A note on consistency improvements of AHP paired comparison data. (English) Zbl 1254.90082

Summary: The Analytic Hierarchy Process (AHP) is a popular multicriteria decision-making approach but the ease of AHP paired comparison data collection entails the problem that consistency restrictions have to be fulfilled for the data evaluation task. Quite a lot of consistency improvement techniques are available, however, this note explains why consistency adjustments are not necessarily helpful for computing acceptable weights for the determination of the underlying overall objective function.


90B50 Management decision making, including multiple objectives
62J15 Paired and multiple comparisons; multiple testing
91B06 Decision theory
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[1] Beynon MJ (2002) An investigation of the role of scale values in the DS/AHP method of multi-criteria decision making. J Multi-Criteria Decis Anal 11(6):327–343 · Zbl 1103.90345
[2] Bozoki S, Fueloep J, Poesz A (2010) On pairwise comparison matrices that can be made consistent by the modification of a few elements. Central Eur J Oper Res 19(2):157–175 · Zbl 1213.90132
[3] Cao D, Leung L, Law J (2008) Modifying inconsistent comparison matrix in analytic hierarchy process: a heuristic approach. Decis Support Syst 44(4):944–953
[4] Carmone FJ, Kara A, Zanakis SH (1997) A monte carlo investigation of incomplete pairwise comparison matrices in AHP. Eur J Oper Res 102(3):538–553 · Zbl 0955.90042
[5] Crawford G, Williams C (1985) A note on the analysis of subjective judgment matrices. J Math Psychol 29(4):387–405 · Zbl 0585.62183
[6] Dadkhah KM, Zahedi F (1993) A mathematical treatment of inconsistency in the analytic hierarchy process. Math Comput Model 17(4–5):111–122 · Zbl 0768.90001
[7] Donegan HA, Dodd FJ, McMaster TBM (1992) A new approach to AHP decision-making. Statistican 41(3):295–302
[8] Dong Y, Xu Y, Li H, Dai M (2008) A comparative study of the numerical scales and the prioritization methods in AHP. Eur J Oper Res 186(1):229–242 · Zbl 1138.91373
[9] Finan JS, Hurley WJ (1999) Transitive calibration of the AHP verbal scale. Eur J Oper Res 112(2):367–372 · Zbl 0939.91036
[10] Gaul W (2012) AHP consistency reconsidered. In: Paper presented at 4th Japanese–German Symposium, Kyoto, Japan · Zbl 1254.90082
[11] Harker PT (1987) Alternative modes of questioning in the analytic hierarchy process. Math Model 9(3–5):353–360 · Zbl 0626.90001
[12] Ho W (2008) Integrated analytic hierarchy process and its applications: a literature review. Eur J Oper Res 186(1):211–228 · Zbl 1146.90447
[13] Ho W, Xu X, Dey PK (2010) Multi-criteria decision making approaches for supplier evaluation and selection: a literature review. Eur J Oper Res 202(1):16–24 · Zbl 1175.90223
[14] Ishizaka A, Labib A (2011) Review of the main developments in the analytic hierarchy process. Expert Syst Appl 38:14336–14345
[15] Ishizaka A, Lusti M (2004) An expert module to improve the consistency of AHP matrices. Int Trans Oper Res 11:97–105 · Zbl 1057.90026
[16] Jensen RE (1984) An alternative scaling method for priorities in hierarchical structures. J Math Psychol 28(3):317–332
[17] Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia
[18] Lia HL, Ma LC (2007) Detecting and adjusting ordinal and cardinal inconsistencies through a graphical and optimal approach in AHP models. Computers Oper Res 34(3):780–798 · Zbl 1125.90029
[19] Liang L, Wang G, Zhang ZHB (2008) Mapping verbal responses to numerical scales in the analytic hierarchy process. Socio-Econ Plan Sci 42(1):46–55
[20] Lin CC, Wang WC, Yu WD (2008) Improving AHP for construction with an adaptive AHP approach (A3). Autom Constr 17(2):180–187
[21] Ma D, Zheng X (1991) 9/9-9/1 scale method of AHP. In: Proceedings of the 2nd International Symposium on the AHP, University of Pittsburgh, vol 1
[22] Meissner M, Decker R, Scholz SW (2010) An adaptive algorithm for pairwise comparison-based preference measurement. J Multi-Criteria Decis Anal 17(3):167–177 · Zbl 05933176
[23] Miller GA (1956) The magical number seven plus or minus two: some limits on our capacity for processing information. Psychol Rev 63:81–97
[24] Netzer O, Toubia O, Bradlow E, Dahan E, Evgeniou T, Feinberg F, Feit E, Hui S, Johnson J, Liechty J, Orlin J, Rao V (2008) Beyond conjoint analysis: advances in preference measurement. Mark Lett 19(3):337–354
[25] Park YH, Ding M, Rao VR (2008) Eliciting preference for complex products: a web-based upgrading method. J Mark Res 45(5):562–574
[26] Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New York
[27] Saaty TL (2003) Decision-making with the AHP: why is the principal eigenvector necessary. Eur J Oper Res 145(1):85–91 · Zbl 1012.90015
[28] Saaty TL, Ozdemir MS (2003) Why the magic number seven plus or minus two. Math Computer Model 38(3–4):233–244 · Zbl 1106.91312
[29] Scholz S, Meissner M, Decker R (2010) Measuring consumer preferences for complex products: a compositional approach based on paired comparisons. J Mark Res 47(4):685–698
[30] Triantaphyllou E, Lootsma FA, Pardalos PM, Mann SH (1994) On the evaluation and application of different scales for quantifying pairwise comparisons in fuzzy sets. J Multi-Criteria Decis Anal 3(3):133–155 · Zbl 0851.90002
[31] Vaidya OS, Kumar S (2006) Analytic hierarchy process: an overview of applications. Eur J Oper Res 169(1):1–29 · Zbl 1077.90542
[32] Vargas L (1982) Reciprocal matrices with random coefficients. Math Model 3(1):69–81 · Zbl 0537.62100
[33] Zeshui X, Cuiping W (1999) A consistency improving method in the analytic hierarchy process. Eur J Oper Res 116(2):443–449 · Zbl 1009.90513
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