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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 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.

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 fuzzy set theory, it is well known that a fuzzy number can be uniquely determined through its position and entropy. Hence, by using the concept of fuzzy entropy the estimators of the fuzzy regression coefficients may be estimated. In the present communication, a fuzzy linear regression (FLR) model with some restrictions in the form of prior information has been considered. The estimators of regression coefficients have been obtained with the help of fuzzy entropy for the restricted/unrestricted FLR model by assigning some weights in the distance function. Some numerical examples have also been provided in order to illustrate the proposed model along with the obtained weighted estimators. Further, in order to compare the performance of unrestricted estimator and restricted estimator, a simulation study has been conducted by using two fundamental criteria of dominance – mean squared error matrix (MSEM) and absolute bias.

In the present paper, a new concept of ‘useful’ fuzzy information, based on utility, is introduced by considering the uncertainties of fuzziness and probabilities of random events. A ‘useful’ fuzzy measure of integrated ambiguity is obtained by integration of fuzzy and probabilistic uncertainties with utility. A new ‘useful’ measure of fuzzy-directed divergence of a fuzzy set from another fuzzy set is proposed and its validity proved. Finally, the constrained optimisation of ‘useful’ fuzzy entropy and ‘useful’ fuzzy-directed divergence is studied.

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 communication, we have implemented the concept of intuitionistic trapezoidal fuzzy numbers (ITFNs) to the study of multiple criteria decision making (MCDM) problem for evaluating the best vendor/supplier whose information take the form of ITFNs. We propose a new algorithm for ITF-MCDM problem where the weights of the involved attributes are supposed to be completely unknown. These weights have been calculated on the basis of the decision maker's qualitative opinion to the attributes with the help of pre-defined linguistic variables and an entropy measure. Finally, the ranking of the vendors/suppliers has been determined by calculating the hamming distance between the ideal alternative and all the available alternatives.

The concept of fuzzy regression model with fuzzy regression coefficients was first introduced in \cite{[1]}, \cite{[2]} and \cite{[3]}. A fuzzy number can be uniquely determined through its position and entropy as described in \cite{[4]}. Hence, by using the concept of fuzzy entropy the estimators of the fuzzy regression coefficients may be estimated. In the present communication, we develop a newer fuzzy linear regression (FLR) model with some restrictions in the form of prior information. We have obtained the estimators of regression coefficients with the help of fuzzy entropy for the restricted FLR model. Applications on some hypothetical numerical examples are provided in order to illustrate the proposed model and the obtained estimators. Cross validation of the regression model has been done based on computation of $R^2$ and using $F$-test. The proposed model may find applications in various areas of actuarial sciences, fuzzy time series analysis, management decision making etc.

The concept of intuitionistic fuzzy sets (IFSs) is considered as a generalization of fuzzy sets and capable of capturing the information that includes some degree of hesitation which is applicable in various fields of research. In the present communication, various existing similarity measures for intuitionistic fuzzy sets have been studied. A new similarity measure for intuitionistic fuzzy sets (IFSs) which calculate the similarity between IFSs on the basis of NTV Metric has been proposed. Further, a computational analysis for medical diagnosis has been performed on the basis of the calculation of similarity from a considered case to all considered illness and values of all symptoms. Based on the comparative analysis, we find that the proposed similarity measure is a better and more robust and gives closer results as compared to the existing similarity measures for IFSs.

The existing fuzzy information system associated with the probabilities of a number of events, does not take into account the effectiveness or importance or utility of these events. In the present communication, we propose a new mathematical model for `useful' fuzzy information system which involves fuzzy information, randomness and utility distribution. The model has been designed in order to study information in a random experiment. Based on the model, a family of fuzzy directed divergence measures has also been introduced and discussed. The parametric information improvement measure in view of fuzziness and utility has been obtained. The monotonic nature of the information measures has been studied with the help of numerical examples with certain number of experimental observations.