Search

In fuzzy set theory, it is well known that a triangular fuzzy number can be uniquely determined through its position and entropies. In the present communication, we extend this concept on triangular intuitionistic fuzzy number for its one-to-one correspondence with its position and entropies. Using the concept of fuzzy entropy the estimators of the intuitionistic fuzzy regression coefficients have been estimated in the unrestricted regression model. An intuitionistic fuzzy weighted linear regression (IFWLR) model with some restrictions in the form of prior information has been considered. Further, the estimators of regression coefficients have been obtained with the help of fuzzy entropy for the restricted/unrestricted IFWLR model by assigning some weights in the distance function.

In the present paper, we analyze the vague reliability of k-out-of-n : G system (particularly, series and parallel system) with independent and non-identically distributed components, where the reliability of the components are unknown. The reliability of each component has been estimated using statistical confidence interval approach. Then we converted these statistical confidence interval into triangular intuitionistic fuzzy numbers. Based on these triangular intuitionistic fuzzy numbers, the reliability of the k-out-of-n : G system has been calculated. Further, in order to implement the proposed methodology and to analyze the results of k-out-of-n : G system, a numerical example has been provided., {"references":["K. Atanassov, \"lntuitionistic fuzzy sets,\" VII ITKR's Session, Deposed in Central Sci.- Techn. Library of Bulg. Acd. of Sci. Sofia, pp. 1677–84, 1983.","K. Atanassov, \"Intuitionistic fuzzy sets,\" Fuzzy Sets and Systems, vol. 20(1), pp.87-96, 1986.","K. Atanassov and G. Gargov, \"Interval-valued intuitionistic fuzzy sets,\" Fuzzy Sets and Systems, vol. 31(3), 343-349, 1989.","RE Barlow and KD. Heidtmann, \"Computing k-out-of-n system reliability,\" IEEE Trans. on Reliability, vol. R-33, pp. 322-323, 1984.","P. Burillo, H. Bustince, and V. Mohedano, \"Some Definition of Intuitionistic Fuzzy Number First Properties,\" Proc. of the First Workshop on Fuzzy based expert systems, pp. 53–55, Sofia, Bulgaria, September 1994.","H. Bustince and P. Burillo, \"Vague sets are intuitionistic fuzzy sets,\" Fuzzy Sets and Systems, vol. 79, 403-405, 1996.","K.Y. Cai and C.Y. Wen, \"Street-Lighting Lamps Replacement: A Fuzzy Viewpoint,\" Fuzzy Sets and Systems, vol. 37, pp. 161-172, 1990.","K.Y. Cai, C.Y. Wen, and M.L. Zhang, \"Fuzzy variables as a Basis for a Theory of Fuzzy Reliability in the Possibility Context,\" Fuzzy Sets and Systems, vol. 42, pp. 145–172, 1991a.","K.Y. Cai, C.Y. Wen and M.L. Zhang, \"Posbist Reliability Behavior of Typical Systems With Two Types of Failures,\" Fuzzy Sets and Systems, 43, 17-32, 1991b.\n[10] K.Y. Cai, C.Y. Wen and M.L. Zhang, (1993), \"Fuzzy States as a Basis for a Theory of Fuzzy Reliability,\" Microelectronic Reliability, vol. 33, 2253-2263, 1993.\n[11] K.Y. Cai, C.Y. Wen and M.L. Zhang, \"Coherent Systems in Profust Reliability Theory,\" in: Reliability and Safety Analyzes Under Fuzziness, Heidelberg: Physica-Verlag, pp. 81-94, 1995.\n[12] C.H. Cheng and D.L. Mon, \"Fuzzy System Reliability Analysis by Interval of Confidence,\" Fuzzy Sets and Systems, vol. 56, pp. 29–35, 1993.\n[13] S.M. Chen, \"Fuzzy System Reliability Analysis using Fuzzy Number Arithmetic Operations,\" Fuzzy Sets and Systems, vol. 64, pp. 31–38, 1994.\n[14] C. Cornelis, K. Atanassov and E.E. Kerre, \"Intuitionistic fuzzy sets and interval-valued fuzzy sets: a critical comparison,\" in: Proceedings of Third European Conference on Fuzzy Logic and Technology (EUSFLAT03), Zittau, Germany, pp. 159-163, 2003.\n[15] G. Deschrijver and E.E. Kerre, \"On the relationship between some extensions of fuzzy set theory,\" Fuzzy Sets and Systems, vol. 133, 227-235, 2003.\n[16] D. Dubois and H. Prade, \"Possibility Theory,\" New York: Plenum Press, 1988.\n[17] Y. Dutuit and A. Rauzy, \"New insights in the assessment of k-out-of-n and related systems,\" Reliability Engineering and System Safety, vol. 72, pp. 303-314, 2001.\n[18] W.L. Gau and D.J. Buehrer, \"Vague sets,\" IEEE Transactions on systems, Man and Cybernetics, vol. 23(2), 610-614, 1993.\n[19] W. Kuo and M.J. Zuo, \"Optimal reliability modeling,\" Wiley, New York, Chapter 7, 258-264, 2003.\n[20] G.S. Mahapatra and T.K. Roy, \"Reliability Evaluation Using Triangular Intuitionistic. Fuzzy Numbers Arithmetic Operations,\" International Journal of Computational and Mathematical Sciences, vol. 3:5, pp. 225–232, 2009.\n[21] K.B. Misra, \"Reliability analysis and prediction: A methodology oriented treatment,\". Elsevier, Amsterdam, 1992.\n[22] T. Onisawa and J. Kacprzyk, \"Reliability and Safety Analyses Under Fuzziness,\" Physica-Verlag (Springer), vol. 4, Heidelberg, 1995. (Book) [23] A.M. Rushdi, \"Utilization of symmetric switching functions in the computation of k-out-of-n system reliability,\" Microelectronics and\nReliability, vol. 26, pp. 973-987, 1986.\n[24] A.M. Rushdi, \"Reliability of k-out-of-n systems,\" In: Misra KB, editor. New trends in system reliability evaluation, vol. 16, pp. 185-227, 1993.\n[25] D. Singer , \"A Fuzzy Set Approach to Fault Tree and Reliability Analysis,\" Fuzzy Sets and Systems, vol. 34, 145–155, 1990.\n[26] J.S. Yao, J.S. Su and T.S. Shing, \"Fuzzy System Reliability Analysis Using Triangular Fuzzy Numbers Based on Statistical Data,\" Journal of Information Science and Engineering, vol. 24, pp. 1521–1535, 2008.\n[27] Y.J. Yang and C. Hinde, \"A new extension of fuzzy sets using rough sets: R-fuzzy sets,\" Information Sciences vol. 180, 354-365, 2010.\n[28] L.A. Zadeh, \"The concept of a linguistic variable and its application to approximate reasoning-I,\" Information Sciences, vol. 8, 199-249, 1975.\n[29] L.A. Zadeh, \"Fuzzy Sets as a Basis for a Theory of Possibility,\" Fuzzy Sets and Systems, vol. 1, pp. 3-28, 1978."]}

In the present paper, we analyze the vague reliability of k-out-of-n : G system (particularly, series and parallel system) with independent and non-identically distributed components, where the reliability of the components are unknown. The reliability of each component has been estimated using statistical confidence interval approach. Then we converted these statistical confidence interval into triangular intuitionistic fuzzy numbers. Based on these triangular intuitionistic fuzzy numbers, the reliability of the k-out-of-n : G system has been calculated. Further, in order to implement the proposed methodology and to analyze the results of k-out-of-n : G system, a numerical example has been provided.

The primary objective of the paper is to propose a new method for solving assignment problem under uncertain situation. In the classical assignment problem ( ), denotes the cost for assigning the job to the person which is deterministic in nature. Here in some uncertain situation, we have assigned a cost in the form of composite relative degree instead of and this replaced cost is in the maximization form. In this paper, it has been solved and validated by the two proposed algorithms, a new mathematical formulation of assignment problem has been presented where the cost has been considered to be an IVIFN and the membership of elements in the set can be explained by positive and negative evidences. To determine the composite relative degree of similarity of the concept of similarity measure and the score function is used for validating the solution which is obtained by Composite relative similarity degree method. Further, hypothetical numeric illusion is conducted to clarify the method's effectiveness and feasibility developed in the study. Finally, conclusion and suggestion for future work are also proposed.

In the present communication, stochastic comparison of a series (parallel) system having heterogeneous components with random lifetimes and series (parallel) system having homogeneous exponential components with random lifetimes has been studied. Further, conditions under which such a comparison is possible has been established., {"references":["N. Balakrishnan and C.R. Rao, Handbook of Statistics 16 - Order\nStatistics: Theory and Methods, Elsevier, New York, 1998a.","N. Balakrishnan and C.R. Rao Handbook of Statistics 17 - Order\nStatistics: Applications, Elsevier, New York, 1998b.","P. Pledger and F. Proschan, Comparison of Order Statistics and of spacings\nfrom Heterogenous distributions, in : J.S. Rustagi (Ed.), Optimizing\nMethods in Statistics, Academic Press, New York, 1971.","S.C. Kochar and J. Rojo, Some new results on stochastic comparisons\nof spacings from heterogeneous exponential distributions, Journal of\nMultivariate Analysis 57, 69-83, 1996.","S.C. Kochar and M. Xu, Stochastic comparisons of parallel systems\nwhen components have proportional hazard rates, Probability in the\nEngineering and Informational Sciences 21, 597-609, 2007.","B. Khaledi and S.C. Kochar, Sample range some stochastic comparisons\nresults, Calcutta Statistical Association Bulletin 50, 283-291, 2000.","C. Genest, S.C. Kochar and M. Xu, On the range of heterogeneous\nsamples, Journal of Multivariate Analysis, 100, 1587-1592, 2009.","P. Zhao and X. Li Stochastic order of sample range from heterogeneous\nexponential random variables, Probability in the Engineering and Informational\nSciences 23, 17-29, 2009.","T. Mao and T. Hu, Equivalent characterizations on orderings of order\nstatistics and sample ranges, Probability in the Engineering and Informational\nSciences, 24, 245-262, 2010.\n[10] M. Xu and N. Balakrishnan, On the convolution of heterogeneous\nBernoulli random variables, Journal of Applied Probability, 48 (3), 877-\n884, 2011.\n[11] S.C. Kochar and M. Xu, Stochastic comparisons of spacings from\nheterogeneous samples, In: Advances in Directional and Linear Statistics\n(Eds., M. Wells and A. Sengupta), 113-129, Springer, New York, 2011a.\n[12] S.C. Kochar and M. Xu, Some unified results on comparing linear\ncombinations of independent gamma random variables, Technical Report,\nIllinois State University, Normal, Illinois, 2011b.\n[13] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life\nTesting, To Begin With, Silver Spring, Maryland, 1981.\n[14] M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications,\nSpringer, New York, 2007.\n[15] A. M¨uller and D. Stoyan, Comparison Methods for Stochastic Models\nand Risks, Wiley Series in Probability and Statistics. John Wiley & Sons\nLtd., Chichester, 2002."]}

In the present communication, we propose an exponential entropy of order β for an intuitionistic fuzzy set and a new intuitionistic fuzzy divergence measure for the intuitionistic fuzzy sets along with their proof of validity. Further, a new algorithm for image edge detection has been framed on the basis of the proposed intuitionistic information measures. The implementation and applicability of the proposed algorithm have been illustrated by taking different sample images. The obtained results have been finally analyzed and found to be better than the existing ones.