Algorithms to Detect and Rectify Multiplicative and Ordinal Inconsistencies of Fuzzy Preference Relations

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Consistency, multiplicative and ordinal, of fuzzy preference relations (FPRs) is investigated. The geometric consistency index (GCI) approximated thresholds are extended to measure the degree of consistency for an FPR. For inconsistent FPRs, two algorithms are devised (1) to find the multiplicative inconsistent elements, and (2) to detect the ordinal inconsistent elements. An integrated algorithm is proposed to improve simultaneously the ordinal and multiplicative consistencies. Some examples, comparative analysis, and simulation experiments are provided to demonstrate the effectiveness of the proposed methods

Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building

The mathematical modelling and representation of Tanino's multiplicative transitivity property to the case of intuitionistic reciprocal preference relations (IRPRs) is derived via Zadeh's extension principle and the representation theorem of fuzzy sets. This result guarantees the correct generalisation of the multiplicative transitivity property of reciprocal preference relations (RPRs), and it allows the multiplicative consistency (MC) property of IRPRs to be defined. The MC property used in decision making problems is threefold: (1) to develop a consistency based procedure to estimate missing values in IRPRs using an indirect chain of alternatives; (2) to quantify the consistency index (CI) of preferences provided by experts; and (3) to build a novel consistency based induced ordered weighted averaging (MC-IOWA) operator that associates a higher contribution in the aggregated value to the more consistent information. These three uses are implemented in developing a consensus model for GDM problems with incomplete IRPRs in which the level of agreement between the experts' individual IRPRs and the collective IRPR, which is referred here as the proximity index (PI), is combined with the CI to design a feedback mechanism to support experts to change some of their preference values using simple advice rules that aim at increasing the level of agreement while, at the same time, keeping a high degree of consistency. In the presence of missing information, the feedback mechanism implements the consistency based procedure to produce appropriate estimate values of the missing ones based on the given information provided by the experts. Under the assumption of constant CI values, the feedback mechanism is proved to converge to unanimous consensus when all experts are provided with recommendations and these are fully implemented. Additionally, visual representation of experts' consensus position within the group before and after implementing their feedback advice is also provided, which help an expert to revisit his evaluations and make changes if considered appropriate to achieve a higher consensus level. Finally, an IRPR fuzzy majority based quantifier-guided non-dominance degree based prioritisation method using the associated score reciprocal preference relation is proposed to obtain the final solution of consensus

On the normalization of a priority vector associated with a reciprocal relation.

In this paper we show that the widely used normalization constraint SUM(i=1,n) wi = 1 does not apply to the priority vectors associated with reciprocal relations, whenever additive transitivity is involved. We show that misleading applications of this type of normalization may lead to unsatisfactory results and we give some examples from the literature. Then, we propose an alternative normalization procedure which is compatible with additive transitivity and leads to better results.reciprocal relation; fuzzy preference relation; priority vector; normalization

Incomplete pairwise comparison and consistency optimization

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.consistency, pairwise comparison matrices

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

A general unified framework for pairwise comparison matrices in multicriterial methods

In a Multicriteria Decision Making context, a pairwise comparison matrix $A=(a_{ij})$ is a helpful tool to determine the weighted ranking on a set $X$ of alternatives or criteria. The entry $a_{ij}$ of the matrix can assume different meanings: $a_{ij}$ can be a preference ratio (multiplicative case) or a preference difference (additive case) or $a_{ij}$ belongs to $[0,1]$ and measures the distance from the indifference that is expressed by 0.5 (fuzzy case). For the multiplicative case, a consistency index for the matrix $A$ has been provided by T.L. Saaty in terms of maximum eigenvalue. We consider pairwise comparison matrices over an abelian linearly ordered group and, in this way, we provide a general framework including the mentioned cases. By introducing a more general notion of metric, we provide a consistency index that has a natural meaning and it is easy to compute in the additive and multiplicative cases; in the other cases, it can be computed easily starting from a suitable additive or multiplicative matrix

Estimating unknown values in reciprocal intuitionistic preference relations via asymmetric fuzzy preference relations

Intuitionistic preference relations are becoming increasingly important in the field of group decision making since they present a flexible and simple way to the experts to provide their preference relations, while at the same time allowing them to accommodate a certain degree of hesitation inherent to all decision making processes. In this contribution, we prove the mathematical equivalence between the set of asymmetric fuzzy preference relations and the set of reciprocal intuitionistic fuzzy preference relations. This result is exploited to tackle the presence of incomplete reciprocal intuitionistic fuzzy preference relation in decision making by developing a consistency driven estimation procedure via the corresponding equivalent incomplete asymmetric fuzzy preference relation

Incomplete interval fuzzy preference relations and their applications

This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n − 1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process

A multi-step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices

This article presents a goal programming framework to solve group decision making problems where decision-makers’ judgments are provided as incomplete interval additive reciprocal comparison matrices (IARCMs). New properties of multiplicative consistent IARCMs are put forward and used to define consistent incomplete IARCMs. A two-step goal programming method is developed to estimate missing values for an incomplete IARCM. The first step minimizes the inconsistency of the completed IARCMs and controls uncertainty ratios of the estimated judgments within an acceptable threshold, and the second step finds the most appropriate estimated missing values among the optimal solutions obtained from the previous step. A weighted geometric mean approach is proposed to aggregate individual IARCMs into a group IARCM by employing the lower bounds of the interval additive reciprocal judgments. A two-step procedure consisting of two goal programming models is established to derive interval weights from the group IARCM. The first model is devised to minimize the absolute difference between the logarithm of the group preference and that of the constructed multiplicative consistent judgment. The second model is developed to generate an interval-valued priority vector by maximizing the uncertainty ratio of the constructed consistent IARCM and incorporating the optimal objective value of the first model as a constraint. Two numerical examples are furnished to demonstrate validity and applicability of the proposed approach