Analytic Hierarchy Process (AHP) Pairwise Matrix with One Missing Value

In order to obtain the results of an Analytic Hierarchy Process (AHP), all of the lower or upper triangle elements of the pairwise matrix need to be filled in. As the number of criteria of an AHP increases, the number of elements of the pairwise matrix increases quadratically. This forces an expert to answer a large number of comparisons. This paper studies and analyzes the characteristics of a pairwise matrix when one of its elements is not available. This is one of the efforts to reduce the number of comparisons that need to be provided by an expert. The results show that a complete pairwise matrix that is consistent tends to have the same characteristics (priority sequence and consistency index) as when it has one missing value. Further research is needed so that the number of comparisons can be decreased while still keeping the pairwise matrix consistent.

Analytic hierarchy process; Pairwise matrix; Consistency index; Missing value

The current paper studies the effects of an incomplete pairwise comparison matrix in an AHP. The study shows that the more consistent a pairwise matrix, the greater the tendency for it to retain its consistency even when one element is missing. This would suggest that the AHP method can be carried out without filling out the entire pairwise comparison matrix, but with one or more missing values. One method that warrants consideration in future research is the approach used to approximate the missing value. In this study, the calculation of Eigenvectors for the pairwise comparison matrix is considered to be missing. It is expected that an approximation of the missing value can enhance the value of the retained percentage of the consistency ratio.

This study was financially supported by Hibah PITTA 2017 from the Directorate of Research and Community Engagement, Universitas Indonesia.

Acharya, V., Sharma, K.S., Gupta, S.K., 2017. Analyzing the Factors in Industrial Automation using Analytic Hierarchy Process. Computers & Electrical Engineering. In Press. https://doi.org/10.1016/j.compeleceng.2017.08.015

Basak, I., Saaty, T.A., 1993. Group Decision Making using the Analytic Hierarchy Process. Mathematical and Computer Modelling, Volume 17(415), pp. 101–109

Bozóki, S., Csató, L., Temesi, J., 2016. An Application of Incomplete Pairwise Comparison Matrices for Ranking Top Tennis Player. European Journal of Operational Research, Volume 248, pp. 211–218

Forman, E.H., 1990. Random Indices for Incomplete Pairwise Comparison Matrices. European Journal of Operational Research, Volume 48, pp. 153–155

Forman, E.H., 1993. Facts and Fictions about the Analytic Hierarchy Process. Mathematical and Computer Modelling, Volume 17(415), pp. 19–26

Harker, P.T., 1987a. Alternative Modes of Questioning in the Analytic Hierarchy Process.  Mathematical Modelling, Volume 9(3-5), pp. 353–360

Harker, P.T., 1987b. Incomplete Pairwise Comparisons in the Analytic Hierarchy Process. Mathematical Modelling, Volume 9(11) pp. 837–848

Ho, W., Ma, X., 2017. The State-of-the-art Integrations and Applications of the Analytic Hierarchy Process. European Journal of Operational Research. In Press. https://doi.org/10.1016/j.ejor.2017.09.007

Ivanco, M., Hou, G., Michaeli, J., 2017. Sensitivity Analysis Method to Address User Disparities in the Analytic Hierarchy Process. Expert Systems with Applications. Volume 90, pp. 111–126

Lambert, J.M., 1991. The Extended Analytic Hierarchy Decision Method. Mathematical and Computer Modelling, Volume 15(11), pp. 141–151

Oliva, G., Setola, R., Scala, A., 2017. Sparse and Distributed Analytic Hierarchy Process. Automatica, Volume 85, pp. 211–220

Saaty, R.W., 1987. The Analityc Hierarchy Process – What it is and How it is Used. Mathematical Modelling, Volume 9(3-5), pp. 161–176

Shen, Y., Hoerl, A. E., McConnell, W., 1992. An Incomplete Design in the Analytic Hierarchy Process. Mathematical and Computer Modelling, Volume 16(5), pp. 121–129

Wedley, W.C., 1993. Consistency Prediction for Incomplete AHP Matrices. Mathematical and Computer Modelling, Volume 17(415), pp. 151–161

Wedley, W.C., Schoner, B., Tang, T.S., 1993. Starting Rules for Incomplete Comparisons in the Analytic Hierarchy Process. Mathematical and Computer Modelling, Volume 17(415), pp. 93–100

Zhu, B., Xu, Z., Zhang, R., Hong, M., 2016. Hesitant Analytic Hierarchy Process. European Journal of Operational Research, Volume 250, pp. 602–614