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In the present work, the adjacency matrix, the energy and the Laplacian energy for a picture fuzzy graph/directed graph have been introduced along with their lower and the upper bounds. Further, in the selection problem of decision making, a methodology for the ranking of the available alternatives has been presented by utilizing the picture fuzzy graph and its energy/Laplacian energy. For the shake of demonstrating the implementation of the introduced methodology, the task of site selection for the hydropower plant has been carried out as an application. The originality of the introduced approach, comparative remarks, advantageous features and limitations have also been studied in contrast with intuitionistic fuzzy and Pythagorean fuzzy information.

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 the present communication, a parametric R-norm fuzzy entropy of a fuzzy variable and subsequently R-norm fuzzy directed divergence measure of fuzzy variables by taking the credibility distribution measure into account have been proposed along with their proof of validity. Based on the proposed information measures, some new results along with its various properties and the monotonic behavior have been studied in detail. Further, some important results and inequalities related to their corresponding noiseless coding theorems have been well established. Some particular cases related to the proposed fuzzy information measures and fuzzy mean codeword length in terms of existing results have also been discussed. In view the monotonic behavior of the fuzzy mean codeword length, an application to chose the appropriate value of the parameter R to reduce the fuzzy mean codeword length significantly for optimal code has been presented with a possible variability.

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 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 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.

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 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.

Equilibrium distribution finds various applications in reliability, stochastic processes, repairable systems and many areas of applied probability such as renewal risk model etc. This paper deals with some of the reliability properties of reliability structures under equilibrium random variable(s). The likelihood ratio ordering for the series and parallel systems has been established under equilibrium. Further, the log-concavity properties of series and parallel systems having independently and identically distributed (i.i.d.) components under the condition of equilibrium have been studied.

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.

Reliability Engineering has an appreciable role in the design and development of a technical system which should be reliable in various applications viz. military application, space exploration etc. In the present paper, we consider the framework of Series and Parallel Systems under the usage of trapezoidal intuitionistic fuzzy sets for the evaluation. An application of the presented framework has also been provided for marine power plant.