An infinite product is expanded to Laurent series by residue theorem.Applying this expansion, the formula for the number of representations of an integer as a sum of eight triangular numbers is easily obtained.
Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygon...Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.展开更多
This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices.Crisp judgments cannot be given for real-life situations,as most of these include some level of fuzzines...This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices.Crisp judgments cannot be given for real-life situations,as most of these include some level of fuzziness and com-plexity.In these situations,judgments are represented by the set of fuzzy numbers.Most of the fuzzy optimization models derive crisp priorities for judgments repre-sented with Triangular Fuzzy Numbers(TFNs)only.They do not work for other types of Triangular Shaped Fuzzy Numbers(TSFNs)and Trapezoidal Fuzzy Numbers(TrFNs).To overcome this problem,a sum of squared error(SSE)based optimization model is proposed.Unlike some other methods,the proposed model derives crisp weights from all of the above-mentioned fuzzy judgments.A fuzzy number is simulated using the Monte Carlo method.A threshold-based constraint is also applied to minimize the deviation from the initial judgments.Genetic Algorithm(GA)is used to solve the optimization model.We have also conducted casestudiesto show the proposed approach’s advantages over the existingmethods.Results show that the proposed model outperforms other models to minimize SSE and deviation from initial judgments.Thus,the proposed model can be applied in various real time scenarios as it can reduce the SSE value upto 29%compared to the existing studies.展开更多
The evaluation of thesis by undergraduate has the characteristics of multi-factor, multi-layer and easy to be affected by subjective factors. To reduce the subjectivity, triangular fuzzy number is used as index set to...The evaluation of thesis by undergraduate has the characteristics of multi-factor, multi-layer and easy to be affected by subjective factors. To reduce the subjectivity, triangular fuzzy number is used as index set to give weight, and on this basis, fuzzy comprehensive evaluation is used to evaluate the quality of graduation thesis. The empirical analysis shows that the combination of triangular fuzzy number and fuzzy comprehensive evaluation has certain practical value in the quality evaluation of graduation thesis.展开更多
The traditional triangular fuzzy fault tree prediction model adopts the linear approximation method.Therefore,the accident prediction error is large.Based on the analysis of the error sources and the fuzzy set,the pre...The traditional triangular fuzzy fault tree prediction model adopts the linear approximation method.Therefore,the accident prediction error is large.Based on the analysis of the error sources and the fuzzy set,the precise calculation method of the event at the top of the fault tree is given.By using the numerical calculation software,an accurate calculation method of nonlinear triangular fuzzy accident prediction was adopted to predict lithium battery air transport fire accidents,and the fuzzy importance of the cause event was calculated.展开更多
This paper discusses the problem of finding a shortest path from a fixed origin s to a specified node t in a network with arcs represented as typical triangular fuzzy numbers (TFN). Because of the characterist...This paper discusses the problem of finding a shortest path from a fixed origin s to a specified node t in a network with arcs represented as typical triangular fuzzy numbers (TFN). Because of the characteristic of TFNs, the length of any path p from s to t , which equals the extended sum of all arcs belonging to p , is also TFN. Therefore, the fuzzy shortest path problem (FSPP) becomes to select the smallest among all those TFNs corresponding to different paths from s to t (specifically, the smallest TFN represents the shortest path). Based on Adamo's method for ranking fuzzy number, the pessimistic method and its extensions - optimistic method and λ combination method, are presented, and the FSPP is finally converted into the crisp shortest path problems.展开更多
In this paper, we present a fuzzy linguistic scale, which is characterized by triangular fuzzy numbers on [1/9, 9], for the comparison between two alternatives, and introduce a possibility degree formula for comparing...In this paper, we present a fuzzy linguistic scale, which is characterized by triangular fuzzy numbers on [1/9, 9], for the comparison between two alternatives, and introduce a possibility degree formula for comparing triangular fuzzy numbers. We utilize the fuzzy linguistic scale to construct a linguistic preference matrix, and propose a fuzzy induced ordered weighted geometric averaging (FIOWGA) operator to aggregate linguistic preference information. A method based on the fuzzy linguistic scale and FIOWGA operator for decision-making problems is presented. Finally, an illustrative example is given to verify the developed method and to demonstrate its feasibility and effectiveness.展开更多
This paper is concerned with a technique for order performance by similarity to ideal solution(TOPSIS)method for fuzzy multi-attribute decision making,in which the information about attribute weights is partly known a...This paper is concerned with a technique for order performance by similarity to ideal solution(TOPSIS)method for fuzzy multi-attribute decision making,in which the information about attribute weights is partly known and the attribute values take form of triangular fuzzy numbers.Considering the fact that the triangular fuzzy TOPSIS results yielded by different distance measures are different from others,a comparative analysis of triangular fuzzy TOPSIS ranking from each distance measure is illustrated with discussion on standard deviation.By applying the most reasonable distance,the deviation degrees between attribute values are measured.A linear programming model based on the maximal deviation of weighted attribute values is established to obtain the attribute weights.Therefore,alternatives are ranked by using TOPSIS method.Finally,a numerical example is given to show the feasibility and effectiveness of the method.展开更多
Many kinds of uncertainties are involved, such as random, fuzzy, grey, unascertained property and so on, in soil erosion process. To exactly predict the non-point source pollution loads, some uncertainties should be t...Many kinds of uncertainties are involved, such as random, fuzzy, grey, unascertained property and so on, in soil erosion process. To exactly predict the non-point source pollution loads, some uncertainties should be taken into consideration. Aiming at the deficiency of present blind number theory being helpless for fuzziness, a novel blind number, i.e. extended-blind number, was introduced by substituting a set of triangular fuzzy numbers （TFNs）, expressed as a-cuts, for interval values in present blind number, and the expected value of extended-blind number was also brought forward by referring to the current blind number theory. On the basis of denoting the parameters of Uni- versal Soil Loss Equation （USLE） as extended-blind parameters, a novel USLE model was established for quantitatively evaluating soil erosion loss and non-point source pollution loads. As a case, the uncertain USLE was employed for predicting the soil erosion loss and non-point source pollution loads of absorbed nitrogen and phosphorus in a dis- trict in the Hangbu-Fengle River basin, in the upstream of Chaohu Lake watershed. The results show that it is feasible in theory to extend blind number into fuzzy environment and reliable on conclusion to apply extended-blind number theory for predicting non-point source pollution loads.展开更多
The objective of this work was to determine the location of emergency material warehouses. For the site selection problem of emergency material warehouses, the triangular fuzzy numbers are respectively demand of the d...The objective of this work was to determine the location of emergency material warehouses. For the site selection problem of emergency material warehouses, the triangular fuzzy numbers are respectively demand of the demand node, the distance between the warehouse and demand node and the cost of the warehouse, a bi-objective programming model was established with minimum total cost of the system and minimum distance between the selected emergency material warehouses and the demand node. Using the theories of fuzzy numbers, the fuzzy programming model was transformed into a determinate bi-objective mixed integer programming model and a heuristic algorithm for this model was designed. Then, the algorithm was proven to be feasible and effective through a numerical example. Analysis results show that the location of emergency material warehouse depends heavily on the values of degree a and weight wl. Accurate information of a certain emergency activity should be collected before making the decision.展开更多
This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solu...This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method （VIM）. This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf. Sci. （2008） 178 1917] along with homotopy perturbation method （HPM） and [Int. Commun. Heat Mass Transfer （2012） 39 30] in the special cases to demonstrate the validity and applicability.展开更多
Purpose–The purpose of this paper is to study a nascent theory and an emerging concept of solving a fully fuzzy linear system(FFLS)with no non negative restrictions on the triangular fuzzy numbers chosen as parameter...Purpose–The purpose of this paper is to study a nascent theory and an emerging concept of solving a fully fuzzy linear system(FFLS)with no non negative restrictions on the triangular fuzzy numbers chosen as parameters.Two new simplified computational methods are proposed to solve a FFLS without any sign restrictions.The first method eliminates the non-negativity constraint from the coefficient matrix while the second method eliminates the constraint of non-negativity on the solution vector.The methods are introduced with an objective to broaden the domain of fuzzy linear systems to encompass a wide range of problems occurring in reality.Design/methodology/approach–The design of numerical methods is motivated by decomposing the fuzzy based linear system into its equivalent crisp linear form which can be further solved by variety of classical methods to solve a crisp linear system.Further the paper investigates Schur complement technique to solve the crisp equivalent of the FFLS.Findings–The results that are obtained reveal interesting properties of a FFLS.By using the proposed methods,the authors are able to check the consistency of the fuzzy linear system as well as obtain the nature of obtained solutions,i.e.trivial,unique or infinite.Further it is also seen that an n£n FFLS may yield finitely many solutions which may not be entirely feasible(strong).Also the methods successfully remove the non-negativity restriction on the coefficient matrix and the solution vector,respectively.Research limitations/implications–Evolving methods with better computational complexity and that which remove the non-negativity restriction jointly on all the parameters are left as an open problem.Originality/value–The proposed methods are new and conceptually simple to understand and apply in several scientific areas where fuzziness persists.The methods successfully remove several constraints that have been employed exhaustively by researchers and thus eventually tend to widen the breadth of applicability and usability of fuzzy linear models in real life situations.Heretofore,the usability of FFLS is largely dormant.展开更多
Cancer is a disease that is rapidly expanding in prevalence all over the world.Cancer cells canmetastasize,or spread,across the body and impact several different cell types.Additionally,the incidence rates of several ...Cancer is a disease that is rapidly expanding in prevalence all over the world.Cancer cells canmetastasize,or spread,across the body and impact several different cell types.Additionally,the incidence rates of several subtypes of cancer have been on the rise in India.The countermeasures for the cancer disease can be taken by determining the specific expansion rate of each type.To rank the various forms of cancer’s rate of progression,we used some of the available data.Numerous studies are available in the literature which show the growth rate of cancer by different techniques.The accuracy of the scheme in determining the highest growth rate may vary due to the variation in the dependent factors.Within the context of this research,the Fuzzy triangular technique for order preference by similarity to ideal solution(TOPSIS),is utilized to rank the various categorizations of cancer with the help of four groups of medical professionals acting in the capacity of decision-makers.The number of decision-makers may variate according to the required accuracy of results.The findings of the three-dimensional Fuzzy TOPSIS analysis categorize each variety of cancer according to the rate at which it spreads over time.Numerical results along with visual representation are presented to examine the efficiency of our proposed work.展开更多
In a two-stage supply chain composed of one supplier and one retailer,the supply chain coordination mechanism in a fuzzy continuous demand environment is researched.A positive triangular fuzzy number is used to model ...In a two-stage supply chain composed of one supplier and one retailer,the supply chain coordination mechanism in a fuzzy continuous demand environment is researched.A positive triangular fuzzy number is used to model the external market demand.Using the method of fuzzy cut sets theory,both fuzzy decentralized and centralized decision-making processes are analyzed,and another model of fuzzy return contract is proposed to help coordinate such supply chain.It is shown that in fuzzy environment there exists a unique solution of the retailer＇s optimal order quantity,the double marginalization problem can be solved by providing different tactics for wholesale pricing and return pricing,and the fuzzy expected profit of each actor can be expected to improve in the return contract.Finally,a numerical example is given to illustrate the models and the solution-seeking process.展开更多
A new fully fuzzy linear programming(FFLP)problem with fuzzy equality constraints is discussed.Using deviation degree measures,the FFLP problem is transformed into a crisp 6-parametric linear programming(LP)problem.Gi...A new fully fuzzy linear programming(FFLP)problem with fuzzy equality constraints is discussed.Using deviation degree measures,the FFLP problem is transformed into a crisp 6-parametric linear programming(LP)problem.Giving the value of deviation degree in each constraint,the 6-fuzzy optimal solution of the FFLP problem can be obtained by solving this LP problem.An algorithm is also proposed to find a balance-fuzzy optimal solution between two goals in conflict:to improve the values of the objective function and to decrease the values of the deviation degrees.A numerical example is solved to illustrate the proposed method.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
Utilizing granular computing to enhance artificial neural network architecture, a newtype of network emerges—thegranular neural network (GNN). GNNs offer distinct advantages over their traditional counterparts: The a...Utilizing granular computing to enhance artificial neural network architecture, a newtype of network emerges—thegranular neural network (GNN). GNNs offer distinct advantages over their traditional counterparts: The ability toprocess both numerical and granular data, leading to improved interpretability. This paper proposes a novel designmethod for constructing GNNs, drawing inspiration from existing interval-valued neural networks built uponNNNs. However, unlike the proposed algorithm in this work, which employs interval values or triangular fuzzynumbers for connections, existing methods rely on a pre-defined numerical network. This new method utilizesa uniform distribution of information granularity to granulate connections with unknown parameters, resultingin independent GNN structures. To quantify the granularity output of the network, the product of two commonperformance indices is adopted: The coverage of numerical data and the specificity of information granules.Optimizing this combined performance index helps determine the optimal parameters for the network. Finally,the paper presents the complete model construction and validates its feasibility through experiments on datasetsfrom the UCIMachine Learning Repository. The results demonstrate the proposed algorithm’s effectiveness andpromising performance.展开更多
Modification of a fuzzy partition often leads to the change of fuzziness of a fuzzy system. Researches on the change of fuzzy entropy of a fuzzy set, responding to shape alteration of membership function, therefore...Modification of a fuzzy partition often leads to the change of fuzziness of a fuzzy system. Researches on the change of fuzzy entropy of a fuzzy set, responding to shape alteration of membership function, therefore, play a significant role in analysis of the change of fuzziness of a fuzzy system because a fuzzy partition consists of a set of fuzzy sets which satisfy some special constraints. This paper has shown several results about entropy changes of a fuzzy set. First, the entropies of two same type of fuzzy sets have a constant proportional relationship which depends on the ratio of the sizes of their support intervals. Second, as for Triangular Fuzzy Numbers （TFNs）, the entropies of any two TFNs which can not be always the same type, also, have a constant proportional relationship which depends on the ratio of the sizes of their support intervals. Hence, any two TFNs with the same sizes of support intervals have the same entropies. Third, concerning two Triangular Fuzzy Sets （TFSs） with same sizes of support intervals and different heights, the relationship of their entropies lies on their height. Finally, we point it out a mistake that Chen＇s assertion that the entropy of resultant fuzzy set of elevation operation is directly to that of the original one while elevation factor just acts as a propartional factor. These results should contribute to the analysis and design of a fuzzy system.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
The distributed and customized 3D printing can be realized by 3D printing services in a cloud manufacturing environment.As a growing number of 3D printers are becoming accessible on various 3D printing service platfor...The distributed and customized 3D printing can be realized by 3D printing services in a cloud manufacturing environment.As a growing number of 3D printers are becoming accessible on various 3D printing service platforms,there raises the concern over the validation of virtual product designs and their manufacturing procedures for novices as well as users with 3D printing experience before physical products are produced through the cloud platform.This paper presents a 3D model to help users validate their designs and requirements not only in the traditional digital 3D model properties like shape and size,but also in physical material properties and manufacturing properties when producing physical products like surface roughness,print accuracy and part cost.These properties are closely related to the process of 3D printing and materials.In order to establish the 3D model,the paper analyzes the model of the 3D printing process selection in the cloud platform.Triangular intuitionistic fuzzy numbers are applied to generate a set of 3D printers with the same process and material.Based on the 3D printing process selection model,users can establish the 3D model and validate their designs and requirements on physical material properties and manufacturing properties before printing physical products.展开更多
文摘An infinite product is expanded to Laurent series by residue theorem.Applying this expansion, the formula for the number of representations of an integer as a sum of eight triangular numbers is easily obtained.
文摘Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.
文摘This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices.Crisp judgments cannot be given for real-life situations,as most of these include some level of fuzziness and com-plexity.In these situations,judgments are represented by the set of fuzzy numbers.Most of the fuzzy optimization models derive crisp priorities for judgments repre-sented with Triangular Fuzzy Numbers(TFNs)only.They do not work for other types of Triangular Shaped Fuzzy Numbers(TSFNs)and Trapezoidal Fuzzy Numbers(TrFNs).To overcome this problem,a sum of squared error(SSE)based optimization model is proposed.Unlike some other methods,the proposed model derives crisp weights from all of the above-mentioned fuzzy judgments.A fuzzy number is simulated using the Monte Carlo method.A threshold-based constraint is also applied to minimize the deviation from the initial judgments.Genetic Algorithm(GA)is used to solve the optimization model.We have also conducted casestudiesto show the proposed approach’s advantages over the existingmethods.Results show that the proposed model outperforms other models to minimize SSE and deviation from initial judgments.Thus,the proposed model can be applied in various real time scenarios as it can reduce the SSE value upto 29%compared to the existing studies.
文摘The evaluation of thesis by undergraduate has the characteristics of multi-factor, multi-layer and easy to be affected by subjective factors. To reduce the subjectivity, triangular fuzzy number is used as index set to give weight, and on this basis, fuzzy comprehensive evaluation is used to evaluate the quality of graduation thesis. The empirical analysis shows that the combination of triangular fuzzy number and fuzzy comprehensive evaluation has certain practical value in the quality evaluation of graduation thesis.
基金supported by Shanghai University New Teacher Training Research Project.
文摘The traditional triangular fuzzy fault tree prediction model adopts the linear approximation method.Therefore,the accident prediction error is large.Based on the analysis of the error sources and the fuzzy set,the precise calculation method of the event at the top of the fault tree is given.By using the numerical calculation software,an accurate calculation method of nonlinear triangular fuzzy accident prediction was adopted to predict lithium battery air transport fire accidents,and the fuzzy importance of the cause event was calculated.
文摘This paper discusses the problem of finding a shortest path from a fixed origin s to a specified node t in a network with arcs represented as typical triangular fuzzy numbers (TFN). Because of the characteristic of TFNs, the length of any path p from s to t , which equals the extended sum of all arcs belonging to p , is also TFN. Therefore, the fuzzy shortest path problem (FSPP) becomes to select the smallest among all those TFNs corresponding to different paths from s to t (specifically, the smallest TFN represents the shortest path). Based on Adamo's method for ranking fuzzy number, the pessimistic method and its extensions - optimistic method and λ combination method, are presented, and the FSPP is finally converted into the crisp shortest path problems.
基金The National Natural Science Foundation of China(79970093) the Ph.D. Dissertation Foundation of Southeast University- NARI-Relays Electric Co. Ltd.
文摘In this paper, we present a fuzzy linguistic scale, which is characterized by triangular fuzzy numbers on [1/9, 9], for the comparison between two alternatives, and introduce a possibility degree formula for comparing triangular fuzzy numbers. We utilize the fuzzy linguistic scale to construct a linguistic preference matrix, and propose a fuzzy induced ordered weighted geometric averaging (FIOWGA) operator to aggregate linguistic preference information. A method based on the fuzzy linguistic scale and FIOWGA operator for decision-making problems is presented. Finally, an illustrative example is given to verify the developed method and to demonstrate its feasibility and effectiveness.
基金supported by the National Natural Science Foundation of China(70473037)the Key Project of National Development and Reform Commission(1009-213011)
文摘This paper is concerned with a technique for order performance by similarity to ideal solution(TOPSIS)method for fuzzy multi-attribute decision making,in which the information about attribute weights is partly known and the attribute values take form of triangular fuzzy numbers.Considering the fact that the triangular fuzzy TOPSIS results yielded by different distance measures are different from others,a comparative analysis of triangular fuzzy TOPSIS ranking from each distance measure is illustrated with discussion on standard deviation.By applying the most reasonable distance,the deviation degrees between attribute values are measured.A linear programming model based on the maximal deviation of weighted attribute values is established to obtain the attribute weights.Therefore,alternatives are ranked by using TOPSIS method.Finally,a numerical example is given to show the feasibility and effectiveness of the method.
基金Under the auspices of Tackling Key Program for Science and Technology of Anhui Province (No. 07010302165)Natural Science Foundation of Anhui Province (No. 050450303)
文摘Many kinds of uncertainties are involved, such as random, fuzzy, grey, unascertained property and so on, in soil erosion process. To exactly predict the non-point source pollution loads, some uncertainties should be taken into consideration. Aiming at the deficiency of present blind number theory being helpless for fuzziness, a novel blind number, i.e. extended-blind number, was introduced by substituting a set of triangular fuzzy numbers （TFNs）, expressed as a-cuts, for interval values in present blind number, and the expected value of extended-blind number was also brought forward by referring to the current blind number theory. On the basis of denoting the parameters of Uni- versal Soil Loss Equation （USLE） as extended-blind parameters, a novel USLE model was established for quantitatively evaluating soil erosion loss and non-point source pollution loads. As a case, the uncertain USLE was employed for predicting the soil erosion loss and non-point source pollution loads of absorbed nitrogen and phosphorus in a dis- trict in the Hangbu-Fengle River basin, in the upstream of Chaohu Lake watershed. The results show that it is feasible in theory to extend blind number into fuzzy environment and reliable on conclusion to apply extended-blind number theory for predicting non-point source pollution loads.
基金Project(71071162)supported by the National Natural Science Foundation of China
文摘The objective of this work was to determine the location of emergency material warehouses. For the site selection problem of emergency material warehouses, the triangular fuzzy numbers are respectively demand of the demand node, the distance between the warehouse and demand node and the cost of the warehouse, a bi-objective programming model was established with minimum total cost of the system and minimum distance between the selected emergency material warehouses and the demand node. Using the theories of fuzzy numbers, the fuzzy programming model was transformed into a determinate bi-objective mixed integer programming model and a heuristic algorithm for this model was designed. Then, the algorithm was proven to be feasible and effective through a numerical example. Analysis results show that the location of emergency material warehouse depends heavily on the values of degree a and weight wl. Accurate information of a certain emergency activity should be collected before making the decision.
基金the UGC, Government of India, for financial support under the Rajiv Gandhi National Fellowship (RGNF)
文摘This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method （VIM）. This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf. Sci. （2008） 178 1917] along with homotopy perturbation method （HPM） and [Int. Commun. Heat Mass Transfer （2012） 39 30] in the special cases to demonstrate the validity and applicability.
文摘Purpose–The purpose of this paper is to study a nascent theory and an emerging concept of solving a fully fuzzy linear system(FFLS)with no non negative restrictions on the triangular fuzzy numbers chosen as parameters.Two new simplified computational methods are proposed to solve a FFLS without any sign restrictions.The first method eliminates the non-negativity constraint from the coefficient matrix while the second method eliminates the constraint of non-negativity on the solution vector.The methods are introduced with an objective to broaden the domain of fuzzy linear systems to encompass a wide range of problems occurring in reality.Design/methodology/approach–The design of numerical methods is motivated by decomposing the fuzzy based linear system into its equivalent crisp linear form which can be further solved by variety of classical methods to solve a crisp linear system.Further the paper investigates Schur complement technique to solve the crisp equivalent of the FFLS.Findings–The results that are obtained reveal interesting properties of a FFLS.By using the proposed methods,the authors are able to check the consistency of the fuzzy linear system as well as obtain the nature of obtained solutions,i.e.trivial,unique or infinite.Further it is also seen that an n£n FFLS may yield finitely many solutions which may not be entirely feasible(strong).Also the methods successfully remove the non-negativity restriction on the coefficient matrix and the solution vector,respectively.Research limitations/implications–Evolving methods with better computational complexity and that which remove the non-negativity restriction jointly on all the parameters are left as an open problem.Originality/value–The proposed methods are new and conceptually simple to understand and apply in several scientific areas where fuzziness persists.The methods successfully remove several constraints that have been employed exhaustively by researchers and thus eventually tend to widen the breadth of applicability and usability of fuzzy linear models in real life situations.Heretofore,the usability of FFLS is largely dormant.
文摘Cancer is a disease that is rapidly expanding in prevalence all over the world.Cancer cells canmetastasize,or spread,across the body and impact several different cell types.Additionally,the incidence rates of several subtypes of cancer have been on the rise in India.The countermeasures for the cancer disease can be taken by determining the specific expansion rate of each type.To rank the various forms of cancer’s rate of progression,we used some of the available data.Numerous studies are available in the literature which show the growth rate of cancer by different techniques.The accuracy of the scheme in determining the highest growth rate may vary due to the variation in the dependent factors.Within the context of this research,the Fuzzy triangular technique for order preference by similarity to ideal solution(TOPSIS),is utilized to rank the various categorizations of cancer with the help of four groups of medical professionals acting in the capacity of decision-makers.The number of decision-makers may variate according to the required accuracy of results.The findings of the three-dimensional Fuzzy TOPSIS analysis categorize each variety of cancer according to the rate at which it spreads over time.Numerical results along with visual representation are presented to examine the efficiency of our proposed work.
基金Sponsored by the National Natural Science Foundation of China (7047106370771010)
文摘In a two-stage supply chain composed of one supplier and one retailer,the supply chain coordination mechanism in a fuzzy continuous demand environment is researched.A positive triangular fuzzy number is used to model the external market demand.Using the method of fuzzy cut sets theory,both fuzzy decentralized and centralized decision-making processes are analyzed,and another model of fuzzy return contract is proposed to help coordinate such supply chain.It is shown that in fuzzy environment there exists a unique solution of the retailer＇s optimal order quantity,the double marginalization problem can be solved by providing different tactics for wholesale pricing and return pricing,and the fuzzy expected profit of each actor can be expected to improve in the return contract.Finally,a numerical example is given to illustrate the models and the solution-seeking process.
基金supported by the National Natural Science Foundation of China(71202140)the Fundamental Research for the Central Universities(HUST:2013QN099)
文摘A new fully fuzzy linear programming(FFLP)problem with fuzzy equality constraints is discussed.Using deviation degree measures,the FFLP problem is transformed into a crisp 6-parametric linear programming(LP)problem.Giving the value of deviation degree in each constraint,the 6-fuzzy optimal solution of the FFLP problem can be obtained by solving this LP problem.An algorithm is also proposed to find a balance-fuzzy optimal solution between two goals in conflict:to improve the values of the objective function and to decrease the values of the deviation degrees.A numerical example is solved to illustrate the proposed method.
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.
基金the National Key R&D Program of China under Grant 2018YFB1700104.
文摘Utilizing granular computing to enhance artificial neural network architecture, a newtype of network emerges—thegranular neural network (GNN). GNNs offer distinct advantages over their traditional counterparts: The ability toprocess both numerical and granular data, leading to improved interpretability. This paper proposes a novel designmethod for constructing GNNs, drawing inspiration from existing interval-valued neural networks built uponNNNs. However, unlike the proposed algorithm in this work, which employs interval values or triangular fuzzynumbers for connections, existing methods rely on a pre-defined numerical network. This new method utilizesa uniform distribution of information granularity to granulate connections with unknown parameters, resultingin independent GNN structures. To quantify the granularity output of the network, the product of two commonperformance indices is adopted: The coverage of numerical data and the specificity of information granules.Optimizing this combined performance index helps determine the optimal parameters for the network. Finally,the paper presents the complete model construction and validates its feasibility through experiments on datasetsfrom the UCIMachine Learning Repository. The results demonstrate the proposed algorithm’s effectiveness andpromising performance.
基金The National Natural Science Foundation of China(No.60474022)
文摘Modification of a fuzzy partition often leads to the change of fuzziness of a fuzzy system. Researches on the change of fuzzy entropy of a fuzzy set, responding to shape alteration of membership function, therefore, play a significant role in analysis of the change of fuzziness of a fuzzy system because a fuzzy partition consists of a set of fuzzy sets which satisfy some special constraints. This paper has shown several results about entropy changes of a fuzzy set. First, the entropies of two same type of fuzzy sets have a constant proportional relationship which depends on the ratio of the sizes of their support intervals. Second, as for Triangular Fuzzy Numbers （TFNs）, the entropies of any two TFNs which can not be always the same type, also, have a constant proportional relationship which depends on the ratio of the sizes of their support intervals. Hence, any two TFNs with the same sizes of support intervals have the same entropies. Third, concerning two Triangular Fuzzy Sets （TFSs） with same sizes of support intervals and different heights, the relationship of their entropies lies on their height. Finally, we point it out a mistake that Chen＇s assertion that the entropy of resultant fuzzy set of elevation operation is directly to that of the original one while elevation factor just acts as a propartional factor. These results should contribute to the analysis and design of a fuzzy system.
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.
基金the National High-Tech Research and Development Plan of China under Grant No.2015AA042101 and Fund of State Key Laboratory of Intelligent Manufacturing System Technology in China.
文摘The distributed and customized 3D printing can be realized by 3D printing services in a cloud manufacturing environment.As a growing number of 3D printers are becoming accessible on various 3D printing service platforms,there raises the concern over the validation of virtual product designs and their manufacturing procedures for novices as well as users with 3D printing experience before physical products are produced through the cloud platform.This paper presents a 3D model to help users validate their designs and requirements not only in the traditional digital 3D model properties like shape and size,but also in physical material properties and manufacturing properties when producing physical products like surface roughness,print accuracy and part cost.These properties are closely related to the process of 3D printing and materials.In order to establish the 3D model,the paper analyzes the model of the 3D printing process selection in the cloud platform.Triangular intuitionistic fuzzy numbers are applied to generate a set of 3D printers with the same process and material.Based on the 3D printing process selection model,users can establish the 3D model and validate their designs and requirements on physical material properties and manufacturing properties before printing physical products.
