Document Type : Research Paper


Department of Industrial Engineering, Facuty of Mechanical Engineering, Jundi-Shapur University of Technology, Dezful, Iran.


The Analytic Hierarchy Process (AHP) which was developed by Saaty is a decision analysis tool. It has been applied to many different decision fields. Acquiring Pairwise Comparison Matrices (PCM) is the main step in AHP and also is frequently used in other multi-criteria decision-making methods. In a real problem when the number of alternatives/criteria to be compared is increased, the number of Pairwise Comparisons (PC) often becomes overwhelming. Since the Decision Maker’s (DM) performance in representing the relative preferences tends to deteriorate in such cases, it is preferred to gather fewer data from each individual DM in the form of pairwise comparisons. Missing values in Pairwise Comparison Matrices (PCM) in AHP is a spreading problem in areas dealing with great or dynamic data. The aim of this paper is to present an efficient mathematical programming model for estimating preference vector of pairwise comparison matrices with missing entries.


Main Subjects

[1]   Alonso, S., Chiclana, F., Herrera, F., Herrera‐Viedma, E., Alcalá‐Fdez, J., & Porcel, C. (2008). A consistency‐based procedure to estimate missing pairwise preference values. International journal of intelligent systems23(2), 155-175.
[2]   Benítez, J., Carrión, L., Izquierdo, J., & Pérez-García, R. (2014). Characterization of consistent completion of reciprocal comparison matrices. Abstract and Applied Analysis.
[3]   Bozóki, S., & Fülöp, J. (2018). Efficient weight vectors from pairwise comparison matrices. European journal of operational research264(2), 419-427.
[4]   Bozóki, S., Fülöp, J., & Koczkodaj, W. W. (2011). An LP-based inconsistency monitoring of pairwise comparison matrices. Mathematical and computer modelling54(1-2), 789-793.
[5]   Bozóki, S., FüLöP, J., & RóNyai, L. (2010). On optimal completion of incomplete pairwise comparison matrices. Mathematical and computer modelling52(1-2), 318-333.
[6]   Chen, K., Kou, G., Tarn, J. M., & Song, Y. (2015). Bridging the gap between missing and inconsistent values in eliciting preference from pairwise comparison matrices. Annals of operations research235(1), 155-175.
[7]   Chen, Q., & Triantaphyllou, E. (2001). Estimating data for multicriteria decision making problems: optimization techniques. Encyclopedia of optimization, 567-576.
[8]   Chiclana, F., Herrera-Viedma, E., Alonso, S., & Herrera, F. (2008). A note on the estimation of missing pairwise preference values: a uninorm consistency based method. International journal of uncertainty, fuzziness and knowledge-based systems16(supp02), 19-32.
[9]   Choo, E. U., & Wedley, W. C. (2004). A common framework for deriving preference values from pairwise comparison matrices. Computers & operations research31(6), 893-908.
[10]Ergu, D., Kou, G., Peng, Y., & Zhang, M. (2016). Estimating the missing values for the incomplete decision matrix and consistency optimization in emergency management. Applied mathematical modelling40(1), 254-267.
[11]Fedrizzi, M., & Giove, S. (2007). Incomplete pairwise comparison and consistency optimization. European journal of operational research183(1), 303-313.
[12]Harker, P. T. (1987). Alternative modes of questioning in the analytic hierarchy process. Mathematical modelling9(3-5), 353-360.
[13]Harker, P. T. (1987). Incomplete pairwise comparisons in the analytic hierarchy process. Mathematical modelling9(11), 837-848.
[14]Kou, G., Ergu, D., Lin, C., & Chen, Y. (2016). Pairwise comparison matrix in multiple criteria decision making. Technological and economic development of economy22(5), 738-765.
[15]Kwiesielewicz, M. (1996). The logarithmic least squares and the generalized pseudoinverse in estimating ratios. European journal of operational research93(3), 611-619.
[16]Oliva, G., Setola, R., & Scala, A. (2017). Sparse and distributed analytic hierarchy process. Automatica85, 211-220.
[17]Saaty, T. L., & Vargas, L. G. (2012). Models, methods, concepts & applications of the analytic hierarchy process (Vol. 175). Springer Science & Business Media.
[18]Shiraishi, S., Obata, T., & Daigo, M. (1998). Properties of a positive reciprocal matrix and their application to AHP. Journal of the operations research society of japan41(3), 404-414.
[19]Siraj, S., Mikhailov, L., & Keane, J. A. (2012). Enumerating all spanning trees for pairwise comparisons. Computers & operations research39(2), 191-199.
[20]Wedley, W. C. (1993). Consistency prediction for incomplete AHP matrices. Mathematical and computer modelling17(4-5), 151-161.
[21]Xu, Y., Patnayakuni, R., & Wang, H. (2013). Logarithmic least squares method to priority for group    decision making with incomplete fuzzy preference relations. Applied mathematical modelling37(4), 2139-2152.