Automatically Detecting Ordinal and Preferential Inconsistencies of Pair-Wised Comparison Matrices

Pair-wised comparison matrix (PCM) is the fundamental of Analytical Hierarchy Process (AHP). Priority vectors can be obtained from PCMs by using some methods. However, the decision maker tends to be inconsistent giving a PCM. Such inconsistencies include cardinal inconsistency and ordinal inconsistency. Different methods would lead to considerably different priority vectors if the PCM is inconsistent. Many methods for dealing with cardinal inconsistency have been proposed. Some of them modify problematic PCM elements based on the priority vector derived from the PCM, which is precisely a chicken and egg situation. The elements of a PCM are mutually related such that any modification would change the final priority vector. Yet, no method guarantees that the priority vector after modification would be closer to the one in the decision maker mind than the original one. Cardinal inconsistency is merely representing the inaccuracy of the decision maker giving the PCM and cannot largely affect the priority. However, ordinal inconsistency needs to be corrected for reflecting the decision maker’s contradictory judgments that might make the priority disbelievable. All currently available curing methods for ordinal inconsistency are defective. Some fail to reveal all ordinal inconsistencies, while some must be carried out manually. Besides, the effectiveness of these inefficient manual methods still requires verification. Therefore, there seems to be no method for ordinal inconsistency that is both efficient and effective can be found in the literature. Furthermore, this study has observed some special inconsistency that is categorized as cardinal inconsistency. Cardinal inconsistency can be overlooked. But this inconsistency that involves the order inconsistency of only two items and might considerably alter the order of some items needs to be identified and cured. The inconsistency that has not been reported in the literature so far will be defined as preferential inconsistency in this study. In conclusion, this study aims to propose linear programming models to automatically identify and cure ordinal and preferential inconsistencies and to test the models through a Monti Carlo simulation.

Project ID:PB10307-0476
External Project ID:MOST103-2410-H182-007