An optimization approach for managing ordinal and cardinal consistencies for pairwise comparison matrices

WU Zhibin, TU Jiancheng, XU Jiuping

Systems Engineering - Theory & Practice ›› 2021, Vol. 41 ›› Issue (5) : 1107-1118.

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Systems Engineering - Theory & Practice ›› 2021, Vol. 41 ›› Issue (5) : 1107-1118. DOI: 10.12011/SETP2020-1812

An optimization approach for managing ordinal and cardinal consistencies for pairwise comparison matrices

  • WU Zhibin, TU Jiancheng, XU Jiuping
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Abstract

Both ordinal consistency and cardinal consistency are able to characterize the rationality degree of individual preferences. In the analytic hierarchy process (AHP), it is difficult to reasonably rank the alternatives when pairwise comparison matrices (PCMs, also called multiplicative preference relations) are significantly cardinal and/or ordinal inconsistent. However, few previous researches have simultaneously considered individual ordinal consistency and cardinal consistency. In this paper, an optimization approach is proposed to solve the individual consistency problems in AHP. Firstly, the conditions of satisfying the ordinal consistency of PCM are analyzed, and the expressions of inequality constraints corresponding to the ordinal consistency conditions are derived, so that the ordinal consistency can be controlled explicitly in the optimization model. Three optimization models are proposed:The first one is used to get a revised preference relation meeting ordinal consistency; the second one is used to obtain a revised preference relation with acceptable cardinal consistency level; and the third one is used to control both kinds of consistencies. Compared with the existing individual inconsistency adjustment methods, the proposed models can directly get the optimal results under the predetermined goals so as to provide a more accurate interactive feedback for decision makers. Finally, the superiority and validity of the proposed models are verified by using a classical example.

Key words

pairwise comparison matrix / ordinal consistency / cardinal consistency / optimization model / analytic hierarchy process

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WU Zhibin , TU Jiancheng , XU Jiuping. An optimization approach for managing ordinal and cardinal consistencies for pairwise comparison matrices. Systems Engineering - Theory & Practice, 2021, 41(5): 1107-1118 https://doi.org/10.12011/SETP2020-1812

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Funding

National Natural Science Foundation of China (71971148, 71671118); Fundamental Research Funds for the Central Universities (SXYPY202103, SCU-BS-PY-202010)
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