×

Interactive consistency correction in the analytic hierarchy process to preserve ranks. (English) Zbl 1465.91043

Summary: The analytic hierarchy process is a widely used multi-criteria decision-making method that involves the construction of pairwise comparison matrices. To infer a decision, a consistent or near-consistent matrix is desired, and therefore, several methods have been developed to control or improve the overall consistency of the matrix. However, controlling the overall consistency does not necessarily prevent having strong local inconsistencies. Local inconsistencies are local distortions which can lead to rank reversal when a new alternative is added or deleted. To address this problem, we propose an algorithm for controlling the inconsistency during the construction of the pairwise comparison matrix. The proposed algorithm assists decision makers whilst entering their judgments and does not allow strong local inconsistencies. This algorithm is based on the transitivity rule and has been verified through statistical simulations. Appropriate thresholds of acceptable evaluations have been inferred from these simulations. We demonstrate that the proposed algorithm is a helpful decision aid to decision makers when entering pairwise comparison judgments.

MSC:

91B06 Decision theory
90B50 Management decision making, including multiple objectives
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Aguarón, J.; Moreno-Jiménez, J., The geometric consistency index: approximated thresholds, Eur. J. Oper. Res., 147, 1, 137-145 (2003) · Zbl 1060.90657
[2] Bana e. Costa, C.; Vansnick, J., A critical analysis of the eigenvalue method used to derive priorities in AHP, Eur. J. Oper. Res., 187, 3, 1422-1428 (2008) · Zbl 1137.91350
[3] Barzilai, J.; Lootsma, F., Power relation and group aggregation in the multiplicative AHP and SMART, J. Multi-Crit. Decis. Anal., 6, 3, 155-165 (1997) · Zbl 0889.90003
[4] Belton, V.; Gear, A., On a shortcoming of Saaty’s method of analytical hierarchies, Omega, 11, 3, 228-230 (1983)
[5] Belton, V.; Gear, T., The legitimacy of rank reversal — A comment, Omega, 13, 3, 143-144 (1985) · doi:10.1016/0305-0483(85)90052-0
[6] Brunelli, M., A survey of inconsistency indices for pairwise comparisons, Int. J. Gen. Syst., 47, 8, 751-771 (2018) · doi:10.1080/03081079.2018.1523156
[7] Brunelli, M.; Canal, L.; Fedrizzi, M., Inconsistency indices for pairwise comparison matrices: a numerical study, Ann. Oper. Res., 211, 1, 493-509 (2013) · Zbl 1286.90076 · doi:10.1007/s10479-013-1329-0
[8] Camerer, C.; Kagel, J.; Roth, A., Individual decision making, The Handbook of Experimental Economics, 587-703 (1995), Princeton: Princeton University Press, Princeton
[9] Cao, D.; Leung, LC; Law, JS, Modifying inconsistent comparison matrix in analytic hierarchy process: a heuristic approach, Decis. Support Syst., 44, 4, 944-953 (2008)
[10] Corbin, R.; Marley, AAJ, Random utility models with equality: an apparent, but not actual, generalization of random utility models, J. Math. Psychol., 11, 3, 274-293 (1974) · Zbl 0296.90004
[11] De Keyser, W.; Peeters, P., A note on the use of PROMETHEE multicriteria methods, Eur. J. Oper. Res., 89, 3, 457-461 (1996) · Zbl 0916.90163
[12] Dyer, J., A clarification of “remarks on the analytic hierarchy process”, Manag. Sci., 36, 3, 274-275 (1990)
[13] Dyer, J., Remarks on the analytic hierarchy process, Manag. Sci., 36, 3, 249-258 (1990)
[14] Ergu, D.; Kou, G.; Peng, Y.; Shi, Y., A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP, Eur. J. Oper. Res., 213, 1, 246-259 (2011) · Zbl 1237.90220 · doi:10.1016/j.ejor.2011.03.014
[15] Ergu, D.; Kou, G.; Shi, Y.; Shi, Y., Analytic network process in risk assessment and decision analysis, Comput. Oper. Res., 8, 8-9 (2013) · Zbl 1348.91086 · doi:10.1016/j.cor.2011.03.005
[16] Figueira, JR; Roy, B., A note on the paper, “ranking irregularities when evaluating alternatives by using some ELECTRE methods”, by Wang and Triantaphyllou, Omega (2008), Omega, 37, 3, 731-733 (2009)
[17] Finan, JS; Hurley, WJ, The analytic hierarchy process: can wash criteria be ignored?, Comput. Oper. Res., 29, 8, 1025-1030 (2002) · Zbl 0995.90046
[18] Forman, E.; Gass, S., The analytic hierarchy process—an exposition, Oper. Res., 49, 4, 469-486 (2001) · Zbl 1163.90300
[19] García-Cascales, S.; Lamata, T., On rank reversal and TOPSIS method, Math. Comput. Model., 56, 5-6, 123-132 (2012) · Zbl 1255.91072 · doi:10.1016/j.mcm.2011.12.022
[20] Gaul, W.; Gastes, D., A note on consistency improvements of AHP paired comparison data, Adv. Data Anal. Classif., 6, 4, 289-302 (2012) · Zbl 1254.90082 · doi:10.1007/s11634-012-0119-x
[21] Golden, B.; Wang, Q.; Golden, B.; Wasil, E.; Harker, P., An alternate measure of consistency, The Analytic Hierarchy Process: Application and Studies, 68-81 (1989), New York: Springer, New York
[22] González-Pachón, J.; Romero, C., A method for dealing with inconsistencies in pairwise comparisons, Eur. J. Oper. Res., 158, 2, 351-361 (2004) · Zbl 1067.90070
[23] Harker, P.; Vargas, L., The theory of ratio scale estimation: saaty’s analytic hierarchy process, Manag. Sci., 33, 11, 1383-1403 (1987)
[24] Harker, P.; Vargas, L., Reply to “remarks on the analytic hierarchy process”, Manag. Sci., 36, 3, 269-273 (1990)
[25] Holder, R., Some comment on the analytic hierarchy process, J. Oper. Res. Soc., 41, 11, 1073-1076 (1990)
[26] Holder, R., Response to Holder’s comments on the analytic hierarchy process: response to the response, J. Oper. Res. Soc., 42, 10, 914-918 (1991)
[27] Ishizaka, A.; Balkenborg, D.; Kaplan, T., Influence of aggregation and measurement scale on ranking a compromise alternative in AHP, J. Oper. Res. Soc., 62, 4, 700-710 (2010)
[28] Ishizaka, A.; Labib, A., Review of the main developments in the analytic hierarchy process, Expert Syst. Appl., 38, 11, 14336-14345 (2011) · doi:10.1016/j.eswa.2011.04.143
[29] Ishizaka, A.; Labib, A., Selection of new production facilities with the group analytic hierarchy process ordering method, Expert Syst. Appl., 38, 6, 7317-7325 (2011) · doi:10.1016/j.eswa.2010.12.004
[30] Ishizaka, A.; López, C., Cost-benefit AHPSort for performance analysis of offshore providers, Int. J. Prod. Res. (2019) · doi:10.1080/00207543.2018.1509393
[31] Ishizaka, A.; Lusti, M., An expert module to improve the consistency of AHP matrices, Int. Trans. Oper. Res., 11, 1, 97-105 (2004) · Zbl 1057.90026
[32] Kendall, MG; Smith, BB, On the method of paired comparisons, Biometrika, 31, 3-4, 324-345 (1940) · Zbl 0023.14803
[33] Keskin, B.; Köksal, C., A hybrid AHP/DEA-AR model for measuring and comparing the efficiency of airports, Int. J. Prod. Perform. Manag., 68, 3, 524-541 (2019) · doi:10.1108/IJPPM-02-2018-0043
[34] Kwiesielewicz, M.; van Uden, E., Inconsistent and contradictory judgements in pairwise comparison method in AHP, Comput. Oper. Res., 31, 5, 713-719 (2004) · Zbl 1048.90121
[35] Linares, P., Are inconsistent decisions better? An experiment with pairwise comparisons, Eur. J. Oper. Res., 193, 2, 492-498 (2009) · Zbl 1180.90155
[36] Lootsma, F., Scale sensitivity in the multiplicative AHP and SMART, J. Multi-Crit. Dec. Anal., 2, 2, 87-110 (1993) · Zbl 0838.90009
[37] Maleki, H.; Zahir, S., A comprehensive literature review of the rank reversal phenomenon in the analytic hierarchy process, J. Multi-Crit. Decis. Anal. (2012) · doi:10.1002/mcda.1479
[38] Mareschal, B., De Smet, Y., Nemery, P.: Rank reversal in the PROMETHEE II method: some new results. Paper presented at the IEEE 2008 International Conference on Industrial Engineering and Engineering Management, Singapore
[39] Millet, I.; Saaty, T., On the relativity of relative measures-accommodating both rank preservation and rank reversals in the AHP, Eur. J. Oper. Res., 121, 1, 205-212 (2000) · Zbl 0948.91015
[40] Monti, S., Carenini, G.: Dealing with the expert inconsistencies: the sooner the better. Paper presented at the IJCAI-95 Workshop: “Building Probabilistic Networks: where do the numbers come from?”, Montreal (1995)
[41] Perez, J.; Jimeno, JL; Mokotoff, E., Another potential shortcoming of AHP, TOP, 14, 1, 99-111 (2006) · Zbl 1110.91010
[42] Pérez, J., Some comments on Saaty’s AHP, Manag. Sci., 41, 6, 1091-1095 (1995) · Zbl 0859.90007
[43] Saaty, T., A scaling method for priorities in hierarchical structures, J. Math. Psychol., 15, 3, 234-281 (1977) · Zbl 0372.62084
[44] Saaty, T., The Analytic Hierarchy Process (1980), New York: McGraw-Hill, New York · Zbl 1176.90315
[45] Saaty, T., Rank generation, preservation and reversal in the analytic hierarchy decision process, Decis. Sci., 18, 2, 157-177 (1987)
[46] Saaty, T., An exposition of the AHP in reply to the paper “remarks on the analytic hierarchy process”, Manag. Sci., 36, 3, 259-268 (1990)
[47] Saaty, T., Response to Holder’s comments on the analytic hierarchy process, J. Oper. Res. Soc., 42, 10, 909-929 (1991)
[48] Saaty, T., Decision-making with the AHP: why is the principal eigenvector necessary?, Eur. J. Oper. Res., 145, 1, 85-91 (2003) · Zbl 1012.90015
[49] Saaty, T., Rank from comparisons and from ratings in the analytic hierarchy/network processes, Eur. J. Oper. Res., 168, 2, 557-570 (2006) · Zbl 1101.90368
[50] Saaty, T.; Sagir, M., An essay on rank preservation and reversal, Math. Comput. Modell. (2008) · doi:10.1016/j.mcm.2008.1008.1001
[51] Saaty, T.; Takizawa, M., Dependence and independence: from linear hierarchies to nonlinear networks, Eur. J. Oper. Res., 26, 2, 229-237 (1986) · Zbl 0599.90067
[52] Saaty, T.; Vargas, L., The legitimacy of rank reversal, Omega, 12, 5, 513-516 (1984)
[53] Saaty, T.; Vargas, L., The analytic hierarchy process: wash criteria should not be ignored, Int. J. Manag. Decis. Mak., 7, 2-3, 180-188 (2006)
[54] Salo, A.; Hamalainen, R., On the measurement of preference in the analytic hierarchy process, J. Multi-Crit. Decis. Anal., 6, 6, 309-319 (1997) · Zbl 1078.91525
[55] Schoner, B.; Wedley, W., Ambiguous criteria weights in AHP: consequences and solutions, Decis. Sci., 20, 3, 462-475 (1989)
[56] Schoner, B.; Wedley, W.; Choo, E., A unified approach to AHP with linking pins, Eur. J. Oper. Res., 64, 3, 384-392 (1993)
[57] Schoner, B.; Wedley, WC; Choo, EU, A rejoinder to forman on AHP, with emphasis on the requirements of composite ratio scales, Decis. Sci., 23, 2, 509-517 (1992)
[58] Sipahi, S.; Timor, M., The analytic hierarchy process and analytic network process: an overview of applications, Manag. Decis., 48, 5, 775-808 (2010)
[59] Siraj, S.; Mikhailov, L.; Keane, J., A heuristic method to rectify intransitive judgments in pairwise comparison matrices, Eur. J. Oper. Res., 216, 2, 420-428 (2012) · Zbl 1237.90121 · doi:10.1016/j.ejor.2011.07.034
[60] Subramanian, N.; Ramanathan, R., A review of applications of Analytic Hierarchy Process in operations management, Int. J. Prod. Econ., 138, 2, 215-241 (2012) · doi:10.1016/j.ijpe.2012.03.036
[61] Tone, K.; Tone, K., A comparative study of AHP and DEA. Theory and applications, Advances in DEA Theory and Applications (2017), Chichester: Wiley, Chichester · Zbl 1139.91354
[62] Triantaphyllou, E., Two new cases of rank reversals when the AHP and some of its additive variants are used that do not occur with the multiplicative AHP, J. Multi-Crit. Decis. Anal., 10, 1, 11-25 (2001) · Zbl 0985.90057
[63] Troutt, M., Rank reversal and the dependence of priorities on the underlying MAV function, Omega, 16, 4, 365-367 (1988)
[64] Tversky, A.; Slovic, P.; Kahneman, D., The cause of preference reversal, Am. Econ. Assoc., 80, 1, 204-217 (1990)
[65] Vargas, L., Comments on Barzilai and Lootsma why the multiplicative AHP is invalid: a practical counterexample, J. Multi-Crit. Decis. Anal., 6, 4, 169-170 (1997)
[66] Vargas, L.G.: A rejoinder. OMEGA 13(4), 249 (1985)
[67] Veni, K.; Rajesh, R.; Pugazhendhi, S., Development of decision making model using integrated AHP and DEA for vendor selection, Proc. Eng., 38, 3700-3708 (2012) · doi:10.1016/j.proeng.2012.06.425
[68] Wang, Y.; Chin, K-S; Luo, Y., Aggregation of direct and indirect judgements in pairwise comparison matrices with a re-examination of the criticisms by Bana e Costa and Vansnick, Inf. Sci., 179, 3, 329-337 (2009) · Zbl 1156.91338
[69] Wang, Y.; Elhag, T., An approach to avoiding rank reversal in AHP, Decis. Support Syst., 42, 3, 1474-1480 (2006)
[70] Wang, Y.; Luo, Y., On rank reversal in decision analysis, Math. Comput. Model., 49, 5-6, 1221-1229 (2009) · Zbl 1165.91353
[71] Wanga, X.; Triantaphyllou, E., Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega, 36, 1, 45-63 (2008)
[72] Xu, Z.; Wei, C., A consistency improving method in the analytic hierarchy process, Eur. J. Oper. Res., 116, 2, 443-449 (1999) · Zbl 1009.90513
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.