Real-life data introduce noise,uncertainty,and imprecision to statistical projects;it is advantageous to consider strategies to overcome these information expressions and processing problems.Neutrosophic(indeterminate...Real-life data introduce noise,uncertainty,and imprecision to statistical projects;it is advantageous to consider strategies to overcome these information expressions and processing problems.Neutrosophic(indeterminate)numbers can flexibly and conveniently represent the hybrid information of the partial determinacy and partial indeterminacy in an indeterminate setting,while a fuzzy multiset is a vital mathematical tool in the expression and processing of multi-valued fuzzy information with different and/or same fuzzy values.If neutrosophic numbers are introduced into fuzzy sequences in a fuzzy multiset,the introduced neutrosophic number sequences can be constructed as the neutrosophic number multiset or indeterminate fuzzy multiset.Motivated based on the idea,this study first proposes an indeterminate fuzzy multiset,where each element in a universe set can be repeated more than once with the different and/or identical indeterminate membership values.Then,we propose the parameterized correlation coefficients of indeterminate fuzzy multisets based on the de-neutrosophication of transforming indeterminate fuzzy multisets into the parameterized fuzzy multisets by a parameter(the parameterized de-neutrosophication method).Since indeterminate decision-making issues need to be handled by an indeterminate decision-making method,a group decision-making method using the weighted parameterized correlation coefficients of indeterminate fuzzy multisets is developed along with decision makers’different decision risks(small,moderate,and large risks)so as to handle multicriteria group decision-making problems in indeterminate fuzzy multiset setting.Finally,the developed group decision-making approach is used in an example on a selection problem of slope design schemes for an open-pit mine to demonstrate its usability and flexibility in the indeterminate group decision-making problem with indeterminate fuzzy multisets.展开更多
In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operatio...In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space B<sup>n</sup> are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.展开更多
文摘Real-life data introduce noise,uncertainty,and imprecision to statistical projects;it is advantageous to consider strategies to overcome these information expressions and processing problems.Neutrosophic(indeterminate)numbers can flexibly and conveniently represent the hybrid information of the partial determinacy and partial indeterminacy in an indeterminate setting,while a fuzzy multiset is a vital mathematical tool in the expression and processing of multi-valued fuzzy information with different and/or same fuzzy values.If neutrosophic numbers are introduced into fuzzy sequences in a fuzzy multiset,the introduced neutrosophic number sequences can be constructed as the neutrosophic number multiset or indeterminate fuzzy multiset.Motivated based on the idea,this study first proposes an indeterminate fuzzy multiset,where each element in a universe set can be repeated more than once with the different and/or identical indeterminate membership values.Then,we propose the parameterized correlation coefficients of indeterminate fuzzy multisets based on the de-neutrosophication of transforming indeterminate fuzzy multisets into the parameterized fuzzy multisets by a parameter(the parameterized de-neutrosophication method).Since indeterminate decision-making issues need to be handled by an indeterminate decision-making method,a group decision-making method using the weighted parameterized correlation coefficients of indeterminate fuzzy multisets is developed along with decision makers’different decision risks(small,moderate,and large risks)so as to handle multicriteria group decision-making problems in indeterminate fuzzy multiset setting.Finally,the developed group decision-making approach is used in an example on a selection problem of slope design schemes for an open-pit mine to demonstrate its usability and flexibility in the indeterminate group decision-making problem with indeterminate fuzzy multisets.
文摘In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space B<sup>n</sup> are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.
