Abstract
In this paper, we propose new methods to represent interdependence among alternative attributes and experts’ opinions by constructing Choquet integral using interval-valued intuitionistic fuzzy numbers. In the sequel, we apply these methods to solve the multiple attribute group decision-making (MAGDM) problems under interval-valued intuitionistic fuzzy environment. First, the concept of interval-valued intuitionistic fuzzy Choquet integral is defined, and some elementary properties are studied in detail. Next, an axiomatic system of interval-valued intuitionistic fuzzy measure is established by delivering a series of mathematical proofs. Then, with fuzzy entropy and Shapely-values in game theory, we propose the interval-valued intuitionistic fuzzy measure development methods in order to form the importance measure of attributes and correlation measure of the experts, respectively. Based on the results of theoretical analysis, a new method is proposed to handle the interval-valued intuitionistic fuzzy group decision making problems. A numerical example illustrates the procedure of the proposed methods and verifies the validity and effectiveness of our new proposed methods.
Article PDF
Avoid common mistakes on your manuscript.
References
L.A. Zadeh, Fuzzy sets, Information and control, 8 (1965) 338–353.
D.-C. Lin, J.-S. Yao, Fuzzy economic production for production inventory, Fuzzy sets and systems, 111 (2000) 465–495.
Z. Hu, C. Rao, Y. Zheng, D. Huang, Optimization Decision of Supplier Selection in Green Procurement under the Mode of Low Carbon Economy, International Journal of Computational Intelligence Systems, 8 (2015) 407–421.
X. Huang, Mean-entropy models for fuzzy portfolio selection, IEEE Transactions on Fuzzy Systems, 16 (2008) 1096–1101.
A. Bonetti, S. Bortot, M. Fedrizzi, R.M. Pereira, A. Molinari, Modelling group processes and effort estimation in project management using the Choquet integral: An MCDM approach, Expert Systems with Applications, 39 (2012) 13366–13375.
I. Saad, S. Hammadi, M. Benrejeb, P. Borne, Choquet integral for criteria aggregation in the flexible jobshop scheduling problems, Mathematics and Computers in Simulation, 76 (2008) 447–462.
D. Wu, G. Zhang, J. Lu, A fuzzy preference treebased recommender system for personalized business-to-business e-services, IEEE Transactions on Fuzzy Systems, 23 (2015) 29–43.
S. Han, J.M. Mendel, A new method for managing the uncertainties in evaluating multi-person multi-criteria location choices, using a perceptual computer, Annals of Operations Research, 195 (2012) 277–309.
W. Pedrycz, From fuzzy data analysis and fuzzy regression to granular fuzzy data analysis, Fuzzy Sets and Systems, 274 (2015) 12–17.
S. Srivastava, M. Singh, V.K. Madasu, M. Hanmandlu, Choquet fuzzy integral based modeling of nonlinear system, Applied Soft Computing, 8 (2008) 839–848.
S. Dhompongsa, A. Kaewkhao, S. Saejung, On topological properties of the Choquet weak convergence of capacity functionals of random sets, Information Sciences, 177 (2007) 1852–1859.
S. Wang, W. Pedrycz, Q. Zhu, W. Zhu, Subspace learning for unsupervised feature selection via matrix factorization, Pattern Recognition, 48 (2015) 10–19.
R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management science, 17 (1970) 141–164.
J. Qin, X. Liu, Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment, Information Sciences, 297 (2015) 293–315.
W. Pedrycz, P. Ekel, R. Parreiras, Fuzzy multicriteria decision-making: models, methods and applications, John Wiley & Sons, 2011.
W. Pedrycz, Allocation of information granularity in optimization and decision-making models: towards building the foundations of granular computing, European Journal of Operational Research, 232 (2014) 137–145.
K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems, 20 (1986) 87–96.
K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy sets and systems, 31 (1989) 343–349.
Z. Xu, R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International journal of general systems, 35 (2006) 417–433.
Z. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems , 15 (2007) 1179–1187.
C.-P. Wei, P. Wang, Y.-Z. Zhang, Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications, Information Sciences, 181 (2011) 4273–4286.
W. Wang, X. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE Transactions on Fuzzy Systems, 20 (2012) 923–938.
D.-F. Li, Mathematical-programming approach to matrix games with payoffs represented by Atanassov’s interval-valued intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systems, 18 (2010) 1112–1128.
B. Farhadinia, A.I. Ban, Developing new similarity measures of generalized intuitionistic fuzzy numbers and generalized interval-valued fuzzy numbers from similarity measures of generalized fuzzy numbers, Mathematical and Computer Modelling, 57 (2013) 812–825.
J. Wu, F. Chiclana, A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions, Applied Soft Computing, 22 (2014) 272–286.
B. Liu, Y. Shen, W. Zhang, X. Chen, X. Wang, An interval-valued intuitionistic fuzzy principal component analysis model-based method for complex multi-attribute large-group decision-making, European Journal of Operational Research, 245 (2015) 209–225.
P. Liu, Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making, IEEE Transactions on Fuzzy Systems, 22 (2014) 83–97.
C. Tan, B. Ma, D.D. Wu, X. Chen, Multi-criteria decision making methods based on interval-valued intuitionistic fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 22 (2014) 469–488.
S.-P. Wan, D.-F. Li, Fuzzy mathematical programming approach to heterogeneous multiattribute decisionmaking with interval-valued intuitionistic fuzzy truth degrees, Information Sciences, 325 (2015) 484–503.
X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to intervalvalued intuitionistic fuzzy group decision making, Applied Soft Computing, 26 (2015) 42–56.
B. Farhadinia, A theoretical development on the entropy of interval-valued fuzzy sets based on the intuitionistic distance and its relationship with similarity measure, Knowledge-Based Systems, 39 (2013) 79–84.
P. Liu, Some geometric aggregation operators based on interval intuitionistic uncertain linguistic variables and their application to group decision making, Applied Mathematical Modelling, 37 (2013) 2430–2444.
J. Wu, F. Chiclana, Non-dominance and attitudinal prioritisation methods for intuitionistic and intervalvalued intuitionistic fuzzy preference relations, Expert Systems with Applications, 39 (2012)13409–13416.
Z.-J. Wang, K.W. Li, An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives, Expert Systems with Applications, 39 (2012) 13509–13516.
M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy sets and Systems, 69 (1995) 279–298.
G. Choquet, Theory of capacities, in: Annales de l’institut Fourier, Institut Fourier, 1954, 131–295.
M. Sugeno, Theory of fuzzy integrals and its applications, Tokyo Institute of Technology, 1974.
R.R. Yager, Choquet aggregation using order inducing variables, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12 (2004) 69–88.
F. Meng, X. Chen, Q. Zhang, Some interval-valued intuitionistic uncertain linguistic Choquet operators and their application to multi-attribute group decision making, Applied Mathematical Modelling, 38 (2014) 2543–2557.
Z. Xu, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences, 180 (2010) 726–736.
Z. Xu, Multi-person multi-attribute decision making models under intuitionistic fuzzy environment, Fuzzy Optimization and Decision Making, 6 (2007) 221–236.
C. Tan, A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integralbased TOPSIS, Expert Systems with Applications, 38 (2011) 3023–3033.
C. Tan, X. Chen, Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making, Expert Systems with Applications, 37 (2010) 149–157.
F. Meng, J. Tang, Interval-Valued Intuitionistic Fuzzy Multiattribute Group Decision Making Based on Cross Entropy Measure and Choquet Integral, International Journal of Intelligent Systems, 28 (2013) 1172–1195.
F. Meng, H. Cheng, Q. Zhang, Induced Atanassov’s interval-valued intuitionistic fuzzy hybrid Choquet integral operators and their application in decision making, International Journal of Computational Intelligence Systems, 7 (2014) 524–542.
Y. Xu, H. Wang, J.M. Merig, Intuitionistic fuzzy Einstein Choquet integral operators for multiple attribute decision making, Technological and Economic Development of Economy, 20 (2014) 227–253.
J. Wu, F. Chen, C. Nie, Q. Zhang, Intuitionistic fuzzyvalued Choquet integral and its application in multicriteria decision making, Information Sciences, 222 (2013) 509–527.
J.-q. Wang, D.-d. Wang, H.-y. Zhang, X.-h. Chen, Multi-criteria group decision making method based on interval 2-tuple linguistic information and Choquet integral aggregation operators, Soft Computing, 19 (2015) 389–405.
M. Xia, Z. Xu, N. Chen, Some hesitant fuzzy aggregation operators with their application in group decision making, Group Decision and Negotiation, 22 (2013) 259–279.
F. Meng, Q. Zhang, H. Cheng, Approaches to multiple-criteria group decision making based on interval-valued intuitionistic fuzzy Choquet integral with respect to the generalized λ-Shapley index, Knowledge-Based Systems, 37 (2013) 237–249.
R. Yang, Z. Wang, P.-A. Heng, K.-S. Leung, Classification of heterogeneous fuzzy data by Choquet integral with fuzzy-valued integrand, IEEE Transactions on Fuzzy Systems, 15 (2007) 931–942.
C. He, Approximation of polygonal fuzzy neural networks in sense of Choquet integral norms, International Journal of Machine Learning and Cybernetics, 5 (2014) 93–99.
M. Grabisch, E. Raufaste, An empirical study of statistical properties of Choquet and Sugeno integrals, IEEE Transactions on Fuzzy Systems, 16 (2008) 839–850.
M. Timonin, Robust optimization of the Choquet integral, Fuzzy Sets and Systems, 213 (2013) 27–46.
J.-L. Marichal, k-intolerant capacities and Choquet integrals, European Journal of Operational Research, 177 (2007) 1453–1468.
I. Kojadinovic, J.-L. Marichal, M. Roubens, An axiomatic approach to the definition of the entropy of a discrete Choquet capacity, Information Sciences, 172 (2005) 131–153.
S. Angilella, S. Greco, B. Matarazzo, Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral, European Journal of Operational Research, 201 (2010) 277–288.
J. Ashayeri, G. Tuzkaya, U.R. Tuzkaya, Supply chain partners and configuration selection: An intuitionistic fuzzy Choquet integral operator based approach, Expert Systems with Applications, 39 (2012) 3642–3649.
H. Bustince, M. Galar, B. Bedregal, A. Kolesarova, R. Mesiar, A new approach to interval-valued Choquet integrals and the problem of ordering in intervalvalued fuzzy set applications, IEEE Transactions on Fuzzy Systems, 21 (2013) 1150–1162.
Z. Wang, G. Klir, Fuzzy measure theory, Springer Science and Business Media, 2013.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).
About this article
Cite this article
Qin, J., Liu, X. & Pedrycz, W. Multi-attribute group decision making based on Choquet integral under interval-valued intuitionistic fuzzy environment. Int J Comput Intell Syst 9, 133–152 (2016). https://doi.org/10.1080/18756891.2016.1146530
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/18756891.2016.1146530