This work analyze singularly perturbed convection-diffusion-reaction (CDR) models with two parame... more This work analyze singularly perturbed convection-diffusion-reaction (CDR) models with two parameters and variable coefficients by developing a mesh-free scheme based on local radial basis function-finite difference (LRBF-FD) approximation. In the evolvement of the scheme, time derivative is discretized by forward finite difference. After that, LRBF-FD approximation is used for spatial discretization, and we obtained a system of linear equations. Then, the obtained linear system is solved by LU decomposition method in MATLAB. For numerical simulation, four singularly perturbed models are pondered to check the efficiency and chastity of the proposed scheme.
A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost ... more A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost and fuzzy time of transportation without according priorities to them is considered. To the best of our knowledge, there is no method in the literature to find efficient solutions of the bi-objective transportation problem under uncertainty. In this paper, a bi-objective transportation problem in an uncertain environment has been formulated. An algorithm has been proposed to find efficient solutions of the bi-objective transportation problem under uncertainty. The proposed algorithm avoids the degeneracy and gives the optimal solution faster than other existing algorithms for the given uncertain transportation problem.
In this study, a new composite algorithm with the help of the finite difference and the modified ... more In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the lite...
The objective of this work is to present a triangular interval type-2 (TIT2) intuitionistic fuzzy... more The objective of this work is to present a triangular interval type-2 (TIT2) intuitionistic fuzzy sets and their corresponding aggregation operators, namely, TIT2 intuitionistic fuzzy weighted averaging, TIT2 intuitionistic fuzzy ordered weighted averaging and TIT2 intuitionistic fuzzy hybrid averaging based on Frank norm operation laws. Furthermore, based on these operators, an approach to multi-criteria decision-making, in which assessments are in the form of TIT2 intuitionistic fuzzy numbers has been developed. A practical example to illustrate the decision-making process has been presented and compared their results with the existing operator results.
Background: Chemical straightening has increased in popularity because of its durability over the... more Background: Chemical straightening has increased in popularity because of its durability over thermal straightening and because it gives the hair a natural smooth, shiny look, more styling options for a longer time. The aim of this study was to evaluate the degree of damage that occur to the hair after chemical straightener treatment by assessing the change in hair porosity and hair cysteic acid content before and after straightener treatment. Methods: Forty hair samples were collected from female volunteers; each sample was divided into 2 groups; group A (before straightening) and group B (after straightening). Hair samples were washed then blow dried after that hair straightener was applied for 30 min then flat iron was done, hair samples were washed and blow dried. Each group was subjected to test their porosity and Fourier transform infrared spectroscopy examination to assess cysteic acid content. Results: Hair porosity showed significant increase after straitening. Cysteic acid and disulfide content showed insignificant difference between both groups. Conclusion: Hair straighteners may cause great damage to hair internal structures. So, proper choice of the material used and assessing the hair characters before straightening is necessary.
Type 2 intuitionistic fuzzy set (T2IFS) is one of the most important concepts to describe fuzzy i... more Type 2 intuitionistic fuzzy set (T2IFS) is one of the most important concepts to describe fuzzy information by providing an additional degree of freedom to the decision maker to decide the decision-making process. The fundamental superiority of the T2IFS, over the intuitionistic fuzzy set, has been its ability to capture the degrees of membership of relevant membership values, where the uncertainty is handled more accurately, and hence accommodate more uncertainties. Keeping these features in mind, this paper explores the theory of T2IFS by defining some families of the similarity measures to measure the degree of similarity between the two or more T2IFSs. For this, the characteristic of each object is measured in terms of degrees of primary membership and non-membership, secondary membership and non-membership and the variance margin function between them. Based on that, several formulations of the similarities measures are defined and their properties are investigated . Also, some fundamental relation between the proposed measures is stated. Further, based on the similarity measure, a group decision-making approach is explored to rank the alternatives. An illustrative example is provided to demonstrate the approach, and the comparative analysis with some of the existing measures is performed to explore the results.
Type-2 intuitionistic fuzzy set (T2IFS) is a powerful and important extension of the classical fu... more Type-2 intuitionistic fuzzy set (T2IFS) is a powerful and important extension of the classical fuzzy set, intuitionistic fuzzy set to measure the vagueness and uncertainty. In a practical decision-making process, there always occurs an inter-relationship among the multi-input arguments. To deal with this point, the motivation of the present paper is to develop some new interval type-2 (IT2) intuitionistic fuzzy aggregation operators which can consider the multi interaction between the input argument. To achieve it, we define a symmetric triangular interval T2IFS (TIT2IFS), its operations, Hamy mean (HM) operator to aggregate the preference of the symmetric TIT2IFS and then shows its applicability through a multi-criteria decision making (MCDM). Several enviable properties and particular cases together with following different parameter values of this operator are calculated in detail. At last a numerical illustration is to given to exemplify the practicability of the proposed techni...
Abstract This work deals to capture the different types of patterns of nonlinear time dependent c... more Abstract This work deals to capture the different types of patterns of nonlinear time dependent coupled reaction-diffusion models. To accomplish this work, a new differential quadrature (DQ) algorithm is developed with the help of modified trigonometric cubic B-spline functions. The stability part of the developed algorithm is studied by matrix stability analysis method. In the experimental part, different types of patterns of Gray–Scott, Schnakenberg, Isothermal Chemical and Brusselator Models are captured which are similar to the existing patterns of the models.
Type-2 fuzzy set (T2FS) is a generalization of the ordinary fuzzy set in which the membership val... more Type-2 fuzzy set (T2FS) is a generalization of the ordinary fuzzy set in which the membership value for each member of the set is itself a fuzzy set. However, it is difficult, in some situations, for the decision-makers to give their preferences towards the object in terms of single or exact number. For handling this, a concept of type-2 intuitionistic fuzzy set (T2IFS) has been proposed and hence under this environment, a family of distance measures based on Hamming, Euclidean and Hausdorff metrics are presented. Some of its desirable properties have also been investigated in details. Finally, based on these measures, a group decision making method has been presented for ranking the alternatives. The proposed measures has been illustrated with a numerical example.
In this article, we have pointed out that some propositions corresponding to the distance measure... more In this article, we have pointed out that some propositions corresponding to the distance measure between the type-2 fuzzy sets (T2FSs) as provided by Singh (Frontiers of Computer Science, 2014, 8(5), 741–752), are incorrect by a counterexample. Further, these propositions have been corrected in the present manuscript by giving a correct relation between the T2FSs and validate it with a numerical example.
This work analyze singularly perturbed convection-diffusion-reaction (CDR) models with two parame... more This work analyze singularly perturbed convection-diffusion-reaction (CDR) models with two parameters and variable coefficients by developing a mesh-free scheme based on local radial basis function-finite difference (LRBF-FD) approximation. In the evolvement of the scheme, time derivative is discretized by forward finite difference. After that, LRBF-FD approximation is used for spatial discretization, and we obtained a system of linear equations. Then, the obtained linear system is solved by LU decomposition method in MATLAB. For numerical simulation, four singularly perturbed models are pondered to check the efficiency and chastity of the proposed scheme.
A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost ... more A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost and fuzzy time of transportation without according priorities to them is considered. To the best of our knowledge, there is no method in the literature to find efficient solutions of the bi-objective transportation problem under uncertainty. In this paper, a bi-objective transportation problem in an uncertain environment has been formulated. An algorithm has been proposed to find efficient solutions of the bi-objective transportation problem under uncertainty. The proposed algorithm avoids the degeneracy and gives the optimal solution faster than other existing algorithms for the given uncertain transportation problem.
In this study, a new composite algorithm with the help of the finite difference and the modified ... more In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the lite...
The objective of this work is to present a triangular interval type-2 (TIT2) intuitionistic fuzzy... more The objective of this work is to present a triangular interval type-2 (TIT2) intuitionistic fuzzy sets and their corresponding aggregation operators, namely, TIT2 intuitionistic fuzzy weighted averaging, TIT2 intuitionistic fuzzy ordered weighted averaging and TIT2 intuitionistic fuzzy hybrid averaging based on Frank norm operation laws. Furthermore, based on these operators, an approach to multi-criteria decision-making, in which assessments are in the form of TIT2 intuitionistic fuzzy numbers has been developed. A practical example to illustrate the decision-making process has been presented and compared their results with the existing operator results.
Background: Chemical straightening has increased in popularity because of its durability over the... more Background: Chemical straightening has increased in popularity because of its durability over thermal straightening and because it gives the hair a natural smooth, shiny look, more styling options for a longer time. The aim of this study was to evaluate the degree of damage that occur to the hair after chemical straightener treatment by assessing the change in hair porosity and hair cysteic acid content before and after straightener treatment. Methods: Forty hair samples were collected from female volunteers; each sample was divided into 2 groups; group A (before straightening) and group B (after straightening). Hair samples were washed then blow dried after that hair straightener was applied for 30 min then flat iron was done, hair samples were washed and blow dried. Each group was subjected to test their porosity and Fourier transform infrared spectroscopy examination to assess cysteic acid content. Results: Hair porosity showed significant increase after straitening. Cysteic acid and disulfide content showed insignificant difference between both groups. Conclusion: Hair straighteners may cause great damage to hair internal structures. So, proper choice of the material used and assessing the hair characters before straightening is necessary.
Type 2 intuitionistic fuzzy set (T2IFS) is one of the most important concepts to describe fuzzy i... more Type 2 intuitionistic fuzzy set (T2IFS) is one of the most important concepts to describe fuzzy information by providing an additional degree of freedom to the decision maker to decide the decision-making process. The fundamental superiority of the T2IFS, over the intuitionistic fuzzy set, has been its ability to capture the degrees of membership of relevant membership values, where the uncertainty is handled more accurately, and hence accommodate more uncertainties. Keeping these features in mind, this paper explores the theory of T2IFS by defining some families of the similarity measures to measure the degree of similarity between the two or more T2IFSs. For this, the characteristic of each object is measured in terms of degrees of primary membership and non-membership, secondary membership and non-membership and the variance margin function between them. Based on that, several formulations of the similarities measures are defined and their properties are investigated . Also, some fundamental relation between the proposed measures is stated. Further, based on the similarity measure, a group decision-making approach is explored to rank the alternatives. An illustrative example is provided to demonstrate the approach, and the comparative analysis with some of the existing measures is performed to explore the results.
Type-2 intuitionistic fuzzy set (T2IFS) is a powerful and important extension of the classical fu... more Type-2 intuitionistic fuzzy set (T2IFS) is a powerful and important extension of the classical fuzzy set, intuitionistic fuzzy set to measure the vagueness and uncertainty. In a practical decision-making process, there always occurs an inter-relationship among the multi-input arguments. To deal with this point, the motivation of the present paper is to develop some new interval type-2 (IT2) intuitionistic fuzzy aggregation operators which can consider the multi interaction between the input argument. To achieve it, we define a symmetric triangular interval T2IFS (TIT2IFS), its operations, Hamy mean (HM) operator to aggregate the preference of the symmetric TIT2IFS and then shows its applicability through a multi-criteria decision making (MCDM). Several enviable properties and particular cases together with following different parameter values of this operator are calculated in detail. At last a numerical illustration is to given to exemplify the practicability of the proposed techni...
Abstract This work deals to capture the different types of patterns of nonlinear time dependent c... more Abstract This work deals to capture the different types of patterns of nonlinear time dependent coupled reaction-diffusion models. To accomplish this work, a new differential quadrature (DQ) algorithm is developed with the help of modified trigonometric cubic B-spline functions. The stability part of the developed algorithm is studied by matrix stability analysis method. In the experimental part, different types of patterns of Gray–Scott, Schnakenberg, Isothermal Chemical and Brusselator Models are captured which are similar to the existing patterns of the models.
Type-2 fuzzy set (T2FS) is a generalization of the ordinary fuzzy set in which the membership val... more Type-2 fuzzy set (T2FS) is a generalization of the ordinary fuzzy set in which the membership value for each member of the set is itself a fuzzy set. However, it is difficult, in some situations, for the decision-makers to give their preferences towards the object in terms of single or exact number. For handling this, a concept of type-2 intuitionistic fuzzy set (T2IFS) has been proposed and hence under this environment, a family of distance measures based on Hamming, Euclidean and Hausdorff metrics are presented. Some of its desirable properties have also been investigated in details. Finally, based on these measures, a group decision making method has been presented for ranking the alternatives. The proposed measures has been illustrated with a numerical example.
In this article, we have pointed out that some propositions corresponding to the distance measure... more In this article, we have pointed out that some propositions corresponding to the distance measure between the type-2 fuzzy sets (T2FSs) as provided by Singh (Frontiers of Computer Science, 2014, 8(5), 741–752), are incorrect by a counterexample. Further, these propositions have been corrected in the present manuscript by giving a correct relation between the T2FSs and validate it with a numerical example.
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