Power harmonic aggregation operator with trapezoidal intuitionistic fuzzy numbers for solving MAGDM problems

Document Type: Research Paper

Authors

Department of Mathematics, Indian Institute of Technology Patna, India

Abstract

Trapezoidal intuitionistic fuzzy numbers (TrIFNs) express abundant and flexible information in a suitable manner and  are very useful to depict the decision information in the procedure of decision making. In this paper, some new aggregation operators, such as, trapezoidal intuitionistic fuzzy weighted power harmonic mean (TrIFWPHM) operator, trapezoidal intuitionistic fuzzy ordered weighted power harmonic mean (TrIFOWPHM) operator, trapezoidal intuitionistic fuzzy induced ordered weighted power harmonic mean (TrIFIOWPHM) operator and trapezoidal intuitionistic fuzzy hybrid power harmonic mean (TrIFhPHM) operator are introduced to aggregate the decision information. The desirable properties of these operators are presented in detail. A prominent characteristic of these operators is that, the aggregated value by using these operators is also a TrIFN. It is observed that the proposed TrIFWPHM operator is the generalization of trapezoidal intuitionistic fuzzy weighted harmonic mean (TrIFWHM) operator, trapezoidal intuitionistic fuzzy weighted arithmetic mean (TrIFWAM) operator, trapezoidal intuitionistic fuzzy weighted geometric mean (TrIFWGM) operator and trapezoidal intuitionistic fuzzy weighted quadratic mean (TrIFWQM) operator, {it i.e.,} we can easily reduce the TrIFWPHM operator to TrIFWHM, TrIFWGM, TrIFWAM and TrIFWQM operators, depending upon the decision situation. Further, we develop an approach to multi-attribute group decision making (MAGDM) problem on the basis of the proposed aggregation operators. Finally, the effectiveness and applicability of our proposed MAGDM model, as well as comparison analysis with other approaches are illustrated with a practical example.

Keywords


[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87{96.
[2] K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and
Systems, 31(3) (1989), 343{349.
[3] K. T. Atanassov, G. Pasi and R. Yager, Intuitionistic fuzzy interpretations of multi-criteria
multi-person and multi-measurement tool decision making, International Journal of Systems
Science, 36(14) (2005), 859{868.
[4] B. F. Baird, Managerial decisions under uncertainty: An introduction to the analysis of
decision making, John Wiley and Sons, 1989.
[5] A. P. G. Beliakov and T. Calvo, Aggregation functions: A guide for practitioners, Springer
Verlag, Berlin Heidelberg, 2007.

[6] F. E. Boran, S. Genc, M. Kurt and D. Akay, A multi-criteria intuitionistic fuzzy group
decision making for supplier selection with topsis method, Expert Systems with Applications,
36(8) (2009), 11363{11368.
[7] P. Burillo, H. Bustince and V. Mohedano, Some de nitions of intuitionistic fuzzy number.
rst properties, In: Proceedings of the 1st Workshop on Fuzzy Based Expert Systems, (1994),
53{55.
[8] M. Casanovas and J. M. Merigo, Fuzzy aggregation operators in decision making with
dempster-shafer belief structure, Expert Systems with Applications, 39(8) (2012), 7138{
7149.
[9] S.-M. Chen and J.-M. Tan, Handling multicriteria fuzzy decision-making problems based on
vague set theory, Fuzzy Sets and Systems, 67(2) (1994), 163{172.
[10] H. Chen, C. Liu and Z. Sheng, Induced ordered weighted harmonic averaging (IOWHA) oper-
ator and its application to combination forecasting method, Chinese Journal of Management
Science, 12(5) (2004), 35{40.
[11] Z. Chen and W. Yang, A new multiple attribute group decision making method in intuition-
istic fuzzy setting, Applied Mathematical Modelling, 35(9) (2011), 4424{4437.
[12] F. Chiclana, F. Herrera and E. Herrera-Viedma, The ordered weighted geometric operator:
Properties and application in mcdm problems, In: Technologies for Constructing Intelligent
Systems, Springer, (2002), 173{183.
[13] G. Choquet, Theory of capacities, In: Annales de linstitut Fourier, 5 (1954), 131{295.
[14] S. Das and D. Guha, Ranking of intuitionistic fuzzy number by centroid point, Journal of
Industrial and Intelligent Information, 1(2) (2013), 107{110.
[15] S. Das, B. Dutta and D. Guha, Weight computation of criteria in a decision-making problem
by knowledge measure with intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set,
Soft Computing, DOI 10.1007/s00500-015-1813-3.
[16] S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical
diagnosis, Fuzzy Sets and Systems, 117(2) (2001), 209{213.
[17] B. Farhadinia, An ecient similarity measure for intuitionistic fuzzy sets, Soft Computing,
18(1) (2014), 85{94.
[18] M. Grabisch and C. Labreuche, A decade of application of the choquet and sugeno integrals
in multi-criteria decision aid, Annals of Operations Research, 175(1) (2010), 247{286.
[19] P. Grzegorzewski, Distances and orderings in a family of intuitionistic fuzzy numbers, In:
EUSFLAT Conference, Citeseer, (2003), 223{227.
[20] S. Gupta and V. Kapoor, Fundamentals of mathematical statistics: A modern approach,
Sultan Chand and Sons, 1982.
[21] W. Jianqiang and Z. Zhong, Aggregation operators on intuitionistic trapezoidal fuzzy number
and its application to multi-criteria decision making problems, Journal of Systems Engineer-
ing and Electronics, 20(2) (2009), 321{326.
[22] A. Kharal, Homeopathic drug selection using intuitionistic fuzzy sets, Homeopathy, 98(1)
(2009), 35{39.
[23] D.-F. Li, A note on using intuitionistic fuzzy sets for fault-tree analysis on printed circuit
board assembly, Microelectronics Reliability, 48(10) (2008), 17{41.
[24] D.-F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application
to madm problems, Computers and Mathematics with Applications, 60(6) (2010), 1557{1570.
[25] A. Matuszak, Di erences-arithmetic-geometric-harmonic-means, The Economist at
large;http://economistatlarge.com/ nance/applied- nance/di erences-arithmetic-geometric-
harmonic-means.
[26] D. Meng and Z. Pei, On weighted unbalanced linguistic aggregation operators in group deci-
sion making, Information Sciences, 223 (2013), 31{41.
[27] F. Meng and X. Chen, Interval-valued intuitionistic fuzzy multi-criteria group decision mak-
ing based on cross entropy and 2-additive measures, Soft Computing, 19(7) (2015), 2071-
2082.
[28] F. Meng, X. Chen and Q. Zhang, Multi-attribute decision analysis under a linguistic hesitant
fuzzy environment, Information Sciences, 267 (2014), 287{305.

[29] F. Meng and X. Chen, Entropy and similarity measure of atanassovs intuitionistic fuzzy sets
and their application to pattern recognition based on fuzzy measures, Pattern Analysis and
Applications, DOI 10.1007/s10044{014{0378{6.
[30] F. Meng, X. Chen and Q. Zhang, Some uncertain generalized shapley aggregation operators
for multi-attribute group decision making, Journal of Intelligent and Fuzzy Systems, DOI:
10.3233/IFS-131069.
[31] G. A. Papakostas, A. G. Hatzimichailidis and V. G. Kaburlasos, Distance and similarity
measures between intuitionistic fuzzy sets: A comparative analysis from a pattern recognition
point of view, Pattern Recognition Letters, 34(14) (2013), 1609{1622.
[32] J. H. Park, J. M. Park and Y. C. Kwun, 2-tuple linguistic harmonic operators and their
applications in group decision making, Knowledge-Based Systems, 44 (2013), 10{19.
[33] M.-H. Shu, C.-H. Cheng and J.-R. Chang, Using intuitionistic fuzzy sets for fault-tree analysis
on printed circuit board assembly, Microelectronics Reliability, 46(12) (2006), 2139{2148.
[34] E. Szmidt and J. Kacprzyk, Intuitionistic fuzzy sets in some medical applications, in: Com-
putational Intelligence, Theory and Applications, Springer, (2001), 148{151.
[35] S.-P. Wan and J.-Y. Dong, Method of intuitionistic trapezoidal fuzzy number for multi-
attribute group decision, Control and Decision, 25(5) (2010), 773{776.
[36] S.-P. Wan, Multi-attribute decision making method based on interval-valued intuitionistic
trapezoidal fuzzy number, Control and Decision, 26(6) (2011), 857{861.
[37] S.Wan, Method based on fractional programming for interval-valued intuitionistic trapezoidal
fuzzy number multi-attribute decision making, Control and Decision, 27(3) (2012), 455{458.
[38] S.-P. Wan, Q.-Y. Wang and J.-Y. Dong, The extended VIKOR method for multi-attribute
group decision making with triangular intuitionistic fuzzy numbers, Knowledge-Based Sys-
tems, 52 (2013), 65{77.
[39] S.-P. Wan, D.-F. Li and Z.-F. Rui, Possibility mean, variance and covariance of triangular
intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24(4) (2013), 847{
858.
[40] S.-P. Wan, Multi-attribute decision making method based on possibility variance coecient of
triangular intuitionistic fuzzy numbers, International Journal of Uncertainty, Fuzziness and
Knowledge-Based Systems, 21(2) (2013), 223{243.
[41] S.-P. Wan and D.-F. Li, Fuzzy linmap approach to heterogeneous MADM considering com-
parisons of alternatives with hesitation degrees, Omega, 41(6) (2013), 925{940.
[42] S.-P. Wan, Power average operators of trapezoidal intuitionistic fuzzy numbers and appli-
cation to multi-attribute group decision making, Applied Mathematical Modelling, 37(6)
(2013), 4112{4126.
[43] S.-P. Wan and D.-F. Li, Atanassovs intuitionistic fuzzy programming method for heteroge-
neous multiattribute group decision making with atanassovs intuitionistic fuzzy truth degrees,
IEEE Tran. On Fuzzy Systems, 22(2) (2014), 300{312.
[44] S.Wan and J. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-
attribute group decision making, Journal of Computer and System Sciences, 80(1) (2014),
237{256.
[45] S.-P. Wan and J.-Y. Dong, Possibility method for triangular intuitionistic fuzzy multi-
attribute group decision making with incomplete weight information, International Journal
of Computational Intelligence Systems, 7(1) (2014), 65{79.
[46] S.-P. Wan and J.-Y. Dong, Power geometric operators of trapezoidal intuitionistic fuzzy
numbers and application to multi-attribute group decision making, Applied Soft Computing,
29 (2015), 153{168.
[47] J.-Q. Wang, R. Nie, H.-Y. Zhang and X.-H. Chen, New operators on triangular intuitionistic
fuzzy numbers and their applications in system fault analysis, Information Sciences, 251
(2013), 79{95.
[48] G. Wei, Some induced geometric aggregation operators with intuitionistic fuzzy information
and their application to group decision making, Applied Soft Computing, 10(2) (2010), 423{
431.

[49] G. Wei, Some arithmetic aggregation operators with intuitionistic trapezoidal fuzzy numbers
and their application to group decision making, Journal of Computers, 5(3) (2010), 345{351.
[50] J. Wu and Q.-W. Cao, Same families of geometric aggregation operators with intuitionistic
trapezoidal fuzzy numbers, Applied Mathematical Modelling, 37(1) (2013), 318{327.
[51] Z. Xu and Q. Da, The ordered weighted geometric averaging operators, International Journal
of Intelligent Systems, 17(7) (2002), 709{716.
[52] Z. Xu and J. Chen, An interactive method for fuzzy multiple attribute group decision making,
Information Sciences, 177(1) (2007), 248{263.
[53] Z. Xu, Intuitionistic fuzzy aggregation operators, IEEE Tran. on Fuzzy Systems, 15(6) (2007),
1179{1187.
[54] Z. Xu, Fuzzy harmonic mean operators, International Journal of Intelligent Systems, 24(2)
(2009), 152{172.
[55] Z. Xu and R. R. Yager, Power-geometric operators and their use in group decision making,
IEEE Tran. on Fuzzy Systems, 18(1) (2010), 94{105.
[56] Y. Xu and H. Wang, The induced generalized aggregation operators for intuitionistic fuzzy
sets and their application in group decision making, Applied Soft Computing, 12(3) (2012),
1168{1179.
[57] R. R. Yager, Connectives and quanti ers in fuzzy sets, Fuzzy Sets and Systems, 40(1) (1991),
39{75.
[58] R. R. Yager, On a semantics for neural networks based on fuzzy quanti ers, International
Journal of Intelligent Systems, 7(8) (1992), 765{786.
[59] R. R. Yager and D. Filev, Fuzzy logic controllers with
exible structures, In: Proceedings
of the 2nd International Conference on Fuzzy Sets and Neural Networks, Tizuka, (1992),
317{320.
[60] R. R. Yager and J. Kacprzyk, The ordered weighted averaging operators:Theory and appli-
cations, Kluwer, Norwell, MA, 1997.
[61] R. R. Yager, The power average operator, IEEE Tran. on Systems and Humans, Man and
Cybernetics, 31(6) (2001), 724{731.
[62] R. R. Yager, Generalized OWA aggregation operators, Fuzzy Optimization and Decision Mak-
ing, 3(1) (2004), 93{107.
[63] R. R. Yager, On generalized bonferroni mean operators for multi-criteria aggregation, Inter-
national Journal of Approximate Reasoning, 50(8) (2009), 1279{1286.
[64] J. Ye, Multicriteria group decision-making method using vector similarity measures for trape-
zoidal intuitionistic fuzzy numbers, Group Decision and Negotiation, 21(4) (2012), 519{530.
[65] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.