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The notion of picture fuzzy set and distance measures are found to be useful in various models and situations in which the human opinions involve the responses of types: yes, abstain, no and refusal. In the present paper, we propose a novel concept of probabilistic distance measure for picture fuzzy sets where the probability of occurrence of an event with respect to picture fuzzy membership and the probability of non-occurrence of the event with respect to picture fuzzy non-membership have been duly taken into consideration. This framework has been clearly addressed through outline of a formulated problem and its probable solution structure along with its proof of validity. Further, the proposed probabilistic distance measure has been utilized to propose an algorithm for solving some classification decision making problems in a more generalized way. Some important illustrative examples related to the problem of classification- building material classification, mineral classification and a decision making problem of financial investment risk have been worked out in order to show the implementation of the proposed methodology. The obtained results have also been compared with the existing approaches of solving the classification problems. The uncertainty feature of the problem has been handled in a more broader sense which reflects the advantage of the introduced approach.

The present communication aims to study the Triangular Intuitionistic Fuzzy Multiple Criteria Decision Making (TIF-MCDM) problem for finding the best option where the phonetic factors for the criteria are pre-characterized. In perspective of this, a new parametric entropy under α-cut/(α,/β)- cut based distance measures has been proposed and implemented for various conceivable estimations of parameters. A survey structure based on a questionnaire for the purchase of a car has also been studied and devised the ranking procedure for the evaluation criteria. Further, a ranking algorithm for TIF-MCDM issue for the accessible options by processing the various distances between the perfect option and all the accessible options has been proposed. An illustrative example to rank the alternatives in view of different opinions has also been included.

In the present manuscript, we introduce a novel concept of T-spherical fuzzy soft set with various important operations and properties. In the field of information theory, an aggregation operator is a structured mathematical function which aggregates all the information received as input and provides a single output entity, which are found to be applicable for various important decision making applications. Some averaging aggregation operators and geometric aggregation operators (weighted, ordered and hybrid) for T-spherical fuzzy soft numbers have been proposed with their various properties. Further, utilizing the proposed aggregation operators of various types along with the properly defined score function/accuracy function, an algorithm for solving a decision making problem has been provided. The proposed methodology has also been well illustrated with the help of a numerical example. Some comparative remarks and advantages of the introduced notion of T-spherical fuzzy soft set and the proposed methodology have been listed for a better motivation and readability.

Eigen fuzzy set of a fuzzy relation often occurs to be invariant under different computational aspects. The present communication introduces the novel concept of eigen spherical fuzzy set of spherical fuzzy relation along with various composition operators for the first time. We have proposed two algorithms to determine the greatest eigen spherical fuzzy sets and least eigen spherical fuzzy sets using the $max-min$ and $min-max$ composition operators respectively and illustrated the steps with the help of flow charts. Further, two numerical examples related to different fields of decision-making problems have been taken into account for illustrating the proposed methodology. The scope of future work in the field of image information retrieval, genetic algorithm for image reconstruction and notion of eigen spherical fuzzy soft sets/matrices has been duly outlined. The comparative remarks and advantages of the proposed eigen spherical fuzzy sets have also been included for a better readability.

The notion of interval-valued Pythagorean fuzzy sets permits four important parameters, i.e., membership degree, non-membership degree, and a pair of values strength of commitment and direction of commitment, to a given set to have an interval value in dealing with imprecise information. In the present communication, a new accuracy function is being provided to overcome the shortcomings of the existing score and available accuracy functions for interval-valued Pythagorean fuzzy sets. The validity of the proposed accuracy function has been discussed in detail through the illustrative examples. Further, a new interval-valued Pythagorean fuzzy $p$-distance measure for interval-valued Pythagorean fuzzy numbers has been proposed and used in context with the existing weighted averaging operators. Finally, in view of the proposed accuracy function, distance measure and weighted averaging operators, a numerical example of multi-criteria decision-making problem has been solved to validate the proposed methodology.

In the present communication, we propose a new approach for detecting copy-move forgery in digital images using statistical moments and two dimensional discrete cosine transform. We first slide a window centered around every pixel of the suspicious image, then each window is passed through two dimensional discrete cosine transform (2 D – DCT) to obtain the quantized coefficient matrix. The low dimensional statistical feature vector of each quantized coefficient matrix is obtained and arranged in a feature matrix F . The columns of F contain 4 - statistical features, i.e., mean Me , variance Var , third order moment skewness S k and fourth order moment kurtosis Kr obtained from the quantized coefficient matrix. In order to make similar windows adjacent, the feature matrix F is lexicographically sorted using radix sort. Finally, a copy-move forgery detection is performed using the adjacent pairs of feature vectors. It has also been observed that the proposed method has the lower dimension feature vector with lower computational complexity.

The shortest path problem is a classical network optimization problem which has a wide range of applications in various fields of science and engineering. In the present communication, an algorithm to find the shortest path and shortest distance in an interval-valued intuitionistic fuzzy graph with nodes and links being crisp but the edge weights will be Interval-valued Intuitionistic Trapezoidal Fuzzy Numbers (IITFNs). Finally, a numerical example has been provided for illustrating the proposed approach.

In this paper, we proposed an algorithm to find the optimal threshold value for denoising an image. A new cost function is designed to find the optimal threshold in every image. The cost function is based on the intuitionistic fuzzy divergence measure of the denoised image and original image. In addition, the intuitionistic fuzzy entropy of denoised image is added to the cost function. This is necessary, because when the algorithm threshold value is decreased, the denoised image is blurred, although its divergence of original image is decreased. When the value of intutionistic fuzzy entropy and intutionistic fuzzy divergence measure are minimum, the sum that is the cost value will also minimum. The threshold for image denoising with a minimum cost value will be the optimal threshold for image denoising. 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.

Reliability Properties of Residual Life Time and Inactivity Time of Series and Parallel SystemThe concepts of residual life time and inactivity time are extensively used in reliability theory for modeling life time data. In this paper we prove some new results on stochastic comparisons of residual life time and inactivity time in series and parallel systems. These results are in addition to the existing results of Li & Zhang (2003) and Li & Lu (2003). We also present sufficient conditions for aging properties of the residual life time and inactivity life time of series and parallel systems. Some examples from Weibull and Gompertz distributions are provided to support the results as well.

the present communication, we discuss some reliability properties of mean inactivity time order and study the preservation of mean inactivity time under some weight function. Weighted Distribution methods arise in the context of research related to reliability, data gathering, ecology, inference, modeling, bio-medicine and several other areas. In general, weighted distributions have wide role for the analysis of life time data.

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. 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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.