Important Facts and Observations about Pairwise Comparisons (the special issue edition)

• Autorzy: Koczkodaj, W. W. , Redlarski, G. , Szybowski, J. , Wajch, E. , Mikhailov, L. , Soltys, M. , Tamazian, G. , Yuen, K. K. F.
• Języki publikacji: EN

Abstrakty

EN

This study has been inspired by numerous requests for clarification from researchers who often confuse Saaty’s Analytic Hierarchy Process (AHP) with the pairwise comparisons (PC) method, taking AHP as the only representation of PC. This study should be regarded as an interpretation and clarification of past investigations of PC. In addition, this article is a reflection on general PC research at a higher level of abstraction: the philosophy of science. It delves into the foundations and implications of pairwise comparisons. Some results of this study are based on a recently published work by Koczkodaj and Szwarc. Finally, open problems have also been reported for future research.

Słowa kluczowe

EN

pairwise comparisons; inconsistency analysis; ratio scale; approximation

• Czasopismo: Fundamenta Informaticae
• Rocznik: 2016
• Tom: Vol. 144, nr 3/4
• Strony: 291--307
• Opis fizyczny: Bibliogr. 60 poz., rys., tab.

Twórcy

autor

Koczkodaj, W. W.

• Laurentian University, 935 Ramsey Lake Road, Sudbury, Ontario P3E 2C6, Canada,

autor

Redlarski, G.

• Gdansk University of Technology, 80-297 Gdansk, Narutowicza st. 11/12, Poland,

autor

Szybowski, J.

• AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland,

autor

Wajch, E.

• Siedlce University of Natural Sciences and Humanities, 3 Maja 54, 08-110 Siedlice, Poland,

autor

Mikhailov, L.

• The University of Manchester, M13 9PL, United Kingdom,

autor

Soltys, M.

• California State University Channel Islands, One University Drive, Camarillo, CA 93012, USA,

autor

Tamazian, G.

• St. Petersburg State University, 198207 Srendiy Prospekt, d. 41, St. Petersburg, Russia,

autor

Yuen, K. K. F.

• Xi’an Jiaotong-Liverpool University, Suzhou 215123, China,

Bibliografia

• [1] Faliszewski P, Hemaspaandra E, Hemaspaandra LA. Using complexity to protect elections. Communications of the ACM. 2010;53(11):74–82. doi:10.1145/1839676.1839696.
• [2] Fechner G. Elements of Psychophysics. Vol. I. Henry Holt editions in psychology. New York : Holt, Rinehart & Winston; 1966.
• [3] de Condorcet N. Essay on the Application of Analysis to the Probability of Majority Decisions. Paris: l’Imprimerie Royale; 1785.
• [4] Thurstone LL. A law of comparative judgment. Psychological Review. 1927;34(4):273.
• [5] Saaty TL. A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology. 1977;15(3):234–281. doi:10.1016/0022-2496(77)90033-5.
• [6] Janicki R, Zhai Y. On a pairwise comparison-based consistent non-numerical ranking. Logic Journal of IGPL. 2012;20(4):667–676. doi:10.1093/jigpal/jzr018.
• [7] Koczkodaj WW, Szybowski J, Wajch E. Inconsistency indicator maps on groups for pairwise comparisons. International Journal of Approximate Reasoning. 2016;69(2):81–90. Available from: http://arxiv.org/abs/1502.06160. doi:http://www.sciencedirect.com/science/article/pii/S0888613X15001747.
• [8] Cavallo B, D’Apuzzo L. A general unified framework for pairwise comparison matrices in multicriterial methods. International Journal of Intelligent Systems. 2009;24(4):377–398. doi:10.1002/int.20329.
• [9] Ramík J. Pairwise comparison matrix with fuzzy elements on alo-group. Information Sciences. 2015;297:236–253. doi:10.1016/j.ins.2014.11.010.
• [10] Kułakowski K. A heuristic rating estimation algorithm for the pairwise comparisons method. Central European Journal of Operations Research. 2013;23(1):187–203. doi:10.1007/s10100-013-0311-x.
• [11] Kułakowski K. Heuristic Rating Estimation Approach to The Pairwise Comparisons Method. Fundamenta Informaticae. 2014;133(4):367–386. doi:10.3233/FI-2014-108.
• [12] Jensen RE. An alternative scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology. 1984;28(3):317–332.
• [13] Koczkodaj WW, Orłowski M. Computing a consistent approximation to a generalized pairwise comparisons matrix. Computers & Mathematics with Applications. 1999;37(3):79–85.
• [14] Koczkodaj WW, Szybowski J. Pairwise comparisons simplified. Applied Mathematics and Computation. 2015;253:387–394.
• [15] Temesi J. Pairwise comparison matrices and the error-free property of the decision maker. Central European Journal of Operations Research. 2011;19(2):239–249. doi:10.1007/s10100-010-0145-8.
• [16] Yuen K, Kam F. Pairwise opposite matrix and its cognitive prioritization operators: comparisons with pairwise reciprocal matrix and analytic prioritization operators. Journal of the Operational Research Society. 2012;63(3):322–338. Available from: http://dx.doi.org/10.1057/jors.2011.33.
• [17] Janicki R. Pairwise Comparisons Based Non-Numerical Ranking. Fundamenta Informaticae. 2009;94(2): 197–217. doi:10.3233/FI-2009-126.
• [18] Maskin E, Amartya Sen with contributions from Kenneth J Arrow PKP Partha Dasgupta, Stiglitz JE. The Arrow Impossibility Theorem. Kenneth J. Arrow Lecture Series. Columbia University Press; 2014. Available from: http://www.jstor.org/stable/10.7312/mask15328.
• [19] Janicki R, Koczkodaj WW. A Weak Order Approach to Group Ranking. Computers & Mathematics With Applications. 1996;32(2):51–59. doi:10.1016/0898-1221(96)00102-2.
• [20] Miller GA. The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychological Review. 1956;63(2):81. Available from: http://dx.doi.org/10.1037/h0043158.
• [21] Clermont KM. Procedures Magical Number-3 - Psychological Bases for Standards of Decision. Cornell Law Review. 1987;72(6):1115–1156. Available from: http://scholarship.law.cornell.edu/facpub/266.
• [22] Kusherbaeva V, Sushkov Y, Tamazian G. Scales and methods for deriving weights in the analytic hierarchy process. Vestnik St Petersburg University: Mathematics. 2011;44(4):282–291. doi:10.3103/S106345411104008X.
• [23] Ma D, Zheng X. 9/9–9/1 scale method of AHP. In: Proceedings of the 2nd International Symposium on the AHP. vol. 1; 1991. p. 197–202.
• [24] Lootsma FA. Scale sensitivity in the multiplicative AHP and SMART. Journal of Multi-Criteria Decision Analysis. 1993;2(2):87–110. doi:10.1002/mcda.4020020205.
• [25] Donegan HA, Dodd FJ, McMaster TBM. A new approach to AHP decision-making. The Statistician. 1992;41(3):295–302. Available from: http://www.jstor.org/stable/2348551. doi:10.2307/2348551.
• [26] Salo AA, Hämäläinen RP. On the measurement of preferences in the analytic hierarchy process. Journal of Multi-Criteria Decision Analysis. 1997;6(6):309–319. doi:10.1002/(SICI)1099-1360(199711)6:6<309::AID-MCDA163>3.0.CO;2-2.
• [27] Ji P, Jiang R. Scale transitivity in the AHP. Journal of the Operational Research Society. 2003;54(8):896–905. Available from: http://www.jstor.org/stable/4101660.
• [28] Beynon M. An analysis of distributions of priority values from alternative comparison scales within AHP. European Journal of Operational Research. 2002;140(1):104–117. doi:10.1016/S0377-2217(01)00221-1.
• [29] Dong Y, Xu Y, Li H, Dai M. A comparative study of the numerical scales and the prioritization methods in AHP. European Journal of Operational Research. 2008;186(1):229–242. doi:10.1016/j.ejor.2007.01.044.
• [30] Elliott MA. Selecting numerical scales for pairwise comparisons. Reliability Engineering & System Safety. 2010;95(7):750–763. doi:10.1016/j.ress.2010.02.013.
• [31] Fülöp J, Koczkodaj WW, Szarek SJ. A different perspective on a scale for pairwise comparisons. In: Transactions on computational collective intelligence I. vol. 6220 of Lecture Notes in Computer Science. Springer; 2010. p. 71–84. doi:10.1007/978-3-642-15034-0_5.
• [32] Fülöp J. A method for approximating pairwise comparison matrices by consistent matrices. Journal of Global Optimization. 2008;42(3):423–442. doi:10.1007/s10898-008-9303-0.
• [33] MacLennan BJ. Principles of Programming Languages: Design, Evaluation, and Implementation. Oxford University Press, Inc.; 1999. ISBN: 0-19-511306-3.
• [34] Hill RJ. A note on inconsistency in paired comparison judgments. American Sociological Review. 1953;18(5):564–566. doi:10.2307/2087442.
• [35] Saaty TL. On the measurement of intengibles. A principal Eigenvector approach to relative measurement derived from paired comparisons. Notices of the American Mathematical Society. 2013;60(2):192–208. doi:10.1090/noti944.
• [36] e Costa CAB, Vansnick JC. A critical analysis of the eigenvalue method used to derive priorities in AHP. European Journal of Operational Research. 2008;187(3):1422–1428. doi:10.1016/j.ejor.2006.09.022.
• [37] Barzilai J. Deriving weights from pairwise comparison matrices. Journal of the Operational Research Society. 1997;48(12):1226–1232. Available from: http://www.jstor.org/stable/3010752. doi:10.2307/3010752.
• [38] Fichtner J. Some thoughts about the mathematics of the analytic hierarchy process. Munich, Germany: Institut für Angewandte Systemforschung u. Operations-Research; 1984. 8403.
• [39] Kendall MG, Smith BB. On the method of paired comparisons. Biometrika. 1940;31(3/4):324–345.
• [40] Gerard HB, Shapiro HN. Determining the degree of inconsistency in a set of paired comparisons. Psychometrika. 1958;23(1):33–46. doi:10.1007/BF02288977.
• [41] Slater P. Inconsistencies in a schedule of paired comparisons. Biometrika. 1961;48(3/4):303–312. Available from: http://www.jstor.org/stable/2332752.
• [42] Koczkodaj WW, Szwarc R. On Axiomatization of Inconsistency Indicators for Pairwise Comparisons. Fundamenta Informaticae. 2014;132:485–500.
• [43] Koczkodaj WW. A new definition of consistency of pairwise comparisons. Mathematical and Computer Modelling. 1993;18(7):79–84.
• [44] KoczkodajWW, Szarek SJ. On distance-based inconsistency reduction algorithms for pairwise comparisons. Logic Journal of IGPL. 2010;18(6):859–869.
• [45] Koczkodaj WW, Almowanes A, Kakiashvili T, Duncan G. Monte Carlo Study of the Random Image Area Estimation by Pairwise Comparisons. Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. 2015;117:271.
• [46] Genest C, Rivest LP. A statistical look at Saaty’s method of estimating pairwise preferences expressed on a ratio scale. Journal of Mathematical Psychology. 1994;38(4):477–496. doi:10.1006/jmps.1994.1034.
• [47] Herman MW, Koczkodaj WW. A Monte Carlo study of pairwise comparisons. Information Processing Letters. 1996;57(1):25–29. doi:10.1016/0020-0190(95)00185-9.
• [48] KoczkodajWW, Kosiek M, Szybowski J, Xu D. Fast convergence of distance-based inconsistency in pairwise comparisons. Fundamenta Informaticae (in print). 2015;137(3):355–367.
• [49] Saaty TL, Vargas LG. Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Mathematical Modelling. 1984;5(5):309–324. doi:10.1016/0270-0255(84)90008-3.
• [50] Kutbi II. A pragmatic pairwise group-decision method for selection of sites for nuclear power plants. Nuclear Engineering and Design. 1987;100(1):49 – 63. ISSN: 0029-5493. Available from: http://www.sciencedirect.com/science/article/pii/0029549387900719. doi:http://dx.doi.org/10.1016/0029-5493(87)90071-9.
• [51] Barbara S, Soltys M. Complex ranking procedures. Fundamenta Informaticae Special Issue on Pairwise Comparisons. 2015;.
• [52] Soltys M. Fair Ranking in Competitive Bidding Procurement: A Case Analysis. Procedia Computer Science. 2014;35(0):1138–1144. doi:10.1016/j.procs.2014.08.207.
• [53] Koczkodaj WW, Kułakowski K, Ligęza A. On the quality evaluation of scientific entities in Poland supported by consistency-driven pairwise comparisons method. Scientometrics. 2014;99(3):911–926.
• [54] Saaty TL, Tran LT. On the invalidity of fuzzifying numerical judgments in the Analytic Hierarchy Process. Mathematical and Computer Modelling. 2007;46(7):962–975. doi:10.1016/j.mcm.2007.03.022.
• [55] Saaty TL. There is no mathematical validity for using fuzzy number crunching in the analytic hierarchy process. Journal of Systems Science and Systems Engineering. 2006;15(4):457–464. doi:10.1007/s11518-006-5021-7.
• [56] Chang DY. Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research. 1996;95(3):649–655. doi:10.1016/0377-2217(95)00300-2.
• [57] Dijkstra EW. Complexity controlled by hierarchical ordering of function and variability. In: Naur P, Randel P, editors. Software Engineering. NATO Science Committee; 1969. p. 181–185. Available from: http://homepages.cs.ncl.ac.uk/brian.randell/NATO/nato1968.PDF.
• [58] HIPO — A Design Aid and Documentation Technique; 1974. Manual No. GC20-1851-0.
• [59] Katzan H. Systems Design and Documentation: An Introduction to the HIPO Method. Computer Science Series. Van Nostrand Reinhold New York; 1976. ISBN: 10:0442242670, 13:978-0442242671.
• [60] Koczkodaj WW. Pairwise Comparisons Rating Scale Paradox. Transactions of Computational Collective Intelligence, XXII, TCCI XXII, LNCS 9655, pp. 1–9, 2016.

Identyfikatory

DOI

10.3233/FI-2016-1336