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Objective: A new variant of energy-constrained Quickest Path Problem (QPP) is being addressed in Intuitionistic Trapezoidal Fuzzy environment. The consideration of energy with QPP enables to compute the path for continuity aware critical applications. This new variant problem called as Energy-constrained Intuitionistic Trapezoidal Fuzzy Quickest Path Problem (EITFQPP) has been considered and the computations for quickest path have been carried out. Methods: As one of the important feature of the proposed model, the weight parameters associated with a given link, e.g., delay, capacity, energy and data which are completely unknown and therefore, based on the real life situations, they may be assumed as intuitionistic trapezoidal fuzzy numbers. An algorithm to solve the EITFQPP for transmitting the fuzzy data in the network, where the nodes are associated with a sufficient amount of energy (imprecise) for continuous data flow, has been proposed. Results: In order to illustrate the implementation of the proposed algorithm, a numerical example of a benchmark network has been provided. The proposed algorithm successfully provides a quickest path using shortest paths computations under the energy constrained approach. Conclusion: The illustration through a numerical example shows the effectiveness of consideration of energy in the selections of set of paths. Finally, some of the possible directions for the future research are also discussed.

The information based on entropy of uncertainty associated with the residual life of a random variable is implemented in various fields, e.g., information theory, statistical physics, reliability t...

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.

In recent years, there is a rapid development in the direction of hydrogen energy industry having primary involvement in automobile transportation fuel. Hydrogen is the most abundant element and serves as a perfect energy carrier. In general, choosing an appropriate hydrogen power plant site is a complex selection multi-criteria decision making (MCDM) problem which involves proper assessment of a location based on various essential criteria, decision maker's expert opinion and other qualitative/quantitative factors. In the present communication, we first incorporate (R,S)-Norm Pythagorean fuzzy entropy and respective discriminant measure in the VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) MCDM techniques and proposed the modified MCDM algorithms in two different stages. Further, the hydrogen power plant site selection problem has been dealt with proper matching of the laid down essential criteria under a wider sense of Pythagorean fuzzy information measures. Such measures have not been utilized in the study of site selection problems of the hydrogen energy resources. In view of the existing literature on the MCDM problems, comparative remarks including the sensitivity analysis and the advantages have been presented for providing the novelty of the proposed methodologies. The presented work proves to be an effective tool for handling the various similar types of selection problems with a desired consistency.

Data envelopment analysis is a mathematical linear programming based method where the overall effectiveness of a set of decision-making units is assessed for accomplishing the task. In this communication, we have projected Pythagorean fuzzy entropy DEA model with the inclusion of the Pythagorean fuzzy information which has a more extensive sense of flexibility in contrast with the intuitionistic fuzzy information. Furthermore, an approach to solve a decision-making problem with Pythagorean fuzzy information has been well presented with a defined set of weights. A numerical example based on evaluation of five government universities has also been provided for illustrating the projected methodology.

The notion of T-spherical fuzzy set is the most recent generalization of intuitionistic fuzzy set available in the literature having the capability to handle the uncertainty, fuzziness and vagueness found in human sense in terms of the four parameters: membership (yes), neutral (abstain), non-membership (no) and refusal (non-participation). In the present communication, the concept of T-spherical fuzzy graph has been introduced along with the operations of product, composition, union, join and complement. Further, two algorithms utilizing the notion of T-spherical fuzzy graphs have been presented for solving the decision-making problems in the field of supply chain management and evaluation problem of service centers. In order to illustrate the actual implementation of the proposed algorithms, numerical examples have also been provided. For the sake of the novelty of the proposed approach, comparison and advantages in contrast with the methodologies of intuitionistic fuzzy set and Pythagorean fuzzy set have also been discussed.

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 the present communication, we have proposed an optimal inventory management system for deteriorating items with linear demand, shortages, variable rate of deterioration and partial backlogging in the fuzzy environment. In real life situations, we certainly face the uncertainty component associated with the different parameters such as demand, raw materials supply, various other inventory costs, rate of deterioration, etc. Uncertainty theory in fuzzy terms has been found to be a useful methodology to describe the inherent subjective indeterminacy. The objective of the present work is to optimise the total average cost function by taking triangular fuzzy numbers into account. The defuzzification methods viz. graded mean, signed distance and centroid have been applied to obtain the results of the proposed model. For a detailed illustration of the proposed model, an example of inventory model has been considered and solved in view of the parameters involved. Further, the sensitivity analysis with respect to the various influencing parameters have been presented with the help of some graphical representations. The paper is finally concluded with a scope for the possible future work.

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.

In the present communication, we study some preservation properties of the Laplace transform ordering and the moment generating function ordering of residual life and inactivity time under the reliability operation of convolution. These results are in addition to the existing results available in the literature.

In the present communication, the conditional Tsallis and Kapur entropies have been studied for their monotonic behaviour. Further, the results on the convolutions of Tsallis and Renyi entropies have been provided.

In the present paper, we introduce the idea of intuitionistic fuzzy variables by means of credibility theory. The mean value of intuitionistic fuzzy variables is obtained with the help of credibility distribution. Then, we develop a more comprehensive ranking method based on mean value of intuitionistic fuzzy variable. Also, we analyze the consistency of the propose ranking method with existing ranking methods.

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 the present communication, we have applied the concept of Triangular Intuitionistic Fuzzy Numbers (TIFNs) to the study of a Multiple Criteria Decision Making (MCDM) problem for finding the best alternative where the linguistic variables for the criteria are intuitively predefined in the form of TIFNs. On the basis of decision maker's qualitative opinions as well as the management's opinions to the criteria, the weight of each criterion is being calculated with the help of parametric entropy and 'useful' parametric entropy under a -- cut/(a, s) -- cut based distance measures for different possible values of parameters. Also, an algorithm for Triangular Intuitionistic Fuzzy Multi-criteria Decision Making (TIF-MCDM) problem where the ranking of the available alternatives by calculating the various distances between the ideal alternative and all the available alternatives has been provided. An illustrative example showing the procedure to rank the alternatives in view of the different opinions has also been provided.

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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