Characterisation of the consistent completion of AHP comparison matrices using graph theory

[EN] Decision-making is frequently affected by uncertainty and/or incomplete information, which turn decision-making into a complex task. It is often the case that some of the actors involved in decision-making are not sufficiently familiar with all of the issues to make the appropriate decisions. In this paper, we are concerned about missing information. Specifically, we deal with the problem of consistently completing an analytic hierarchy process comparison matrix and make use of graph theory to characterize such a completion. The characterization includes the degree of freedom of the set of solutions and a linear manifold and, in particular, characterizes the uniqueness of the solution, a result already known in the literature, for which we provide a completely independent proof. Additionally, in the case of nonuniqueness, we reduce the problem to the solution of nonsingular linear systems. In addition to obtaining the priority vector, our investigation also focuses on building the complete pairwise comparison matrix, a crucial step in the necessary process (between synthetic consistency and personal judgement) with the experts. The performance of the obtained results is confirmed.Benítez López, J.; Carpitella, S.; Certa, A.; Izquierdo Sebastián, J. (2019). Characterisation of the consistent completion of AHP comparison matrices using graph theory. Journal of Multi-Criteria Decision Analysis. 26(1-2):3-15. https://doi.org/10.1002/mcda.1652S315261-2Benítez, J., Carrión, L., Izquierdo, J., & Pérez-García, R. (2014). Characterization of Consistent Completion of Reciprocal Comparison Matrices. Abstract and Applied Analysis, 2014, 1-12. doi:10.1155/2014/349729Benítez, J., Delgado-Galván, X., Gutiérrez, J. A., & Izquierdo, J. (2011). Balancing consistency and expert judgment in AHP. Mathematical and Computer Modelling, 54(7-8), 1785-1790. doi:10.1016/j.mcm.2010.12.023Benítez, J., Delgado-Galván, X., Izquierdo, J., & Pérez-García, R. (2011). Achieving matrix consistency in AHP through linearization. Applied Mathematical Modelling, 35(9), 4449-4457. doi:10.1016/j.apm.2011.03.013Benítez, J., Delgado-Galván, X., Izquierdo, J., & Pérez-García, R. (2015). Consistent completion of incomplete judgments in decision making using AHP. Journal of Computational and Applied Mathematics, 290, 412-422. doi:10.1016/j.cam.2015.05.023Benítez, J., Delgado-Galván, X., Izquierdo, J., & Pérez-García, R. (2012). Improving consistency in AHP decision-making processes. Applied Mathematics and Computation, 219(5), 2432-2441. doi:10.1016/j.amc.2012.08.079Benítez, J., Izquierdo, J., Pérez-García, R., & Ramos-Martínez, E. (2014). A simple formula to find the closest consistent matrix to a reciprocal matrix. Applied Mathematical Modelling, 38(15-16), 3968-3974. doi:10.1016/j.apm.2014.01.007Beynon, M., Curry, B., & Morgan, P. (2000). The Dempster–Shafer theory of evidence: an alternative approach to multicriteria decision modelling. Omega, 28(1), 37-50. doi:10.1016/s0305-0483(99)00033-xBozóki, 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. doi:10.1016/j.ejor.2015.06.069Bozóki, S., Fülöp, J., & Rónyai, L. (2010). On optimal completion of incomplete pairwise comparison matrices. Mathematical and Computer Modelling, 52(1-2), 318-333. doi:10.1016/j.mcm.2010.02.047Certa, A., Enea, M., Galante, G. M., & La Fata, C. M. (2013). A Multistep Methodology for the Evaluation of Human Resources Using the Evidence Theory. International Journal of Intelligent Systems, 28(11), 1072-1088. doi:10.1002/int.21617Crawford, G., & Williams, C. (1985). A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(4), 387-405. doi:10.1016/0022-2496(85)90002-1Dong, M., Li, S., & Zhang, H. (2015). Approaches to group decision making with incomplete information based on power geometric operators and triangular fuzzy AHP. Expert Systems with Applications, 42(21), 7846-7857. doi:10.1016/j.eswa.2015.06.007Ergu, D., Kou, G., Peng, Y., Li, F., & Shi, Y. (2014). Data Consistency in Emergency Management. International Journal of Computers Communications & Control, 7(3), 450. doi:10.15837/ijccc.2012.3.1386Ergu, 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 Modelling, 40(1), 254-267. doi:10.1016/j.apm.2015.04.047Floricel, S., Michela, J. L., & Piperca, S. (2016). Complexity, uncertainty-reduction strategies, and project performance. International Journal of Project Management, 34(7), 1360-1383. doi:10.1016/j.ijproman.2015.11.007Forman, E., & Peniwati, K. (1998). Aggregating individual judgments and priorities with the analytic hierarchy process. European Journal of Operational Research, 108(1), 165-169. doi:10.1016/s0377-2217(97)00244-0Guitouni, A., & Martel, J.-M. (1998). Tentative guidelines to help choosing an appropriate MCDA method. European Journal of Operational Research, 109(2), 501-521. doi:10.1016/s0377-2217(98)00073-3Harker, P. T. (1987). Alternative modes of questioning in the analytic hierarchy process. Mathematical Modelling, 9(3-5), 353-360. doi:10.1016/0270-0255(87)90492-1Ho, W. (2008). Integrated analytic hierarchy process and its applications – A literature review. European Journal of Operational Research, 186(1), 211-228. doi:10.1016/j.ejor.2007.01.004Homenda, W., Jastrzebska, A., & Pedrycz, W. (2016). Multicriteria decision making inspired by human cognitive processes. Applied Mathematics and Computation, 290, 392-411. doi:10.1016/j.amc.2016.05.041Hsu, W.-K. K., Huang, S.-H. S., & Tseng, W.-J. (2016). Evaluating the risk of operational safety for dangerous goods in airfreights – A revised risk matrix based on fuzzy AHP. Transportation Research Part D: Transport and Environment, 48, 235-247. doi:10.1016/j.trd.2016.08.018Hua, Z., Gong, B., & Xu, X. (2008). A DS–AHP approach for multi-attribute decision making problem with incomplete information. Expert Systems with Applications, 34(3), 2221-2227. doi:10.1016/j.eswa.2007.02.021Karanik, M., Wanderer, L., Gomez-Ruiz, J. A., & Pelaez, J. I. (2016). Reconstruction methods for AHP pairwise matrices: How reliable are they? Applied Mathematics and Computation, 279, 103-124. doi:10.1016/j.amc.2016.01.008Kubler, S., Robert, J., Derigent, W., Voisin, A., & Le Traon, Y. (2016). A state-of the-art survey & testbed of fuzzy AHP (FAHP) applications. Expert Systems with Applications, 65, 398-422. doi:10.1016/j.eswa.2016.08.064Liu, S., Chan, F. T. S., & Ran, W. (2016). Decision making for the selection of cloud vendor: An improved approach under group decision-making with integrated weights and objective/subjective attributes. Expert Systems with Applications, 55, 37-47. doi:10.1016/j.eswa.2016.01.059Lolli, F., Ishizaka, A., Gamberini, R., & Rimini, B. (2017). A multicriteria framework for inventory classification and control with application to intermittent demand. Journal of Multi-Criteria Decision Analysis, 24(5-6), 275-285. doi:10.1002/mcda.1620Massanet, S., Vicente Riera, J., Torrens, J., & Herrera-Viedma, E. (2016). A model based on subjective linguistic preference relations for group decision making problems. Information Sciences, 355-356, 249-264. doi:10.1016/j.ins.2016.03.040Ortiz-Barrios, M. A., Aleman-Romero, B. A., Rebolledo-Rudas, J., Maldonado-Mestre, H., Montes-Villa, L., De Felice, F., & Petrillo, A. (2017). The analytic decision-making preference model to evaluate the disaster readiness in emergency departments: The A.D.T. model. Journal of Multi-Criteria Decision Analysis, 24(5-6), 204-226. doi:10.1002/mcda.1629Pandey, A., & Kumar, A. (2016). A note on ‘‘Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP”. Information Sciences, 346-347, 1-5. doi:10.1016/j.ins.2016.01.054Qazi, A., Quigley, J., Dickson, A., & Kirytopoulos, K. (2016). Project Complexity and Risk Management (ProCRiM): Towards modelling project complexity driven risk paths in construction projects. International Journal of Project Management, 34(7), 1183-1198. doi:10.1016/j.ijproman.2016.05.008Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15(3), 234-281. doi:10.1016/0022-2496(77)90033-5Saaty, T. L. (2008). Relative measurement and its generalization in decision making why pairwise comparisons are central in mathematics for the measurement of intangible factors the analytic hierarchy/network process. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 102(2), 251-318. doi:10.1007/bf03191825Seiti, H., Tagipour, R., Hafezalkotob, A., & Asgari, F. (2017). Maintenance strategy selection with risky evaluations using RAHP. Journal of Multi-Criteria Decision Analysis, 24(5-6), 257-274. doi:10.1002/mcda.1618Shiraishi, 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. doi:10.15807/jorsj.41.404Srdjevic, 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 & Information Sciences, 8(2), 459-467. doi:10.12785/amis/080202Vaidya, O. S., & Kumar, S. (2006). Analytic hierarchy process: An overview of applications. European Journal of Operational Research, 169(1), 1-29. doi:10.1016/j.ejor.2004.04.028Van Laarhoven, P. J. M., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11(1-3), 229-241. doi:10.1016/s0165-0114(83)80082-7van Uden , E. 2002 Estimating missing data in pairwise comparison matrices Texts in Operational and Systems Research in the Face to Challenge the XXI Century, Methods and Techniques in Information Analysis and Decision Making Academic Printing House WarsawVargas, L., De Felice, F., & Petrillo, A. (2017). Editorial journal of multicriteria decision analysis special issue on «Industrial and Manufacturing Engineering: Theory and Application using AHP/ANP». Journal of Multi-Criteria Decision Analysis, 24(5-6), 201-202. doi:10.1002/mcda.1632Wang, T.-C., & Chen, Y.-H. (2008). Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP. Information Sciences, 178(19), 3755-3765. doi:10.1016/j.ins.2008.05.028Wang, Z.-J., & Tong, X. (2016). Consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations. Information Sciences, 361-362, 29-47. doi:10.1016/j.ins.2016.04.047Wang, H., & Xu, Z. (2016). Interactive algorithms for improving incomplete linguistic preference relations based on consistency measures. Applied Soft Computing, 42, 66-79. doi:10.1016/j.asoc.2015.09.058Weiss-Cohen, L., Konstantinidis, E., Speekenbrink, M., & Harvey, N. (2016). Incorporating conflicting descriptions into decisions from experience. Organizational Behavior and Human Decision Processes, 135, 55-69. doi:10.1016/j.obhdp.2016.05.005Xu, Y., Chen, L., Rodríguez, R. M., Herrera, F., & Wang, H. (2016). Deriving the priority weights from incomplete hesitant fuzzy preference relations in group decision making. Knowledge-Based Systems, 99, 71-78. doi:10.1016/j.knosys.2016.01.047Zhang, H. (2016). Group decision making based on multiplicative consistent reciprocal preference relations. Fuzzy Sets and Systems, 282, 31-46. doi:10.1016/j.fss.2015.04.009Zhang, H. (2016). Group decision making based on incomplete multiplicative and fuzzy preference relations. Applied Soft Computing, 48, 735-744. doi:10.1016/j.asoc.2016.07.04

Integer programming modeling on group decision making with incomplete hesitant fuzzy linguistic preference relations

© 2013 IEEE. Complementing missing information and priority vector are of significance important aspects in group decision making (GDM) with incomplete hesitant fuzzy linguistic preference relations (HFLPRs). In this paper, an integer programming model is developed based on additive consistency to estimate missing values of incomplete HFLPRs by using additive consistency. Once the missing values are complemented, a mixed 0-1 programming model is established to derive the priority vectors from complete HFLPRs, in which the underlying idea of the mixed 0-1 programming model is the probability sampling in statistics and minimum deviation between the priority vector and HFLPR. In addition, we also propose a new GDM approach for incomplete HFLPRs by integrating the integer programming model and the mixed 0-1 programming model. Finally, two case studies and comparative analysis detail the application of the proposed models

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