Abstract
This chapter deals with incomplete pairwise comparisons. A brief literature review is followed by a summary of priority deriving methods and inconsistency indices, both modified from complete pairwise comparisons or proposed for the incomplete case only. Also, the inconsistency in the Best-Worst Method is included.
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References
Aczél, J., & Saaty, T. L. (1983). Procedures for synthesizing ratio judgments. Journal of Mathematical Psychology, 27, 93–102.
Ágoston, K. C., & Csató, L. (2022). Inconsistency thresholds for incomplete pairwise comparison matrices. Omega, 108, 102576.
Aguarón, J., & Moreno-Jiménez, J. M. (2003). The geometric consistency index: Approximated threshold. European Journal of Operational Research, 147(1), 137–145.
Alrasheedi, M. A. (2019). Incomplete pairwise comparative judgments: Recent developments and a proposed method. Decision Science Letters, 8(3), 261–274.
Barzilai, J. (1998). Consistency measures for pairwise comparison matrices. Journal of Multi-Criteria Decision Analysis, 7(3), 123–132.
Bozóki, S., Csató, L., & Temesi, J. (2016). An application of incomplete pairwise comparison matrices for ranking top tennis players. European Journal of Operational Research, 248(1), 211–218.
Bozóki, S., Fülöp, J., & Rónyai, L. (2010). On optimal completion of incomplete pairwise comparison matrices. Mathematical and Computer Modelling, 52(1–2), 318–333.
Bozóki, S., Fülöp, J., & Koczkodaj, W. W. (2011). An LP-based inconsistency monitoring of pairwise comparison matrices. Mathematical and Computer Modelling, 54(1–2), 789–793. https://doi.org/10.1016/j.mcm.2011.03.026
Brunelli, M., & Rezaei, J. (2019). A multiplicative best–worst method for multi-criteria decision making. Operations Research Letters, 47(1), 12–15. https://doi.org/10.1016/j.orl.2018.11.008
Carmone, F., Kara, A., & Zanakis, S. (1997). A Monte Carlo investigation of incomplete pairwise comparison matrices in AHP. European Journal of Operational Research, 102, 538–553.
Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms (3rd ed.). MIT Press.
Crawford, G. B. (1987). The geometric mean procedure for estimating the scale of a judgement matrix. Mathematical Modelling, 9(3), 327–334. https://doi.org/10.1016/0270-0255(87)90489-1
Crawford, G., & Williams, C. (1985). A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(4), 387–405.
Csató, L. (2013). Ranking by pairwise comparisons for Swiss-system tournaments. Central European Journal of Operations Research, 21, 783–803. https://doi.org/10.1007/s10100-012-0261-8
Elo, A. E. (1967). The proposed USCF rating system, its development, theory, and applications. Chess Life, 22(8), 242–247.
Fedrizzi, M., & Giove, S. (2007). Incomplete pairwise comparison and consistency optimization. European Journal of Operational Research, 183, 303–313.
Forman, E. H. (1990). Random indices for incomplete pairwise comparison matrices. European Journal of Operational Research, 48(1), 153–155. https://doi.org/10.1016/0377-2217(90)90072-J
Harker, P. T. (1987). Alternative modes of questioning in the analytic hierarchy process. Mathematical Modelling, 9(3), 353–360.
Harker, P. T. (1987). Incomplete pairwise comparisons in the analytic hierarchy process. Mathematical Modelling, 9(11), 837–848.
Haseli, G., Sheikh, R., & Sana, S. S. (2020). Base-criterion on multi-criteria decision-making method and its applications. International Journal of Management Science and Engineering Management, 15(2), 79–88. https://doi.org/10.1080/17509653.2019.1633964
Hu, Y.-C., & Tsai, J.-F. (2006). Backpropagation multi-layer perceptron for incomplete pairwise comparison matrices in analytic hierarchy process. Applied Mathematics and Computation, 180, 53–62.
Ishizaka, A., & Labib, A. (2011). Review of the main developments in the analytic hierarchy process. Expert Systems with Applications, 38(11), 14336–14345. https://doi.org/10.1016/j.eswa.2011.04.143
Ishizaka, A., & Siraj, S. (2020). Interactive consistency correction in the analytic hierarchy process to preserve ranks. Decisions in Economics and Finance, 43, 443–464. https://doi.org/10.1007/s10203-020-00309-4
Kendall, M. (1948). Rank correlation methods. Charles Griffin & Company Ltd.
Kulakowski, K. (2019). On the geometric mean method for incomplete pairwise comparisons. CoRR, abs/1905.04609.
Kulakowski, K., & Szybowski, J. (2014). The new triad based inconsistency indices for pairwise comparisons. Procedia Computer Science, 35, 1132–1137.
Kulakowski, K., & Talaga, D. (2020). Inconsistency indices for incomplete pairwise comparisons matrices. International Journal of General Systems, 49(2), 174–200. https://doi.org/10.1080/03081079.2020.1713116
Leal, J. E. (2019). AHP-express: A simplified version of the analytical hierarchy process method. MethodsX, 7, 100748. https://doi.org/10.1016/j.mex.2019.11.021
Liang, F., Brunelli, M., & Rezaei, J. (2020). Consistency issues in the best worst method: Measurements and thresholds. Omega, 96, 102175. https://doi.org/10.1016/j.omega.2019.102175
Merris, R. (1994). Laplacian matrices of graphs: A survey. Linear Algebra and Its Applications, 197–198, 143–176.
Movafaghpour, M. A. (2019). An efficient nonlinear programming method for eliciting preference weights of incomplete comparisons. Journal of Applied Research on Industrial Engineering, 6(2), 131–138.
Obata, T. Shiraishi, S., Daigo, M., & Nakajima, N. (1999). Assessment for an incomplete comparison matrix and improvement of an inconsistent comparison: Computational experiments. In ISAHP. Kobe.
Oliva, G., Setola, R., & Scala, A. (2017). Sparse and distributed analytic hierarchy process. Automatica, 85, 211–220.
Peng, Y., Kou, G., Ergu, D., & Shi, Y. (2011). Handling missing survey data in AHP. In XI Symposium ISAHP. Sorrento, Naples, Italy.
Quarteroni, A., Sacco, R., & Saleri, F. (2000). Numerical mathematics. Springer.
Ra, J. W. (1999). Chainwise paired comparisons. Decision Sciences, 30, 581–599. https://doi.org/10.1111/j.1540-5915.1999.tb01623.x
Ramík, J. (2013). Incomplete pairwise comparison matrix and its application to ranking of alternatives. Multiple Criteria Decision Making, 8, 114–128.
Rezaei, J. (2015). Best-worst multi-criteria decision-making method. Omega, 53, 40–57.
Rezaei, J. (2016). Best-worst multi-criteria decision-making method: Some properties and a linear model. Omega, 64, 126–130. https://doi.org/10.1016/j.omega.2015.12.001
Salo, A. A., & Hämäläinen, R. (1995). Preference programming through approximate ratio comparisons. European Journal of Operational Research, 82(3), 458–475.
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 Japan, 41(3), 404–414.
Siraj, S., Mikhailov, L., & Keane, J. (2012). A heuristic method to rectify intransitive judgments in pairwise comparison matrices. European Journal of Operational Research, 216, 420–428. https://doi.org/10.1016/j.ejor.2011.07.034
Srdjevic, B., Srdjevic, Z., & Blagojevic, B. (2014). First-level transitivity rule method for filling in incomplete pair-wise comparison matrices in the analytic hierarchy process. Applied Mathematics and Information Sciences, 8(2), 459–467.
Szybowski, J., Kulakowski, K., & Prusak, A. (2020). New inconsistency indicators for incomplete pairwise comparisons matrices. Mathematical Social Sciences, 108, 138–145.
Wedley, W. C. (1993). Consistency prediction for incomplete AHP matrices. Mathematical and Computer Modelling, 17(4–5), 151–161.
Zermelo, E. (1929). Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 29(1), 436–460. https://doi.org/10.1007/BF01180541
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Mazurek, J. (2023). Inconsistency of Incomplete Pairwise Comparisons Matrices. In: Advances in Pairwise Comparisons. Multiple Criteria Decision Making. Springer, Cham. https://doi.org/10.1007/978-3-031-23884-0_6
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DOI: https://doi.org/10.1007/978-3-031-23884-0_6
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