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FTRANSFORM FOR NUMERICAL SOLUTION OF TWOPOINT BOUNDARY VALUE PROBLEM
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We propose a fuzzybased approach aiming at finding numerical solutions to some classical problems. We use the technique of Ftransform to solve a secondorder ordinary differential equation with boundary conditions. We reduce the problem to a system of linear equations and make experiments that demonstrate applicability of the proposed method. We estimate the order of accuracy of the proposed method. We show that the Ftransformbased approach does not only extend the set of its applications, but has a certain advantage in the solution of illposed problems.
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13


Irina
Perfilieva
University of Ostrava, Centre of Excellence IT4Innovations, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic
University of Ostrava, Centre of Excellence
Czech Republic
irina.perfilieva@osu.cz


Petra
Stevuliakova
University of Ostrava, Centre of Excellence IT4Innovations,
Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03
Ostrava 1, Czech Republic
University of Ostrava, Centre of Excellence
Czech Republic


Radek
Valasek
University of Ostrava, Centre of Excellence IT4Innovations, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1,
Czech Republic
University of Ostrava, Centre of Excellence
Czech Republic
Ftransform
Differential equation
Boundary value problem
Second order differential equation
[[1] V. I. Arnold and R. Cook, Ordinary Differential Equations, Springer Textbook, Springer,##Berlin Heidelberg, 1992. ISBN 9783540548133.##[2] U. Ascher, Collocation for twopoint boundary value problems revisited, SIAM J. Numer.##Anal. 23 (1986), 596{609.##[3] K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., John Wiley, Iowa, USA,##[4] N. S. Bakhvalov, N. P. Zhidkov and G. M. Kobelkov, Numerical Methods, Nauka, Moskva,##(1987), 600 p.##[5] W. Chen and Y. Schen, Approximate solution for a class of secondorder ordinary differenci##tal equations by the fuzzy transform, Intelligent and Fuzzy Systems. 27 (2014), 73{82.##[6] H. Keller, Numerical Methods for TwoPoint Boundary Value Problems, Ginn (Blaisdell),##Boston, 1968.##[7] A. Khastan, I. Perfilieva and Z. Alijani, A new fuzzy approximation method to Cauchy prob##lems by fuzzy transform, Fuzzy Sets and Systems, 288 (2016), 75{95. ##[8] A. Khastan, Z. Alijani and I. Perfilieva, Fuzzy transform to approximate solution of two##point boundary value problems, Mathematical Methods in the Applied Sciences. (2016),##http://dx.doi.org/10.1002/mma.3832.##[9] I. Perfilieva, Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems, 157 (2006),##[10] I. Perfilieva, Fuzzy approach to solution of differential equations with imprecise data: appli##cation to reef growth problem, in: R.V. Demicco, G.J. Klir (Eds.), Fuzzy Logic in Geology,##Academic Press, Amsterdam, (2003), 275300.##[11] I. Perfilieva, M. Holcapek and V. Kreinovich, A new reconstruction from the Ftransform##components, Fuzzy Sets and Systems, 288 (2016), 3{25.##[12] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer, New York, 2007.##[13] P. Stevuliakova, I. Perfilieva and R. Valasek, R., A New Approach to Fuzzy Boundary Value##Problem, in: World Scientific Proceedings Series on Computer Engineering and Information##Science, 10 (2016), 276{281.##]
DIAGNOSIS OF BREAST LESIONS USING THE LOCAL CHANVESE MODEL, HIERARCHICAL FUZZY PARTITIONING AND FUZZY DECISION TREE INDUCTION
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2
Breast cancer is one of the leading causes of death among women. Mammography remains today the best technology to detect breast cancer, early and efficiently, to distinguish between benign and malignant diseases. Several techniques in image processing and analysis have been developed to address this problem. In this paper, we propose a new solution to the problem of computer aided detection and interpretation for breast cancer. In the proposed approach, a Local ChanVese (LCV) model is used for the mass lesion segmentation step to isolate a suspected abnormality in a mammogram. In the classification step, we propose a twostep process: firstly, we use the hierarchical fuzzy partitioning (HFP) to construct fuzzy partitions from data, instead of using the only human information, available from expert knowledge, which are not sufficiently accurate and confronted to errors or inconsistencies. Secondly,fuzzy decision tree induction are proposed to extract classification knowledge from a set of featurebased examples. Fuzzy decision trees are first used to determine the class of the abnormality detected (welldefined mass, illdefined mass, architectural distortion, or speculated masses), then, to identify the Severity of the abnormality, which can be benign or malignant. The proposed system is tested by using the images from Mammographic Image Analysis Society[MIAS] database. Experimental results show the efficiency of the proposed approach, resulting in an accuracy rate of 87, a sensitivity of 82.14%, and good specificity of 91.42
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15
40


Fouzia
Boutaouche
laboraoire SIMPA, Departement d'informatique, Faculte des
mathematiques et d'informatique, Universite des sciences et de la technologie d'Oran "Mohamed BOUDIAF", USTOMB; BP 1505 El M'naouer 31000, Oran, Algerie
laboraoire SIMPA, Departement d'informatique,
Algeria
boutaouchef@netcourrier.com


Nacéra
Benamrane
laboratoire SIMPA, Departement d'informatique, Faculte des
mathematiques et d'informatique, Universite des sciences et de la technologie d'Oran "Mohamed BOUDIAF", USTOMB; BP 1505 El M'naouer 31000, Oran, Algerie
laboratoire SIMPA, Departement d'informatique
Algeria
Breast cancer
Mass segmentation
Local ChanVese model fuzzy decision tree
Fuzzy partitioning
Computeraided detection
[[1] S. N. Acho and W. I. D. Rae, Dependence of shapebased descriptors and mass segmentation##areas on initial contour placement using the chanvese method on digital mammograms,##Computational and Mathematical Methods in Medicine, 2015 (2015), 116.##[2] American Cancer Society, Cancer facts and figures, Atlanta, Ga: American Cancer Society,##(2013), 160.##[3] R. Bellotti, A completely automated CAD system for mass detection in a large mammographic##database, Medical Physics, 33(8) (2006), 3066–3075.##[4] L. Breslo and D. Aha, Simplifying decision trees: a survey, The Knowledge Engineering##Review, 12(1) (1997), 140.##[5] L. F. A. Campos, A. C Silva and A. K. Barros, Diagnosis of breast cancer in digital mammograms##using independent component analysis and neural networks, X Iberoamerican, Conference##on Pattern Recognition, Havana, Lecture notes in computer science, 3773 (2005),##460–469. ##[6] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Process, 10(2)##(2001), 266277.##[7] R. Crandall, Image segmentation using Chan Vese algorithm, ECE532 Project fall, 2009.##[8] A. Keles and Y. Ugur,Expert system based on neurofuzzy rules for diagnosis breast cancer,##Expert Syst Appl., 38(5)(2011), 5719–5726.##[9] U. Khan, H. Shin, J. P. Choi and M. Kim, Weighted fuzzy decision trees for prognosis of##breast cancer survivability, Proc of the Australasian Data Mining Conferenre, Glenelg, South##Australia, 7(3) (2008), 141152.##[10] Z. Lei and K. Ardrew Chan, An artificial intelligent algorithm for tumor detection in screening##mammogram, IEEE Trans. on Medical Imaging, 20(7) (2009), 559567.##[11] M. Leonardo de Oliveira, G. Braz Junior, C. S. .Aristofanes, A. Cardoso de Paiva and M.##Gattass, Detection of masses in digital mammograms using Kmeans and support vector##machine, Electronic Letters on Computer Vision and Image Analysis, 8(2) (2009), 3950.##[12] A. M.Maciej. Y.J.Lo, P.B.Harrawood and D. G Tourassc, Mutual informationbased template##matching scheme for detection of breast masses: From mammography to digital breast##tomosynthesis, Journal of Biomedical Informatics 44(5) (2011), 815823.##[13] C. Marsala, Apprentissage inductif en pr´esence de donn´ees impr´ecises : construction et##utilisation d’arbres de d´ecision flous, Th`ese de doctorat, Universit´e de Paris 6, (1988).##[14] C. Marsala, Fuzzy decision trees to help flexible querying, KYBERNETICA, 36 (2006), 689–##[15] L. Martins, A. Dos Santos, A. Silva and A. Paiva,Classification of normal, benign and malignant##tissues using cooccurrence matrix and bayesian neural network in mammographic##images, Proceedings of the Ninth Brazilian Symposium on Neural Networks, (2006), 479–486.##[16] A. Materka and M. Strzelecki, Texture analysis methods, A review, COST B11 Technical##Report, LodzBrussels: Technical University of Lodz, (1998), 911.##[17] G. H. B. Miranda and J. C Felipe,Computeraided diagnosis system based on fuzzy logic for##breast cancer categorization, Computers in Biology and Medicine, 64(1) (2015), 33434.##[18] J. I. Mohamed, M. Ahmadi and A. S. A. Maher, An efficient automatic mass classification##method in digitized mammograms using artificial neural network, International Journal of##Artificial Intelligence and Application (IJAIA), 1(3) (2010), 113.##[19] E. Molins, F. Macia and F. Ferrer, Association between radiologists’ experience and accuracy##in interpreting screening mammograms, BMC Health Serv Res., 8(91) (2008), 110.##[20] S. K. Murthy, Automatic construction of decision trees from data: a multidisciplinary survey,##Data Min Knowl Disc, 2(4) (1998), 345389.##[21] C. Olaru anf L. Wehenkel, A complete fuzzy decision tree technique, Fuzzy Sets and Systems,##138(2) (2003), 221254.##[22] A. Oliver, J. Freixenet, R Mart´ı et al., A novel breast tissue density classification methodology,##IEEE Transactions on Information Technology in Biomedicine, 12(1) (2008), 55–65.##[23] S. Osher and N. Paragios, Geometric level set methods in imaging, vision and Graphics,##SpringerVerlag, 2003.##[24] G. Palma, G. Peters, S. Muller and I. Bloch, Masses classification using fuzzy active contours##and fuzzy decision tree, SPIE Symposium on Medical Imaging, San Diego, CA, USA,##6915(2008), 691509.1691509.11.##[25] o. Pitchumani Angayarkanni and N. Banu Kamal, Association rule mining based decision tree##induction for efficient detection of cancerous masses in mammogram, International Journal##of Computer Applications, 31(6) (2011), 15.##[26] P. Rahmati, A. Adler and G. Hamarneh, Mammography segmentation with maximum likelihood##active contours, Medical Image Analysis, 16(6) (2012), 1167–1186.##[27] R. Ramani and N. Suthanthira Vanitha, Computer aided detection of tumours in mammograms,##international .Journal of Image, Graphics and Signal Processing, 6(4) (2014), 5459.##[28] M. Ramdani, Syst`eme d’induction formelle `a base de connaissances impr´ecises, Th`ese de##doctorat, Paris 6, LIP6, 1994.##[29] R. S. Safavian and D. Landgrebe, A survey of decision tree classifier methodology, IEEE##Transactions on Systems, Man, and Cybernetics, 3(21) (1991), 660–674. ##[30] G. Saborta, Probabilit´es, Analyse des donn´ees et Statistique, Ed. Technip, 1990.##[31] M. S. Salve and A. Chakkarwar, Classification of mammographic images using Gabor Wavelet##and discrete wavelet transform, International Journal of Advanced Research in Electronics##and Communication Engineering (IJARECE), 2(5) (2013).##[32] G. Serge and B. Charnomordic, Generating an interpretable family Of fuzzy partitions, IEEE##Transactions on Fuzzy Systems, 12(3) (2004), 324– 335.##[33] G. Serge, Induction de r`egles floues interpr´etables, Th`ese de Doctorat, INSA Toulouse,##France, 2001.##[34] S. Shanthi and M. BhaskaraR, Intuistionistic fuzzy Cmeans and decision tree approach for##breast cancer detection and classification, European Journal of Scientific research, 66(I3)##(2011), 345351.##[35] J. Suckling, J. Parker, D. R. Dance et al., The mammographic image analysis society digital##mammogram database, Excerpta Medica International Congress Series, (1069) (1994), 375##[36] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling##and control, IEEE Trans. Syst. Man Cybern, 15 (1985), 116–132.##[37] H. D. Thanh, Mesures de discrimination et leurs applications en apprentissage inductif,##Th`ese de doctorat, Universit´e de Paris 6, 2007.##[38] X. F. Wang, D. S. Huang and H. Xu, An efficient local Chan–vese model for image segmentation,##Pattern Recognition, 43(3) (2010), 603618.##[39] Y. Wu, O. Alagoz, M. U. S. Ayvaci, A. Munoz del Rio, A. D. J. Vanness, R. Woods and E. S.##Burnsise, A comprehensive methodology for determining the most informative mammographic##features, Journal of digital imaging, 26(5) (2013), 941947.##[40] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338353.##]
INFORMATION MEASURES BASED TOPSIS METHOD FOR MULTICRITERIA DECISION MAKING PROBLEM IN INTUITIONISTIC FUZZY ENVIRONMENT
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In the fuzzy set theory, information measures play a paramount role in several areas such as decision making, pattern recognition etc. In this paper, similarity measure based on cosine function and entropy measures based on logarithmic function for IFSs are proposed. Comparisons of proposed similarity and entropy measures with the existing ones are listed. Numerical results limpidly betoken the efficiency of these measures over others. An intuitionistic fuzzy weighted measures (IFWM) with TOPSIS method for multicriteria decision making (MCDM) quandary is introduced to grade the alternatives. This approach is predicated on entropy and weighted similarity measures for IFSs. An authentic case study is discussed to rank the four organizations. To compare the different rankings, a portfolio selection problem is considered. Various portfolios have been constructed and analysed for their risk and return. It has been examined that if the portfolios are developed using the ranking obtained with proposed method, the return is increased with slight increment in risk.
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Arunodaya Raj
Mishra
Department of Mathematics, ITM University, Gwalior
474001, M. P., India
Department of Mathematics, ITM University,
India


Pratibha
Rani
Department of Mathematics, Jaypee University of Engineering and
Technology, Guna473226, M. P., India
Department of Mathematics, Jaypee University
India
pratibha138@gmail.com
Fuzzy set
Intuitionistic fuzzy set
Entropy
Similarity measure
TOPSIS
MCDM
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Mishra, On trigonometric fuzzy information measures, ARPN Journal##of Science and Technology, 05 (2015), 145152.##[8] D. S. Hooda, A. R. Mishra and D. Jain, On generalized fuzzy mean code word lengths,##American Journal of Applied Mathematics, 02 (2014), 127134.##[9] C. C. Hung and L. H. Chen, A fuzzy TOPSIS decision making model with entropy weight##under intuitionistic fuzzy environment, In: Proceedings of the international multi conference##of engineers and computer scientists (IME CS2009), 01 (2009), 1316.##[10] W. L. Hung and M. S. Yang, Fuzzy entropy on intuitionistic fuzzy sets, International Journal##of Intelligent Systems, 21 (2006), 443451.##[11] W. L. Hung and M. S. Yang, On similarity measures between intuitionistic fuzzy sets, Inter##national Journal of Intelligent Systems, 23 (2008), 364383.##[12] W. L. Hung and M. S. Yang, Similarity measures of intuitionistic fuzzy sets based on Haus##dor distance, Pattern Recognition Letters, 25 (2004), 16031611.##[13] C. L. Hwang and K. S. Yoon, Multiple attribute decision making: methods and applications,##Berlin: SpringerVerlag, 1981. ##[14] D. Joshi and S. Kumar, Intuitionistic fuzzy entropy and distance measure based TOPSIS##method for multicriteria decision making, Egyptian informatics journal, 15 (2014), 97104.##[15] A. Jurio, D. Paternain, H. Bustince, C. Guerra and G. Beliakov, A construction method of##attanassov's intuitionistic fuzzy sets for image processing, In: Proceedings of the Fifth IEEE##Conference on Intelligent Systems, 01 (2010), 337342.##[16] D. F. Li, Relative ratio method for multiple attribute decision making problems, International##Journal of Information Technology & Decision Making, 08 (2010), 289311.##[17] D. Li and C. Cheng, New similarity measures of intuitionistic fuzzy sets and application to##pattern recognition, Pattern Recognition Letters, 23 (2003), 221225.##[18] J. Li, G. Deng, H. Li and W. Zeng, The relationship between similarity measure and entropy##of intuitionistic fuzzy sets, Information Science, 188 (2012), 314321.##[19] F. Li, Z. H. Lu and L. J. Cai, The entropy of vague sets based on fuzzy sets, J. Huazhong##Univ. Sci. Tech., 31 (2003), 2425.##[20] Z. Z. Liang and P. F. Shi, Similarity measures on intuitionistic fuzzy sets, Pattern Recognition##Letters, 24 (2003), 26872693.##[21] L. Lin, X. H. Yuan and Z. Q. Xia, Multicriteria fuzzy decisionmaking methods based on##intuitionistic fuzzy sets, J. Comp. Syst. Sci., 73 (2007), 8488.##[22] H. W. Liu and G. J. Wang, Multicriteria decisionmaking methods based on intuitionistic##fuzzy sets, European Journal of Operational Research, 179 (2007), 220233.##[23] H. M. Markowitz, Foundations of portfolio theory, J. Finance, 469 (1991), 469471.##[24] H. M. Markowitz, Portfolio selection, J. Finance, 01 (1952), 7791.##[25] A. R. Mishra, Intuitionistic fuzzy information measures with application in rating of township##development, Iranian Journal of Fuzzy Systems, 13(3) (2016), 4970.##[26] A. R. Mishra, D. Jain and D. S. Hooda, Exponential intuitionistic fuzzy information measure##with assessment of service quality, International journal of fuzzy systems, 19(3) (2017), 788##[27] A. R. Mishra, D. Jain and D. S. Hooda, Intuitionistic fuzzy similarity and information mea##sures with physical education teaching quality assessment, Proceedings of the Second Inter##national Conference on Computer and Communication Technologies, Advances in Intelligent##Systems and Computing, 379 (2016), 387399.##[28] A. R. Mishra, D. Jain and D. S. Hooda, On fuzzy distance and induced fuzzy information##measures, Journal of Information and Optimization Sciences, 37 (2) (2016), 193211.##[29] A. R. Mishra, D. Jain and D. S. Hooda, On logarithmic fuzzy measures of information and##discrimination, Journal of Information and Optimization Sciences, 37 (2) (2016), 213231.##[30] A. R. Mishra, D. S. Hooda and D. Jain, On exponential fuzzy measures of i07.##[31] A. R. Mishra, D. S. Hooda and D. Jain, Weighted trigonometric and hyperbolic fuzzy infor##mation measures and their applications in optimization principles, International Journal Of##Computer And Mathematical Sciences, 03 (2014), 6268.##[32] H. B. Mitchell, On the DengfengChuntian similarity measure and its application to pattern##recognition, Pattern Recognition Letters, 24 (2003), 31013104.##[33] S. M. Mousavi, H. Gitinavard and B. Vahdani, Evaluating construction projects by a new##group decisionmaking model based on intuitionistic fuzzy logic concepts, International Jour##nal of Engineering, 28 (2015), 13131319.##[34] S. M. Mousavi and B. Vahdani, Crossdocking location selection in distribution systems: a##new intuitionistic fuzzy hierarchical deci109.##[35] S. M. Mousavi, S. Mirdamadi, S. Siadat, J. Dantan and R. TavakkoliMoghaddam, An intu##itionistic fuzzy grey model for selection problems with an application to the inspection plan##ning in manufacturing rms, Engineering Applications of Articial Intelligence, 39 (2015),##[36] S. M. Mousavi, B. Vahdani and S. Sadigh Behzadi, Designing a model of intuitionistic fuzzy##VIKOR in multiattribute group decisionmaking problems, Iranian Journal of Fuzzy Systems,##13(1) (2016), 4565. ##[37] O. Parkash, P. K. Sharma and R. Mahajan, New measures of weighted fuzzy entropy and##their applications for the study of maximum weighted fuzzy entropy principle, Information##Sciences, 178 (2008), 23892395.##[38] B. Soylu, Integrating PROMETHEE II with tchebyche function for multi criteria decision##making, International Journal of Information Technology & Decision Making, 09 (2010),##[39] E. Szmidt and J. Kacprzyk, Entropy for Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems,##118 (2011), 467477.##[40] B. Vahdani, M. Salimi and S. M. Mousavi, A new compromise decision making model based##on TOPSIS and VIKOR for solving multiobjective largescale programming problems with a##block angular structure under uncertainty, International Journal of Engineering Transactions##B: Applications, 27 (2014), 16731680.##[41] I. K. Vlachos and G. D. Sergiagis, Intuitionistic fuzzy information { Application to pattern##recognition, Pattern Recognition Lett., 28 (2007), 197206.##[42] X. Z. Wang, B. De Baets and E. Kerre, A comparative study of similarity measures, Fuzzy##Sets and Systems, 73 (1995), 259268.##[43] P. Z. Wang, Fuzzy Sets and Its Applications, Shanghai Science and Technology Press, Shang##hai, 1983.##[44] C. P. Wei and Y. Zhang, Entropy measures for intervalvalued intuitionistic fuzzy sets and##their application in group decision making, Mathematical Problems in Engineering, Article##ID 563745, 2015 (2015), 0113.##[45] C. P. Wei, P. Wang and Y. Zhang, Entropy, similarity measure of intervalvalued intuition##istic fuzzy sets and their applications, Information Sciences, 181 (2011), 42734286.##[46] Z. B.Wu and Y. H. Chen, The maximizing deviation method for group multiple attribute deci##sion making under linguistic environment, Fuzzy Sets and Systems, 158 (2007), 16081617.##[47] M. M. Xia and Z. S. Xu, Entropy/cross entropybased group decision making under intu##itionistic fuzzy environment, Information Fusion, 13 (2012), 3147.##[48] Z. H. Xu, Intuitionistic preference relations and their application in group decision making,##Information Sciences, 177 (2007), 23632379.##[49] Z. H. Xu, J. Chen and J. J.Wu, Clustering algorithm for intuitionistic fuzzy sets, Information##Sciences, 178 (2008), 37753790.##[50] Z. S. Xu and Q. L. Da, The ordered weighted geometric averaging operators, International##Journal of Intelligent Systems, 17 (2002), 709716.##[51] Z. S. Xu and X. Q. Cai, Non linear optimization models for multiple attribute group decision##making with intuitionistic fuzzy information, International Journal of Intelligent Systems, 25##(2010), 489513.##[52] R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision##making, IEEE Transactions on Systems, Man, and Cybernetics, 18 (1988), 183190.##[53] J. Ye, Two eective measures of intuitionistic fuzzy entropy, Computing, 87 (2010), 5562.##[54] J. Ye, Fuzzy decisionmaking method based on the weighted correlation coecient under##intuitionistic fuzzy environment, European Journal of Operational Research, 205 (2010),##[55] Z. Yue, Extension of TOPSIS to determine weight of decision maker for group decision##making problems with uncertain information, Exp. Syst. Appl., 39 (2012), 63436350.##[56] L. A. Zadeh, Fuzzy sets, Information and Control, 08 (1965), 338353.##[57] L. A. Zadeh, Is there a need for fuzzy logic?, Information Sciences, 178 (2008), 27512779.##]
MULTIPERIOD CREDIBILITIC MEAN SEMIABSOLUTE DEVIATION PORTFOLIO SELECTION
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In this paper, we discuss a multiperiod portfolio selection problem with fuzzy returns. We present a new credibilitic multiperiod mean semi absolute deviation portfolio selection with some real factors including transaction costs, borrowing constraints, entropy constraints, threshold constraints and risk control. In the proposed model, we quantify the investment return and risk associated with the return rate on a risky asset by its credibilitic expected value and semi absolute deviation. Since the proposed model is a nonlinear dynamic optimization problem with path dependence, we design a novel forward dynamic programming method to solve it. Finally, we provide a numerical example to demonstrate the performance of the designed algorithm and the application of the proposed model.
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86


Peng
Zhang
School of Economics and Management, South China Normal University,
Guangzhou 510006, P. R. China
School of Economics and Management, South
China
zhangpeng300478@aliyun.com
Finance
Multiperiod portfolio selection
Mean semiabsolute deviation
Entropy constraints
The forward dynamic programming method
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Kapur, Maximum Entropy Models in Science and Engineering, Wiley Eastern Limited,##New Delhi, (1990), 428{436. ##[13] M. Koksalan and C. T. Sakar, An interactive approach to stochastic programmingbased##portfolio optimization, To appear in Annals of Operations Research, 245(1{2) (2016), 47{##[14] H. Konno and H. Yamazaki, Mean absolute portfolio optimisation model and its application##to Tokyo stock market, Management Science ,37(5) (1991), 519{531.##[15] C. J. Li and Z. F. Li, Multiperiod portfolio optimization for assetCliability management with##bankrupt control, Applied Mathematics and Computation, 218(22) (2012), 11196{11208.##[16] D. Li and W. L. Ng, Optimal dynamic portfolio selection: multiperiod meanCvariance for##mulation, Mathematical Finance, 10(3) (2000), 387{406.##[17] X. Li, Z. Qin and S. Kar, Meanvarianceskewness model for portfolio selection with fuzzy##returns, European Journal of operational Research, 202(1)(2010), 239{247.##[18] D. Lien and Y. K. Tse, Hedging downside risk: futures vs options, Internat. Rev. Econom.##Finance, 10(2) (2001), 159{169.##[19] B. Liu, Inequalities and convergence concepts of fuzzy and rough variables, Fuzzy Optimization##and Decision Making, 2 (2) (2003), 87{100.##[20] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE##Transactions on Fuzzy Systems, 10(4) (2002), 445{450.##[21] Y. J. Liu and W. G. Zhang, A multiperiod fuzzy portfolio optimization model with minimum##transaction lots, European Journal of Operational Research, 242 (3) (2015), 933{941 .##[22] Y.J. Liu, W. G. Zhang and Q. Zhang, Credibilistic multiperiod portfolio optimization model##with bankruptcy control and affine recourse, Applied Soft Computing, 38(3) (2016), 890{906.##[23] R. Mansini, W. Ogryczak and M. G. Speranza, Conditional value at risk and related lin##ear programming models for portfolio optimization, Annals of Operations Research ,152(1)##(2007), 227{256.##[24] H. M. Markowitz, Portfolio selection, Journal of Finance, 7(1) (1952),77{91.##[25] Y. Simaan, Estimation risk in portfolio selection: The mean variance model and the mean##absolute deviation model, Management Science, 43(10) (1997), 1437{1446.##[26] M.G. Speranza, Linear programming model for portfolio optimization, Finance, 14(1) (1993),##[27] S. Stevenson, Emerging markets, downside risk and the asset allocation decision, Emerging##Markets Rev., 2(1) (2001), 50{66.##[28] A. B. Terol, B. P. Gladish, M. A. Parra and M. V. R. Ura, Fuzzy compromise programming##for portfolio selection, Applied Mathematics and Computation, 173(1) (2006), 251{264##[29] J. H. Van Binsbergen and M. Brandt, Solving dynamic portfolio choice problems by recursing##on optimized portfolio weights or on the value function?, Computational Economics ,29(34)##(2007), 355{367.##[30] E. Vercher, J. Bermudez and J. Segura, Fuzzy portfolio optimization under downside risk##measures, Fuzzy Sets and Systems, 158(7) (2007), 769{782.##[31] S.Y. Wang and S.S. Zhu, On fuzzy portfolio selection problem, Fuzzy Optimization and##Decision Making, 1(14) (2002),361{377##[32] H. L. Wu and Z. F. Li, Multiperiod meanCvariance portfolio selection with regime switching##and a stochastic cash ##ow, Insurance: Mathematics and Economics, 50(3) (2012), 371{384.##[33] W. Yan and S. R. Li, A class of multiperiod semivariance portfolio selection with a four##factor futures price model, Journal of Applied Mathematics and Computing, 29(12) (2009),##[34] W. Yan, R. Miao and S.R. Li, Multiperiod semivariance portfolio selection: Model and##numerical solution, Applied Mathematics and Computation, 194(1)(2007), 128{134##[35] M. Yu, S. Takahashi, H. Inoue and S. Y. Wang, Dynamic portfolio optimization with risk##control for absolute deviation model, European Journal of Operational Research, 201(2)##(2010), 349{364.##[36] M. Yu and S. Y. Wang, Dynamic optimal portfolio with maximum absolute deviation model,##Journal of Global Optimization, 53(2) (2012), 363{380.##[37] L. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1(328)##(1978), 61{72. ##[38] L. Zadeh, A theory of approximate reasoning, in: J. Hayes, D. Michie, R.M. Thrall (Eds.),##Mathematical Frontiers of the Social and Policy Sciences, Westview Press, Boulder, Colorado,##(1979), 69{129.##[39] W. G. Zhang, Y. J. Liu and W. J. Xu, A new fuzzy programming approach for multiperiod##portfolio Optimization with return demand and risk control, Fuzzy Sets and Systems, 246(4)##(2014), 107{126.##[40] W. G. Zhang and Y. J. Liu, Credibilitic meanvariance model for multiperiod portfolio se##lection problem with risk control, OR Spectrum, 36(1) (2014), 113{132.##[41] W.G. Zhang, Y. J. Liu and W. J. Xu, A possibilistic meansemivarianceentropy model##for multiperiod portfolio selection with transaction costs, European Journal of Operational##Research, 222(2) (2012), 341{349.##[42] W. G. Zhang, Y. L. Wang, Z. P. Chen and Z. K. Nie, Possibilistic meanCvariance models##and ecient frontiers for portfolio selection problem, Information Sciences, 177(13) (2007),##2787{2801.##[43] P. Zhang and W. G. Zhang, Multiperiod mean absolute deviation fuzzy portfolio selection##model with risk control and cardinality constraints, Fuzzy Sets and Systems, 255(2) (2014),##[44] W. G. Zhang, X. L. Zhang and W. L. Xiao, Portfolio selection under possibilistic meanC##variance utility and a SMO algorithm, European Journal of Operational Research, 197(2)##(2009), 693{700.##[45] S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection:##a generalized meanCvariance formulation, IEEE Transactions on Automatic Control, 49(3)##(2004), 447{457.##]
ON CONVERGENCE THEOREMS FOR FUZZY HENSTOCK INTEGRALS
2
2
The main purpose of this paper is to establish different types of convergence theorems for fuzzy Henstock integrable functions, introduced by Wu and Gong cite{wu:hiff}. In fact, we have proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrable functions and fuzzy monotone convergence theorem. Finally, a necessary and sufficient condition under which the pointwise limit of a sequence of fuzzy Henstock integrable functions is fuzzy Henstock integrable has been established.
1

87
102


B. M.
Uzzal Afsan
Department of Mathematics, Sripat Singh College, Jiaganj742123, Murshidabad, West Bengal, India
Department of Mathematics, Sripat Singh College,
India
Fuzzy number
Fuzzy number function
Fuzzy Henstock integral
Fuzzy monotone sequence
[[1] R. G. Bartle, A convergence theorem for generalized Riemann integrals, Real Anal. Exchange,##20(2) (199495), 119{124.##[2] B. Bongiorno, L. Di Piazza and K. Musia l, A decomposition theorem for the fuzzy Henstock##integral (I), Fuzzy Sets and Systems, 200 (2012), 36{47.##[3] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986),##[4] Z. Gong, On the problem of characterizing derivatives for the fuzzyvalued functions (II):##almost everywhere differentiability and strong Henstock integral, Fuzzy Sets and Systems,##145 (2004), 381{393.##[5] Z. Gong and Y. Shao, The controlled convergence theorems for the strong Henstock integrals##of fuzzynumbervalued functions, Fuzzy Sets and Systems, 160 (2009), 1528{1546.##[6] Z. Gong and L. Wang, The HenstockStieltjes integral for fuzzynumbervalued functions,##Inform. Sci., 188 (2012), 276{297.##[7] Z. GuangQuan, Fuzzy continuous function and its properties, Fuzzy Sets and Systems, 43##(1991), 159{171.##[8] R. Henstock, Theory of Integration, Butterworths, London, 1963.##[9] J. Kurzweil, Generalized ordinary dierential equations and continuous dependence on a##parameter, Czechoslovak Math. J., 7(82) (1957), 418{446.##[10] Ma Ming, On embedding problem of fuzzy number space: Part 4, Fuzzy Sets and Systems,##58 (1993), 185{193.##[11] K. Musia l, A decomposition theorem for Banach space valued fuzzy Henstock integral, Fuzzy##Sets and Systems, 259 (2015), 21{28.##[12] C. Wu and Z. Gong, On Henstock integral of fuzzynumbervalued functions, Fuzzy Sets and##Systems, 120 (2001), 523{532.##[13] C. Wu and Ma Ming, On embedding problem of fuzzy number space: Part 1, Fuzzy Sets and##Systems, 44 (1991), 33{38.##]
CREDIBILITY THEORY ORIENTED PREFERENCE INDEX FOR RANKING FUZZY NUMBERS
2
2
This paper suggests a novel approach for ranking the most applicable fuzzy numbers, i.e. $LR$fuzzy numbers. Applying the $alpha$optimistic values of a fuzzy number, a preference criterion is proposed for ranking fuzzy numbers using the Credibility index. The main properties of the proposed preference criterion are also studied. Moreover, the proposed method is applied for ranking fuzzy numbers using targetrankbased methods. Some numerical examples are used to illustrate the proposed ranking procedure. The proposed preference criterion is also examined in order to compare with some common methods and the feasibility and effectiveness of the proposed ranking method is cleared via some numerical comparisons.
1

103
117


Gholamreza
Hesamian
Department of Statistics, Payame Noor University,, Tehran
193953697, Iran
Department of Statistics, Payame Noor University,,
Iran
ghesamian@math.iut.ac.ir


Farid
Bahrami
Department of Mathematical Sciences,, Isfahan University of Technology,, Isfahan 8415683111, Iran
Department of Mathematical Sciences,, Isfahan
Iran
f.ahmadi@math.iut.ac.ir
Credibility index
$alpha$optimistic values
Robustness
Reciprocity
Fuzzy target
[[1] S. Abbasbandy and B. Asady, Ranking of fuzzy numbers by sign distance, Information Sciences,##176 (2006), 240524016.##[2] B. Asady and A. Zendehnam, Ranking fuzzy numbers by distance minimization, Applied##Mathematical Modelling, 31(2007), 25892598.##[3] R. G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley Interscience, 1995.##[4] G. Bortolan and R. A. Degani, Review of some methods for ranking fuzzy subsets, Fuzzy Sets##and Systems, 15 (1985), 119.##[5] S. Chanas, M. Delgado, J. L. Verdegay and M. A. Vila, Ranking fuzzy interval numbers in##the setting of random sets, Information Sciences, 69 (1993), 201217.##[6] S. Chanas and P. Zielinski, Ranking fuzzy interval numbers in the setting of random sets##further results, Information Sciences, 117 (1999), 191200.##[7] P. T. Chang and E. S. Lee, Ranking of fuzzy sets based on the concept of existence, Computers##and Mathematics with Applications, 27 (1994), 121.##[8] S. H. Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and##Systems, 17 (1985), 113129.##[9] S. J. Chen and S. M. Chen, Fuzzy risk analysis based on the ranking of generalized trapezoidal##fuzzy numbers, Applied Intelligence, 26 (2007), 111. ##[10] L. H. Chen and H. W. Lu, An approximate approach for ranking fuzzy numbers based on left##and right dominance, Computers and Mathematics with Applications, 41 (2001), 15891602.##[11] T. C. Chu and C. T. Tsao, Ranking fuzzy numbers with an area between the centroid point##and original point, Computers and Mathematics with Applications, 43 (2002), 111117.##[12] L. Decampos and G. A. Mu~noz, A subjective approach for ranking fuzzy numbers, Fuzzy Sets##and Systems, 29 (1989), 145153.##[13] M. Delgado, J. L. Verdegay and M. A. Vila, A procedure for ranking fuzzy numbers using##fuzzy relations, Fuzzy Sets and Systems, 26 (1988), 4962.##[14] Y. Deng, Z. Zhenfu and L. Qi, Ranking fuzzy numbers with an area method using radius of##gyration, Computers and Mathematics with Applications, 51 (2006), 11271136.##[15] P. Diamond, and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientic, Singapore 1994.##[16] D. Dubois and H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Information##Sciences, 30 (1983), 183224.##[17] D. Dubois and H. Prade, Possibility theory: an approach to computerized processing of##uncertainty, Plenum Press, New York, 1998.##[18] P. Y. Ekel, I. V. Kokshenev, R. O. Parreiras, G. B. Alves and P. M. N. Souza, Fuzzy set##based models and methods of decision making and power engineering problems, Engineering,##5 (2013), 4151.##[19] R. Ezzati, T. Allahviranloo, S. Khezerloo and M. Khezerloo, An approach for ranking of##fuzzy numbers, Expert Systems with Applications, 39 (2012), 690695.##[20] G. Hesamian and M. Shams, Parametric testing statistical hypotheses for fuzzy random vari##ables, Soft Computing, 20 (2015), 15371548.##[21] V. H. Huynh, Y. Nakamori and J. Lawry, A probabilitybased approach to comparison of fuzzy##numbers and applications to targetoriented decision making, IEEE Transactions on Fuzzy##Systems, 16 (2008), 371387.##[22] R. Jain, Decision making in the presence of fuzzy variables, IEEE Transactions on Systems,##Man, and Cybernetics , 6 (1976), 698703.##[23] K. Kim and K. S. Park, Ranking fuzzy numbers with index of optimism, Fuzzy Sets and##Systems, 35 (1990), 143150.##[24] A. Kumar, P. Singh, P. Kaur and A. Kaur, A new approach for ranking of LR type generalized##fuzzy numbers, Expert Systems with Applications, 38 (2011), 10906{10910.##[25] H. L. Kwang and J. H. Lee, A method for ranking fuzzy numbers and its application to##decisionmaking, IEEE T. Fuzzy Syst., 7 (1999), 677685.##[26] K. H. Lee, First Course on Fuzzy Theory and Applications, SpringerVerlag, Berlin, 2005.##[27] E. S. Lee and R. J. Li, Comparison of fuzzy numbers based on the probability measure of##fuzzy events, Computers and Mathematics with Applications, 15(1988), 887896.##[28] D. F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application##to MADM problems, Computers and Mathematics with Applications, 60 (2010), 15571570.##[29] T. S. Liou and M. J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and##Systems, 50 (1992), 247255.##[30] B. Liu, Uncertainty Theory, SpringerVerlag, Berlin, 2004.##[31] X. W. Liu and S. L. Han, Ranking fuzzy numbers with preference weighting function expec##tations, Computers and Mathematics with Applications, 49 (2005), 17311753.##[32] M. Modarres and S. S. Nezhad, Ranking fuzzy numbers by preference ratio, Fuzzy Sets and##Systems, 118 (2001), 429436.##[33] J. Peng, H. Liu and G. Shang, Ranking fuzzy variables in terms of credibility measure,##Proceedings of the 3th international conference on Fuzzy Systems and Knowledge Discovery,##Xi'an, China, (2006), 2428.##[34] S. Rezvani, Ranking generalized exponential trapezoidal fuzzy numbers based on variance,##Applied Mathematics and Computation, 262 (2015), 191{198.##[35] J. Saade and H. Schwarzlander, Ordering fuzzy sets over the real line: an approach based on##decision making under uncertainty, Fuzzy Sets and Systems, 50 (1992), 237246.##[36] H. Sun and J. Wu, A new approach for ranking fuzzy numbers based on fuzzy simulation##analysis method, Applied Mathematics and Computation, 174 (2006), 755767. ##[37] L. Tran and L. Duckstein, Comparison of fuzzy numbers using a fuzzy distance measure,##Fuzzy Sets and Systems, 130 (2002), 331341.##[38] E. Valvis, A new linear ordering of fuzzy numbers on subsets of F(R), Fuzzy Optimazation##and Decision Making, 8 (2009), 141163.##[39] Y. M. Wang, Centroid defuzzication and the maximizing set and minimizing set ranking##based on alpha level sets, Computers and Industrial Engineering, 57 (2009), 228236.##[40] X. Wang and E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I),##Fuzzy Sets and Systems, 118 (2001), 375385.##[41] X. Wang and E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (II),##Fuzzy Sets and Systems, 118 (2001), 387405.##[42] Y. J. Wang and H. S. Lee, The revised method of ranking fuzzy numbers with an area between##the centroid and original points, Computers and Mathematics with Applications, 55 (2008),##20332042.##[43] Z. X. Wang, Y. J. Liu, Z. P. Fan and B. Feng, Ranking LRfuzzy number based on deviation##degree, Information Science, 179 (2009), 20702077.##[44] Y.M. Wang and Y. Luo, Area ranking of fuzzy numbers based on positive and negative ideal##points, Computers and Mathematics with Applications, 58 (2009), 17691779.##[45] Z. X. Wang and Y. N. Mo, Ranking fuzzy numbers based on ideal solution, Fuzzy Information##and Engineering, 2 (2010), 2736.##[46] P. Xu, X. Su, J. Wu, X. Sun, Y. Zhang and Y. Deng, A note on ranking generalized fuzzy##numbers, Expert Systems with Applications, 39 (2012), 64546457.##[47] Y. Yuan, Criteria for evaluating fuzzy ranking methods, Fuzzy Sets and Systems, 43 (1991),##]
SOME PROBABILISTIC INEQUALITIES FOR FUZZY RANDOM VARIABLES
2
2
In this paper, the concepts of positive dependence and linearlypositive quadrant dependence are introduced for fuzzy random variables. Also,an inequality is obtained for partial sums of linearly positive quadrant dependent fuzzy random variables. Moreover, a weak law of large numbers is established for linearly positive quadrant dependent fuzzy random variables. Weextend some well known inequalities to independent fuzzy random variables.Furthermore, a weak law of large numbers for independent fuzzy random variables is stated and proved.
1

119
134


Hamed
Ahmadzade
Department of Statistics, University of Sistan and Baluchestan,
Zahedan, Iran
Department of Statistics, University of Sistan
Iran
ahmadzadeh.h.63@gmail.com


Mohammad
Amini
Department of Statistics, Faculty of Mathematical Sciences,
Ferdowsi University of Mashhad, Mashhad 91775, Iran
Department of Statistics, Faculty of Mathematical
Iran
mamini48@yahoo.com


Seyed Mahmoud
Taheri
Faculty of Engineering Science, College of Engineering,
University of Tehran, Tehran, Iran
Faculty of Engineering Science, College of
Iran
sm_taheri@yahoo.com


Abolghasem
Bozorgnia
Department of Statistics, Khayyam University, Mashhad,
Iran
Department of Statistics, Khayyam University,
Iran
a.bozorgnia@khayyam.ac.ir
Fuzzy random variable
Linearly Positive Quadrant Dependence
Independence
Law of Large Numbers
[[1] H. Agahi and E. Eslami, A general inequality of Chebyshev type for semi(co)normed fuzzy##integrals, Soft Computing, 15 (2011), 771{780.##[2] H. Agahi, A. F. Franulic and S. M. Vaezpour, Fatou's lemma for Sugeno integral, Applied##Mathematics and Computation, 217(13) (2011), 6092{6096.##[3] H. Ahmadzade, M. Amini, S. M. Taheri and A. Bozorgnia, Some limit theorems for indepen##dent fuzzy random variables, Thai Journal of Mathematics, 12 (2014), 537{548.##[4] H. Ahmadzade, M. Amini, S. M. Taheri and A. Bozorgnia, Some moment inequalities for##fuzzy martingales and their applications, Journal Uncertainty Analysis and Applications, 2##(2014), 1{14.##[5] H. Ahmadzade, M. Amini, S. M. Taheri and A. Bozorgnia, Maximal inequalities and some##convergence theorems for fuzzy random variables, Kybernetika, 52(2) (2016), 307{328.##[6] H. Ahmadzade, M. Amini, S. M. Taheri and A. Bozorgnia, Negative Dependence for Fuzzy##Random Variables: Basic Denitions and Some Limit Theorems, Filomat, 30(9) (2016),##2535{2549.##[7] T. Birkel, A functional central limit theorem for positively dependent random variables, Jour##nal of Multivariate Analysis, 44 (1993), 314{320.##[8] P. Diamond and P. Kloeden, Metric space of fuzzy sets, Fuzzy Sets and Systems, 35 (1990),##[9] Y. Feng, An approach to generalize laws of large numbers for fuzzy random variables, Fuzzy##Sets and Systems, 128 (2002), 237{245.##[10] Y. Feng, L. Hu and H. Shu, The variance and covariance of fuzzy random variables and their##applications, Fuzzy Sets and Systems, 120 (2001), 487{497.##[11] K. Fu and L. X. Zhang, Strong limit theorems for random sets and fuzzy random sets with##slowly varying weights, Information Sciences, 178 (2008), 2648{2660.##[12] A. Gut, Probability: A Graduate Course, Springer, New York, 2005.##[13] L. Hongxing, Probability representations of fuzzy systems, Science China Information Sci##ences, 49 (2006), 339{363.##[14] S. Y. Joo, Strong law of large numbers for tight fuzzy random variables, Journal of the Korean##Statistical Society, 31 (2002), 129{140.##[15] S. Y. Joo, Y. K. Kim and J. S. Kwon, On Chung's type law of large numbers for fuzzy random##variables, Statistics and Probability Letteres, 74 (2005), 67{75.##[16] S. Y. Joo, S. S. Lee and Y. H. Yoo, A strong law of large numbers for stationary fuzzy random##variables, Journal of the Korean Statistical Society, 30 (2001), 153161.##[17] E. P. Klement, M. L. Puri and D. A. Ralescu, Limit theorems for fuzzy random variables,##Proceedings of the Royal Society of London A, 407 (1986), 171182.##[18] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy LogicTheory and Applications, PrenticeHall,##Upper Saddle River, NJ, 1995.##[19] R. Korner, On the variance of fuzzy random variables, Fuzzy Sets and Systems 92 (1997),##[20] V. Kratschmer, Probability theory in fuzzy sample spaces, Metrika, 60 (2004), 167{180.##[21] R. Kruse and K. D. Meyer, Statistics with vague data, Reidel Publishing Company, Dordrecht,##[22] H. Kwakernaak, Fuzzy random variables I. Information Sciences, 15 (1978), 1{29.##[23] Z. Lin and Z. Bai, Probability inequalities, Springer, New York, 2010.##[24] Y. K. Liu and B. Liu, Fuzzy Random Variables: A Scalar Expected Value Operator, Fuzzy##Optimization and Decision Making, 2 (2003), 143{160.##[25] S. Louhichi, Rosenthal's inequality for LPQD sequences, Statistics and Probability Letters,##42 (1992), 139144.##[26] M. Miyakoshi and M. Shimbo, A strong law of large numbers for fuzzy random variables.##Fuzzy Sets and Systems 12 (1984), 133142.##[27] R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Siam,##Philadelphia, 2009.##[28] C. M. Newman, Asymptotic independence and limit theorems for positively and negatively##dependent random variables, Statistics and Probability, 5 (1984), 127{140.##[29] H. T. Nguyen, T. Wang and B. Wu, On probabilistic methods in fuzzy theory, International##Journal of Intelligent Systems, 19 (2004), 99109.##[30] M. L. Puri and D. A. Ralescu, Fuzzy random variables. Journal of Mathematical Analysis##and Applications, 114 (1986), 402{422.##[31] M. Rong, N. K. Mahapatra and M. Maiti, A multiobjective wholesalerretailers inventory##distribution model with controllable leadtime based on probabilistic fuzzy set and triangular##fuzzy number, Applied Mathematical Modelling, 32 (2008), 2670{2685.##[32] B. Sadeghpour Gildeh and S. Rahimpour, A Fuzzy Bootstrap Test for the Mean with Dp;q##distance, Fuzzy Information and Engineering, 4 (2011), 351{358.##[33] Q. M. Shao, H. Yu, Weak convergence for weighted empirical processes of dependent se##quences, Annals of Probability, 24 (1996), 2052{2078.##[34] R. L. Taylor, L. Seymour and Y. Chen, Weak laws of large numbers for fuzzy random sets,##Nonlinear Analysis, 47 (2001), 1245{125.##[35] R. Viertl. Statistical Methods for Fuzzy Data, John Wiley and Sons, Chichester, 2011.##[36] H. C. Wu, The central limit theorems for fuzzy random variables, Information Sciences. 120,##(1999), 239256.##[37] H. C. Wu, The laws of large numbers for fuzzy random variables, Fuzzy Sets and Systems.##116 (2000), 245262.##[38] H. C. Wu, Fuzzy Bayesian system reliability assessment based on exponential distribution,##Applied Mathematical Modelling, 30 (2006), 509{530.##[39] L. A. Zadeh, Fuzzy sets, Information and Control, 8, (1965), 338{353.##]
ROBUSTNESS OF THE TRIPLE IMPLICATION INFERENCE METHOD BASED ON THE WEIGHTED LOGIC METRIC
2
2
This paper focuses on the robustness problem of full implication triple implication inference method for fuzzy reasoning. First of all, based on strong regular implication, the weighted logic metric for measuring distance between two fuzzy sets is proposed. Besides, under this metric, some robustness results of the triple implication method are obtained, which demonstrates that the triple implication method possesses a good behavior of robustness.
1

135
148


Jun
Li
School of Science, Lanzhou University of Technology, Lanzhou 730050,
Gansu, China
School of Science, Lanzhou University of
China
lijun@lut.cn


Chao
Fu
School of Science, Lanzhou University of Technology, Lanzhou 730050,
Gansu, China
School of Science, Lanzhou University of
China
fuchao45612@sina.com
Robustness
Triple implication method
Weighted logic metric
Weighted logic similarity degree
Fuzzy reasoning
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ON STRATIFIED LATTICEVALUED CONVERGENCE SPACES
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In this paper we provide a common framework for different stratified $LM$convergence spaces introduced recently. To this end, we slightly alter the definition of a stratified $LMN$convergence tower space. We briefly discuss the categorical properties and show that the category of these spaces is a Cartesian closed and extensional topological category. We also study the relationship of our category to the categories of stratified $L$topological spaces and of enriched $LM$fuzzy topological spaces.
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Gunther
Jager
School of Mechanical Engineering, University of Applied Sciences
Stralsund, D18435 Stralsund, Germany
School of Mechanical Engineering, University
Germany
g.jager@ru.ac.za, gunther.jaeger@fhstralsund.de
Latticevalued convergence
$LM$convergence space
Stratified $LMN$convergence tower space
Stratified $LM$filter
Stratified $L$topological space
Enriched $LM$fuzzy topological space
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Persiantranslation Vol.14, No.6
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