Anal

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The article describes an approach, ensuring higher credibility of expert estimation results, based on specific order of pair‐wise comparisons. The order of pair‐wise comparisons is, in its turn, based on the distance between estimated objects in the ranking. According to the suggested approach (and some human psychophysiological features), the most ordinally distant objects should be compared before ordinally closer ones. In order to empirically confirm this assumption, a special experiment involving real experts has been conducted. The results of the experiment indicate that if objects are presented to the expert for comparison in the suggested order, then in the majority of cases relative weights of objects, obtained using eigenvector method, most adequately reflect this expert's priorities. Moreover, pair‐wise comparison matrices constructed using the suggested comparison order tend to be slightly more consistent. The suggested approach to re‐ordering of pair‐wise comparisons can be applied as part of the AHP algorithm in weakly structured subject domains, influenced by multiple intangible criteria. It also provides conceptual basis for reduction of the number of pair‐wise comparisons, required to obtain credible results, in AHP without loss or distortion of expert data. It can also be used for modification of combinatorial pair‐wise comparison aggregation method, based on spanning tree enumeration. And, finally, it will improve the overall multi‐criteria decision‐making process in diverse subject domains, characterized by high uncertainty levels.

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Filling in pattern designs for incomplete pairwise comparison matrices: (Quasi-)regular graphs with minimal diameter

Cited by 17 publications

References 34 publications

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

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

Optimal filling in sequences for incomplete pairwise comparison matrices

Significance of the order of pair‐wise comparisons in Analytic Hierarchy Process: an experimental study

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