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In this paper, we partially modify our definition of Hyperbolic Fuzzy Set (HFS), where we have mentioned about the indeterminateness degree of HFSs.

In this paper, we review recent advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixing models. After a general introduction to BSS and ICA, we discuss in more detail uniqueness and separability issues, presenting some new results. A fundamental difficulty in the nonlinear BSS problem and even more so in the nonlinear ICA problem is that they provide non-unique solutions without extra constraints, which are often implemented by using a suitable regularization. In this paper, we explore two possible approaches. The first one is based on structural constraints. Especially, post-nonlinear mixtures are an important special case, where a nonlinearity is applied to linear mixtures. For such mixtures, the ambiguities are essentially the same as for the linear ICA or BSS problems. The second approach uses Bayesian inference methods for estimating the best statistical parameters, under almost unconstrained models in which priors can be easily added. In the later part of this paper, various separation techniques proposed for post-nonlinear mixtures and general nonlinear mixtures are reviewed.

Fuzziness is inherent in decision data and decision making process. In this paper, interval-valued intuitionistic fuzzy set is used to capture fuzziness in multi-criteria decision making problems. The purpose of this paper is to develop a new method for solving multi-criteria decision making problem in interval-valued intuitionistic fuzzy environments. First, we introduce and discuss the concept of interval-valued intuitionistic fuzzy point operators. Using the interval-valued intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an interval-valued intuitionistic fuzzy set. Then, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker's requirement. Furthermore, a series of new score functions are defined for multi-criteria decision making problem based on the interval-valued intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.

The present paper proposes an improved score function for solving multi-criteria decision-making (MCDM) problem with partially known weight information, In it, the preferences related to criteria are taken in the form of interval-valued Pythagorean fuzzy sets. Based on these preferences and an improved score function, a score matrix has been formulated and then a linear programming based method has been proposed to solve MCDM problems with unknown attribute weights. Some generalized properties have also been proven for justification. Illustrative examples have been given for showing the superiority of the approach with the other existing functions in the decision-making process.

Recently, multi-attribute decision making (MADM) approaches concerning decision maker’s psychological behaviors have received increasing attention, but few of them have taken all the criteria interactions (positive, negative and dependent interactions) into consideration. In this paper, we combine the TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) method with the 2-additive fuzzy measure and Choquet integral theory to demonstrate how criteria interactions can be determined and further extend it into intuitionistic fuzzy environment. To begin with, we propose the novel score function and accuracy function to compare the difference among intuitionistic fuzzy sets, which have been proven to be more effective and rational than the existing measure functions. Next, we construct the nonlinear programming model based on maximum-entropy principal to obtain the optimal criteria interactions. Further, 2-additive Choquet-based dominance degree is defined whereby we put forward the 2-additive Choquet integral-based TODIM method under intuitionistic fuzzy environment to handle more challenging MADM problems. Finally, we present results of a didactic example, which concerns selection of suppliers for a manufacturing company, to evaluate the validity and rationality of proposed approach.

There have been several works aimed at the determination of the shape of the hazard rate when lifetime is treated as continuous. In the present work, we establish some theorems, that enable the identification of the nature of discrete hazard rates. Further, some applications of the results to construct new discrete bathtub-shaped distributions are also proposed.

Here, we appraise the reliability for numerous complex structures (series structure, parallel structure and bridge structure) using accuracy and score function under fuzzy environment. The main focus of this effort is to address an advanced technique for fuzzy reliability evaluation of various complex systems having different arrangements by treating reliability of the unit/component as an interval valued intuitionistic hesitant fuzzy element. This technique helps to handle uncertainty and hesitancy in multi-attribute group decision-making related issues, specially when information occurs in interval form in fuzzy set. A numerical illustration is also included to demonstrate the proposed technique.

Combining multiple information retrieval (IR) systems has been shown to improve performance over individual systems. However, it remains a challenging problem to determine when and how a set of individual systems should to be combined. In this paper, we investigate these issues using combinatorial fusion analysis and five data sets provide by TREC 2, 3, 4, 5, and 6. In particular, we compare the performance of combining six IR systems selected by random choice vs. by performance measurement from these five TREC data sets. Two experiments are conducted, which include: (1) combination of two systems and their performance outcome in terms of performance ratio and cognitive diversity, and (2) combinatorial fusion of t-systems, t = 2 to 6, using both score and rank combinations and exploration of the effect of diversity on the performance outcome. It is demonstrated in both experiments that combination of two or more systems improves the performance more significantly when the systems are selected by performance evaluation than those selected by random choice. Our work provides a distinctive method of system selection for the combination of multiple retrieval systems.

The COVID-19 epidemic has drastically altered the global landscape and thus organizations are finding it difficult to provide a modern and productive work environment to their stakeholders. As a result, personnel have shifted from in-person interactions to working remotely via video conferencing tools (VCTs). Owing to the onset of COVID-19 outbreak, usage of VTCs has gained great interest among people, especially student fraternity and employees as it reduces the hassle of direct interaction. But selecting the right platform for virtual interaction can be challenging, as it involves a number of factors in order to reduce expenses and maximize productivity. The present treatise has focused on devising a mathematical approach to identify the best VCT by structurizing the problem as a multi-criteria group decision making (MCGDM) model. To carry out the research, the notions of two popular techniques, namely Method based on the Removal Effects of Criteria (MEREC) and Weighted Aggregated Sum Product Assessment (WASPAS) have been extended under Picture fuzzy environment with the aid of Picture fuzzy weighted averaging operator to deal with MCGDM situation. Furthermore, to overcome limitations of existing Picture fuzzy score function (PFSF), a novel PFSF has been developed and its significant properties are analyzed through theoretical and numerical justifications, showing its superiority over the existing one. The proposed PFSF can not only overcome the limitations of the existing SF, but also can rank any type of Picture fuzzy numbers irrespective of their membership and non-membership grades. Thereafter, the extended MCGDM techniques have been integrated with the aid of the proposed novel PFSF and applied to assess the best VCT. The proposed technique has identified Google Meet as the best VCT for virtual communication. Consistency and robustness of proposed technique have been validated through comparative and sensitivity analysis.

Intuitionistic fuzzy set plays a vital role in data analysis and decision-making problems. In this paper, we propose an enhanced and versatile method of forecasting using the concept of intuitionistic fuzzy time series (FTS) based on their score function. The developed method has been presented in the form of simple computational steps of forecasting instead of complicated max–min compositions operator of intuitionistic fuzzy sets to compute the relational matrix $R$. Also, the proposed method is based on the maximum score and minimum accuracy function of intuitionistic fuzzy numbers (IFNs) to fuzzify the historical time series data. Further intuitionistic fuzzy logical relationship groups are defined and also provide a forecasted value and lies in an interval and is more appropriate rather than a crisp value. Furthermore, the proposed method has been implemented on the historical student enrollments data of University of Alabama and obtains the forecasted values which have been compared with the existing methods to show its superiority. The suitability of the proposed model has also been examined to forecast the movement of share market price of State Bank of India (SBI) at Bombay Stock Exchange (BSE). The results of the comparison of MSE and MAPE indicate that the proposed method produces more accurate forecasting results.

This paper’s core objective is to introduce a novel notion called hyperbolic fuzzy set (HFS) where, the grades follow the stipulation that the product of optimistic and pessimistic degree must be less than or equal to one (1), rather than their sum not exceeding one (1) as in case of IFSs. The concept of HFS originates from a hyperbola, which provides extreme flexibility to the decision makers in the representation of vague and imprecise information. It is observed that IFSs, Pythagorean fuzzy sets (PFSs), and q-rung orthopair fuzzy sets (Q-ROFSs) often failed to express the uncertain information properly under some specific situations, while HFS tends to overcome such limitations by being applicable under those perplexed situations too. In this paper, we first define some basic operational laws and few desirable properties of HFSs. Second, we define a novel score function, accuracy function, and also establish some of their properties. Third, a novel similarity and distance measure is proposed for HFSs that are capable of distinguishing between different physical objects or alternatives based on the grounds of “similitude degree” and “farness coefficient”, respectively. Later, the advantages of all of these newly defined measures have been showcased by performing a meticulous comparative analysis. Finally, these measures have been successfully applied in various COVID-19 associated problems such as medical decision-making, antivirus face-mask selection, efficient sanitizer selections, and effective medicine selection for COVID-19. The final results obtained with our newly defined measures comply with several other existing methods that we considered and the decision strategy adopted is simple, logical, and efficient. The significant findings of this study are certain to aid the healthcare department and other frontline workers to take necessary measures to reduce the intensity of the coronavirus transmission, so that we can hopefully progress toward the end of this ruthless pandemic.

In this paper, we define the basic concept of triangular neutrosophic cubic hesitant fuzzy number and their properties. We develop a triangular neutrosophic cubic hesitant fuzzy ordered weighted arithmetic averaging (TNCHFOWAA) operator and a triangular neutrosophic cubic hesitant fuzzy ordered weighted geometric averaging (TNCHFOWGA) operator to aggregate triangular neutrosophic cubic hesitant fuzzy number (TNCHFN) information and investigate their properties. Furthermore, a multiple attribute decision-making method based on the TNCHFOWAA operator and triangular neutrosophic cubic hesitant fuzzy ordered weighted geometric (TNCHFOWG) operator and the score function of TNCHFN is established under a TNCHFN environment. Finally, an illustrative example of investment alternatives is given to demonstrate the application and effectiveness of the developed approach.

Conditional probability distribution models have been widely used in economics and finance. In this chapter, we introduce two closely related popular methods to estimate conditional distribution models—Maximum Likelihood Estimation (MLE) and Quasi-MLE (QMLE). MLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is correctly specified, and QMLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is misspecified. Because the score function is an MDS and the dynamic Information Matrix (IM) equality holds when a conditional distribution model is correctly specified, the asymptotic properties of MLE is analogous to those of the OLS estimator when the regression disturbance is an MDS with conditional homoskedasticity, and we can use the Wald test, LM test and Likelihood Ratio (LR) test for hypothesis testing, where the LR test is analogous to the J · F test statistic. On the other hand, when the conditional distribution model is misspecified, the score function has mean zero, but it may no longer be an MDS and the dynamic IM equality may fail. As a result, the asymptotic properties of QMLE are analogous to those of the OLS estimator when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. Robust Wald tests and LM tests can be constructed for hypothesis testing, but the LR test can no longer be used, for a reason similar to the failure of the F-test statistic when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. We discuss methods to test the MDS property of the score function, and the dynamic IM equality, and correct specification of a conditional distribution model.

Spherical fuzzy sets are the latest extension of the ordinary fuzzy sets, which are based on three independent parameters, namely membership degree, non-membership degree and hesitancy degree. The squared sum of these parameters must be equal or less than “1”. Several papers have been published on the arithmetic operations, aggregation operators, defuzzification operations and many others in the literature. However, there are still gaps in these operations for different types of spherical fuzzy numbers such as triangular, trapezoidal, and left-right (LR) fuzzy numbers. In this paper, we focus on score and accuracy functions for several types of spherical fuzzy numbers by illustrating numerical examples.

A problem of parameter estimation for Markov-Additive Processes of arrivals is considered. It is supposed that empirical data are aggregated: only overall numbers of arrivals of various classes are observed in a long time interval. Maximum likelihood method is used for estimation: score function is derived and the gradient method is used for the optimization. Numerical example is considered.