Abstract

The analytic hierarchy process (AHP) is a structured technique for dealing with complex decision- making and is already used to solve many group decision problems. This paper used the Cauchy-Schwarz inequality to improve the AHP algorithm that was developed by Escobar et al. (2004). They provided an upper bound for the group geometric consistency index in order to prove that the group geometric consistency index is less than the maximum of each individual geometric consistency indexes. Although they proposed a useful and novel AHP method, the upper bound estimation still could be improved to provide better group consistency estimation accuracy. This paper proposed a new upper bound estimation that would be able to function in a situation where there are some individual decision makers, whose geometric consistency indexes are greater than the threshold that is proposed by Aguarón and Moreno-Jiménez. The experiment results showed that this paper provided a robust and better estimation. The purposes of this study are as follow; first, this study used Cauchy-Schwarz inequality to improve the synthesized method in AHP method, and to achieve better upper bound estimation. Second, numerical examples are provided to illustrate the findings. Our relative error is 7% of that by Escobar and others to indicate the accuracy. Third, this paper showed that even if the weights changed, the proposed method is still robust with different combinations for decision makers. This study modified one entry of the comparison matrix. The results showed that our estimation performed well on most cases (16 of 17, about 94%) by the sensitivity analysis. Finally, two existing papers with group decision problem were examined with our findings to indicate 6 of 9 are predictable by our upper bound. We also provide a reasonable explanation why the other 3 of 9 cannot be predicted by ours.

Key words: Analytic hierarchy process, group decisions, geometric consistency index.