In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem （Minx（F）EntP） and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the ...In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem （Minx（F）EntP） and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the set of membership functions satisfying the given moment constraints generated by given moment functions it is required to choose the membership function that is closest to a priori membership function in the sense of cross fuzzy entropy measure. The existence of solution of formulated problem is proved by virtue of concavity property of cross fuzzy entropy measure, the implicit function theorem and Lagrange multipliers method. Moreover, Generalized Cross Fuzzy Entropy Optimization Methods in the form of MinMinx（F）EntM and MaxMinx（F）EntM are suggested on the basis of primary phase of minimizing cross fuzzy entropy measure for fixed moment vector function and on the definition of the special functional with Minx（F）Ent values of cross fuzzy entropy measure. Next phase for obtaining mentioned distributions consists of optimization of defined functional with respect to moment vector functions. Distributions obtained by mentioned methods are defined as （MinMinx（F）Ent）m and （MaxMinx（F）Ent）m distributions.展开更多
In this paper a fuzzy transportation problem under a fuzzy environment is solved using octagonal fuzzy numbers.The transportation problem is significant and has been widely studied in the field of applied mathematics ...In this paper a fuzzy transportation problem under a fuzzy environment is solved using octagonal fuzzy numbers.The transportation problem is significant and has been widely studied in the field of applied mathematics to solve a system of linear equations in many applications in science.Systems of concurrent linear equations play a vital major role in operational research.The main perspective of this research paper is to find out the minimum amount of transportation cost of some supplies through a capacitated network formerly the availability and the demand notes are octagonal fuzzy numbers.Octagonal fuzzy numbers are used and showed a membership function.To illustrate this method,a fuzzy transportation problem is solved by using octagonal fuzzy numbers using the ranking technique.It is shown that it is the best optimal solution and it is demonstrated with a numerical example.展开更多
The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as ...The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as exact values.But in actual practice because of different errors and incomplete information,the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers.This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems(FNEPs)where involved parameters are fuzzy numbers viz.triangular and trapezoidal.Based on the parametric form,fuzzy numbers have been transformed into family of standard intervals.Further due to the presence of interval overestimation problem in standard interval arithmetic,affine arithmetic based approach has been implemented.In the proposed method,the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach.Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure.Lastly,fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function.Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.展开更多
In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single...In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single objective function from the fuzzy multi-objective linear programming problems. At first, a numerical example of solving fuzzy multi-objective linear programming problem has been provided to validate the maximum risk reduction by the proposed method. The proposed method has been applied to assess the risk of damage due to natural calamities like flood, cyclone, sidor, and storms at the coastal areas in Bangladesh. The proposed method of solving the fuzzy multi-objective linear programming problems by the statistical method has been compared with the Chandra Sen’s method. The numerical results show that the proposed method maximizes the risk reduction capacity better than Chandra Sen’s method.展开更多
In the real world,one of the most common problems in project management is the unpredictability of resources and timelines.An efficient way to resolve uncertainty problems and overcome such obstacles is through an ext...In the real world,one of the most common problems in project management is the unpredictability of resources and timelines.An efficient way to resolve uncertainty problems and overcome such obstacles is through an extended fuzzy approach,often known as neutrosophic logic.Our rigorous proposed model has led to the creation of an advanced technique for computing the triangular single-valued neutrosophic number.This innovative approach evaluates the inherent uncertainty in project durations of the planning phase,which enhances the potential significance of the decision-making process in the project.Our proposed method,for the first time in the neutrosophic set literature,not only solves existing problems but also introduces a new set of problems not yet explored in previous research.A comparative study using Python programming was conducted to examine the effectiveness of responsive and adaptive planning,as well as their differences from other existing models such as the classical critical path problem and the fuzzy critical path problem.The study highlights the use of neutrosophic logic in handling complex projects by illustrating an innovative dynamic programming framework that is robust and flexible,according to the derived results,and sets the stage for future discussions on its scalability and application across different industries.展开更多
A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming probl...A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming problem can be converted into the single objective function by various methods as Chandra Sen’s method, weighted sum method, ranking function method, statistical averaging method. In this paper, Chandra Sen’s method and statistical averaging method both are used here for making single objective function from multi-objective function. Two multi-objective programming problems are solved to verify the result. One is numerical example and the other is real life example. Then the problems are solved by ordinary simplex method and fuzzy programming method. It can be seen that fuzzy programming method gives better optimal values than the ordinary simplex method.展开更多
As a classic NP-hard problem in machine learning and computational geometry,the k-means problem aims to partition the given dataset into k clusters according to the minimal squared Euclidean distance.Different from k-...As a classic NP-hard problem in machine learning and computational geometry,the k-means problem aims to partition the given dataset into k clusters according to the minimal squared Euclidean distance.Different from k-means problem and most of its variants,fuzzy k-means problem belongs to the soft clustering problem,where each given data point has relationship to every center point.Compared to fuzzy k-means problem,fuzzy k-means problem with penalties allows that some data points need not be clustered instead of being paid penalties.In this paper,we propose an O(αk In k)-approximation algorithm based on seeding algorithm for fuzzy k-means problem with penalties,whereαinvolves the ratio of the maximal penalty value to the minimal one.Furthermore,we implement numerical experiments to show the effectiveness of our algorithm.展开更多
The main aim of this paper is to develop an approach based on trapezoidal fuzzy numbers to optimize transportation problem in fuzzy environment.The present algorithm has representation of availability,demand and trans...The main aim of this paper is to develop an approach based on trapezoidal fuzzy numbers to optimize transportation problem in fuzzy environment.The present algorithm has representation of availability,demand and transportation cost as trapezoidal fuzzy numbers.This algorithm is found quicker in terms of runtime as comparison to fuzzy VAM discussed in[Kaur A.,Kumar A.,A new method for solving fuzzy transportation problem using ranking function,Appl.Math.Model.35:5652–5661,2011;Ismail Mohideen S.,Senthil Kumar P.,A comparative study on transportation problem in fuzzy environment,Int.J.Math.Res.2:151–158,2010].On the other hand this technique gives much better results than some classical methods like north-west corner and least cost method.Another benefit of this algorithm is that for certain transportation problems it directly gives optimal solution.It is one of the simplest methods to apply and perceive.Practical usefulness of the new method over other existing methods is demonstrated with two numerical examples.展开更多
Conflicts between two or more parties arise for various reasons andperspectives. Thus, resolution of con-flicts frequently relies on some form of negotiation. Thispaper presents a general problem-solving framework for...Conflicts between two or more parties arise for various reasons andperspectives. Thus, resolution of con-flicts frequently relies on some form of negotiation. Thispaper presents a general problem-solving framework for modeling multi-issue multilateral negotiationusing fuzzy constraints. Agent negotiation is formulated as a distributed fuzzy constraintsatisfaction problem (DFCSP). Fuzzy constrains are thus used to naturally represent each agent''sdesires involving imprecision and human conceptualization, particularly when lexical imprecision andsubjective matters are concerned. On the other hand, based on fuzzy constraint-basedproblem-solving, our approach enables an agent not only to systematically relax fuzzy constraints togenerate a proposal, but also to employ fuzzy similarity to select the alternative that is subjectto its acceptability by the opponents. This task of problem-solving is to reach an agreement thatbenefits all agents with a high satisfaction degree of fuzzy constraints, and move towards the dealmore quickly since their search focuses only on the feasible solution space. An application tomultilateral negotiation of a travel planning is provided to demonstrate the usefulness andeffectiveness of our framework.展开更多
文摘In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem （Minx（F）EntP） and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the set of membership functions satisfying the given moment constraints generated by given moment functions it is required to choose the membership function that is closest to a priori membership function in the sense of cross fuzzy entropy measure. The existence of solution of formulated problem is proved by virtue of concavity property of cross fuzzy entropy measure, the implicit function theorem and Lagrange multipliers method. Moreover, Generalized Cross Fuzzy Entropy Optimization Methods in the form of MinMinx（F）EntM and MaxMinx（F）EntM are suggested on the basis of primary phase of minimizing cross fuzzy entropy measure for fixed moment vector function and on the definition of the special functional with Minx（F）Ent values of cross fuzzy entropy measure. Next phase for obtaining mentioned distributions consists of optimization of defined functional with respect to moment vector functions. Distributions obtained by mentioned methods are defined as （MinMinx（F）Ent）m and （MaxMinx（F）Ent）m distributions.
文摘In this paper a fuzzy transportation problem under a fuzzy environment is solved using octagonal fuzzy numbers.The transportation problem is significant and has been widely studied in the field of applied mathematics to solve a system of linear equations in many applications in science.Systems of concurrent linear equations play a vital major role in operational research.The main perspective of this research paper is to find out the minimum amount of transportation cost of some supplies through a capacitated network formerly the availability and the demand notes are octagonal fuzzy numbers.Octagonal fuzzy numbers are used and showed a membership function.To illustrate this method,a fuzzy transportation problem is solved by using octagonal fuzzy numbers using the ranking technique.It is shown that it is the best optimal solution and it is demonstrated with a numerical example.
文摘The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as exact values.But in actual practice because of different errors and incomplete information,the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers.This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems(FNEPs)where involved parameters are fuzzy numbers viz.triangular and trapezoidal.Based on the parametric form,fuzzy numbers have been transformed into family of standard intervals.Further due to the presence of interval overestimation problem in standard interval arithmetic,affine arithmetic based approach has been implemented.In the proposed method,the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach.Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure.Lastly,fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function.Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.
文摘In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single objective function from the fuzzy multi-objective linear programming problems. At first, a numerical example of solving fuzzy multi-objective linear programming problem has been provided to validate the maximum risk reduction by the proposed method. The proposed method has been applied to assess the risk of damage due to natural calamities like flood, cyclone, sidor, and storms at the coastal areas in Bangladesh. The proposed method of solving the fuzzy multi-objective linear programming problems by the statistical method has been compared with the Chandra Sen’s method. The numerical results show that the proposed method maximizes the risk reduction capacity better than Chandra Sen’s method.
文摘In the real world,one of the most common problems in project management is the unpredictability of resources and timelines.An efficient way to resolve uncertainty problems and overcome such obstacles is through an extended fuzzy approach,often known as neutrosophic logic.Our rigorous proposed model has led to the creation of an advanced technique for computing the triangular single-valued neutrosophic number.This innovative approach evaluates the inherent uncertainty in project durations of the planning phase,which enhances the potential significance of the decision-making process in the project.Our proposed method,for the first time in the neutrosophic set literature,not only solves existing problems but also introduces a new set of problems not yet explored in previous research.A comparative study using Python programming was conducted to examine the effectiveness of responsive and adaptive planning,as well as their differences from other existing models such as the classical critical path problem and the fuzzy critical path problem.The study highlights the use of neutrosophic logic in handling complex projects by illustrating an innovative dynamic programming framework that is robust and flexible,according to the derived results,and sets the stage for future discussions on its scalability and application across different industries.
文摘A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming problem can be converted into the single objective function by various methods as Chandra Sen’s method, weighted sum method, ranking function method, statistical averaging method. In this paper, Chandra Sen’s method and statistical averaging method both are used here for making single objective function from multi-objective function. Two multi-objective programming problems are solved to verify the result. One is numerical example and the other is real life example. Then the problems are solved by ordinary simplex method and fuzzy programming method. It can be seen that fuzzy programming method gives better optimal values than the ordinary simplex method.
基金Higher Educational Science and Technology Program of Shandong Province(No.J17KA171)Natural Science Foundation of Shandong Province(No.ZR2020MA029).
文摘As a classic NP-hard problem in machine learning and computational geometry,the k-means problem aims to partition the given dataset into k clusters according to the minimal squared Euclidean distance.Different from k-means problem and most of its variants,fuzzy k-means problem belongs to the soft clustering problem,where each given data point has relationship to every center point.Compared to fuzzy k-means problem,fuzzy k-means problem with penalties allows that some data points need not be clustered instead of being paid penalties.In this paper,we propose an O(αk In k)-approximation algorithm based on seeding algorithm for fuzzy k-means problem with penalties,whereαinvolves the ratio of the maximal penalty value to the minimal one.Furthermore,we implement numerical experiments to show the effectiveness of our algorithm.
文摘The main aim of this paper is to develop an approach based on trapezoidal fuzzy numbers to optimize transportation problem in fuzzy environment.The present algorithm has representation of availability,demand and transportation cost as trapezoidal fuzzy numbers.This algorithm is found quicker in terms of runtime as comparison to fuzzy VAM discussed in[Kaur A.,Kumar A.,A new method for solving fuzzy transportation problem using ranking function,Appl.Math.Model.35:5652–5661,2011;Ismail Mohideen S.,Senthil Kumar P.,A comparative study on transportation problem in fuzzy environment,Int.J.Math.Res.2:151–158,2010].On the other hand this technique gives much better results than some classical methods like north-west corner and least cost method.Another benefit of this algorithm is that for certain transportation problems it directly gives optimal solution.It is one of the simplest methods to apply and perceive.Practical usefulness of the new method over other existing methods is demonstrated with two numerical examples.
文摘Conflicts between two or more parties arise for various reasons andperspectives. Thus, resolution of con-flicts frequently relies on some form of negotiation. Thispaper presents a general problem-solving framework for modeling multi-issue multilateral negotiationusing fuzzy constraints. Agent negotiation is formulated as a distributed fuzzy constraintsatisfaction problem (DFCSP). Fuzzy constrains are thus used to naturally represent each agent''sdesires involving imprecision and human conceptualization, particularly when lexical imprecision andsubjective matters are concerned. On the other hand, based on fuzzy constraint-basedproblem-solving, our approach enables an agent not only to systematically relax fuzzy constraints togenerate a proposal, but also to employ fuzzy similarity to select the alternative that is subjectto its acceptability by the opponents. This task of problem-solving is to reach an agreement thatbenefits all agents with a high satisfaction degree of fuzzy constraints, and move towards the dealmore quickly since their search focuses only on the feasible solution space. An application tomultilateral negotiation of a travel planning is provided to demonstrate the usefulness andeffectiveness of our framework.
