A Numerical Comparison of Iterative Algorithms for Inconsistency Reduction in Pairwise Comparisons

The aim of this paper is to compare selected iterative algorithms for inconsistency reduction in pairwise comparisons by Monte Carlo simulations. We perform simulations for pairwise comparison matrices of the order <inline-formula> <tex-math notation="LaTeX">$n = 4 $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n = 8 $ </tex-math></inline-formula> with the initial inconsistency <inline-formula> <tex-math notation="LaTeX">$0.10 < CR < 0.80 $ </tex-math></inline-formula> and entries drawn from Saaty’s fundamental scale. Subsequently, we evaluate the algorithms’ performance with respect to four measures that express the degree of original preference preservation. Our results indicate that no algorithm outperforms all other algorithms with respect to every measure of preference preservation. The Xu and Wei’s algorithm is the best with regard to the preservation of an original priority vector and the ranking of objects, the Step-by-Step algorithm best preserves the original preferences expressed in the form of a pairwise comparison matrix, and the algorithm of Szybowski keeps the most matrix entries unchanged during inconsistency reduction.

[1]  Kang Xu,et al.  A direct consistency test and improvement method for the analytic hierarchy process , 2020, Fuzzy Optimization and Decision Making.

[2]  Konrad Kulakowski,et al.  The New Triad based Inconsistency Indices for Pairwise Comparisons , 2014, KES.

[3]  Matthias Ehrgott,et al.  Multiple criteria decision analysis: state of the art surveys , 2005 .

[4]  Caroline Maria de Miranda Mota,et al.  Exploring Multicriteria Elicitation Model Based on Pairwise Comparisons: Building an Interactive Preference Adjustment Algorithm , 2019, Mathematical Problems in Engineering.

[5]  Jinghui Gao,et al.  A new method for modification consistency of the judgment matrix based on genetic ant algorithm , 2011, 2011 International Conference on Multimedia Technology.

[6]  Michele Fedrizzi,et al.  Inconsistency indices for pairwise comparison matrices: a numerical study , 2013, Annals of Operations Research.

[7]  János Fülöp,et al.  On reducing inconsistency of pairwise comparison matrices below an acceptance threshold , 2013, Central Eur. J. Oper. Res..

[8]  Waldemar W. Koczkodaj,et al.  On distance-based inconsistency reduction algorithms for pairwise comparisons , 2010, Log. J. IGPL.

[9]  Joaquín Izquierdo,et al.  Balancing consistency and expert judgment in AHP , 2011, Math. Comput. Model..

[10]  M. Bohanec,et al.  The Analytic Hierarchy Process , 2004 .

[11]  Han-Lin Li,et al.  Detecting and adjusting ordinal and cardinal inconsistencies through a graphical and optimal approach in AHP models , 2007, Comput. Oper. Res..

[12]  José Ignacio Peláez,et al.  A Method for Improving the Consistency of Judgements , 2002, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[13]  Chu-Sing Yang,et al.  Ant algorithm for modifying an inconsistent pairwise weighting matrix in an analytic hierarchy process , 2014, Neural Computing and Applications.

[14]  Jiří Mazurek,et al.  On the inconsistency of pairwise comparisons: an experimental study , 2017 .

[15]  Ashkan Negahban,et al.  Optimizing consistency improvement of positive reciprocal matrices with implications for Monte Carlo Analytic Hierarchy Process , 2018, Comput. Ind. Eng..

[16]  Jaroslav Ramík,et al.  Some new properties of inconsistent pairwise comparisons matrices , 2019, Int. J. Approx. Reason..

[17]  M. T. Lamata,et al.  A new measure of consistency for positive reciprocal matrices , 2003 .

[18]  T. L. Saaty.  A Scaling Method for Priorities in Hierarchical Structures , 1977 .

[19]  Waldemar W. Koczkodaj,et al.  Convergence of Inconsistency Algorithms for the Pairwise Comparisons , 1996, Inf. Process. Lett..

[20]  Jacek Szybowski,et al.  The improvement of data in pairwise comparison matrices , 2018, KES.

[21]  János Fülöp,et al.  On pairwise comparison matrices that can be made consistent by the modification of a few elements , 2011, Central Eur. J. Oper. Res..

[22]  María Teresa Lamata,et al.  Consistency in the Analytic Hierarchy Process: a New Approach , 2006, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[23]  Michele Fedrizzi,et al.  Axiomatic properties of inconsistency indices for pairwise comparisons , 2013, J. Oper. Res. Soc..

[24]  John A. Keane,et al.  Inconsistency reduction in decision making via multi-objective optimisation , 2017, Eur. J. Oper. Res..

[25]  J. Rezaei.  Best-worst multi-criteria decision-making method: Some properties and a linear model , 2016 .

[26]  Joaquín Izquierdo,et al.  A simple formula to find the closest consistent matrix to a reciprocal matrix , 2014 .

[27]  Zeshui Xu,et al.  A consistency improving method in the analytic hierarchy process , 1999, Eur. J. Oper. Res..

[28]  Jiří Mazurek,et al.  A New Step-by-Step (SBS) Algorithm for Inconsistency Reduction in Pairwise Comparisons , 2020, IEEE Access.

[29]  Konrad Kulakowski,et al.  A Concurrent Inconsistency Reduction Algorithm for the Pairwise Comparisons Method , 2015, ICAISC.

[30]  Valdecy Pereira,et al.  Nonlinear programming applied to the reduction of inconsistency in the AHP method , 2015, Ann. Oper. Res..

[31]  Sławomir Jarek.  Removing Inconsistency in Pairwise Comparisons Matrix in the AHP , 2016 .

[32]  R. Lee,et al.  Inconsistency Adjustment in the AHP Using the Complete Transitivity Convergence Algorithm , 2006, 2006 IEEE International Conference on Systems, Man and Cybernetics.

[33]  Ian Lerche,et al.  How Many Monte Carlo Simulations Does One Need to Do? , 2005 .

[34]  Yacine Ouzrout,et al.  Optimal Inconsistency Repairing of Pairwise Comparison Matrices Using Integrated Linear Programming and Eigenvector Methods , 2014, Mathematical Problems in Engineering.

[35]  J. Mazurek,et al.  Some notes on the properties of inconsistency indices in pairwise comparisons , 2018 .

[36]  Ding Xu,et al.  Fast Convergence of Distance-based Inconsistency in Pairwise Comparisons , 2015, Fundam. Informaticae.

[37]  Gang Kou,et al.  A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP , 2011, Eur. J. Oper. Res..

[38]  Joaquín Izquierdo,et al.  Improving consistency in AHP decision-making processes , 2012, Appl. Math. Comput..

[39]  Zan Gao,et al.  Pairwise View Weighted Graph Network for View-based 3D Model Retrieval , 2020, SIGIR.

[40]  Weili Guan,et al.  A Pairwise Attentive Adversarial Spatiotemporal Network for Cross-Domain Few-Shot Action Recognition-R2 , 2020, IEEE Transactions on Image Processing.

[41]  Dong Cao,et al.  Modifying inconsistent comparison matrix in analytic hierarchy process: A heuristic approach , 2008, Decis. Support Syst..

[42]  Gang Kou,et al.  Enhancing data consistency in decision matrix: Adapting Hadamard model to mitigate judgment contradiction , 2014, Eur. J. Oper. Res..

[43]  W. W. Koczkodaj.  A new definition of consistency of pairwise comparisons , 1993 .

[44]  László Csató,et al.  Axiomatizations of inconsistency indices for triads , 2018, Annals of Operations Research.

[45]  Matteo Brunelli,et al.  Studying a set of properties of inconsistency indices for pairwise comparisons , 2015, Ann. Oper. Res..

[46]  Abbas Mardani,et al.  Multiple criteria decision-making techniques and their applications – a review of the literature from 2000 to 2014 , 2015 .