A Similarity Measure-based Optimization Model for Group Decision Making with Multiplicative and Fuzzy Preference Relations
AbstractGroup decision making (GDM) problem based on different preference relations aims to obtain a collective opinion based on various preference structures provided by a group of decision makers (DMs) or experts, those who have varying backgrounds and interests in real world. The decision process in proposed question includes three steps: integrating varying preference structures, reaching consensus opinion, selecting the best alternative. Two major approaches: preference transformation and optimization methods have been developed to deal with the issue in first step. However, the transformation processes causes information lose and existing optimization methods are so computationally complex that it is not easy to be used by management practice. This study proposes a new consistency-based method to integrate multiplicative and fuzzy preference relations, which is based on a cosine similarity measure to derive a collective priority vector. The basic idea is that a collective priority vector should be as similar per column as possible to a pairwise comparative matrix (PCM) in order to assure the group preference has highest consistency for each decision makers. The model is computationally simple, because it can be solved using a Lagrangian approach and obtain a collective priority vector following four simple steps. The proposed method can further used to derive priority vector of fuzzy AHP. Using three illustrative examples, the effectiveness and simpleness of the proposed model is demonstrated by comparison with other methods. The results show that the proposed model achieves the largest cosine values in all three examples, indicating the solution is the nearest theoretical perfectly consistent opinion for each decision makers.
 E. Herrera-Viedma, F. Herrera, F. Chiclana (2002), A consensus model for multiperson decision making with different preference structures, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 32(3):394-402.
 F. Chiclana, F. Herrera, E. Herrera-Viedma (1998), Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems, 97:33-48.
 M.P. Brady, S.T. Wu (2010), The aggregation of preferences in groups: Identity, responsibility, and polarization, Journal of Economic Psychology, 31(6):950-963.
 F. Chiclana, F. Herrera, E. Herrera-Viedma (2001), Integrating multiplicative preference relations in a multipurpose decision making model based on fuzzy preference relations, Fuzzy Sets and Systems, 122(2):277-291.
 M. Delgado, F. Herrera, E. Herrera-Viedma, L. Marffnez (1998), Combining numerical and linguistic information in group decision making, Information Sciences, 107:177-194.
 Z.P. Fan, J. Ma, Y.P. Jiang, Y.H. Sun, L. Ma (2006), A goal programming approach to group decision making based on multiplicative preference and fuzzy preference relations, European Journal of Operational Research, 174:311-321.
 J. Ma, Z. P. Fan, Y. P. Jiang, J. Y. Mao (2006), An optimization approach to multiperson decision making based on different formats of preference information, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 36(5):876-889.
 Z.S. Xu, X.Q. Cai, S. S. Liu (2011), Nonlinear programming model integrating different preference structures,IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 41(1):169-177.
 Y.C. Dong, H.J. Zhang, E. Herrera-Viedma (2016), Integrating experts' weights generated dynamically into the consensus reaching process and its applications in managing noncooperative behaviors, Decision Support Systems, 84:1-15.
 X.H. Xu, Z.J. Du, X.H Chen (2015), Consensus model for multi-criteria large-group emergency decision making considering non-cooperative behaviors and minority opinions. Decision Support Systems, 79:150-160.
 Y.C. Dong, H.J. Zhang (2014), Multiperson decision making with different preference representation structures: A direct consensus framework and its properties, Knowledge-Based Systems, 58:45-57.
 Y.C. Dong, N. Luo, H.M. Liang, Consensus building in multiperson decision making with heterogeneous preference representation structures: A perspective based on prospect theory, Applied Soft Computing, (2015)35:898-910.
 I. Palomares, L.Martínez, F. Herrera (2014), A Consensus Model to Detect and Manage Noncooperative Behaviors in Large-Scale Group Decision Making, IEEE Transactions on Fuzzy Systems, 22:516-530
 X. Chen, H. Zhang, Y. Dong (2015), The fusion process with heterogeneous preference structures in group decision-making: A survey,Information Fusion, 24:72-83.
 Z.B. Wu, J.P. Xu (2012), A consistency and consensus based decision support model for group decision-making with multiplicative preference relations, Decision Support Systems, 52:757-767.
 Y. C. Dong, G. Q. Zhang, W. C. Hong, Y. F. Xu (2010), Consensus models for AHP group decision-making under row geometric mean prioritization method, Decision Support Systems, 49:281-289.
 Y.J. Xu, R. Patnayakuni, H.M. Wang (2013), The ordinal consistency of a fuzzy preference relation, Information Sciences, 224:152-164.
 Y.J. Xu, K.W. Li, H.M. Wang (2013), Distance-based consensus models for fuzzy and multiplicative preference relations, Information Sciences, 253:56-73.
 L.A. Yu, K.K. Lai(2011), A distance-based group decision-making methodology for multiperson multi-criteria emergency decision support, Decision Support Systems, 51(2):307-315.
 K.J. Arrow (1963), Social Choice and Individual Values, New York: Wiley, 1963.
 A.K. Sen (1970), Collective Choice and Social Welfare, HoldenDay, San Francisco, CA, 1970.
 W.J.M. Kickert (1978), Fuzzy Theories on Decision-Making, Dordrecht:Nijho, 1978.
 T. L. Saaty (1980), The Analytic Hierarchy Process, New York, NY, USA: McGraw-Hill, 1980.
 J. Kacprzyk, M. Fedrizzi (1990), Multiperson Decision-Making Models Using Fuzzy Sets and Possibility Theory, Dordrecht: Kluwer Academic Publishers, 1990.
 F. Chiclana, F. Herrera, E. Herrera-Viedma (2002), A note on the interval consistency of various preference representations, Fuzzy Sets and Systems, 131:75-78.
 Z.S. Xu (2007), Multiple-attribute group decision making with different formats of preference information on attributes, IEEE Transactions on Systems Man and Cybernetics Part B Cybernetics, 37(6):1500-1511.
 Y.M. Wang, Z.P. Fan, Z.S. Hua (2007), A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations, European Journal of Operational Research, 182:356-366.
 T. L. Saaty (1986), Axiomatic foundation of the analytic hierarchy process, Management Sciences, 32(7):841-855.
 J. Aguarón, J. M. Moreno-Jiménez (2003), The geometric consistency index approximated thresholds, European Journal of Operational Research, 147(1):137-145
 M.T. Escobar, J. Aguar on, J.M. Moreno-Jiménez (2004), A note on AHP group consistency for the row geometric, European Journal of Operational Research, 153:318-322.
 D. Ergu, G. Kou, Y. Peng, Y. Shi (2011), A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP, European Journal of Operational Research, 213:246-259.
 C. S. Lin, G. Kou, D. Ergu (2013), A statistical approach to measure the consistency level of the pairwise comparison matrix, European Journal of Operational Research, 65(9):1380- 1386.
 C. S. Lin, G. Kou, D. Ergu (2013), An improved statistical approach for consistency test in AHP, Annals of Operations Research 211(1):289-299.
 C. S. Lin, G. Kou (2015), Bayesian revision of the individual pair-wise comparison matrices under consensus in AHP-GDM, Applied Soft Computing 35:802-811.
 T. L. Saaty (1977), A scaling method for priorities in a hierarchical structure, Journal of Mathematical Psychology, 15(3):234-281.
 A. T. W. Chu, R. E. Kalaba, K. Spingarn (1979), A comparison of two methods for determining the weights of belonging to fuzzy sets, Journal of Optimization Theory and Applications, 27:531-538.
 G. Crawford, C. Williams (1985), A note on the analysis of subjective judgment matrices, Journal of Mathematical Psychology, 29:387-405.
 C. S. Lin, G. Kou, D. Ergu (2013), A heuristic approach for deriving the priority vector in AHP, Applied Mathematical Modelling, 37:5828-5836.
 G. Kou, C.S. Lin (2014), A cosine maximization method for the priority vector derivation in AHP, European Journal of Operational Research, 235(1):225-232.
 J.A. Gomez-Ruiz, M. Karanik, J. I. Peláez (2010), Estimation of missing judgments in AHP pairwise matrices using a neural network-based model, Applied Mathematics and Computation, 216:2959-2975.
 T. Tanino (1984), Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems, 12:117-131.
 E. Herrera-Viedma, F. Herrera, F. Chiclana, M. Luque (2004), Some issues on consistency of fuzzy preference relations, European Journal of Operational Research, 154:98-109.
 G.Q. Zhang, Y.C. Dong, Y.F. Xu (2012), Linear optimization modeling of consistency issues in group decision making based on fuzzy preference relations, Expert Systems with Applications, 39:2415-2420.
 L. Mikhailov (2003), Deriving priorities from fuzzy pairwise comparison judgements,Fuzzy Sets and Systems,134:365-385.
 Z.S. Xu, Q.L. Da (2005), A least deviation method to obtain a priority vector of a fuzzy preference relation,European Journal of Operational Research, 64:206-216.
 J. Kacprzyk (1986), Group decision making with a fuzzy linguistic majority, Fuzzy Sets and Systems, 18:105-118.
 M. Roubens (1989), Some properties of choice functions based on valued binary relatios, European Journal of Operational Research, 40:309-321.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
ONLINE OPEN ACCES: Acces to full text of each article and each issue are allowed for free in respect of Attribution-NonCommercial 4.0 International (CC BY-NC 4.0.
You are free to:
-Share: copy and redistribute the material in any medium or format;
-Adapt: remix, transform, and build upon the material.
The licensor cannot revoke these freedoms as long as you follow the license terms.
DISCLAIMER: The author(s) of each article appearing in International Journal of Computers Communications & Control is/are solely responsible for the content thereof; the publication of an article shall not constitute or be deemed to constitute any representation by the Editors or Agora University Press that the data presented therein are original, correct or sufficient to support the conclusions reached or that the experiment design or methodology is adequate.