ORIGINAL_ARTICLE
Cover vol. 14, no. 1, February 2017
http://ijfs.usb.ac.ir/article_3088_dd2a325fd028a1ecba3efff967ef6955.pdf
2017-03-01T11:23:20
2017-09-24T11:23:20
0
10.22111/ijfs.2017.3088
ORIGINAL_ARTICLE
Group Generalized Interval-valued Intuitionistic Fuzzy Soft Sets and Their Applications in\\ Decision Making
Interval-valued intuitionistic fuzzy sets (IVIFSs) are widely used to handle uncertainty and imprecision in decision making. However, in more complicated environment, it is difficult to express the uncertain information by an IVIFS with considering the decision-making preference. Hence, this paper proposes a group generalized interval-valued intuitionistic fuzzy soft set (G-GIVIFSS) which contains the basic description by interval-valued intuitionistic fuzzy soft set (IVIFSS) on the alternatives and a group of experts' evaluation of it. It contributes the following threefold: 1) A generalized interval-valued intuitionistic fuzzy soft set (GIVIFSS) is proposed by introducing an interval-valued intuitionistic fuzzy parameter, which reflects a new and senior expert's opinion on the basic description. The operations, properties and aggregation operators of GIVIFSS are discussed. 2) Based on GIVIFSS, a G-GIVIFSS is then proposed to reduce the impact of decision-making preference by introducing more parameters by a group of experts. Its important operations, properties and the weighted averaging operator are also defined. 3) A multi-attribute group decision making model based on G-GIVIFSS weighted averaging operator is built to solve the group decision making problems in the more universal IVIF environment, and two practical examples are taken to validate the efficiency and effectiveness of the proposed model.
http://ijfs.usb.ac.ir/article_3034_c764374b831ad45afb14a759482b14ed.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
1
21
10.22111/ijfs.2017.3034
Group decision making
Interval-valued intuitionistic fuzzy set
Generalized interval-valued intuitionistic fuzzy soft set
Soft set
Hua
Wu
sunshinesmilewh@gmail.com
true
1
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China and University of Chinese Academy of Sciences, Beijing, China.
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China and University of Chinese Academy of Sciences, Beijing, China.
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China and University of Chinese Academy of Sciences, Beijing, China.
LEAD_AUTHOR
Xiuqin
Su
suxiuqin@opt.ac.cn
true
2
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China
AUTHOR
[1] M. Agarwal, K. Biswas and M. Hanmandlu, Generalized intuitionistic fuzzy soft sets with
1
applications in decision-making, Applied Soft Computing, 13(8) (2013), 3552{3566.
2
[2] M. Ali, F. Feng, X. Liu, W. Min and M. Shabir, On some new operations in soft set theory,
3
Computers & Mathematics with Applications, 57(9) (2009), 1547{1553.
4
[3] M. Ali and M. Shabir, Logic connective for soft sets and fuzzy soft sets, IEEE Trancsactions
5
on Fuzzy Systems, 22(6) (2014), 1431{1442.
6
[4] K. Atanassov, Intuitionistic fuzzy sets, Intuitionistic Fuzzy Sets, 20(1) (1986), 87{96.
7
[5] K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets Systems,
8
31(3) (1989), 343{349.
9
[6] L. Chen and C. Tu, Dominance-Based Ranking Functions for Interval-Valued Intuitionistic
10
Fuzzy Sets, IEEE Trancsactions on Cybernetics, 44(8) (2014), 1269{1282.
11
[7] S. Das and S. Kar, Group decision making in medical system: an intuitionistic fuzzy soft set
12
approach, Applied Soft Computing, 24 (2014), 196{211.
13
[8] B. Dinda and T. Bera and T. Samanta, Generalised intuitionistic fuzzy soft sets and its
14
application in decision making, arXiv preprint arXiv:1010.2468, 2010.
15
[9] S. Ebrahimnejad and H. Hashemi and S. Mousavi and B. Vahdani, A new interval-valued
16
intuitionistic fuzzy model to group decision making for the selection of outsourcing providers,
17
Economic Computation and Economic Cybernetics Studies and Research, 49(2) (2015), 269{
18
[10] F. Feng and C. Li and B. Davvaz and M. Ali, Soft sets combined with fuzzy sets and rough
19
sets: a tentative approach, Soft Computing, 14(9) (2010), 899{911.
20
[11] H. Hashemi, J. Bazargan, S. Mousavi and B. Vahdani, An extended compromise ratio model
21
with an application to reservoir
22
ood control operation under an interval-valued intuitionistic
23
fuzzy environment, Applied Mathematical Modelling, 38(14) (2014), 3495{3511.
24
[12] Y. Jiang, Y. Tang,Q. Chen, H. Liu and J. Tang, Interval-valued intuitionistic fuzzy soft sets
25
and their properties, Computers & Mathematics with Applications, 60(3) (2010), 906{918.
26
[13] F. Jin, L. Pei, H. Chen and L. Zhou, Interval-valued intuitionistic fuzzy continuous weighted
27
entropy and its application to multi-criteria fuzzy group decision making, Knowledge-Based
28
Systems, 59 (2014), 132{141.
29
[14] P. Liu, Some hamacher aggregation operators based on the interval-valued intuitionistic fuzzy
30
numbers and their application to group decision making, IEEE Transaction on Fuzzy Systems,
31
22(1) (2014), 83{97.
32
[15] X. Ma, H. Qin, N. Sulaiman, T. Herawan and J. Abawajy, The parameter reduction of
33
the interval-valued fuzzy soft sets and its related algorithms, IEEE trancsactions on Fuzzy
34
Systems, 22(1) (2014), 57{71.
35
[16] P. Maji, R. Biswas and A. Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics, 9(3) (2001),
36
[17] P. Maji, R. Biswas and A. Roy, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics,
37
9(3) (2001), 677{692.
38
[18] P. Maji and R. Biswas and A. Roy, Soft set theory, Computers & Mathematics with Appli-
39
cations, 45(4) (2003), 555{562.
40
[19] P. Majumdar and S. Samanta, Similarity measure of soft sets, New Mathematics and Natural
41
Computation, 4(1) (2008), 1{12.
42
[20] P. Majumdar and S. Samanta, Generalised fuzzy soft sets, Computers & Mathematics with
43
Applications, 59(4) (2010), 1425{1432.
44
[21] F. Meng, X. Chen and Q. Zhang, Some interval-valued intuitionistic uncertain linguistic
45
Choquet operators and their application to multi-attribute group decision making, Applied
46
Mathematical Modelling, 38(9) (2014), 2543{2557.
47
[22] D. Molodtsov, Soft set theory-rst results, Computers & and Mathematics with Applications,
48
37(4) (1999), 19{31.
49
[23] S. Mousavi, B. Vahdani and S. Behzadi, Designing a model of intuitionistic fuzzy VIKOR
50
in multi-attribute group decision-making problems, Iranian Journal of Fuzzy Systems, 13(1)
51
(2016), 45{65.
52
[24] S. Mousavi and B. Vahdani, Cross-docking location selection in distribution systems: a new
53
intuitionistic fuzzy hierarchical decision model, International Journal of Computational In-
54
telligence Systems, 9(1) (2016), 91{109.
55
[25] S. Mousavi and H. Gitinavard and B. Vahdani, Evaluating construction projects by a new
56
group decision-making model based on intuitionistic fuzzy logic concepts, International Jour-
57
nal of Engineering-Transactions C: Aspects, 28(9) (2015), 1312{1319.
58
[26] A. Roy and P. Maji, A fuzzy soft set theoretic approach to decision making problems, Journal
59
of Computational and Applied Mathematics, 203(2) (2007), 412{418.
60
[27] B. Vahdani, S. Mousavi, R. Tavakkoli-Moghaddam and H. Hashemi, A new design of the
61
elimination and choice translating reality method for multi-criteria group decision-making in
62
an intuitionistic fuzzy environment, Applied Mathematical Modelling, 37(4) (2013), 1781{
63
[28] W. Wang and X. Liu, The multi-attribute decision making method based on interval-valued
64
intuitionistic fuzzy Einstein hybrid weighted geometric operator, Computers & Mathematics
65
with Applications, 66(10) (2013), 1845{1856.
66
[29] H.Wu and X. Su, Threat assessment of aerial targets based on group generalized intuitionistic
67
fuzzy soft sets, Control and Decision, 30(8) (2015), 1462{1468.
68
[30] J. Wu and F. Chiclana, A risk attitudinal ranking method for interval-valued intuitionistic
69
fuzzy numbers based on novel attitudinal expected score and accuracy functions, Applied Soft
70
Computing, 22 (2014), 272{286.
71
[31] Z. Xu, Method for aggregating interval-valued intuitionistic fuzzy information and their ap-
72
plication to decision making, Control and Decision, 22(2) (2007), 215{219.
73
[32] Z. Xu, A method based on distance measure for interval-valued intuitionistic fuzzy group
74
decision making, Information Sciences, 180(1) (2010), 181{190.
75
[33] Z. Xu and J. Chen, An approach to group decision making based on intervalvalued intuition-
76
istic judgment matrices, Systems Engineering-Theory & Practice, 27(4) (2007), 126{133.
77
[34] Z. Yue, A group decision making approach based on aggregating interval data into interval-
78
valued intuitionistic fuzzy information, Applied Mathematical Modelling, 38(2) (2014), 683{
79
[35] Z. Yue and Y. Jia, A group decision making model with hybrid intuitionistic fuzzy informa-
80
tion, Computers & Industrial Engineering, 87 (2015), 202{212.
81
[36] L. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.
82
[37] X. Zhang and Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS ap-
83
proach to interval-valued intuitionistic fuzzy group decision making, Applied Soft Computing,
84
26 (2015), 42{56.
85
ORIGINAL_ARTICLE
Soft Computing Based on a Modified MCDM Approach under Intuitionistic Fuzzy Sets
The current study set to extend a new VIKOR method as a compromise ranking approach to solve multiple criteria decision-making (MCDM) problems through intuitionistic fuzzy analysis. Using compromise method in MCDM problems contributes to the selection of an alternative as close as possible to the positive ideal solution and far away from the negative ideal solution, concurrently. Using Atanassov intuitionistic fuzzy sets (A-IFSs) may simultaneously express the degree of membership and non-membership to decision makers (DMs) to describe uncertain situations in decision-making problems. The proposed intuitionistic fuzzy VIKOR indicates the degree of satisfaction and dissatisfaction of each alternative with respect to each criterion and the relative importance of each criterion, respectively, by degrees of membership and non-membership. Thus, the ratings for the importance of criteria, DMs, and alternatives are in linguistic variables and expressed in intuitionistic fuzzy numbers. Using IFS aggregation operators and with respect to subjective judgment and objective information, the most suitable alternative is indicated among potential alternatives. Moreover, practical examples illustrate the procedure of the proposed method.
http://ijfs.usb.ac.ir/article_3035_56c511a22322a3f72ffb46e35bb130df.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
23
41
10.22111/ijfs.2017.3035
Multiple criteria decision making (MCDM)
Decision makers (DMs)
Atanassov intuitionistic fuzzy sets (A-IFSs)
Intuitionistic fuzzy numbers
M. R.
Shahriari
true
1
Faculty of Management, South Tehran Branch, Islamic Azad University, Tehran, Iran
Faculty of Management, South Tehran Branch, Islamic Azad University, Tehran, Iran
Faculty of Management, South Tehran Branch, Islamic Azad University, Tehran, Iran
LEAD_AUTHOR
[1] B. A. Aghajani, M. Osanloo and B. Karimi, Deriving preference order of open pit minese-
1
quipment through MCDM methods: Application ofmodied VIKOR metho, Expert Systems
2
with Applications, 38(3) (2011), 2550-2556.
3
[2] M. Amiri, S. A. Ayazi, L. Olfat, and J. Siahkali Moradi, Group decision making process for
4
supplier selection with VIKOR under fuzzy circumstance case study: an Iranian car parts
5
supplier, International Bulletin of Business Administration, 10(6) (2011), 66-75.
6
[3] M. Amiri, M. Zandieh, B. Vahdani, R. Soltani and V. Roshanaei, An integrated eigenvector
7
DEATOPSIS methodology for portfolio risk evaluation in the FOREX spot market, Expert
8
Systems with Applications, 37(1) (2010), 509-516.
9
[4] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems,20(1) (1986), 87-96.
10
[5] S. Bandyopadhyay, P. Kumar Nayak and M. Pal, Solution of matrix game in intuitionistic
11
fuzzy environment, JCER, 3 (2013), 84-89.
12
[6] F. E. Boran, S. Genc, M. Kurt and D. Akay, A multi-criteria intuitionistic fuzzy group deci-
13
sion making for supplier selection with TOPSIS method, Expert Systems with Applications,
14
36(8) (2009), 11363-11368.
15
[7] D. Bouyssou, Using DEA as a tool for MCDM: some remarks, Journal of the operational
16
Research Society, 50(9) (1999), 974-978.
17
[8] C. L. Chang, A modied VIKOR method for multiple criteria analysis, Environmental monitoring
18
and assessment, 168(1-4) (2010), 339-344.
19
[9] C. T. Chen, A fuzzy approach to select location of the distribution center, Fuzzy sets and
20
systems,118(1) (2001), 65-73.
21
[10] S. M. Chen and C. H. Wang, A generalized model for prioritized multi-criteria decision
22
making systems, Expert Systems with Applications, 36(3) (2009), 4773-4783.
23
[11] T. C. Chu, Facility location selection using fuzzy TOPSIS under group decision, International
24
journal of uncertainty, fuzziness and knowledge-based systems, 10(6) (2002), 687-701.
25
[12] T. C. Chu, Selecting plant location via a fuzzy TOPSIS approach, The International Journal
26
of Advanced Manufacturing Technology, 20(11) (2002), 859-864.
27
[13] S. Ebrahimnejad, S. M. Mousavi, R. Tavakkoli-Moghaddam, H. Hashemi and B. Vahdani, A
28
novel two-phase group decision-making approach for construction project selection in a fuzzy
29
environment, Applied Mathematical Modelling, 36(9) (2012), 4197-4217.
30
[14] S. Ebrahimnejad, H. Hashemi, S. M. Mousavi and B. Vahdani, A new interval-valued intu-
31
itionistic fuzzy model to group decision making for the selection of outsourcing providers,
32
Journal of Economic Computation and Economic Cybernetics Studies and Research, 49(2)
33
(2015), 269-290.
34
[15] P. Grzegorzewski, Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy
35
sets based on the Hausdor metric, Fuzzy sets and systems, 148(2) (2004), 319-328.
36
[16] H. Hashemi, J. Bazargan, S. M. Mousavi and B. Vahdani An extended compromise ratio model
37
with an application to reservoir
38
ood control operation under an interval-valued intuitionistic
39
fuzzy environment, Applied Mathematical Modelling, 38(14) (2014), 3495-3511.
40
[17] A. Jahan, M. Faizal, I. Md Yusof, S. M. Sapuan, and M. Bahraminasab, A comprehensive
41
VIKOR method for material selection, Materials and Design, 32(3) (2011), 1215-1221.
42
[18] L. James JH, C. Y. Tsai, R. H. Lin and and G. H. Tzeng, A modifed VIKOR multiple-criteria
43
decision method for improving domestic airlines service quality, Journal of Air Transport
44
Management, 17(2) (2011), 57-61.
45
[19] G. S. Liang and M. J. J. Wang, A fuzzy multi-criteria decision-making method for facility
46
site selection, The International Journal of Production Research, 29(11) (1991), 2313-2330.
47
[20] S. M. Mousavi, F. Jolai, R. Tavakkoli-Moghaddam and B. Vahdani, A fuzzy grey model based
48
on the compromise ranking for multi-criteria group decision making problems in manufac-
49
turing systems, Journal of Intelligent and Fuzzy Systems, 24(4) (2013), 819-827.
50
[21] V. Mohagheghi, S. M. Mousavi and B. Vahdani, A new optimization model for project port-
51
folio selection under interval-valued fuzzy environment, Arabian Journal for Science and
52
Engineering, 40(11) (2015), 3351-3361.
53
[22] S. M. Mousavi, H. Gitinavard and B. Vahdani, Evaluating construction projects by a new
54
group decision-making model based on intuitionistic fuzzy logic concepts, International Journal
55
of Engineering-Transactions C: Aspects, 28(9) (2015), 1312-1319.
56
[23] S. M. Mousavi, B. Vahdani, M. Amiri, R. Tavakkoli-Moghaddam, S. Ebrahimnejad and
57
M. Amiri A multi-stage decision making process for multiple attributes analysis under an
58
interval-valued fuzzy environment, International Journal of Advanced Manufacturing Technology,
59
64(9-12) (2013), 1263-1273.
60
[24] S. M. Mousavi B. Vahdani and S. Sadigh Behzadi, Designing a model of intuitionistic fuzzy
61
VIKOR in multi-attribute group decision-making problems, Iranian Journal of Fuzzy Systems,
62
13(1) (2016), 45-65.
63
[25] S. M. Mousavi and B. Vahdani Cross-docking location selection in distribution systems: a
64
new intuitionistic fuzzy hierarchical decision model, International Journal of Computational
65
Intelligence Systems, 9(1) (2016), 91-109.
66
[26] S. M. Mousavi, B. Vahdani, R. Tavakkoli-Moghaddam and N. Tajik, Soft computing based
67
on a fuzzy grey compromise solution approach with an application to the selection problem of
68
material handling equipment, International Journal of Computer Integrated Manufacturing,
69
27(6) (2014), 547-569.
70
[27] S. Opcorvic, Multicriteria optimization of Civil engineering systems, Faculty of Civil Engineering,
71
2(1) (1998), 5-21.
72
[28] S. Opricovic and G. H. Tzeng, Multi-criteria planning of post-earthquake sustainable recon-
73
struction, omputer-Aided Civil and Infrastructure Engineering, 17(3) (2002), 211-220.
74
[29] S. Opricovic and G. H. Tzeng, The compromise solution by MCDM methods: A comparative
75
analysis of VIKOR and TOPSIS, European Journal of Operational Research, 156(2) (2004),
76
[30] S. Opricovic and G. H. Tzeng, Extended VIKOR method in comparison with outranking
77
methods, European Journal of Operational Research, 178(2) (2007), 514-529.
78
[31] V. Roshanaei, B. Vahdani, S. M. Mousavi, M. Mousakhani and G. Zhang, CAD/CAM system
79
selection: A multi-component hybrid fuzzy MCDM model, Arabian Journal for Science and
80
Engineering, 38(9) (2013), 2579-2594 .
81
[32] M. Salimi, B. Vahdani, S. M . Mousavi and R. Tavakkoli-Moghaddam, A new method based on
82
multi-segment decision matrix for solving decision making problems, Scientia Iranica, 20(6)
83
(2013), 2259-2274.
84
[33] J. R. San Cristobal, Multi-criteria decision-making in the selection of a renewable energy
85
project in Spain: The VIKOR method, Renewable Energy, 36(2) (2011), 498-502.
86
[34] B. Vahdani, H. Hadipour, J. S. Sadaghiani and M. Amiri, Extension of VIKOR method
87
based on interval-valued fuzzy sets, The International Journal of Advanced Manufacturing
88
Technology, 47(9-12) (2010), 1231-1239.
89
[35] B. Vahdani and H. Hadipour, Extension of the ELECTRE method based on interval-valued
90
fuzzy sets, Soft Computing, 15(3) (2011), 569-579.
91
[36] B. Vahdani, M. Salimi, and M. Charkhchian, A new FMEA method by integrating fuzzy
92
belief structure and TOPSIS to improve risk evaluation process, The International Journal
93
of Advanced Manufacturing Technology, 77(1-4) (2015), 357-368.
94
[37] B. Vahdani, M. Zandieh and A. Alem-Tabriz, Supplier selection by balancing and ranking
95
method, Journal of Applied Sciences, 8(19) (2008), 3467-3472.
96
[38] B. Vahdani, M. Zandieh, and R. Tavakkoli-Moghaddam, Two novel FMCDM methods for
97
alternative-fuel buses selection, Applied Mathematical Modelling, 35(3) (2011), 1396-1412.
98
[39] B. Vahdani, S. M. Mousavi and S. Ebrahimnejad, Soft computing-based preference selection
99
index method for human resource management, Journal of Intelligent and Fuzzy Systems,
100
26(1) (2014), 393-403.
101
[40] B. Vahdani, H. Hadipour and R. Tavakkoli-Moghaddam, Soft computing based on interval
102
valued fuzzy ANP-A novel methodology, Journal of Intelligent Manufacturing, 23(5) (2012),
103
1529-1544.
104
[41] B. Vahdani, M. Salimi and S. M. Mousavi, A new compromise decision-making model based
105
on TOPSIS and VIKOR for solving multi-objective large-scale programming problems with a
106
block angular structure under uncertainty, International Journal of Engineering Transactions
107
B: Applications,27(11) (2014), 1673-1680.
108
[42] B. Vahdani, M. Salimi and S. M. Mousavi, A compromise decision-making model based on
109
VIKOR for multi-objective large-scale nonlinear programming problems with a block angular
110
structure under uncertainty, Scientia Iranica. Transaction E, Industrial Engineering,22(6)
111
(2015), 2571-2584.
112
[43] B. Vahdani, S. M. Mousavi and R. Tavakkoli-Moghaddam, Group decision making based
113
on novel fuzzy modied TOPSIS method, Applied Mathematical Modelling, 35(9) (2011),
114
4257-4269.
115
[44] B. Vahdani, S. M. Mousavi, R. Tavakkoli-Moghaddam and H. Hashemi, A new design of the
116
elimination and choice translating reality method for multi-criteria group decision making
117
in an intuitionistic fuzzy environment, Applied Mathematical Modeling ,37(4) (2013), 1781-
118
[45] B, Vahdani, S. M. Mousavi, H. Hashemi, M. Mousakhani and R. Tavakkoli-Moghaddam, A
119
new compromise solution method for fuzzy group decision-making problems with an appli-
120
cation to the contractor selection, Engineering Applications of Articial Intelligence, 26(2)
121
(2012), 779-788.
122
[46] B. Vahdani and M. Zandieh Selecting suppliers using a new fuzzy multiple criteria deci-
123
sion model: the fuzzy balancing and ranking method, International Journal of Production
124
Research, 48(18) (2010) 5307-5326.
125
[47] B. Vahdani, S. M. Mousavi, R. Tavakkoli-Moghaddam, A. Ghodratnama and M. Mohammadi,
126
Robot selection by a multiple criteria complex proportional assessment method un-
127
der an interval-valued fuzzy environment, International Journal of Advanced Manufacturing
128
Technology, 73(5-8) (2014), 687-697.
129
[48] B. Vahdani, R. Tavakkoli-Moghaddam, S. M. Mousavi and A. Ghodratnama, Soft computing
130
based on new interval-valued fuzzy modied multi-criteria decision-making method, Applied
131
Soft Computing, 13(1) (2013), 165-172.
132
[49] Y. Y. Wu and D. J. Yu, Extended VIKOR for Multi-criteria decision making problems under
133
intuitionistic environment, 18th International Conference on Management Science & Engineering,
134
[50] Z. Xu, An overview of methods for determining OWA weights, International Journal of Intelligent
135
Systems,20(8) (2005), 843-865.
136
[51] Z. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy
137
sets, International Journal of General Systems, 35(4) (2006), 417-433.
138
[52] Z. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems, 15(6)
139
(2007), 1179-1187.
140
[53] D. Yong, Plant location selection based on fuzzy TOPSIS, The International Journal of Advanced
141
Manufacturing Technology ,28(7-8) (2006), 839-844.
142
[54] L. A. Zadeh, Fuzzy sets, Information and control , 8(3) (1965), 338-353.
143
ORIGINAL_ARTICLE
Support vector regression with random output variable and probabilistic constraints
Support Vector Regression (SVR) solves regression problems based on the concept of Support Vector Machine (SVM). In this paper, a new model of SVR with probabilistic constraints is proposed that any of output data and bias are considered the random variables with uniform probability functions. Using the new proposed method, the optimal hyperplane regression can be obtained by solving a quadratic optimization problem. The proposedmethod is illustrated by several simulated data and real data sets for both models (linear and nonlinear) with probabilistic constraints.
http://ijfs.usb.ac.ir/article_3036_7fa269af1f035bda8d625aada763cf7a.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
43
60
10.22111/ijfs.2017.3036
Probabilistic constraints
Support Vector Machine
Support Vector Regression
Quadratic programming
Probability function
Monte Carlo simulation
Maryam
Abaszade
true
1
Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran
AUTHOR
Sohrab
Effati
s-effati.profcms@um.ac.ir
true
2
Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
LEAD_AUTHOR
[1] A. R. Arabpour and M. Tata, Estimating the parameters of a fuzzy linear regression model,
1
Iranian Journal of Fuzzy Systems, 5(2) (2008), 1–19.
2
[2] K. Bache and M. Lichman, UCI machine learning repository, Available on-line at:
3
http://archive.ics.uci.edu/ml/machine-learning-databases, 2013.
4
[3] A. Ben-Tal, S. Bhadra, C. Bhattacharyya and J. S. Nath, Chance constrained uncertain
5
classification via robust optimization, Math. Program., 127(1) (2011), 145–173.
6
[4] P. Bosch, J. Lopez, H. Ramirez and H. Robotham, Support vector machine under uncertainty:
7
an application for hydroacoustic classification of fish-schools in Chile, Expert Systems with
8
Applications, 40 (2013), 4029–4034.
9
[5] K. D. Brabanter, J. D. Brabanter, J. A. K. Suykens and B. D. Moor, Approximate confidence
10
and prediction intervals for least squares support vector regression, IEEE Transactions on
11
Neural Networks, 22 (2011), 110–120.
12
[6] E. Carrizosa, J. E. Gordillo and F. Plastria, Kernel support vector regression with imprecise
13
output, Dept. MOSI. Vrije Univ. Brussel. Belgium. Tech. Rep., Available on-line at:
14
http://www.optimization-online.org/DB_FILE/2008/01/1896.pdf, 2008.
15
[7] E. Carrizosa, J. E. Gordillo and F. Plastria, Support vector regression for imprecise
16
data, Dept. MOSI. Vrije Univ. Brussel. Belgium. Tech. Rep., Available on-line at:
17
http://www.optimization-online.org/DB_HTML/2007/11/1826.html, 2007.
18
[8] J. H. Chiang and P. Y. Hao, Support vector learning mechanism for fuzzy rule-based modeling:
19
a new approach, IEEE Trans. Fuzzy Syst., 12(1) (2004), 1–12.
20
[9] H. Drucker, Ch. J. C. Burges, L. Kaufman, A. Smola and V. Vapnik, Support vector regression
21
machines, Adv. Neural Inform. Process. Syst., 9 (1997), 155–161.
22
[10] B. Efron, Bootstrap methods: Another look at the jackknife, Annals of Statistics, 7 (1979)
23
1–26.
24
[11] A. Farag and R. M. Mohamed, Classification of multispectral data using support vector machines
25
approach for density estimation, IEEE Seventh International Conference on Intell.
26
Eng. Syst., (2003), 4–6.
27
[12] J. B. Gao, S. R. Gunn, C. J. Harris and M. Brown, A probabilistic framework for SVM
28
regression and error bar estimation, Machine Learning, 46 (2002), 71–89.
29
[13] P. Y. Hao and J. H. Chiang, A fuzzy model of support vector regression machine, International
30
Journal of Fuzzy Systems, 9(1) (2007), 45–49.
31
[14] H. P. Huang and Y. H. Liu, Fuzzy support vector machines for pattern recognition and data
32
mining, International Journal of Fuzzy Systems, 4 (2002), 826–835.
33
[15] G. Huang, S. Song, C. Wu and K. You, Robust support vector regression for uncertain input
34
and output data, IEEE Transactions on Neural Networks and Learning Systems, 23(11)
35
(2012), 1690–1700.
36
[16] R. K. Jayadeva, R. Khemchandani and S. Chandra, Twin support vector machines for pattern
37
classification, IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(5)
38
(2007), 905–910.
39
[17] Y. Jinglin, H. X. Li and H. Yong, A probabilistic SVM based decision system for pain diagnosis,
40
Expert Systems with Applications, 38 (2011), 9346–9351.
41
[18] A. F. Karr, probability, Springer, New york, (1993), 52–74.
42
[19] M. A. Kumar and M. Gopal, Least squares twin support vector machines for pattern classification,
43
Expert Systems with Applications, 36(4) (2009), 7535–7543.
44
[20] J. T. Y. Kwok, The evidence framework applied to support vector machines, IEEE Transactions
45
on Neural Networks, 11 (2000), 1162–1173.
46
[21] G. R. G. Lanckriet, L. E. Ghaoui, Ch. Bhattacharyya and M. I. Jordan, A robust minimax
47
approach to classification, J. Mach. Learn. Res., 3 (2002), 555–582.
48
[22] Y. J. Lee and S. Y. Huang, Reduced support vector machines: a statistical theory, IEEE
49
Transactions on Neural Networks, 18 (2007), 1–13.
50
[23] H. Li, J. Yang, G. Zhang and B. Fan, Probabilistic support vector machines for classification
51
of noise affected data, Information Sciences, 221 (2013), 60–71.
52
[24] C. F. Lin and S. D. Wang, Fuzzy support vector machine, IEEE Transactions on Neural
53
Networks, 13 (2002), 464–471.
54
[25] W. Y. Liu, K. Yue and M. H. Gao, Constructing probabilistic graphical model from predicate
55
formulas for fusing logical and probabilistic knowledge, Information Sciences, 181(18) (2011),
56
3828–3845.
57
[26] M. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming,
58
Linear Algebra Its Appl., 284 (1998), 193–228.
59
[27] O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, (1969), 69–75.
60
[28] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM J. Optim.,
61
2 (1992), 575–601.
62
[29] X. Peng, TSVR: an efficient twin support vector machine for regression, Neural Networks.,
63
23(3) (2010), 365–372.
64
[30] J. C. Platt, Probabilistic outputs for support vector machines and comparisons to regularized
65
likelihood methods, Advances in Large Margin Classifiers, 10(3) (1999), 61–74.
66
[31] Z. Qi, Y. Tian and Y. Shi, Robust twin support vector machine for pattern classification,
67
Pattern Recognition, 46 (2013), 305–316.
68
[32] H. Sadoghi Yazdi, S. Effati and Z. Saberi, The probabilistic constraints in the support vector
69
machine, App. Math. Comput., 194 (2007), 467–479.
70
[33] P. K. Shivaswamy, Ch.Bhattacharyya and A.J.Smola, Second order cone programming approaches
71
for handling missing and uncertain data,J. Mach. Learn. Res.,7 (2006), 1283-1314.
72
[34] P. Sollich, Bayesian methods for support vector machines: evidence and predictive class
73
probabilities, Machine Learning, 46 (2002), 21–52.
74
[35] J. A. K. Suykens and J. Vandewalle, Least squares support vector machine classifiers, Neural
75
Processing Letters, 9(3) (1999), 293–300.
76
[36] T. B. Trafalis and S. A. Alwazzi, Support vector regression with noisy data: a second order
77
cone programming approach, Int. J. General Syst., 36 (2007), 237–250.
78
[37] T. B. Trafalis and R. C. Gilbert, Robust classification and regression using support vector
79
machines, Eur. J. Oper. Res., 173(3) (2006), 893–909.
80
[38] URL http://www.csie.ntu.edu.tw/ cjlin/libsvmtools/datasets/regression.html.
81
[39] URL http://www.dcc.fc.up.pt/ ltorgo/Regression/DataSets.html.
82
[40] URL http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml.
83
[41] V. Vapnik, The nature of statistical learning theory, Springer-Verlag, New York, (1995),
84
123-146, 181-186.
85
[42] V. Vapnik, S. Golowich and A. Smola, Support vector method for multivariate density estimation,
86
Adv. Neural Inform. Process. Syst., 12 (1999), 659–665.
87
[43] Y. Xu and L. Wang, A weighted twin support vector regression, Knowledge-Based Syst., 33
88
(2012), 92–101.
89
[44] Y. Xu, W. Xi, X. Lv and R. Guo, An improved least squares twin support vector machine,
90
Journal of information and computational science, 9(4) (2012), 1063–1071.
91
[45] X. Yang, L. Tan and L. He, A robust least squares support vector machine for regression and
92
classification with noise, Neurocomputing, 140 (2014), 41–52.
93
ORIGINAL_ARTICLE
A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers
In this paper, we determine necessary and sufficient Tauberian conditions under which convergence in Pringsheim's sense of a double sequence of fuzzy numbers follows from its $(C,1,1)$ summability. These conditions are satisfied if the double sequence of fuzzy numbers is slowly oscillating in different senses. We also construct some interesting double sequences of fuzzy numbers.
http://ijfs.usb.ac.ir/article_3037_e93bbd0f6b9452e82d1a57a8403f41d7.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
61
75
10.22111/ijfs.2017.3037
Fuzzy numbers
Double sequences
Slow oscillation
Summability $(C
1
1)$
Tauberian theorems
Ibrahim
Canak
ibrahimcanak@yahoo.com
true
1
Department of Mathematics, Ege University, 35100, Izmir, Turkey
Department of Mathematics, Ege University, 35100, Izmir, Turkey
Department of Mathematics, Ege University, 35100, Izmir, Turkey
AUTHOR
Umit
Totur
utotur@adu.edu.tr
true
2
Department of Mathematics, Adnan Menderes University, 09100, Aydin,
Turkey
Department of Mathematics, Adnan Menderes University, 09100, Aydin,
Turkey
Department of Mathematics, Adnan Menderes University, 09100, Aydin,
Turkey
LEAD_AUTHOR
Zerrin
Onder
zerrin.onder11@gmail.com
true
3
Department of Mathematics, Ege University, 35100, Izmir, Turkey
Department of Mathematics, Ege University, 35100, Izmir, Turkey
Department of Mathematics, Ege University, 35100, Izmir, Turkey
AUTHOR
[1] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, Berlin, 2013.
1
[2] _I. C.
2
anak, Tauberian theorems for Cesaro summability of sequences of fuzzy number, J. Intell.
3
Fuzzy Syst., 27(2) (2014), 937{942.
4
[3] _I. C.
5
anak, On Tauberian theorems for Cesaro summability of sequences of fuzzy numbers, J.
6
Intell. Fuzzy Syst., 30(5) (2016), 2657{2662.
7
[4] _I. C.
8
anak, Holder summability method of fuzzy numbers and a Tauberian theorem, Iranian
9
Journal of Fuzzy Systems, 11(4) (2014), 87{93.
10
anak, Some conditions under which slow oscillation of a sequence of fuzzy numbers follows
11
from Cesaro summability of its generator sequence, Iranian Journal of Fuzzy Systems, 11(4)
12
(2014) 15{22.
13
[6] D. Dubois and H. Prade, Fuzzy sets and systems: Theory and applications, Academic Press,
14
New York-London, 1980.
15
[7] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18(1) (1986)
16
[8] M. Matlako, Sequences of fuzzy numbers, BUSEFAL, 28 (1986), 28{37.
17
[9] F. Moricz, Tauberian theorems for Cesaro summable double sequences, Studia Math., 110
18
(1994), 83{96.
19
[10] F. Moricz, Necessary and sufficient Tauberian conditions, under which convergence follows
20
from summability (C; 1), Bull. London Math. Soc., 26 (1994), 288{294.
21
[11] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst., 33 (1989), 123{126.
22
[12] E. Savas.
23
, A note on double sequences of fuzzy numbers, Turkish J. Math., 20 (1996), 175{178.
24
[13] P. V. Subrahmanyam, Cesaro summability of fuzzy real numbers, J. Anal., 7 (1999), 159{168.
25
[14] B. C. Tripathy and A. J. Dutta, On fuzzy real-valued double sequence spaces, Soochow J.
26
Math., 32(4) (2006), 509{520.
27
[15]O. Talo and F. Bas.ar, On the slowly decreasing sequences of fuzzy numbers, Abstr. Appl.
28
Anal., Article ID 891986 (2013), doi:10.1155/2013/891986, 1-7.
29
[16] O. Talo and C. C.
30
akan, On the Cesaro convergence of sequences of fuzzy numbers, Appl.
31
Math. Lett., 25 (2012), 676{681.
32
[17] O. Talo and C. C.
33
akan, Tauberian theorems for statistically (C; 1)-convergent sequences of
34
fuzzy numbers, Filomat, 28(4) (2014), 849{858.
35
[18] B. C. Tripathy and A. Baruah, Norlund and Riesz mean of sequences of fuzzy real numbers,
36
Appl. Math. Lett., 23 (2010), 651{655.
37
[19] B. C. Tripathy and A. J. Dutta, Bounded variation double sequence space of fuzzy real
38
numbers, Comput. Math. Appl., 59(2) (2010), 1031{1037.
39
[20] B. C. Tripathy and B. Sarma, Double sequence spaces of fuzzy numbers dened by Orlicz
40
function, Acta Math. Sci. Ser. B Engl. Ed., 31(1) (2011), 134{140.
41
[21] B. C. Tripathy and M. Sen, On lacunary strongly almost convergent double sequences of fuzzy
42
numbers, An. Univ. Craiova Ser. Mat. Inform., 42(2) (2015), 254{259.
43
[22] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 29{44.
44
ORIGINAL_ARTICLE
Some topological properties of spectrum of fuzzy submodules
Let $R$ be a commutative ring with identity and $M$ be an$R$-module. Let $FSpec(M)$ denotes the collection of all prime fuzzysubmodules of $M$. In this regards some basic properties of Zariskitopology on $FSpec(M)$ are investigated. In particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$FSpec(M)$ is a $T_0-$space or Hausdorff.
http://ijfs.usb.ac.ir/article_3038_cf0537ee7303573d6949705f2d57737a.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
77
87
10.22111/ijfs.2017.3038
Fuzzy prime submodule
Fuzzy prime spectrum
Zariski topology
Irreducible subset
R.
Ameri
rez_ameri@yahoo.com
true
1
School of Mathematics, Statistics and Computer Science, College of
Sciences, University of Tehran, Teheran, Iran
School of Mathematics, Statistics and Computer Science, College of
Sciences, University of Tehran, Teheran, Iran
School of Mathematics, Statistics and Computer Science, College of
Sciences, University of Tehran, Teheran, Iran
LEAD_AUTHOR
R.
Mahjoob
ra−mahjoob@yahoo.com
true
2
Department of Mathematics, Semnan University, Semnan, Iran
Department of Mathematics, Semnan University, Semnan, Iran
Department of Mathematics, Semnan University, Semnan, Iran
AUTHOR
[1] R. Ameri, Some properties of zariski topology of multiplication modules, Houston Journal of
1
Mathematics, 36(2) (2009), 337-344.
2
[2] R. Ameri and R. Mahjoob, Prime spectrum of L-Submodules, Fuzzy Sets and Systems, 159(9)
3
(2008), 1107-1115.
4
[3] R. Ameri and R. Mahjoob, Zariski topology on the spectrum of prime L-submodules, Soft
5
Comput., 12(9) (2008), 901-908.
6
[4] S. K. Bhambri, R. Kumar and P. Kumar,Fuzzy prime submodules and radical of a fuzzy
7
submodules, Bull. Cal. Math. Soc., 87 (1993), 163-168.
8
[5] V. N. Dixit, R. Kummar and N. Ajmal,Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy
9
Sets and Systems, 44 (1991), 127-138.
10
[6] J. A. Goguen, L-fuzzy sets, Journal Math. Appl., 18 (1967) 145-174.
11
[7] H. Hadji-Abadi and M. M. Zahedi, Some results on fuzzy prime spectrum of a ring, Fuzzy
12
Sets and Systems, 77 (1996), 235-240.
13
[8] R. Kumar, Fuzzy prime spectrum of a ring, Fuzzy Sets and Systems, 46 (1992), 147-154.
14
[9] R. Kumar and J. K. Kohli,Fuzzy prime spectrum of a ring II, Fuzzy Sets and Systems, 59
15
(1993), 223-230.
16
[10] H. V. Kumbhojkar,Some comments on spectrum of prime fuzzy ideals of a ring, Fuzzy Sets
17
and Systems, 85 (1997), 109-114.
18
[11] H. V. Kumbhojkar,Spectrum of prime fuzzy ideals, Fuzzy Sets and Systems, 62 (1994), 101-
19
[12] Chin. Pi. Lu,Prime submodules of modules, Comm. Math. Univ., 33 (1987), 61-69.
20
[13] Chin.Pi. Lu, The zariski topology on the spectrum of a modules, Houston Journal of Mathe-
21
matics, 25(3) (1999), 417-432.
22
[14] Chin.Pi. Lu,Spectra of modules, Comm. in Algebra, 23(10) (1995) 3741-3752.
23
[15] R. L. McCasland, M. E. Moore and P. F. Smith,On the Spectrum of Modules Over a Com-
24
mutative Ring, Communications in Algebra, 25(1) (1997), 79-103.
25
[16] John. N. Mordeson and D. S. Malik,Fuzzy Commutative Algebra, World Scientic Publishing
26
Co. Pet. Ltd, 1998.
27
[17] T. K. Mukherjee and M. K. Sen,On fuzzy ideals of a ring I; Fuzzy Sets and systems, 21
28
(1987), 99-104.
29
[18] C. V. Negoita and D. A. Ralescu, Application of fuzzy systems analysis, Basel and Stuttgart,
30
Birkhauser Verlag; New York, Wiley- Halstead, (1975), pp. 191.
31
[19] F. Z. Pan, Fuzzy nitely generated modules, Fuzzy Sets and Systems, 21 (1987), 105-113.
32
[20] R. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
33
[21] F. I. Sidky, On radical of fuzzy submodules and primary fuzzy submodules, Fuzzy Sets and
34
Systems, 119 (2001), 419-425.
35
[22] L. A. Zadeh, Fuzzy sets, Inform and Control, 8 (1965), 338-353.
36
ORIGINAL_ARTICLE
ON LOCAL HUDETZ g-ENTROPY
In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.
http://ijfs.usb.ac.ir/article_3041_ca853dd5bb21099ad707b1cc56ee9624.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
89
97
10.22111/ijfs.2017.3041
$g$-entropy
$g$-ergodic decomposision
Hudetz correction
M.
Rahimi
m10.rahimi@gmail.com
true
1
Department of Mathematics, Faculty of Science, University of Qom, Qom,
Iran
Department of Mathematics, Faculty of Science, University of Qom, Qom,
Iran
Department of Mathematics, Faculty of Science, University of Qom, Qom,
Iran
LEAD_AUTHOR
[1] D. Dumitrescu, Measure-preserving transformation and the entropy of a fuzzy partition, 13th
1
Linz Seminar on Fuzzy Set Theory, (Linz, 1991), 25-27.
2
[2] D. Dumitrescu, Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl.,
3
176 (1993), 359-373.
4
[3] D. Dumitrescu, Entropy of a fuzzy process, Fuzzy Sets and Systems, 55 (1993), 169-177.
5
[4] D. Dumitrescu, Entropy of fuzzy dynamical systems, Fuzzy Sets and Systems, 70 (1995),
6
[5] T. Hudetz, Space-time dynamical entropy of quantum systems, Lett. Math. Phys., 16 (1988),
7
[6] T. Hudetz, Algebraic topological entropy, In: Eds., G. A. Leonov et al. Eds., Nonlinear
8
Dynamics and Quantum Dynamical Systems, (Akademie Verlag, Berlin, 1990), 110-124.
9
[7] D. Markechova, The entropy on F-quantum spaces, Math. Slovaca, 40 (1990), 177-190.
10
[8] D. Markechova, The entropy of fuzzy dynamical systems and generators, Fuzzy Sets and
11
Systems, 48 (1992), 351-363.
12
[9] D. Markechova, Entropy of complete fuzzy partitions, Math. Slovaca, 43(1) (1993), 1-10.
13
[10] D. Markechova, A note to the Kolmogorov-Sinaj entropy of fuzzy dynamical systems, Fuzzy
14
Sets and Systems, 64 (1994), 87-90.
15
[11] R. Mesiar and J. Rybarik, Entropy of fuzzy partitions: A general model, Fuzzy Sets and
16
Systems, 99 (1998), 73-79.
17
[12] M. Rahimi and A. Riazi, On local entropy of fuzzy partitions, Fuzzy Sets and Systems, 234
18
(2014), 97-108
19
[13] M. Rahimi and A. Riazi, Fuzzy entropy of action of semi-groups, Math. Slovaca, 66(5)
20
(2016), 1157-1168.
21
[14] M. Rahimi, A local approach to g-entropy, Kybernetica, 51(2) (2015), 231-245.
22
[15] B. Riecan, On a type of entropy of dynamical systems, Tatra Mountains Math. Publ., 1
23
(1992), 135-140.
24
[16] B. Riecan and D. Markechova, The entropy of fuzzy dynamical systems, general scheme and
25
generators, Fuzzy Sets and Systems, 96 (1998), 191-199.
26
[17] B. Riecan, On the g-entropy and its Hudetz correction, Kybernetika, 38(4) (2002), 493-500.
27
[18] J. Rybarik, The entropy of the Q-F-dynamical systems, Busefal, 48 (1991), 24-26.
28
[19] J. Rybarik, The entropy based on pseudoarithmetical operations, Tatra Mountains Math.
29
Publ., 6 (1995), 157-164.
30
ORIGINAL_ARTICLE
Probabilistic Normed Groups
In this paper, we introduce the probabilistic normed groups. Among other results, we investigate the continuityof inner automorphisms of a group and the continuity of left and right shifts in probabilistic group-norm. We also study midconvex functions defined on probabilistic normed groups and give some results about locally boundedness of such functions.
http://ijfs.usb.ac.ir/article_3045_07ffef6232373ba99af3aa077fbf129e.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
99
113
10.22111/ijfs.2017.3045
Probabilistic normed groups
Invariant probabilistic metrics
Distributional-slowly varying functions
Midconvex functions
Kourosh
Nourouzi
nourouzi@kntu.ac.ir
true
1
Faculty of Mathematics, K.N.Toosi University of Technology,
P.O.Box 16315-1618, Tehran, Iran.
Faculty of Mathematics, K.N.Toosi University of Technology,
P.O.Box 16315-1618, Tehran, Iran.
Faculty of Mathematics, K.N.Toosi University of Technology,
P.O.Box 16315-1618, Tehran, Iran.
LEAD_AUTHOR
Alireza
Pourmoslemi
a_pourmoslemy@pnu.ac.ir
true
2
Department of Mathematics, Payame Noor University, P.O.BOX
19395-3697, Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O.BOX
19395-3697, Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O.BOX
19395-3697, Tehran, Iran.
AUTHOR
[1] C. Alsina, B. Schweizer and A. Sklar, On the denition of a probabilistic normed space,
1
Aequationes Math, 46(2) (1993), 91{98.
2
[2] N. H. Bingham and A. J. Ostaszewski, Normed versus topological groups: dichotomy and
3
duality, Dissertationes Math, 472 (2010), 138p.
4
[3] G. Birkho, A note on topological groups, Compositio Math, 3 (1936), 427{430.
5
[4] D. R. Farkas, The algebra of norms and expanding maps on groups, J. Algebra, 133(2)
6
(1990), 386{403.
7
[5] M. Frechet, Sur quelques points du calcul fonctionnel, Rendiconti de Circolo Matematico di
8
Palermo, 22 (1906), 1{74.
9
[6] S. Kakutani, Uber die Metrisation der topologischen Gruppen, (German) Proc. Imp. Acad,
10
12(4) (1936), 82{84. (also in Selected Papers, Vol. 1, ed. R. Robert Kallman, Birkhuser,
11
(1986), 60{62.)
12
[7] V. L. Klee, Invariant metrics in groups (solution of a problem of Banach), Proc. Amer.
13
Math. Soc, 3 (1952), 484{487.
14
[8] E. Klement, R. Mesiar and E. Pap, Triangular norms, Trends in Logica{Studia Logica Library,
15
Kluwer Academic Publishers, Dordrecht, 8 (2000).
16
[9] M. Kuczma, An introduction to the theory of functional equations and inequalities, Cauchy's
17
equation and Jensen's inequality, Second edition, Birkhauser Verlag, Basel, 2009.
18
[10] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U. S. A, 28 (1942), 535{537.
19
[11] A. A. Pavlov, Normed groups and their application to noncommutative dierential geometry,
20
J. Math. Sci., 113(5) (2003), 675{682.
21
[12] B. J. Pettis, On continuity and openness of homomorphisms in topological groups, Ann. of
22
Math, 52(2) (1950), 293{308.
23
[13] B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland Series in Probability
24
and Applied Mathematics, North-Holland Publishing Co., New York, 1983.
25
[14] A. N. Serstnev, On the concept of a stochastic normalized space, (Russian), Dokl. Akad.
26
Nauk SSSR, 149 (1963), 280{283.
27
[15] D. A. Sibley, A metric for weak convergence of distribution functions, Rocky Mountain J.
28
Math, 1(3) (1971), 427{430.
29
ORIGINAL_ARTICLE
Implications, coimplications and left semi-uninorms on a complete lattice
In this paper, we firstly show that the $N$-dual operation of the right residual implication, which is induced by a left-conjunctive right arbitrary $\vee$-distributive left semi-uninorm, is the right residual coimplication induced by its $N$-dual operation. As a dual result, the $N$-dual operation of the right residual coimplication, which is induced by a left-disjunctive right arbitrary $\wedge$-distributive left semi-uninorm, is the right residual implication induced by its $N$-dual operation. Then, we demonstrate that the $N$-dual operations of the left semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left semi-uninorms. Finally, we reveal the relationships between conjunctive right arbitrary $\vee$-distributive left semi-uninorms induced by implications and disjunctive right arbitrary $\wedge$-distributive left semi-uninorms induced by coimplications, where both implications and coimplications satisfy the neutrality principle.
http://ijfs.usb.ac.ir/article_3046_01bd93593faf02d51d0a59def6a543fe.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
115
130
10.22111/ijfs.2017.3046
Fuzzy connective
Implication
Coimplication
Left semi-uninorm
Neutrality principle
Yuan
Wang
yctuwangyuan@163.com
true
1
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
AUTHOR
Keming
Tang
tkmchina@126.com
true
2
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
AUTHOR
Zhudeng
Wang
zhudengwang2004@163.com
true
3
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, People's Republic of China
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, People's Republic of China
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, People's Republic of China
LEAD_AUTHOR
[1] M. Baczynski and B. Jayaram, Fuzzy implications, Studies in Fuzziness and Soft Computing,
1
Springer, Berlin, 231 (2008).
2
[2] S. Burris and H. P. Sankappanavar, A course in universal algebra, World Publishing Corporation,
3
Beijing 1981.
4
[3] B. De Baets, Coimplicators, the forgotten connectives, Tatra Mountains Mathematical Publications,
5
12 (1997), 229{240.
6
[4] B. De Baets and J. Fodor, Residual operators of uninorms, Soft Computing, 3 (1999), 89{100.
7
[5] F. Durante, E. P. Klement, R. Mesiar and C. Sempi, Conjunctors and their residual impli-
8
cators: characterizations and construction methods, Mediterranean Journal of Mathematics,
9
4 (2007), 343{356.
10
[6] P. Flondor, G. Georgescu and A. Lorgulescu, Pseudo-t-norms and pseudo-BL-algebras, Soft
11
Computing, 5 (2001), 355{371.
12
[7] J. Fodor and T. Keresztfalvi, Nonstandard conjunctions and implications in fuzzy logic,
13
International Journal of Approximate Reasoning, 12 (1995), 69{84.
14
[8] J. C. Fodor and M. Roubens, Fuzzy preference modelling and multicriteria decision support,
15
Theory and Decision Library, Series D: System Theory, Knowledge Engineering and Problem
16
Solving, Kluwer Academic Publishers, Dordrecht, 1994.
17
[9] J. Fodor, R. R. Yager and A. Rybalov, Structure of uninorms, International Journal of
18
Uncertainty, Fuzziness and Knowledge-Based Systems, 5 (1997), 411{427.
19
[10] D. Gabbay and G. Metcalfe, Fuzzy logics based on [0; 1)-continuous uninorms, Archive for
20
Mathematical Logic, 46 (2007), 425{449.
21
[11] G. Gratzer, Lattice theory: foundation, Birkhauser, Springer Basel AG, 2011.
22
[12] S. Jenei and F. Montagna, A general method for constructing left-continuous t-norms, Fuzzy
23
Sets and Systems, 136 (2003), 263{282.
24
[13] H. W. Liu, Semi-uninorm and implications on a complete lattice, Fuzzy Sets and Systems,
25
191 (2012), 72{82.
26
[14] Z. Ma and W. M.Wu, Logical operators on complete lattices, Information Sicences, 55 (1991),
27
[15] M. Mas, M. Monserrat and J. Torrens, On left and right uninorms, International Journal of
28
Uncertainty, Fuzziness and Knowledge-Based Systems, 9 (2001), 491{507.
29
[16] M. Mas, M. Monserrat and J. Torrens, On left and right uninorms on a nite chain, Fuzzy
30
Sets and Systems, 146 (2004), 3{17.
31
[17] M. Mas, M. Monserrat and J. Torrens, Two types of implications derived from uninorms,
32
Fuzzy Sets and Systems, 158 (2007), 2612{2626.
33
[18] D. Ruiz and J. Torrens, Residual implications and co-implications from idempotent uninorms,
34
Kybernetika, 40 (2004), 21{38.
35
[19] Y. Su and H. W. Liu, Characterizations of residual coimplications of pseudo-uninorms on a
36
complete lattice, Fuzzy Sets and Systems, 261 (2015), 44{59.
37
[20] Y. Su and Z. D. Wang, Pseudo-uninorms and coimplications on a complete lattice, Fuzzy
38
Sets and Systems, 224 (2013), 53{62.
39
[21] Y. Su and Z. D. Wang, Constructing implications and coimplications on a complete lattice,
40
Fuzzy Sets and Systems, 247 (2014), 68{80.
41
[22] Y. Su, Z. D. Wang and K. M. Tang, Left and right semi-uninorms on a complete lattice,
42
Kybernetika, 49 (2013), 948{961.
43
[23] Z. D. Wang and J. X. Fang, Residual operators of left and right uninorms on a complete
44
lattice, Fuzzy Sets and Systems, 160 (2009), 22{31.
45
[24] Z. D. Wang and J. X. Fang, Residual coimplicators of left and right uninorms on a complete
46
lattice, Fuzzy Sets and Systems, 160 (2009), 2086{2096.
47
[25] R. R. Yager, Uninorms in fuzzy system modeling, Fuzzy Sets and Systems, 122 (2001),
48
[26] R. R. Yager, Defending against strategic manipulation in uninorm-based multi-agent decision
49
making, European Journal of Operational Research, 141 (2002), 217{232.
50
[27] R. R. Yager and A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80
51
(1996), 111{120.
52
ORIGINAL_ARTICLE
Structural properties of fuzzy graphs
Matroids are important combinatorial structures and connect close-lywith graphs. Matroids and graphs were all generalized to fuzzysetting respectively. This paper tries to study connections betweenfuzzy matroids and fuzzy graphs. For a given fuzzy graph, we firstinduce a sequence of matroids from a sequence of crisp graph, i.e.,cuts of the fuzzy graph. A fuzzy matroid, named graph fuzzy matroid,is then constructed by using the sequence of matroids. An equivalentdescription of graphic fuzzy matroids is given and their propertiesof fuzzy bases and fuzzy circuits are studied.
http://ijfs.usb.ac.ir/article_3048_b36fae2bb90f788b2a7a140802ec8baa.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
131
144
10.22111/ijfs.2017.3048
Fuzzy graph
Partial fuzzy subgraph
Cycle
Fuzzy matroid
Xiaonan
Li
xnli@xidian.edu.cn
true
1
School of Mathematics and Statistics, Xidian University, Xi'an, 710071,
Shaanxi, China
School of Mathematics and Statistics, Xidian University, Xi'an, 710071,
Shaanxi, China
School of Mathematics and Statistics, Xidian University, Xi'an, 710071,
Shaanxi, China
LEAD_AUTHOR
Huangjian
Yi
yhj255@163.com
true
2
School of Information and Technology, Northwest University, Xi'an,
710069, Shaanxi, China
School of Information and Technology, Northwest University, Xi'an,
710069, Shaanxi, China
School of Information and Technology, Northwest University, Xi'an,
710069, Shaanxi, China
AUTHOR
[1] P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letters, 6 (1987), 297-
1
[2] K. R. Bhutani and A. Rosenfeld, Strong arcs in fuzzy graphs, Information Sciences, 152
2
(2003), 319-322.
3
[3] K. R. Bhutani and A. Rosenfeld, Fuzzy end nodes in fuzzy graphs, Information Sciences, 152
4
(2003), 323-326.
5
[4] K. R. Bhutani and A. Rosenfeld, On M-strong fuzzy graphs, Information Sciences, 155 (2003),
6
[5] M. Blue, B. Bush and J. Puckett, Unied approach to fuzzy graph problems, Fuzzy Sets and
7
Systems, 125 (2002), 355-368.
8
[6] R. Goetschel and W. Voxman, Fuzzy matroids, Fuzzy Sets and Systems, 27 (1988), 291-302.
9
[7] R. Goetschel and W. Voxman, Bases of fuzzy matroids, Fuzzy Sets and Systems, 31 (1989),
10
[8] R. Goetschel and W. Voxman, Fuzzy circuits, Fuzzy Sets and Systems, 32 (1989), 35-43.
11
[9] R. Goetschel and W. Voxman, Fuzzy matroids and a greedy algorithm, Fuzzy Sets and Sys-
12
tems, 37 (1990), 201-213.
13
[10] R. Goetschel and W. Voxman, Fuzzy rank functions, Fuzzy Sets and Systems, 42 (1991),
14
[11] C. E. Huang, Graphic and representable fuzzifying matroids, Proyecciones Journal of Math-
15
ematics, 29 (2010), 17-30.
16
[12] X. N. Li, S. Y. Liu and S. G. Li, Connecttedness of rened GV-fuzzy matroids, Fuzzy Sets
17
and Systems, 161 (2010), 2709-2723.
18
[13] X. N. Li and H. J. Yi, Axioms for fuzzy bases of H fuzzy matroids, Journal of Intelligent and
19
Fuzzy Systems, 29 (2015), 1995-2001.
20
[14] S.G. Li, X. Xin, Y. L. Li, Closure axioms for a class of fuzzy matroids and co-towers of
21
matroids, Fuzzy Sets and Systems, 158 (2007), 1246-1257.
22
[15] L. X. Lu and W. W. Zheng, Categorical relations among matroids, fuzzy matroids and fuzzi-
23
fying matroids, Iranian Journal of Fuzzy Systems, 7(1) (2010), 81-89.
24
[16] J. N. Mordeson and P. S. Nair, Fuzzy graphs and fuzzy hypergraphs, Physica-Verlag, 2000.
25
[17] J. N. Mordeson and J. N. Peng, Operators on fuzzy graphs, Information Sciences, 79 (1994),
26
[18] J. G. Oxley, Matroid Theory, Oxford University Press, New York, 1992.
27
[19] A. Rosenfeld, Fuzzy graphs, In: L. A. Zadeh, K. S. Fu, M. Shimura(Eds.), Fuzzy sets and
28
Their Applications to Cognitive and Decision Processes, Academic Press, New York, (1975),
29
[20] F. G. Shi, A new approach to the fuzzication of matroids, Fuzzy Sets and Systems, 160
30
(2009), 696-705.
31
[21] M. S. Sunitha and A. Vijayakumar, A characterization of fuzzy trees, Information Sciences,
32
113 (1999), 293-300.
33
[22] H. Whitney, On the abstract properties of linear dependence, American Journal of Mathe-
34
matics, 57 (1935), 509-533.
35
[23] W. Yao, Basis axioms and circuits axioms for fuzzifying matroids, Fuzzy Sets and Systems,
36
161 (2010), 3155-3165.
37
ORIGINAL_ARTICLE
M-FUZZIFYING INTERVAL SPACES
In this paper, we introduce the notion of $M$-fuzzifying interval spaces, and discuss the relationship between $M$-fuzzifying interval spaces and $M$-fuzzifying convex structures.It is proved that the category {\bf MYCSA2} can be embedded in the category {\bf MYIS} as a reflective subcategory, where {\bf MYCSA2} and {\bf MYIS} denote the category of $M$-fuzzifying convex structures of $M$-fuzzifying arity $\leq 2$ and the category of $M$-fuzzifying interval spaces, respectively. Under the framework of $M$-fuzzifying interval spaces, subspaces and product spaces are presented and some of their fundamental properties are obtained.
http://ijfs.usb.ac.ir/article_3050_199fd4b0125cf88df6ef7f1e2066dd5c.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
145
162
10.22111/ijfs.2017.3050
$M$-fuzzifying interval spaces
$M$-fuzzifying convex structures
$M$-fuzzifying interval preserving functions
Subspaces
Product spaces
Zhen-Yu
Xiu
xyz198202@163.com
true
1
College of Applied Mathematics, Chengdu University of Information
Technology, Chengdu 610000, P.R. China
College of Applied Mathematics, Chengdu University of Information
Technology, Chengdu 610000, P.R. China
College of Applied Mathematics, Chengdu University of Information
Technology, Chengdu 610000, P.R. China
LEAD_AUTHOR
Fu-Gui
Shi
fugushi@bit.edu.cn
true
2
chool of Mathematics and Statistics, Beijing Institute of Technology,
Beijing 100081, P.R. China
chool of Mathematics and Statistics, Beijing Institute of Technology,
Beijing 100081, P.R. China
chool of Mathematics and Statistics, Beijing Institute of Technology,
Beijing 100081, P.R. China
AUTHOR
[1] M. Berger, Convexity, American Mathematical Monthly, 97(8) (1990), 650{678.
1
[2] P. Dwinger, Characterizations of the complete homomorphic images of a completely distribu-
2
tive complete lattice I, Indagationes Mathematicae (Proceedings), 85 (1982), 403{414.
3
[3] J. Eckho, Helly, Radon, and Caratheodory type theorems, Handbook of convex geometrry,
4
Vol. A, B, North-Holland (1993), 389{448.
5
[4] J. M. Fang, Sums of L-fuzzy topological spaces, Fuzzy Sets and Systems, 157 (2005), 739{754.
6
[5] G. Gierz, et al., Continuous lattices and domains, Encyclopedia of Mathematics and its
7
Applications, 93, Cambridge University Press, Cambridge, 2003.
8
[6] T. E. Gantner, R. C. Steinlage and R. H. Warren, Compactness in fuzzy topological spaces,
9
J. Math. Anal. Appl., 62 (1978), 547{562.
10
[7] H. L. Huang and F. G. Shi, L-fuzzy numbers and their properties, Information Sciences, 178
11
(2008), 1141{1151.
12
[8] U. Hohle, Probabilistsche Metriken auf der Menge nicht negativen verteilungsfunktionen,
13
Aequationes Math., 18 (1978), 345{356.
14
[9] W. Kubis, Abstract convex structures in topology and set theory, PhD thesis, University of
15
Silesia Katowice, 1999.
16
[10] N. N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets and Systems, 27 (1988),
17
[11] Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 1641 (2009), 22{37.
18
[12] C. V. Negoita and D. A. Ralescu, Applications of fuzzy sets to systems analysis, Interdisciplinary
19
Systems Research Series, vol. 11, Birkhauser, Basel, Stuttgart and Halsted Press,
20
New York, 1975.
21
[13] S. Philip, Studies on fuzzy matroids and related topics, PhD thesis, Cochin University of
22
Science and Technology, 2010.
23
[14] S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and compacti-
24
cations, in: U.oHle, S.E. Rodabaugh (Eds.), Mathematics of fuzzy sets: logic, topology, and
25
measure theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers,
26
Dordrecht, (1999), 481{552.
27
[15] M. V. Rosa, On fuzzy topology, fuzzy convexity spaces, and fuzzy local convexity, Fuzzy Sets
28
and Systems, 62 (1994), 97{100.
29
[16] M. V. Rosa, A study of fuzzy convexity with special reference to separation properties, PhD
30
thesis, Cochin University of Science and Technology, 1994.
31
[17] F. G. Shi, Theory of L-nested sets and L-nested sets and its applications, Fuzzy Systems
32
and Mathematics, in Chinese, 4 (1995), 65{72.
33
[18] F. G. Shi, L-fuzzy sets and prime element nested sets, Journal of Mathematical Research
34
and Exposition, in Chinese, 16 (1996), 398{402,
35
[19] F. G. Shi, Theory of molecular nested sets and its applications, Journal of Yantai Teachers
36
University (Natural Science), in Chinese, 1 (1996), 33{36.
37
[20] F. G. Shi, L-fuzzy relation and L-fuzzy subgroup, Journal of Fuzzy Mathematics, 8 (2000),
38
[21] F. G. Shi and Z. Y. Xiu, A new approach to the fuzzication of convex structures, Journal
39
of Applied Mathematics, Article ID 249183, (2014).
40
[22] F. G. Shi, (L;M)-fuzzy metric spaces, Indian Journal of mathematics, 39 (1991), 303{321.
41
[23] F. G. Shi and E. Q. Li, The restricted hull operator of M-fuzzifying convex structures, Journal
42
of Intelligent and Fuzzy Systems., 30(1) (2016), 409{421.
43
[24] M. L. J. Van de Vel, Theory of Convex Structures, North Holland, N. Y., 1993.
44
[25] G. J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992),
45
ORIGINAL_ARTICLE
COUNTING DISTINCT FUZZY SUBGROUPS OF SOME RANK-3 ABELIAN GROUPS
In this paper we classify fuzzy subgroups of a rank-3 abelian group $G = \mathbb{Z}_{p^n} + \mathbb{Z}_p + \mathbb{Z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in \cite{mur:01}. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorphic maximal chains of subgroups and (v) classes of isomorphic fuzzy subgroups of $G$. Illustrative examples are provided.
http://ijfs.usb.ac.ir/article_3051_d9c53364056080d2ad2e5d2af478926e.pdf
2017-02-28T11:23:20
2017-09-24T11:23:20
163
181
10.22111/ijfs.2017.3051
Equivalence
Fuzzy subgroup
Maximal chain
Keychain
Distinguishing factor
Isomorphism
Isaac K.
Appiah
true
1
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
AUTHOR
B. B.
Makamba
bbmakamba@ufh.ac.za
true
2
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
LEAD_AUTHOR
[1] S. Branimir and A. Tepavcevic, A note on a natural equivalence relation on fuzzy power set,
1
Fuzzy Sets and Systems, 148 (2004), 201-210.
2
[2] G. Calugareanu, The total number of subgroups of a nite abelian group, Sci.Math.Jpn. 60
3
(2004), 157-167.
4
[3] C. Degang, J. Jiashang, W. Congxin and E. C. C. Tsang, Some notes on equivalent fuzzy
5
sets and fuzzy subgroups, Fuzzy Sets and systems, 152 (2005), 403-409.
6
[4] J. B. Fraleigh, A rst course in abstract algebra, Addison-Wesley Publishing Co., 1982.
7
[5] A. Iranmanesh and H. Naraghi, The connection between some equivalence relations on fuzzy
8
groups, Iranian Journal of Fuzzy systems, 8(5) (2011), 69-80.
9
[6] A. Jain, Fuzzy subgroups and certain equivalence relations, Iranian Journal of Fuzzy Systems,
10
3(2) (2006), 75-91.
11
[7] V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets and
12
Systems, 123 (2001), 259-264.
13
[8] V. Murali and B. B. Makamba, Counting the number of subgroups of an abelian group of
14
nite order, Fuzzy Sets and Systems, 144 (2004), 459-470.
15
[9] V. Murali and B. B. Makamba, Fuzzy subgroups of abelian groups, Far East J.Maths.Sci.
16
(FEJM), 14(1) (2004), 113-125.
17
[10] V. Murali and B. B. Makamba, Equivalence and isomorphism of fuzzy subgroups of abelian
18
groups, Journal of Fuzzy Mathematics, 16(2) (2008), 351-360.
19
[11] O. Ndiweni and B. B. Makamba, Classication of fuzzy subgroups of a dihedral group of
20
order 2pqr for distinct primes p, q and r, International Jounal of Mathematical Sciences and
21
Engineering Applications, 6(4) (2012), 159-174.
22
[12] S. Ngcibi, Case studies of equivalent fuzzy subgroups of nite abelian groups, Thesis, Rhodes
23
Univ., Grahamstown, 2001.
24
[13] S. Ngcibi, V. Murali and B. B. Makamba, Fuzzy subgroups of rank two abelian p-Groups,
25
Iranian Journal of Fuzzy Systems, 7(2) (2010), 149-153.
26
[14] J. M. Oh, An explicit formula for the number of fuzzy subgroups of a nite abelian p-group
27
of rank two, Iranian Journal of Fuzzy Systems, 10(6) (2013), 125-135.
28
[15] M. Pruszyriska, M. Dudzicz, On isomorphism between Finite Chains, Jounal of Formalised
29
Mathematics 12 (2003), 1-2.
30
[16] S. Ray, Isomorphic fuzzy groups II, Fuzzy Sets and Systems, 50 (1992), 201-207.
31
[17] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512 - 517.
32
[18] V. N. Shokeuv, An expression for the number of subgroups of a given order of a nite p-group,
33
Mathematical notes of the Academy of Sciences of the USSR, 12(5) (1972), 774-778.(Transl
34
from Matematicheskie Zametki,12(5) (1972), 561-568).
35
[19] T. Stehling, On computing the number of subgroups of a nite abelian group, Combinatorica,
36
12(4) (1992), 475-479.
37
[20] M. Tarnauceanu M and L. Bentea, On the number of Subgroups of nite abelian Groups,
38
Fuzzy Sets and Systems, 159 (2008), 1084 - 1096.
39
[21] M. Tarnauceanu, An arithmetic method of counting the subgroups of a nite abelian group,
40
Bull. Math.Soc.Sci.Math.Roumanie Tome, 53(101) No. 4, (2010), 373-386.
41
[22] A. C. Volf, Counting fuzzy subgroups and chains of subgroups, Fuzzy Systems and Articial
42
Intelligence, 10(3) (2004), 191 - 200.
43
[23] Y. Zhang and K. Zou, A note on an equivalence relation on fuzzy subgroups, Fuzzy Sets and
44
Systems, 95 (1992), 243-247.
45
ORIGINAL_ARTICLE
Persian-translation vol. 14, no. 1, February 2017
http://ijfs.usb.ac.ir/article_3089_96891c6b02f1f2f16c91eec6e6a77e02.pdf
2017-03-01T11:23:20
2017-09-24T11:23:20
185
195
10.22111/ijfs.2017.3089