2018
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INVENTORY MODEL WITH DEMAND AS TYPE2 FUZZY NUMBER: A FUZZY DIFFERENTIAL EQUATION APPROACH
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An inventory model is formulated with type2 fuzzy parameters under trade credit policy and solved by using Generalized Hukuhara derivative approach. Representing demand parameter of each expert's opinion is a membership function of type1 and thus, this membership function again becomes fuzzy. The final opinion of all experts is expressed by a type2 fuzzy variable. For this present problem, to get corresponding defuzzified values of the triangular type2 fuzzy demand parameters, first critical value (CV)based reduction methods are applied to reduce corresponding type1 fuzzy variables which becomes pentagonal in form. After that $alpha$ cut of a pentagonal fuzzy number is used to construct the upper $alpha$ cut and lower $alpha$ cut of the fuzzy differential equation. Different cases are considered for fuzzy differential equation: gH(i) differentiable and gH(ii) differentiable systems. The objective of this paper is to find out the optimal time so as to minimize the total inventory cost. The considered problem ultimately reduces to a multiobjective problem which is solved by weighted sum method and global criteria method. Finally the model is solved by generalised reduced gradient method using LINGO (13.0) software. The proposed model and technique are lastly illustrated by providing numerical examples. Results from two methods are compared and some sensitivity analyses both in tabular and graphical forms are presented and discussed. The effects of total cost with respect to the change of demand related parameter ($beta$), holding cost parameter ($r$), unit purchasing cost parameter ($p$), interest earned $(i_e)$ and interest payable $(i_p)$ are discussed. We also find the solutions for type1 and crisp demand as particular cases of type2 fuzzy variable. This present study can be applicable in many aspects in many real life situations where type1 fuzzy set is not sufficient to formulate the mathematical model. From the numerical studies, it is observed that under both gH(i) and gH(ii) cases, total cost of the system gradually reduces for the subcases  1.1, 1.2 and 1.3 depending upon the positions of N(trade credit for customer) and M (trade credit for retailer) with respect to T (time period).
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Bijoy Krishna
Debnath
Department of Mathematics, National Institute of Technology, Agartala, 799046, India
Department of Mathematics, National Institute
India


Pinki
Majumder
Department of Mathematics, National Institute of Technology,
Agartala, 799046, India
Department of Mathematics, National Institute
India
pinki.mjmdr@redimail.com


Uttam Kumar
Bera
Department of Mathematics, National Institute of Technology, Agartala, 799046, India
Department of Mathematics, National Institute
India
berauttam@yahoo.co.in


Manoranjan
Maiti
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721102, India
Department of Applied Mathematics with Oceanology
India
mmmaiti2005@yahoo.co.in
EOQ model
Delay in payment
Type2 fuzzy demand
$alpha$cut of pentagonal number
[[1] T. Allahviranloo and M. Afshar Kermani, Numerical methods for fuzzy linear partial dif##ferential equations under new definition for derivative, Iranian Journal Fuzzy Systems, 7(3)##(2010), 33{50.##[2] S. Arshed, On existence and uniqueness of solution of fuzzy fractional differential equations,##Iranian Journal Fuzzy Systems, 10(6) (2013), 137{151.##[3] B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzynumbervalued func##tions with applications to fuzzy differential equations, Fuzzy Sets Syst., 151(4) (2005), 581{##[4] B. Bede and L. Stefanini, Generalized differentiability of fuzzyvalued functions, Fuzzy Sets##and Systems, 230(5) (2013), 119{141.##[5] K. J. Chung and L. E. CardenasBarron, The simplied solution procedure for deteriorating##items under stockdependent demand and twolevel trade credit in the supply chain manage##ment, Appl. Math. Model., 37(7) (2013), 4653{4660.##[6] K. J. Chung, L. E. CardenasBarron and P. S. Ting, An inventory model with noninstan##taneous receipt and exponentially deteriorating items for an integrated three layer supply##chain system under two levels of trade credit, Int. J. Prod. Eco., 155(5) (2014), 310{317.##[7] S. C. Chen, L. E. CardenasBarron and J. T. Teng, Retailer's economic order quantity when##the supplier offers conditionally permissible delay in payments link to order quantity, Int. J.##Prod. Econ., 155(3) (2014), 284{291.##[8] L. E. CardenasBarron, K. J. Chung and G. TrevioGarza, Celebrating a century of the##economic order quantity model in honor of For Whitman Harris, Int. J. Prod. Econ., 155(7)##(2014), 1{7.##[9] B. Das, N. K. Mahapatra and M. Maiti, Initialvalued first order fuzzy differential equation##in Bilevel inventory model with fuzzy demand, Math. Model. Anal., 13(4) (2008), 493{512.##[10] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press,##New York, 1980.##[11] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci., 9(6) (1978), 613{##[12] R. Ezzati, K. Maleknejad, S. Khezerloo and M. Khezerloo Convergence, Consistency and##stability in fuzzy differential equations, Iranian Journal Fuzzy Systems, 12(3) (2015), 95{##[13] S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, J.##Oper. Res. Soc., 36(4) (1985), 335{338.##[14] P. Guchhait, M. K. Maiti and M. Maiti, A production inventory model with fuzzy produc##tion and demand using fuzzy differential equation: An interval compared genetic algorithm##approach, Eng. Appl. Artif. Intel., 26(7) (2013), 766{778.##[15] Y. F. Huang, Supply chain model for the Retailer's ordering policy under two levels of delay##payments derived algebraically, Opsearch, 44(8) (2007), 366{377.##[16] N. N. Karnik and J. M. Mendel, Centroid of a type2 fuzzy set, Information Sciences, 132(6)##(2001), 195{220.##[17] A. Kandel and W. J. Byatt, Fuzzy differential equations. In Proceedings of the International##Conference on Cybernetics and Society, Tokyo, November 1978, 1213{1216.##[18] F. Liu, An efficient centroid typereduction strategy for general type2 fuzzy logic system,##Information Sciences, 178(7) (2008), 2224{2236.##[19] P. Majumder, U. K. Bera and M. Maiti, An EPQ model for twowarehouse in unremitting##release pattern with two level trade credit period concerning both supplier and retailer, Appl.##Math. Comput., 274(6) (2016), 430{458.##[20] M. Mizumoto and K. Tanaka, Fuzzy sets of type2 under algebraic product and algebraic sum,##Fuzzy Sets and Systems, 5(3) (1981), 277{280.##[21] J. S. Martnez, R. I. John, D. Hissel and M. C. Pera, A surveybased type2 fuzzy logic system##for energy management in hybrid electrical vehicles, Information Sciences, 190(9) (2012),##[22] J. M. Mendel and R. I. John, Type2 fuzzy sets made simple, IEEE Transactions on Fuzzy##Systems, 10(2) (2002), 307{315.##[23] M. K. Maiti and M. Maiti, Fuzzy inventory model with two warehouses under possibility##constraints, Fuzzy Sets Syst, 157(8) (2006), 52{73.##[24] S. M. Mousavi, S. Hajipour and N. N. Aalikar, A multiproduct multiperiod inventory##control problem under inflation and discount: a parametertuned particle swarm optimization##algorithm, Int. J. Adv. Manuf. Tech., 33(4) (2013), 1{18.##[25] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki, N. Alikar, A. Bahreininejad and H. Metselaar,##Two parametertuned metaheuristics for a discounted inventory control problem in a fuzzy##environment, Information Sciences, 276(8) (2014), 42{62.##[26] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki and M. Tavana, A biobjective inventory optimiza##tion model under inflation and discount using tuned Paretobased algorithms: NSGAII,##NRGA, and MOPSO, Applied Soft Computing, 43(6) (2016), 57{72.##[27] S. M. Mousavi, A. Bahreininejad, N. Musa and F. Yusof, A modified particle swarm opti##mization for solving the integrated location and inventory control problems in a twoechelon##supply chain network, J. intell. Manuf., 23(4) (2014), 1{16.##[28] L. Y. Ouyang, C. H. Hob and C. H. Su, An optimization approach for joint pricing and##ordering problem in an integrated inventory system with ordersize dependent trade credit,##Comput. Indust. Eng., 57(7) (2009), 920{930.##[29] S. Pal, M. K. Maiti and M. Maiti, An EPQ model with price discounted promotional demand##in an imprecise planning horizon via Genetic Algorithm, Comput. Indust. Eng., 57(6) (2009),##[30] S. H. R. Pasandideh, S. T. A. Niaki and S. M. Mousavi, Two metaheuristics to solve a multi##item multiperiod inventory control problem under storage constraint and discounts, Int. J.##Adv. Manuf. Technol., 69(7) (2013), 1{14.##[31] T. Pathinathan and K. Ponnivalavan, Pentagonal fuzzy numbers, Int. J. Comput. Algm.,##3(4) (2014), 1003{1005.##[32] R. Qin, Y. K. Liu and Z. Q. Liu, Methods of critical value reduction for type2 fuzzy variables##and their applications, J. Comput. Appl. Math., 235(7) (2011), 1454{1481. ##[33] S. Sharan, S. P. Tiwary and V. K. Yadav, Interval type2 fuzzy rough sets and interval type2##fuzzy closure spaces, Iranian Journal of Fuzzy Systems, 12(3) (2015), 113{125.##[34] N. H. Shah and L. E. CardenasBarron, Retailer's decision for ordering and credit policies##for deteriorating items when a supplier offers orderlinked credit period or cash discount,##Appl. Math. Comp., 259(5) (2015), 569{578.##[35] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of intervalvalued functions##and interval differential equations, Nonlinear Analysis, 71(4) (2009), 1311{1328.##[36] B. Sarkar, S. Saren and L. E. CardenasBarron, An inventory model with tradecredit policy##and variable deterioration for fixed lifetime products, Ann. Oper. Res., 229(1) (2015), 677{##[37] S. Tiwari, L. E. CardenasBarron, A. Khanna and C. K. Jaggi, Impact of trade credit and##inflation on retailer's ordering policies for noninstantaneous deteriorating items in a two##warehouse environment, Int. J. Prod. Econ., 176(3) (2016), 154{169.##[38] J. Wu, F. B. Alkhateeb, J. T. Teng and L. E. CardenasBarron, Inventory models for dete##riorating items with maximum lifetime under downstream partial trade credits to creditrisk##customers by discounted cashflow analysis, Int. J. Prod. Eco., 171(1) (2016), 105{115.##[39] J. Wu, L. Y. Ouyang, L. E. CardenasBarron and S. K. Goyal, Optimal credit period and lot##size for deteriorating items with expiration dates under twolevel trade credit financing, Eur.##J. Oper. Res., 237(3) (2014), 898{908.##[40] P. S. You, S. Ikuta and Y. C. Hsieh, Optimal ordering and pricing policy for an inventory##system with trial periods, Appl. Math. Model., 34(4) (2010), 3179{3188.##[41] L. A. Zadeh, The concept of a linguistic variable and its application to approximate resoning##I, Information Sciences, 8(2) (1975), 199{249.##[42] L. A. Zadeh, The concept of a linguistic variable and its application to approximate resoning##II, Information Sciences, 8(2) (1975), 301{357.##]
SHAPLEY FUNCTION BASED INTERVALVALUED INTUITIONISTIC FUZZY VIKOR TECHNIQUE FOR CORRELATIVE MULTICRITERIA DECISION MAKING PROBLEMS
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Intervalvalued intuitionistic fuzzy set (IVIFS) has developed to cope with the uncertainty of imprecise human thinking. In the present communication, new entropy and similarity measures for IVIFSs based on exponential function are presented and compared with the existing measures. Numerical results reveal that the proposed information measures attain the higher association with the existing measures, which demonstrate their efficiency and reliability. To deal with the interactive characteristics among the elements in a set, Shapley weighted similarity measure based on proposed similarity measure for IVIFSs is discussed via Shapley function. Thereafter, the linear programming model for optimal fuzzy measure is originated for incomplete information about the weights of the criteria and thus, the optimal weight vector is obtained in terms of Shapley values. Further, the VIKOR technique is discussed for correlative multicriteria decision making problems under intervalvalued intuitionistic fuzzy environment. Finally, an example of investment problem is presented to exemplify the application of the proposed technique under incomplete and uncertain information situation.
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Pratibha
Rani
Department of Mathematics, Jaypee University of Engineering and
Technology, Guna473226, M. P., India
Department of Mathematics, Jaypee University
India
pratibha138@gmail.com


Divya
Jain
Department of Mathematics, Jaypee University of Engineering and
Technology, Guna473226, M. P., India
Department of Mathematics, Jaypee University
India
divya.jain@juet.ac.in


D. S.
Hooda
Guru Jambheshwar University of Science and Technology, Hisar125001,
Haryana, India
Guru Jambheshwar University of Science and
India
Fuzzy set
Intervalvalued intuitionistic fuzzy set
Entropy
Similarity measure
MCDM
VIKOR
[[1] M. D. Ansari, S. P. Ghrera and A. R. Mishra, Texture feature extraction using intuitionis##tic fuzzy local binary pattern, Journal of Intelligent Systems, doi: 10.1515/jisys20160155,##[2] K. T. Atanassov and G. Gargov, Intervalvalued intuitionistic fuzzy sets, Fuzzy Sets and##Systems, 31 (1989), 343349.##[3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 8796.##[4] A. Awasthi and G. Kannan, Green supplier development program selection using NGT##and VIKOR under fuzzy environment, Computers and Industrial Engineering, 91 (2016),##[5] P. Burillo and H. Bustince, Entropy on intuitionistic fuzzy sets and on intervalvalued fuzzy##sets, Fuzzy Sets and Systems, 78 (1996), 305316.##[6] L. Chen and Z. S. Xu, A new prioritized multicriteria outranking method: the prioritized##PROMETHEE, Journal of Intelligent and Fuzzy Systems, 29 (2015), 20992110.##[7] Q. Chen, Z. S. Xu, S. S. Liu and X. H. Yu, A method based on intervalvalued intuitionistic##fuzzy entropy for multiple attribute decision making, Information, 13 (2010), 6777.##[8] A. De Luca and S. Termini, A definition of a nonprobabilistic entropy in the setting of fuzzy##sets theory, Information and Control, 20 (1972), 301312.##[9] S. Ebrahimnejad, H. Hashemi, S. M. Mousavi and B. Vahdani, A new intervalvalued in##tuitionistic fuzzy model to group decision making for the selection of outsourcing providers,##Journal of Economic Computation and Economics Cybernetics Studies and Research, 49##(2015), 269290.##[10] S. Ebrahimnejad, S. M. Mousavi, R. TavakkoliMoghaddam, H. Hashemi and B. Vahdani, A##novel twophase group decisionmaking approach for construction project selection in a fuzzy##environment, Applied Mathematical Modelling, 36 (2012), 41974217.##[11] H. Gitinavard, S. M. Mousavi and B. Vahdani, A new multicriteria weighting and rank##ing model for group decisionmaking analysis based on intervalvalued hesitant fuzzy sets to##selection problems, Neural Computing and Applications, 27 (2016), 15931605. ##[12] H. Gitinavard, S. M. Mousavi, B. Vahdani and A. Siadat, A distancebased decision model##in intervalvalued hesitant fuzzy setting for industrial selection problems, Scientia Iranica E,##23 (2016), 19281940.##[13] H. Hashemi, J. Bazargan and S. M. Mousavi, A compromise ratio method with an application##to water resources management: an intuitionistic fuzzy set, Water Resources Management,##27 (2013), 20292051.##[14] H. Hashemi, J. Bazargan, S. M. Mousavi and B. Vahdani, An extended compromise ratio##model with an application to reservoir flood control operation under an intervalvalued intu##itionistic fuzzy environment, Applied Mathematical Modelling, 38 (2014), 34953511.##[15] D. S. Hooda, A. R. Mishra and D. Jain, On generalized fuzzy mean code word lengths,##American Journal of Applied Mathematics, 2 (2014), 127134.##[16] D. S. Hooda and A. R. Mishra, On trigonometric fuzzy information measures, ARPN Journal##of Science and Technology, 5 (2015), 145152.##[17] W. L. Hung and M. S. Yang, Fuzzy entropy on intuitionistic fuzzy sets, International Journal##of Intelligent Systems, 21 (2006), 443451.##[18] W. L. Hung and M. S. Yang, On similarity measures between intuitionistic fuzzy sets, Inter##national Journal of Intelligent Systems, 23 (2008), 364383.##[19] W. L. Hung and M. S. Yang, Similarity measures of intuitionistic fuzzy sets based on Haus##dorff distance, Pattern Recognition Letters, 25 (2004), 16031611.##[20] R. A. Krohling and A. G. C. Pacheco, Intervalvalued intuitionistic fuzzy TODIM, Procedia##Computer Science, 31 (2014), 236244.##[21] D. Li and C. Cheng, New similarity measures of intuitionistic fuzzy sets and application to##pattern recognition, Pattern Recognition Letters, 23 (2002), 221225.##[22] Z. Liang and P. Shi, Similarity measures on intuitionistic fuzzy sets, Pattern Recognition##Letters, 24 (2003), 26872693.##[23] H. Liao, Z. S. Xu and X. J. Zeng, Hesitant fuzzy linguistic VIKOR method and its application##in qualitative multiple criteria decision making, IEEE Transactions on Fuzzy Systems, 23##(2015), 13431355.##[24] X. D. Liu, S. H. Zhang and F. L. Xiong, Entropy and subsethood for general intervalvalued##intuitionistic fuzzy sets, In: L. Wang, Y. Jin (Eds.), FSKD, SpringerVerlag, Berlin Heidel##berg LNAI, 3613 (2005), 4252.##[25] P. Liu and L. Zhang, An extended multiple criteria decision making method based on##neutrosophic hesitant fuzzy information, Journal of Intelligent and Fuzzy Systems, doi:##10.3233/JIFS16136, (2016).##[26] F. Meng and X. Chen, Entropy and similarity measure for Atannasov's intervalvalued in##tuitionistic fuzzy sets and their application, Fuzzy Optimization and Decision Making, 15##(2016), 75101.##[27] A. R. Mishra, Intuitionistic fuzzy information measures with application in rating of township##development, Iranian Journal of Fuzzy Systems, 13(3) (2016), 4970.##[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 (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 (2016), 213231.##[30] 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, 3 (2014), 6268.##[31] A. R. Mishra, D. S. Hooda and Divya Jain, On exponential fuzzy measures of information##and discrimination, International Journal of Computer Applications, 119 (2015), 17.##[32] A. R. Mishra, D. Jain and D. S. Hooda, Intuitionistic fuzzy similarity and information mea##sures with physical education teaching quality assessment, proceeding of IC3T2015, Springer##Advances in intelligent systems and computing series11156, 379 (2016), 387399.##[33] 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),##[34] A. R. Mishra, P. Rani and D. Jain, Information measures based TOPSIS method for mul##ticriteria decision making problem in intuitionistic fuzzy environment, Iranian Journal of##Fuzzy Systems, 14(6) (2017), 4163.##[35] S. M. Mousavi, F. Jolai, R. TavakkoliMoghaddam and B. Vahdani, A fuzzy grey model based##on the compromise ranking for multicriteria group decision making problems in manufac##turing systems, Journal of Intelligent and Fuzzy Systems, 24 (2013), 819827.##[36] S. M. Mousavi, S. A. Torabi and R. TavakkoliMoghaddam, A hierarchical group decision##making approach for new product selection in a fuzzy environment, Arabian Journal of Science##and Engineering, 38 (2013), 32333248.##[37] S. M. Mousavi, B. Vahdani, R. TavakkoliMoghaddam and N. Tajik, Soft computing based##on a fuzzy grey compromise solution approach with an application to the selection problem of##material handling equipment, International Journal of Computer Integrated Manufacturing,##27 (2014), 547569.##[38] 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, Transactions C: Aspects, 28 (2015), 13121319.##[39] S. M. Mousavi, B. Vahdani, H. Gitinavard and H. Hashemi, Solving robot selection problem##by a new intervalvalued hesitant fuzzy multiattributes group decision method, International##Journal of Industrial Mathematics, 8 (2016), 231240.##[40] S. M. Mousavi and B. Vahdani, Crossdocking location selection in distribution systems: a##new intuitionistic fuzzy hierarchical decision model, International Journal of Computational##Intelligence Systems, 9 (2016), 91109.##[41] T. Murofushi, A technique for reading fuzzy measures (I): The shapley value with respect to##a fuzzy measure, In 2nd Fuzzy Workshop, Nagoaka, Japan, (1992), 3948.##[42] S. Opricovic, Multicriteria optimization of civil engineering systems, University of Belgrade,##Belgrade, Serbia, 2 (1998), 521.##[43] N. R. Pal and S. K. Pal, Object background segmentation using new definitions of entropy,##IEEE Proceedings, 136 (1989), 284295.##[44] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, 27##(1948), 379423.##[45] L. S. Shapley, A Value for nperson game. In H. Kuhn & A. Tucker (Eds.), Contributions to##the theory of games, Princeton, Princeton University Press, II (1953), 307317.##[46] S. Singh, O. Ezutah Udoncy. M. Siti Nurmaya, M. Abu Mahat and K. Y. Wong, Strategy##selection for sustainable manufacturing with integrated ahpvikor method under interval##valued fuzzy environment, The International Journal of Advanced Manufacturing Technology,##84 (2016), 547563.##[47] M. Sugeno, Theory of fuzzy integral and its application, Doctorial Dissertation, Tokyo Insti##tute of Technology, (1974), 3055.##[48] E. Szmidt and J. Kacprzyk, A concept of similarity for intuitionistic fuzzy sets and its##application in group decision making, In: Proceedings of International Joint Conference on##Neural Networks & IEEE International Conference on Fuzzy Systems, Budapest, Hungary,##(2004), 2529.##[49] E. Szmidt and J. Kacprzyk, Analysis of similarity measures for Atanassov's intuitionistic##fuzzy sets, In:Proceedings IFSA/EUSFLAT, (2009), 14161421.##[50] E. Szmidt and J. Kacprzyk, Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems,##118 (2001), 467477.##[51] B. Vahdani, S. M. Mousavi, H. Hashemi, M. Mousakhani and R. TavakkoliMoghaddam, A##new compromise solution method for fuzzy group decisionmaking problems with an applica##tion to the contractor selection, Engineering Applications of Artificial Intelligence, 26 (2013),##[52] B. Vahdani, S. M. Mousavi, R. TavakkoliMoghaddam and H. Hashemi, A new design of##the elimination and choice translating reality method for multiplecriteria group decision##making in an intuitionistic fuzzy environment, Applied Mathematical Modelling, 37 (2013),##17811799. ##[53] R. Vetschera and A. Teixeira De Almeida, A PROMETHEEbased approach to portfolio##selection problems, Computers and Operations Research, 39 (2012), 10101020.##[54] R. Verma and B. D. Sharma, Exponential entropy on intuitionistic fuzzy sets, Kybernetika,##49 (2013), 114127.##[55] C. P. Wei, P. Wang and Y. Z. Zhang, Entropy, similarity measure of intervalvalued intu##itionistic fuzzy sets and their applications, Information Sciences, 181 (2011), 42734286.##[56] C. P. Wei and Y. Z. Zhang, Entropy measures for intervalvalued intuitionistic fuzzy sets and##their applications in group decisionmaking, Mathematical Problems in Engineering, 2015##(2015), 113.##[57] K. L. Wu and M. S. Yang, Alternative Cmeans clustering algorithms, Pattern Recognition##Letters, 32 (2002), 22672278.##[58] Y. Wu, K. Chen, B. Zeng H. Xu and Y. Yang, Supplier selection in nuclear power industry##with extended VIKOR method under linguistic information, Applied Soft Computing, 48##(2016), 444457.##[59] Z. S. Xu, An overview of distance and similarity measures of intuitionistic measures, Interna##tional Journal of Uncertainty, Fuzziness and KnowledgeBased systems, 16 (2008), 529555.##[60] Z. S. Xu, On similarity measures of intervalvalued intuitionistic fuzzy sets and their appli##cation to pattern recognitions, Journal of Southeast University, 23 (2007), 139143.##[61] J. Ye, Fuzzy cross entropy of intervalvalued intuitionistic fuzzy sets and its optimal decision##making method based on the weights of alternatives, Experts Systems with Applications, 38##(2011), 61796183.##[62] J. Ye, Multicriteria decisionmaking method using the dice similarity measure based on the##rreduct intuitionistic fuzzy sets of intervalvalued intuitionistic fuzzy sets, Applied Mathe##matical Modelling, 36 (2012), 44664472.##[63] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338356.##[64] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning##I, Information Sciences, 8 (1975), 199249.##[65] X. Zhao, T. Zou, S. Yang and M. Yang, Extended VIKOR method with fuzzy crossentropy##of intervalvalued intuitionistic fuzzy sets, Proceedings of the 2nd International Conference##on Computer and Information Application, (2012), 10931096.##]
ITERATIVE METHOD FOR SOLVING TWODIMENSIONAL NONLINEAR FUZZY INTEGRAL EQUATIONS USING FUZZY BIVARIATE BLOCKPULSE FUNCTIONS WITH ERROR ESTIMATION
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In this paper, we propose an iterative procedure based on two dimensionalfuzzy blockpulse functions for solving nonlinear fuzzy Fredholm integralequations of the second kind. The error estimation and numerical stabilityof the proposed method are given in terms of supplementary Lipschitz condition.Finally, illustrative examples are included in order to demonstrate the accuracyand convergence of the proposed method.
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55
76


Shokrollah
Ziari
Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran
Department of Mathematics, Firoozkooh Branch,
Iran
shok_ziari@yahoo.com
Two dimensional nonlinear fuzzy Fredholm integral equations of the second kind
Two dimensional fuzzy blockpulse functions
Supplementary Lipschitz condition
[[1] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm##fuzzy integral equations of the second kind, Chaos Solitons Fractals, 31(1) (2007), 138146.##[2] R. P. Agrawal, D. Oregan and V. Lakshmikantham, Fuzzy Volterra Integral Equations: A##Stacking Theorem Approach, Applicable Analysis: An International Journal, 83(5) (2004),##[3] G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory, Springer, Berlin (2010). ##[4] E. Babolian, H. Sadeghi Goghary and S. Abbasbandy, Numerical solution of linear Fredholm##fuzzy integral equations of the second kind by Adomian method, Applied Mathematics and##Computation, 161 (2005), 733744.##[5] M. Baghmisheh and R. Ezzati, Numerical solution of nonlinear fuzzy Fredholm integral equa##tions of the second kind using hybrid of blockpulse functions and Taylor series, Advances in##Difference Equations, DOI 10.1186/s1366201503897, 51 (2015), 115.##[6] K. Balachandran and P. Prakash, Existence of solutions of nonlinear fuzzy Volterra Fredholm##integral equations, Indian Journal of Pure and Applied Mathematics, 33 (2002), 329343.##[7] K. Balachandran and K. Kanagarajan, Existence of solutions of general nonlinear fuzzy##VolterraFredholm integral equations, Journal of Applied Mathematics and Stochastic Anal##ysis, 3 (2005), 333343.##[8] B. Bede and S. G. Gal, Quadrature rules for integrals of fuzzynumbervalued functions, Fuzzy##Sets and Systems, 145 (2004), 359380.##[9] A. M. Bica, Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm##integral equations, Information Science, 178 (2008), 12791292.##[10] A. M. Bica and C. Popescu, Approximating the solution of nonlinear Hammerstein fuzzy##integral equations, Fuzzy Sets and Systems, 245 (2014), 117.##[11] A. M. Bica and C. Popescu, Fuzzy trapezoidal cubature rule and application to two##dimensional fuzzy Fredholm integral equations, Soft Computing, 21(5) (2017), 12291243.##[12] A. M. Bica, S. Ziari, Iterative numerical method for solving fuzzy Volterra linear integral##equations in two dimensions, Soft Computing, 21(5) (2017), 10971108.##[13] P. Diamond, Theory and applications of fuzzy Volterra integral equations, IEEE Transactions##on Fuzzy Systems, 10(1) (2002), 97102.##[14] D. Dubois and H. Prade, Fuzzy numbers: an overview, In: Analysis of Fuzzy Information,##CRC Press, BocaRaton, (1) (1987), 339.##[15] R. Ezzati and S. Ziari, Numerical solution and error estimation of fuzzy Fredholm integral##equation using fuzzy Bernstein polynomials, Aust. J. Basic Appl. Sci., 5(9) (2011), 20722082.##[16] R. Ezzati and S. Ziari, Numerical solution of nonlinear fuzzy Fredholm integral equations##using iterative method, Applied Mathematics and Computation, 225 (2013), 3342.##[17] R. Ezzati and S. Ziari, Numerical solution of twodimensional fuzzy Fredholm integral equa##tions of the second kind using fuzzy bivariate Bernstein polynomials, Int. J. Fuzzy Systems,##15(1) (2013), 8489.##[18] R. Ezzati and S. M. Sadatrasoul, Application of bivariate fuzzy Bernstein polynomials to##solve twodimensional fuzzy integral equations, Soft Computing, 21(14) (2017), 38793889.##[19] J. X. Fang and Q. Y. Xue, Some properties of the space fuzzyvalued continuous functions##on a compact set, Fuzzy Sets Systems, 160 (2009), 16201631.##[20] M. A. Fariborzi Araghi and N. Parandin, Numerical solution of fuzzy Fredholm integral##equations by the Lagrange interpolation based on the extension principle, Soft Computing,##15 (2011), 24492456.##[21] M. Friedman, M. Ma and A. Kandel, Numerical solutions of fuzzy differential and integral##equations, Fuzzy Sets and Systems, 106 (1999), 3548.##[22] M. Friedman, M. Ma and A. Kandel, Solutions to fuzzy integral equations with arbitrary##kernels, International Journal of Approximate Reasoning, 20 (1999), 249262.##[23] S. G. Gal, Approximation theory in fuzzy setting, In: Anastassiou, GA (ed.) Handbook of##AnalyticComputational Methods in Applied Mathematics, Chapman & Hall/CRC Press,##Boca Raton, (2000), 617666.##[24] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986),##[25] L. T. Gomes, L. C. Barros and B. Bede, Fuzzy Differential Equations in Various Approaches,##Springer (2015).##[26] Z. H. Jiang and W. Schanfelberger, BlockPulse Functions and Their Applications in Control##Systems, Springer, Berlin (1992).##[27] C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley,##New York (1975). ##[28] J. Y. Park and H. K. Han, Existence and uniqueness theorem for a solution of fuzzy Volterra##integral equations, Fuzzy Sets and Systems, 105 (1999), 481488.##[29] J. Y. Park and J. U. Jeong, On the existence and uniqueness of solutions of fuzzy Volttera##Fredholm integral equations, Fuzzy Sets and Systems, 115 (2000), 425431.##[30] S. M. Sadatrasoul and R. Ezzati, Iterative method for numerical solution of twodimensional##nonlinear fuzzy integral equations, Fuzzy Sets and Systems, 280 (2015), 91106.##[31] S. M. Sadatrasoul and R. Ezzati, Numerical solution of twodimensional nonlinear Hammer##stein fuzzy integral equations based on optimal fuzzy quadrature formula, Journal of Compu##tational and Applied Mathematics, 292 (2016), 430446.##[32] P. V. Subrahmanyam and S. K. Sudarsanam, A note on fuzzy Volterra integral equations,##Fuzzy Sets and Systems, 81 (1996), 237240.##[33] C.Wu, S. Song and H.Wang, On the basic solutions to the generalized fuzzy integral equation,##Fuzzy Sets and Systems, 95 (1998), 255260.##[34] C. Wu and Z. Gong, On Henstock integral of fuzzynumbervalued functions (I), Fuzzy Sets##and Systems, 120 (2001), 523532.##[35] S. Ziari, R. Ezzati and S. Abbasbandy, Numerical solution of linear fuzzy Fredholm inte##gral equations of the second kind using fuzzy Haar wavelet, In: Advances in Computational##Intelligence. Communications in Computer and Information Science, 299 (2012), 7989.##[36] S. Ziari and A. M. Bica, New error estimate in the iterative numerical method for nonlinear##fuzzy HammersteinFredholm integral equations, Fuzzy Sets and Systems, 295 (2016), 136##]
SOME SIMILARITY MEASURES FOR PICTURE FUZZY SETS AND THEIR APPLICATIONS
2
2
In this work, we shall present some novel process to measure the similarity between picture fuzzy sets. Firstly, we adopt the concept of intuitionistic fuzzy sets, intervalvalued intuitionistic fuzzy sets and picture fuzzy sets. Secondly, we develop some similarity measures between picture fuzzy sets, such as, cosine similarity measure, weighted cosine similarity measure, settheoretic similarity measure, weighted settheoretic cosine similarity measure, grey similarity measure and weighted grey similarity measure. Then, we apply these similarity measures between picture fuzzy sets to building material recognition and minerals field recognition. Finally, two illustrative examples are given to demonstrate the efficiency of the similarity measures for building material recognition and minerals field recognition.
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77
89


Guiwu
Wei
School of Business, Sichuan Normal University, Chengdu, 610101, P.R.
China
School of Business, Sichuan Normal University,
China
weiguiwu@163.com
Picture fuzzy set
Cosine similarity measure
Settheoretic similarity measure
Grey similarity measure
Building material recognition
Minerals field recognition
[[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 8796.##[2] K. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33 (1989), 3746.##[3] K. Atanassov and G. Gargov, Intervalvalued intuitionistic fuzzy sets, Fuzzy Sets and Sys##tems, 31 (1989), 343349.##[4] K. Atanassov, Operators over intervalvalued intuitionistic fuzzy sets, Fuzzy Sets and Sys##tems, 64(2) (1994), 159174.##[5] A. Bhattacharya, On a measure of divergence of two multinomial populations, Sankhya, 7##(1946), 401406.##[6] H. Bustince, and P. Burillo, Correlation of intervalvalued intuitionistic fuzzy sets, Fuzzy##Sets and Systems, 74(2) (1995), 237244.##[7] S. M. Chen, S. H. Cheng and C. H. Chiou, Fuzzy multiattribute group decision making based##on intuitionistic fuzzy sets and evidential reasoning methodology, Information Fusion, 27##(2016), 215227.##[8] T. Y. Chen,The inclusionbased TOPSIS method with intervalvalued intuitionistic fuzzy sets##for multiple criteria group decision making, Applied Soft Computing, 26 (2015), 5773.##[9] T. Y. Chen, An intervalvalued intuitionistic fuzzy permutation method with likelihoodbased##preference functions and its application to multiple criteria decision analysis, Applied Soft##Computing, 42 (2016), 390409.##[10] B. Cuong, Picture fuzzy setsfirst results. part 1, In: Seminar "NeuroFuzzy Systems with##Applications", Institute of Mathematics, Hanoi, 2013.##[11] X. P. Jiang and G. W. Wei, Some Bonferroni mean operators with 2tuple linguistic infor##mation and their application to multiple attribute decision making, Journal of Intelligent and##Fuzzy Systems, 27 (2014), 21532162.##[12] D. F. Li and C. Cheng, New similarity measures of intuitionistic fuzzy sets and application##to pattern recognition, Pattern Recognition Letters, 23 (13) (2002), 221225.##[13] D. F. Li, TOPSISBased nonlinearprogramming methodology for multiattribute decision##making with intervalvalued intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systems,##18 (2010), 299311. ##[14] D. F. Li and H. P. Ren, Multiattribute decision making method considering the amount and##reliability of intuitionistic fuzzy information, Journal of Intelligent and Fuzzy Systems, 28(4)##(2015), 18771883.##[15] R. Lin, G. W. Wei, H. J. Wang and X. F. Zhao, Choquet integrals of weighted triangular fuzzy##linguistic information and their applications to multiple attribute decision making, Journal##of Business Economics and Management, 15(5)(2014), 795809.##[16] R. Lin, X. F. Zhao, H. J.Wang and G. W.Wei, Hesitant fuzzy linguistic aggregation operators##and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy##Systems, 27 (2014), 4963.##[17] R. Lin, X. F. Zhao and G. W. Wei, Models for selecting an ERP system with hesitant fuzzy##linguistic information, Journal of Intelligent and Fuzzy Systems, 26(5) (2014), 21552165.##[18] M. Lu and G. W. Wei, Models for multiple attribute decision making with dual hesitant fuzzy##uncertain linguistic information, International Journal of Knowledgebased and Intelligent##Engineering Systems, 20(4) (2016), 217227.##[19] L. D. Miguel, H. Bustince, J. Fernndez, E. Indurin, A. Kolesrov and R. Mesiar, Construction##of admissible linear orders for intervalvalued Atanassov intuitionistic fuzzy sets with an##application to decision making, Information Fusion, 27 (2016), 189197.##[20] G. Salton and M. J. McGill, Introduction to Modern Information Retrieval, McGrawHill##Book Company, New York, 1983.##[21] P. Singh, Correlation coefficients for picture fuzzy sets, Journal of Intelligent & Fuzzy Sys##tems, 27 (2014), 28572868.##[22] L. Son, DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets,##Expert System with Applications, 2 (2015), 5166.##[23] Y. Tang, L. L. Wen and G. W. Wei, Approaches to multiple attribute group decision making##based on the generalized Dice similarity measures with intuitionistic fuzzy information, In##ternational Journal of Knowledgebased and Intelligent Engineering Systems, 21(2) (2017),##[24] P. H. Thong and L. H. Son, A new approach to multivariables fuzzy forecasting using picture##fuzzy clustering and picture fuzzy rules interpolation method, in: 6th International Conference##on Knowledge and Systems Engineering, Hanoi, Vietnam, (2015), 679690.##[25] N. T. Thong, HIFCF: An effective hybrid model between picture fuzzy clustering and intu##itionistic fuzzy recommender systems for medical diagnosis expert systems with applications,##Expert Systems with Applications, 42(7) (2015), 36823701.##[26] H. J. Wang, X. F. Zhao and G. W. Wei, Dual Hesitant Fuzzy Aggregation Operators in##Multiple Attribute Decision Making, Journal of Intelligent and Fuzzy Systems, 26(5) (2014),##22812290.##[27] G. W. Wei, Some geometric aggregation functions and their application to dynamic multiple##attribute decision making in intuitionistic fuzzy setting, International Journal of Uncertainty,##Fuzziness and Knowledge Based Systems, 17(2) (2009), 179196.##[28] G. W. Wei, Some induced geometric aggregation operators with intuitionistic fuzzy informa##tion and their application to group decision making, Applied Soft Computing, 10(2) (2010),##[29] G. W. Wei, GRA method for multiple attribute decision making with incomplete weight##information in intuitionistic fuzzy setting, Knowledgebased Systems, 23(3) (2010), 243247.##[30] G. W.Wei, Gray relational analysis method for intuitionistic fuzzy multiple attribute decision##making, Expert Systems with Applications, 38 (2011), 1167111677.##[31] G. W. Wei and X. F. Zhao, Some induced correlated aggregating operators with intuitionistic##fuzzy information and their application to multiple attribute group decision making, Expert##Systems with Applications, 39 (2) (2012), 20262034.##[32] G. W. Wei, H. J. Wang and R. Lin, Application of correlation coefficient to intervalvalued##intuitionistic fuzzy multiple attribute decision making with incomplete weight information,##Knowledge and Information Systems, 26(2) (2011), 337349. ##[33] G. W. Wei, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision##making with incomplete weight information, International Journal of Fuzzy Systems, 17(3)##(2015), 484489.##[34] G. W. Wei, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in mul##tiple attribute decision making, International Journal of Machine Learning and Cybernetics,##7(6) (2016), 10931114.##[35] G. W. Wei, Picture fuzzy crossentropy for multiple attribute decision making problems,##Journal of Business Economics and Management, 17(4) (2016), 491502.##[36] G. W. Wei, Picture 2tuple linguistic Bonferroni mean operators and their application to##multiple attribute decision making, International Journal of Fuzzy System, 19(4) (2017),##[37] G. W. Wei, F. E. Alsaadi, T. Hayat and A. Alsaedi, Hesitant fuzzy linguistic arithmetic ag##gregation operators in multiple attribute decision making, Iranian Journal of Fuzzy Systems,##13(4) (2016), 116.##[38] G. W. Wei, F. E. Alsaadi, T. Hayat and A. Alsaedi, A linear assignment method for multi##ple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure, International##Journal of Fuzzy Systems, 19(3) (2017), 607614.##[39] G. W. Wei, F. E. Alsaadi, T. Hayat and A. Alsaedi, Projection models for multiple attribute##decision making with picture fuzzy information, International Journal of Machine Learning##and Cybernetics, DOI: 10.1007/s1304201606041, 2016.##[40] G. W. Wei, F. E. Alsaadi, T. Hayat and A. Alsaedi, Picture 2tuple linguistic aggregation##operators in multiple attribute decision making, Soft Computing, 22(3) (2018), 9891002.##[41] G. W. Wei, R. Lin, X. F. Zhao and H. J. Wang, An approach to multiple attribute deci##sion making based on the induced Choquet integral with fuzzy number intuitionistic fuzzy##information, Journal of Business Economics and Management, 15(2) (2014), 277298.##[42] G. W. Wei, R. Lin and H. J. Wang, Distance and similarity measures for hesitant interval##valued fuzzy sets, Journal of Intelligent and Fuzzy Systems, 27(1) (2014), 1936.##[43] G. W. Wei, X. R. Xu and D. X. Deng, Intervalvalued dual hesitant fuzzy linguistic geo##metric aggregation operators in multiple attribute decision making, International Journal of##Knowledgebased and Intelligent Engineering Systems, 20(4) (2016), 189196##[44] G. W. Wei, H. J. Wang, X. F. Zhao and R. Lin, Hesitant triangular fuzzy information##aggregation in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems,##26(3) (2014), 12011209.##[45] G. W. Wei and X. F. Zhao, Some induced correlated aggregating operators with intuitionistic##fuzzy information and their application to multiple attribute group decision making, Expert##Systems with Applications, 39(2) (2012), 20262034.##[46] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transations on Fuzzy Systems,##15(6) (2007), 11791187.##[47] Z. S. Xu, On correlation measures of intuitionistic fuzzy sets, Lecture Notes in Computer##Science, 4224 (2006), 1624.##[48] Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy##sets, International Journal of General System, 35 (2006), 417433.##[49] Z. S. Xu and X. Q. Cai, Intuitionistic Fuzzy Information Aggregation: Theory and Applica##tions, Science Press, 2008.##[50] Z. S. Xu and J. Chen, An overview of distance and similarity measures of intuitionistic fuzzy##sets, International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, 16(4)##(2008), 529555.##[51] J. Ye, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Math##ematical and Computer Modelling, 53(1) (2011), 9197.##[52] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338356.##[53] X. F. Zhao, Q. X. Li and G. W. Wei, Some prioritized aggregating operators with linguistic##information and their application to multiple attribute group decision making, Journal of##Intelligent and Fuzzy Systems, 26(4) (2014), 16191630. ##[54] X. F. Zhao, R. Lin and G. W. Wei, Hesitant triangular fuzzy information aggregation based##on einstein operations and their application to multiple attribute decision making, Expert##Systems with Applications, 41(4) (2014), 10861094.##[55] X. F. Zhao and G. W. Wei, Some intuitionistic fuzzy einstein hybrid aggregation operators##and their application to multiple attribute decision making, KnowledgeBased Systems, 37##(2013), 472479.##[56] L. Y. Zhou, R. Lin, X. F. Zhao and G. W. Wei, Uncertain linguistic prioritized aggregation##operators and their application to multiple attribute group decision making, International##Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, 21(4) (2013), 603627.##[57] B. Zhu, Z. S. Xu and M. M. Xia, Hesitant fuzzy geometric Bonferroni means, Information##Sciences, 205(1) (2012), 7285.##]
A NEW APPROACH FOR PARAMETER ESTIMATION IN FUZZY LOGISTIC REGRESSION
2
2
Logistic regression analysis is used to model categorical dependent variable. It is usually used in social sciences and clinical research. Human thoughts and disease diagnosis in clinical research contain vagueness. This situation leads researchers to combine fuzzy set and statistical theories. Fuzzy logistic regression analysis is one of the outcomes of this combination and it is used in situations where the classical logistic regression assumptions' are not satisfied. Also it can be used if the observations or their relations are vague. In this study, a model called “Fuzzy Logistic Regression Based on Revised Tanaka's Fuzzy Linear Regression Model” is proposed. In this regard, the methodology and formulation of the proposed model is explained in detail and the revised Tanaka's regression model is used to estimate the parameters. The Revised Tanaka's Regression model is an extension of Tanaka's Regression Model in which the objection function is developed. An application is performed on birth weight data set. Also, an application of diabetes data set used in Pourahmad et al.'s study was conducted via our proposed data set. The validity of the model is shown by the help of goodness – of –fit criteria called Mean Degree Memberships (MDM).
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91
102


GULTEKIN
ATALIK
Department of Statistics, Anadolu University, Eskisehir, Turkey and Department of Statistics, Amasya University, Amasya,Turkey
Department of Statistics, Anadolu University,
Turkey
gultekinatalik@anadolu.edu.tr


Sevil
Senturk
Department of Statistics, Anadolu University, Eskisehir, Turkey
Department of Statistics, Anadolu University,
Turkey
Fuzzy logistic regression
Revised Tanaka regression model
MDM criteria
[[1] G. Atalik, A New Approach for Parameter Estimation in Fuzzy Logistic Regression and##an Application, Master of Science Thesis, Anadolu University, Graduate School of Sciences,##Eskisehir (2014).##[2] H. Bircan, Lojistik Regresyon Analizi: Tp Verileri zerine Bir Uygulama, Kocaeli niversitesi##Sosyal Bilimler Enstits Dergisi, 8(1) (2004), 185208.##[3] R. M. Dom, S. A. Kareem, A. Razak and B. Abidin, A learning system prediction method##using fuzzy regression, In Proceedings of the International MultiConference of Engineers and##Computer Scientists, Hong Kong, China, (2008), 1921. ##[4] Y. Q. He, L. K. Chan and M. L. Wu, Balancing productivity and consumer satisfaction##for profitability: statistical and fuzzy regression analysis, European Journal of Operational##Resarch, 176(1) (2007), 252263.##[5] S. S. Hirve and B. R. Ganatra, Determinants of low birth weight: a commun,ty based prospec##tive cohort study, Indian Pediatrics, 31(10) (1994), 12211225.##[6] D. W. Hosmer and S. Lemeshow, Applied Logistic Regression, John Wiley and Sons, New##York, 2000.##[7] D. G. Kleinbaum and M. Klein, Logistic Regression, A SelfLearning Text (Second Edition##ed.), Springer  Verlag , New York, 2002.##[8] E. Kirimi and S. Pence, The affects of smoking during pregnancy to fetus and plasental##development, Van Medical Journal, 6(1) (1999), 2830.##[9] D. C. Montgomery, E. A. Peck and G. G. Vining, Introduction to Linear Regression Analysis,##John Wiley and Sons, New York, 2001.##[10] P. Nagar and S. Srivastava, Adaptive fuzzy regression model for the prediction of dichotomous##response variables using cancer data: a case study, Journal of Applied Mathematics, Statistics##and Informatics, 4(2) (2008), 183191.##[11] M. Namdari, A. Abadi, S. M. Taheri, M. Rezaei, M. Kalantari and N. Omidvar, Effect of##folic acid on appetite in children: Ordinal logistic and fuzzy logistic regressions, Nutrition,##30(3) (2014), 274278.##[12] M. Namdari, J. H. Yoon, A. Abadi, S. M. Taheri and S. H. Choi, Fuzzy Logistic Regression##with Least Absolute Deviations Estimators, Soft Computing, 19(4) (2015), 909917.##[13] S. Pourahmad, S. M. T. Ayatollahi and S. M. Taheri, Fuzzy logistic regression: a new possi##bilistic model and its application in clinical vague status, Iranian Journal of Fuzzy Systems,##8(1) (2011), 117.##[14] S. Pourahmad, S. M. Ayatollahi and S. M. Taheri, Fuzzy logistic regression based on the##least squares approach with application in clinical studies, Computers and Mathematics with##Applications, 62(9) (2011), 33533365.##[15] H. Tanaka, S. Uejima and K. Asai, Lineer regression analysis with fuzzy model, IEEE Transactions##On Systems, Man, and Cybernetics, 12(6) (1982), 903907.##[16] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##[17] L. A. Zadeh, Discussion: probability theory and fuzzy logic are complementary rather than##competitive, Technometrics, 37(3) (1995), 271276.##]
QUANTALEVALUED GAUGE SPACES
2
2
We introduce a quantalevalued generalization of approach spaces in terms of quantalevalued gauges. The resulting category is shown to be topological and to possess an initially dense object. Moreover we show that the category of quantalevalued approach spaces defined recently in terms of quantalevalued closures is a coreflective subcategory of our category and, for certain choices of the quantale, is even isomorphic to our category. Finally, the category of quantalevalued metric spaces is shown to be coreflectively embedded in our category.
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103
122


Gunther
Jager
University of Applied Sciences Stralsund, D18435 Stralsund, Germany
University of Applied Sciences Stralsund,
Germany
g.jager@ru.ac.za, gunther.jaeger@fhstralsund.de


Wei
Yao
Hebei University of Science and Technology, 050054 Shijiazhuang, P.R.China
Hebei University of Science and Technology,
China
yaowei0516@163.com
$L$gauge space
$L$approach space
$L$metric space
Continuity space
[[1] S. Abramsky and A. Jung, Domain Theory, in: S. Abramsky, D.M. Gabby, T.S.E. Maibaum##(Eds.), Handbook of Logic and Computer Science, Vol. 3, Claredon Press, Oxford, 1994.##[2] J. Adamek., H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New##York, 1989.##[3] R. C. Flagg, Quantales and continuity spaces, Algebra Univers., 37 (1997), 257 { 276.##[4] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, Continuous##Lattices and Domains, Cambridge University Press, Cambridge, 2003.##[5] J. Gutierrez Garca, On stratfiied Lvalued filters induced by >filters, Fuzzy Sets and Systems##157 (2006), 813 { 819.##[6] D. Hofmann, G. J. Seal and W. Tholen, Monoidal Topology  A Categorical Approach to##Order, Metric and Topology, Cambridge University Press, Cambridge, 2014.##[7] U. Hohle, Commutative, residuated lmonoids, in: Nonclassical logics and their applications##to fuzzy subsets (U. Hohle, E.P. Klement, eds.), Kluwer, Dordrecht (1995), 53 { 106.##[8] G. Jager, A convergence theory for probabilistic metric spaces, Quaestiones Math., 38 (2015),##[9] G. Jager, Probabilistic approach spaces, Math. Bohemica, 142(3) (2017), 277298.##[10] H. Lai and W. Tholen, Quantalevalued topological spaces via closure and convergence, Topology##Appl., 230 (2017), 599620.##[11] R. Lowen, Approach spaces: A common supercategory of TOP and MET, Math. Nachr., 141##(1989), 183 { 226.##[12] R. Lowen, Approach Spaces: The Missing Link in the TopologyUniformityMetric Triad,##Clarendon Press, Oxford, 1997.##[13] R. Lowen, Index Analysis, SpringerVerlag, London, 2015. ##[14] G. Preuss, Foundations of Topology. An Approach to Convenient Topology, Kluwer Academic##Publishers, Dordrecht, 2002.##[15] S. SamingerPlatz and C. Sempi, A primer on triangle functions I, Aequationes Math., 76##(2008), 201 { 240.##[16] S. SamingerPlatz and C. Sempi, A primer on triangle functions II, Aequationes Math., 80##(2010), 239 { 261.##[17] B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.##[18] W. Yao and B. Zhao, Kernel systems on Lordered sets, Fuzzy Sets and Systems 182 (2011),##101 { 109.##[19] W. Yao, A duality between categories and algebraic categories, Electronic Notes in Theoretical##Computer Science, 301 (2014), 153 { 168.##]
ON TRUNCATED MEASURES OF INCOME INEQUALITY FROM A FUZZY PERSPECTIVE
2
2
In most statistical analysis, inequality or extent of variation in income isrepresented in terms of certain summary measures. But some authors arguedthat the concept of inequality is vague and thus cannot be measured as anexact concept. Therefore, fuzzy set theory provides naturally a useful toolfor such circumstances. In this paper we have introduced a realvalued fuzzymethod of illustrating the measures of income inequality in truncated randomvariables based on the case where the conditional events are vague. Toguarantee certain relevant properties of these measures, we first selectedthree main families of measures and obtained their closed formulas, thenused two simulated and real data set to illustrate the usefulness of derivedresults.
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123
137


Reza
Pourmousa
Department of Statistics, Faculty of Mathematics ~and Computer
Shahid Bahonar University of Kerman
Kerman, Iran
Department of Statistics, Faculty of Mathematics
Iran
pourm@uk.ac.ir
Measures of income inequality
Gini index
Fuzzy event
Membership function
Truncated distribution
[[1] B. C. Arnold, Majorization and the Lorenz order, Lecture notes in statistics 43, Springer,##Berlin and New York, 1987.##[2] F. Belzunce, J. Candel and J. M. Ruiz, Ordering of truncated distributions through concen##tration curves, Sankhya A, 57 (1995), 375383.##[3] N. Bhattacharya, A property of the Pareto distribution, Sankhya B, 25 (1963), 195196.##[4] M. Bonetti, C. Gigliarano and P. Muliere, The Gini concentration test for survival data,##Lifetime Data Analysis, 15 (2009), 493518.##[5] G. M. Cordeiro, The Kumaraswamy generalized halfnormal distribution for skewed positive##data, Journal of Data Science, 10 (2012), 195224.##[6] Y. Dodge, The Oxford Dictionary of Statistical Terms, OUP, 2003.##[7] O. Elteto, E. Frigyes, New income inequality measures as efficient tools for causal analysis##and planning, Econometrica, 36 (1968), 383396.##[8] C. Gini, On the measure of concentration with special reference to incom and statistics,##Colorado College Publication, General Series, 208 (1936), 7379.##[9] K. Hanada, A Formula of Gini's concentration ratio and its applications to life tables, Journal##of Japanese Statistical Society, 19 (1983), 293325.##[10] N. Kakwani, On a class of poverty measures, Econometrica, 48 (1980), 437446.##[11] C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences,##Wiley, Hoboken, 2003. ##[12] D. Kundu and H. Howlader, Bayesian Inference and prediction of the inverse Weibull dis##tribution for TypeII censored data, Computational Statistics and Data Analysis, 54 (2010),##15471558.##[13] J. Kupka and S. Loo, The hazard and vitality measures of aging, Journal of Applied Proba##bility, 26 (1989), 532542.##[14] J. Lawry, Modelling and Reasoning with Vague Concepts, Springer, 2006.##[15] M.O. Lorenz, Methods of measuring the moncentration of wealth, Journal of the american##statistical association, 70(9) (1905), 209217.##[16] T. S. K. Moothathu, A characterization of power function distribution through a property of##the Lorenz curve. Sankhya, The Indian Journal of Statistics, Series B, 48 (1986), 262265.##[17] J. K. Ord, G. P. Patil and C. Taillie, Truncated distributions and measures of income in##equality. Sankhya, The Indian Journal of Statistics, Series B, 45 (1983), 413430.##[18] T. J. Ross, J. M. Booker and W. J. Parkinson, Fuzzy Logic and Probability Applications,##SIAM, Philadelphia, 2002.##[19] V. M. Shkolnikov, E. E. Andreev and A. Z. Begun, Gini coeffcient as a life table func##tion: Computation from discrete data, decomposition of differences and empirical examples,##Demographic Research, 8 (2003), 305358.##[20] L. M. Surhone, M. T. Timpledon and S. F. Marseken, Truncated Distribution, Betascript##Publishing, 2010.##[21] S. Yitzhaki, On an extension of the Gini inequality index, International economic review, 24##(1983), 617628.##[22] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338353.##[23] L.A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and##Applications, 23(1968), 421427##]
A NEW APPROACH IN FAILURE MODES AND EFFECTS ANALYSIS BASED ON COMPROMISE SOLUTION BY CONSIDERING OBJECTIVE AND SUBJECTIVE WEIGHTS WITH INTERVALVALUED INTUITIONISTIC FUZZY SETS
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Failure modes and effects analysis (FMEA) is a wellknown risk analysis approach that has been conducted to distinguish, analyze and mitigate serious failure modes. It demonstrates the effectiveness and the ability of understanding and documenting in a clear manner; however, the FMEA has weak points and it has been criticized by some authors. For example, it does not consider relative importance among three risk factors (i.e., $ O, S $ and $ D $). Different sequences of $ O $, $ S $ and $ D $ may result in exactly the same value of risk priority number (RPN), but their semantic risk implications may be totally different and these three risk factors are difficult to be precisely expressed. This study introduces a new intervalvalued intuitionistic fuzzy (IVIF)decision approach based on compromise solution concept that defeats the above weak points and improves the traditional FMEA's results. This study firstly employs both subjective and objective weights in the decision process simultaneously. Secondly, there are two kinds of subjective weights performed in the study: aggregated weights obtained by experts' assessments as well as entropy measure. Thirdly, this approach is defined under an IVIFenvironment to ensure that the evaluation information would be preserved, and the uncertainties could be handled during the computations. Hence, it considers uncertainty in experts' judgments as well as reduces the probability of obtaining two ranking orders with the same value. Finally, the alternatives are ranked with a new collective index according to the compromise solution concept. To show the effectiveness of the proposed approach, two practical examples are solved from the recent literature in engineering applications. The proposed decision approach has an acceptable performance. Also, its advantages have been mentioned in comparison with other decision approaches.
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139
161


Z.
Hajighasemi
Department of Industrial Engineering, Faculty of Engineering, Shahed University, Tehran, Iran
Department of Industrial Engineering, Faculty
Iran
z.hajighasemi99@gmail.com


S. Meysam
Mousavi
Department of Industrial Engineering, Faculty of Engineering,
Shahed University, Tehran, Iran
Department of Industrial Engineering, Faculty
Iran
Failure modes and effects analysis (FMEA)
Compromise solution concept
Intervalvalued intuitionistic fuzzy sets (IVIFSs)
Subjective and objective weights
Horizontal directional drilling (HDD) machine
Tanker equipment
[[1] K. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Sys##tems, 31 (1989), 343{349.##[2] K. T. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets and##Systems, 64 (1994), 159{174. ##[3] G. Baek, S. Kim, S. Cheon, H. Suh and D. Lee, Prioritizing for failure modes of dynamic##positioning system using fuzzyFMEA, Journal of Korean Institute of Intelligent Systems, 25##(2015), 174{179.##[4] F. E. Boran, S. Genc M. Kurt and D. Akay, A multicriteria intuitionistic fuzzy group decision##making for supplier selection with topsis method, Expert Systems with Applications, 36##(2009), 11363{11368.##[5] M. Braglia, Mafma: multiattribute failure mode analysis, International Journal of Quality##& Reliability Management, 17 (2000), 1017{1033.##[6] N. Chanamool and T. Naenna, Fuzzy FMEA application to improve decisionmaking process##in an emergency department, Applied Soft Computing, 43 (2016), 441{453.##[7] C. L. Chang, C. C. Wei and Y. H. Lee, Failure mode and effects analysis using fuzzy method##and grey theory, Kybernetes, 28 (1999), 1072{1080.##[8] K. H. Chang and C. H. Cheng, A risk assessment methodology using intuitionistic fuzzy set##in FMEA, International Journal of Systems Science, 41 (2010), 1457{1471.##[9] K. H. Chang, C. H., Cheng and Y. C. Chang, Reprioritization of failures in a silane supply##system using an intuitionistic fuzzy set ranking technique, Soft Computing, 14 (2010), 285{##[10] T. Y. Chen, A prioritized aggregation operatorbased approach to multiple criteria decision##making using intervalvalued intuitionistic fuzzy sets: a comparative perspective, Information##Sciences, 281 (2014), 97{112.##[11] T. Y. Chen, The inclusionbased topsis method with intervalvalued intuitionistic fuzzy sets##for multiple criteria group decision making, Applied Soft Computing, 26 (2015), 57{73.##[12] K. S. Chin, Y. M. Wang, G. K. K. Poon and J. B. Yang, Failure mode and effects analysis##by data envelopment analysis, Decision Support Systems, 48 (2009), 246{256.##[13] Y. Du, H. Mo, X. Deng, R. Sadiq and Y. Deng, A new method in failure mode and eects##analysis based on evidential reasoning, International Journal of System Assurance Engineer##ing and Management, 5 (2014), 1{10.##[14] H.R. Feili, N. Akar, H. Lotfizadeh, M. Bairampour and S. Nasiri, Risk analysis of geothermal##power plants using failure modes and effects analysis (FMEA) technique, Energy Conversion##and Management, 72 (2013), 69{76.##[15] H. Hashemi, J. Bazargan, S. M. Mousavi and B. Vahdani, An extended compromise ratio##model with an application to reservoir ood control operation under an intervalvalued intu##itionistic fuzzy environment, Applied Mathematical Modelling, 38 (2014), 3495{3511.##[16] M. S. Kirkire, S. B. Rane and J. R. Jadhav, Risk management in medical product development##process using traditional FMEA and fuzzy linguistic approach: a case study, Journal of##Industrial Engineering International, 11 (2015), 595{611.##[17] M. Kumru and P. Y. Kumru, Fuzzy FMEA application to improve purchasing process in a##public hospital, Applied Soft Computing, 13 (2013), 721{733.##[18] R. J. Kuo, Y. H. Wu and T. S. Hsu, Integration of fuzzy set theory and TOPSIS into HFMEA##to improve outpatient service for elderly patients in taiwan, Journal of the Chinese Medical##Association, 75 (2012), 341{348.##[19] Z. Li and K. C. Kapur, Some perspectives to define and model reliability using fuzzy sets,##Quality Engineering, 25 (2013), 136{150.##[20] H. C. Liu, L. Liu, Q. H. Bian, Q. L. Lin, N. Dong and P. C. Xu, Failure mode and effects##analysis using fuzzy evidential reasoning approach and grey theory, Expert Systems with##Applications, 38 (2011), 4403{4415.##[21] H. C. Liu, L. Liu and P. Li, Failure mode and effects analysis using intuitionistic fuzzy hybrid##weighted euclidean distance operator, International Journal of Systems Science, 45 (2014),##2012{2030.##[22] H. C. Liu, J. X. You, M. M. Shan and L. N. Shao, Failure mode and effects analysis using##intuitionistic fuzzy hybrid TOPSIS approach, Soft Computing, 19 (2015), 1085{1098.##[23] P. Liu, L. He and X. Yu, Generalized hybrid aggregation operators based on the 2dimension##uncertain linguistic information for multiple attribute group decision making, Group Decision##and Negotiation, 25 (2016), 103{126. ##[24] P. Liu and F. Jin, A multiattribute group decisionmaking method based on weighted geomet##ric aggregation operators of intervalvalued trapezoidal fuzzy numbers, Applied Mathematical##Modelling, 36 (2012), 2498{2509.##[25] P. Liu and Y. Liu, An approach to multiple attribute group decision making based on intu##itionistic trapezoidal fuzzy power generalized aggregation operator, International Journal of##Computational Intelligence Systems, 7 (2014), 291{304.##[26] P. Liu and Y. Wang, Multiple attribute decisionmaking method based on singlevalued neu##trosophic normalized weighted bonferroni mean, Neural Computing and Applications, 25##(2014), 2001{2010.##[27] P. Liu and X. Yu, 2dimension uncertain linguistic power generalized weighted aggregation##operator and its application in multiple attribute group decision making, KnowledgeBased##Systems, 57 (2014), 69{80.##[28] P. Liu, X. Zhang and F. Jin, A multiattribute group decisionmaking method based on##intervalvalued trapezoidal fuzzy numbers hybrid harmonic averaging operators, Journal of##Intelligent & Fuzzy Systems, 23 (2012), 159{168.##[29] N. Rachieru, N. Belu and D. C. Anghel, Evaluating the risk of failure on injection pump##using fuzzy FMEA method, Applied Mechanics and Materials, 657 (2014), 976{980.##[30] S.M. SeyedHosseini N. Safaei and M. Asgharpour, Reprioritization of failures in a system##failure mode and effects analysis by decision making trial and evaluation laboratory technique,##Reliability Engineering & System Safety, 91 (2006), 872{881.##[31] F. Smarandache, Neutrosophy: neutrosophic probability, set, and logic: analytic synthesis##& synthetic analysis, American Research Press, USA, ISBN(s): 1879585634, 28(1) (1998),##[32] B. Vahdani, M. Salimi and M. Charkhchian, A new FMEA method by integrating fuzzy belief##structure and TOPSIS to improve risk evaluation process, International Journal of Advanced##Manufacturing Technology, 77 (2015), 357{368.##[33] Z. Wang, K. W. Li and W. Wang, An approach to multiattribute decision making with##intervalvalued intuitionistic fuzzy assessments and incomplete weights, Information Sciences,##179 (2009), 3026{3040.##[34] D.Woods, A failure mode and effects analysis (FMEA) from operating room setup to incision##for living donor liver transplantation, In 2015 Apha Annual Meeting & Expo, 5(2) (2015),##[35] N. Xiao, H. Z. Huang, Y. Li, L. He and T. Jin, Multiple failure modes analysis and weighted##risk priority number evaluation in FMEA, Engineering Failure Analysis, 18 (2011), 1162{##[36] Z. Xu, An overview of methods for determining OWA weights, International Journal of In##telligent Systems, 20 (2005), 843865.##[37] Z. Xu, Methods for aggregating intervalvalued intuitionistic fuzzy information and their##application to decision making, Control and Decision, 22 (2007), 215{225.##[38] R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision##making, IEEE Transactions on Systems, Man, and Cybernetics, 18 (1988), 183{190.##[39] J. Ye, Multicriteria fuzzy decisionmaking method based on a novel accuracy function un##der intervalvalued intuitionistic fuzzy environment, Expert Systems with Applications, 36##(2009), 6899{6902.##[40] J. Ye, Multicriteria fuzzy decisionmaking method using entropy weightsbased correlation##coefficients of intervalvalued intuitionistic fuzzy sets, Applied Mathematical Modelling, 34##(2010), 3864{3870.##[41] T. M. Yeh and L. Y. Chen, Fuzzybased risk priority number in fmea for semiconductor wafer##processes, International Journal of Production Research, 52 (2014), 539{549.##[42] D. Yu, Y. Wu and T. Lu, Intervalvalued intuitionistic fuzzy prioritized operators and their##application in group decision making KnowledgeBased Systems, 30 (2012), 57{66.##[43] S. X. Zeng, C. M. Tam and V. W. Tam, Integrating safety, environmental and quality risks##for project management using a fmea method, Engineering Economics, 66 (2015), 44{52. ##[44] X. Zhang and Z. Xu, Soft computing based on maximizing consensus and fuzzy topsis ap##proach to intervalvalued intuitionistic fuzzy group decision making, Applied Soft Computing,##26 (2015), 4256.##[45] Q. Zhou and V. V. Thai, Fuzzy and grey theories in failure mode and effect analysis for##tanker equipment failure prediction, Safety Science, 83 (2016), 74{79.##]
ON SOMEWHAT FUZZY AUTOMATA CONTINUOUS FUNCTIONS IN FUZZY AUTOMATA TOPOLOGICAL SPACES
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In this paper, the concepts of somewhat fuzzy automata continuous functions and somewhat fuzzy automata open functions in fuzzy automata topological spaces are introduced and some interesting properties of these functions are studied. In this connection, the concepts of fuzzy automata resolvable spaces and fuzzy automata irresolvable spaces are also introduced and their properties are studied.
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163
178


N.
Krithika
Department of Mathematics, Sri Sarada College for Women, Salem,
Tamilnadu, India
Department of Mathematics, Sri Sarada College
India


B.
Amudhambigai
Department of Mathematics, Sri Sarada College forWomen, Salem,
Tamilnadu, India
Department of Mathematics, Sri Sarada College
India
Somewhat fuzzy automata continuous function
Somewhat fuzzy automata open function
Fuzzy automata resolvable and fuzzy automata irresolvable space
[[1] K. K. Azad, On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity,##J. Math. Anal. Appl., 82 (1981), 14{32.##[2] K. K. Azad, Fuzzy Hausdroff spaces and fuzzy perfect mappings, J. Math. Anal. Appl., 82##(1981), 297{305. ##[3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182{190.##[4] P. Das, A fuzzy topology associated with a fuzzy finite state machine, Fuzzy Sets and Systems,##105(3) (1999), 469{479.##[5] D. H. Foster, Fuzzy topological groups, J. Math. Anal. Appl., 67 (1979), 549{564.##[6] K. R. Gentry and H. B. Hoyle III, Somewhat continuous functions, Czech. Math. J., 21 (96)##(1971), 5{12.##[7] M. Ghorani and M. M. Zahedi, Characterizations of complete residuated latticevalued finite##tree automata, Fuzzy Sets and Systems, 199 (2012) 28{46.##[8] E. Hewitt, A problem in set theoretic topology, Duke Math. J., 10 (1943), 309{333.##[9] J. Ignjatovic, M. Ciric and S. Bogdanovic, Determinization of fuzzy automata with member##ship values in complete residuated lattices, Information Sciences, 178 (2008), 164{180.##[10] J. Ignjatovic, M. Ciric and V. Simovic, Fuzzy relation equations and subsystems of fuzzy##transition systems, KnowledgeBased Systems, 38 (2013), 48{61.##[11] Y. M. Li, A categorical approach to latticevalued fuzzy automata, Fuzzy Sets and Systems,##157 (2006), 855{864.##[12] Y. M. Li, Finite automata theory with membership values in lattices, Information Sciences,##181 (2011), 1003{1017.##[13] P. Li and Y. M. Li, Algebraic properties of LAlanguages, Information Sciences, 176 (2006),##3232{3255.##[14] Z. H. Li, P. Li and Y. M. Li, The relationships among several types of fuzzy automata,##Information Sciences, 176 (2006), 2208{2226.##[15] D. S. Malik and J. N. Mordeson, Algebraic fuzzy automata theory, Arabian J. Sci, Eng., 25##(2000), 23{50.##[16] D. S. Malik, J. N. Mordeson and M. K. Sen, On subsystems of fuzzy finite state machines,##Fuzzy Sets and Systems, 68 (1994), 83{92.##[17] P. V. Ramakrishnan and V. Lakshmana Gomathi Nayagam, Nearly fuzzy Hausdor spaces,##Indian J. Pure Appl. Math., 31(5) (2000), 695{712.##[18] A. K. Srivastava and S. P. Tiwari, A topology for fuzzy automata, Proc. AFSS International##Conference on Fuzzy Systems, Lecture Notes in Artificial Intelligence, SpringerVerlag, 2275##(2002), 485{491.##[19] A. K. Srivastava and S. P. Tiwari, On relationships among fuzzy approximation operators,##fuzzy topology, and fuzzy automata, Fuzzy sets and systems, 138 (2003), 197{204.##[20] G. Thangaraj and G. Balasubramanian, On somewhat fuzzy continuous functions, J. Fuzzy##Math., 11(2) (2003), 725{736.##[21] S. P. Tiwari and S. Sharan, Fuzzy automata based on latticeordered monoids with algebraic##and topological aspects, Fuzzy Information and Engineering, 2 (2012), 155{164.##[22] S. P. Tiwari, A. K. Singh and S. Sharan, Fuzzy automata based on latticeordered monoid##and associated topology, Journal of Uncertain Systems, 6(1) (2012), 51{55.##[23] S. P. Tiwari, A. K. Singh, S. Sharan and V. K. Yadav, Bifuzzy core of fuzzy automata, Iranian##Journal of Fuzzy Systems, 12(2) (2015), 63{73.##[24] W. G.Wee, On generalizations of adaptive algorithm and application of the fuzzy sets concept##to pattern classification , Ph. D. Thesis, Purdue University, 1967.##[25] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338{353.##]
Persiantranslation vol. 15, no. 1, February 2018
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