In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d<sup>∞</sup><sub>H</sub> . The results are based on the result ...In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d<sup>∞</sup><sub>H</sub> . The results are based on the result of single-valued random variables obtained by Taylor [1] and set-valued random variables obtained by Li Guan [2].展开更多
In this paper, we shall represent a strong law of large numbers (SLLN) for weighted sums of set- valued random variables in the sense of the Hausdorff metric dH, based on the result of single-valued random variable ob...In this paper, we shall represent a strong law of large numbers (SLLN) for weighted sums of set- valued random variables in the sense of the Hausdorff metric dH, based on the result of single-valued random variable obtained by Taylor [1].展开更多
The character and an algorithm about DRVIP（ discrete random variable with interval probability） and the secured kind DRVFP （discrete random variable with crisp event-fuzzy probability） are researched. Using the fu...The character and an algorithm about DRVIP（ discrete random variable with interval probability） and the secured kind DRVFP （discrete random variable with crisp event-fuzzy probability） are researched. Using the fuzzy resolution theorem, the solving mathematical expectation of a DRVFP can be translated into solving mathematical expectation of a series of RVIP. It is obvious that solving mathematical expectation of a DRVIP is a typical linear programming problem. A very functional calculating formula for solving mathematical expectation of DRVIP was obtained by using the Dantzig＇s simplex method. The example indicates that the result obtained by using the functional calculating formula fits together completely with the result obtained by using the linear programming method, but the process using the formula deduced is simpler.展开更多
Mathematic description about the second kind fuzzy random variable namely the random variable with crisp event-fuzzy probability was studied. Based on the interval probability and using the fuzzy resolution theorem, t...Mathematic description about the second kind fuzzy random variable namely the random variable with crisp event-fuzzy probability was studied. Based on the interval probability and using the fuzzy resolution theorem, the feasible condition about a probability fuzzy number set was given,go a step further the definition and characters of random variable with fuzzy probability (RVFP) and the fuzzy distribution function and fuzzy probability distribution sequence of the RVFP were put forward. The fuzzy probability resolution theorem with the closing operation of fuzzy probability was given and proved. The definition and characters of mathematical expectation and variance of the RVFP were studied also. All mathematic description about the RVFP has the closing operation for fuzzy probability,as a result, the foundation of perfecting fuzzy probability operation method is laid.展开更多
The random variable with fuzzy probability caused by fuzziness of probability density function was studied. The basic concepts/definitions and calculating methods of the interval/fuzzy probability density function, th...The random variable with fuzzy probability caused by fuzziness of probability density function was studied. The basic concepts/definitions and calculating methods of the interval/fuzzy probability density function, the random variable with fuzzy density function (RVFDF) and its distribution function, mathematical expectation and variance are given and same theorems related to the RVFDF are proved.展开更多
By constructing a mcan-square performance index in the case of fuzzy random variable, the optimal estimation theorem for unknown fuzzy state using the fuzzy observation data are given. The state and output of linear d...By constructing a mcan-square performance index in the case of fuzzy random variable, the optimal estimation theorem for unknown fuzzy state using the fuzzy observation data are given. The state and output of linear discrete-time dynamic fuzzy system with Gaussian noise are Gaussian fuzzy random variable sequences. An approach to fuzzy Kalman filtering is discussed. Fuzzy Kalman filtering contains two parts： a real-valued non-random recurrence equation and the standard Kalman filtering.展开更多
For random vibration of airborne platform, the accurate evaluation is a key indicator to ensure normal operation of airborne equipment in flight. However, only limited power spectral density（PSD） data can be obtaine...For random vibration of airborne platform, the accurate evaluation is a key indicator to ensure normal operation of airborne equipment in flight. However, only limited power spectral density（PSD） data can be obtained at the stage of flight test. Thus, those conventional evaluation methods cannot be employed when the distribution characteristics and priori information are unknown. In this paper, the fuzzy norm method（FNM） is proposed which combines the advantages of fuzzy theory and norm theory. The proposed method can deeply dig system information from limited data, which probability distribution is not taken into account. Firstly, the FNM is employed to evaluate variable interval and expanded uncertainty from limited PSD data, and the performance of FNM is demonstrated by confidence level, reliability and computing accuracy of expanded uncertainty. In addition, the optimal fuzzy parameters are discussed to meet the requirements of aviation standards and metrological practice. Finally, computer simulation is used to prove the adaptability of FNM. Compared with statistical methods, FNM has superiority for evaluating expanded uncertainty from limited data. The results show that the reliability of calculation and evaluation is superior to 95%.展开更多
In this paper, we propose a novel approach for Fuzzy random-valued Optimization. The main idea behind our approach consists of taking advantage of interplays between fuzzy random variables and random sets in a way to ...In this paper, we propose a novel approach for Fuzzy random-valued Optimization. The main idea behind our approach consists of taking advantage of interplays between fuzzy random variables and random sets in a way to get an equivalent stochastic program. This helps avoiding pitfalls due to severe oversimplification of the reality. We consider a numerical example that shows the efficiency of the proposed method.展开更多
A new prioritization method in the analytic hierarchy process （AHP）, which improves the group fuzzy preference programming （GFPP） method, is proposed. The fuzzy random theory is applied in the new prioritization m...A new prioritization method in the analytic hierarchy process （AHP）, which improves the group fuzzy preference programming （GFPP） method, is proposed. The fuzzy random theory is applied in the new prioritization method. By modifying the principle of decision making implied in the GFPP method, the improved group fuzzy preference programming （IGFPP） method is formulated as a fuzzy linear programming problem to maximize the average degree of the group satisfaction with all possible group priority vectors. The IGFPP method inherits the advantages of the GFPP method, and solves the weighting trouble existed in the GFPP method. Numerical tests indicate that the IGFPP method performs more effectively than the GFPP method in the case of very contradictive comparison judgments from decision makers.展开更多
For structure system with fuzzy input variables as well as random ones, a new importance measure system is presented for evaluating the effects of the two kinds of input variables on the output response. Based on the ...For structure system with fuzzy input variables as well as random ones, a new importance measure system is presented for evaluating the effects of the two kinds of input variables on the output response. Based on the fact that the fuzziness of the output response is determined by that of the input variable, the presented measure system defines the importance measures which evaluate the effect of the fuzzy input variable. And for the random input variable, the importance measure system analyzes its effect from two aspects, i.e. its effect on the central distribution position and that on the fuzzy degree of the membership function of the output response. Taking the effects of the two kinds of input variables on the first moment and second one of the output response into account, the definitions of the importance measures of the input variables are given and their engineering significations are demonstrated. Combining with the advantages of the point estimates of Zhao and Ono, a solution of the proposed importance measures is provided. Several examples show that the proposed measure system is comprehensive and reasonable, and the proposed solution can improve computational efficiency considerably with acceptable precision.展开更多
Tuned mass dampers (TMD) are well known as one of the most widely adopted devices in vibration control passive strategies. In the past few decades,many methods have been developed to find the optimal parameters of a T...Tuned mass dampers (TMD) are well known as one of the most widely adopted devices in vibration control passive strategies. In the past few decades,many methods have been developed to find the optimal parameters of a TMD installed on a structure and subjected to a random base excitation process,but most of them are usually based on an implicit assumption that all of the structural parameters are deterministic. However,in many real cases this simplification is unacceptable,so robust optimal design criteria becomes aviable alternative to better support engineers in the design process. In Robust Design Optimization (RDO) approaches,indeed the solution must be able to not only minimize the performance but also to limitits variation induced by uncertainty. Most of the currently available RDO methods are based on a probabilistic description of the model uncertainty,even if in many cases they are not able to explicitly include the influence of all the possible sources of uncertainties. Therefore,in this study,a fuzzy version of the robust TMD design optimization problem is proposed. The consistency of the fuzzy approach is studied with respect to the available non-probabilistic formulations reported in the literature and an application to an example of a robust design of a linear TMD subjected to base random vibrations in the presence of fuzzy uncertainties. The results show that the proposed fuzzy-based approach is able to give a set of optimal solutions both in terms of structural efficiency and sensitivity to mechanical and environmental uncertainties.展开更多
The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial mar...The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.展开更多
There are some mathematical models(see Example2.4)and analogous results in standard martingale theorywhich can not be described by the usual fuzzy martingaletheory because of the lack of corresponding semi-orderin the...There are some mathematical models(see Example2.4)and analogous results in standard martingale theorywhich can not be described by the usual fuzzy martingaletheory because of the lack of corresponding semi-orderin the fuzzy number space(E^n,D).In this paper,asuitable semi-order in the fuzzy number space(E^n,D)and the semi-order fuzzy supermartingale and submar-tingale are introduced,the charaterlstics of semi-ordersupermartingales and submartingales,as well as theDood’s stopping theorem for them(the bounded stoppingtimes theorem and the general stopping times theoremfor a class of closable semi-order fuzzy supermartin-gales and submartingales)are established.展开更多
Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are ext...Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are extended. In order to prove results of this paper, the notion of fuzzy martingale difference sequences is also introduced.展开更多
The notions of fuzzy (super) pramart are introduced. Then the completeness and separability of metric space are discussed. A necessary and sufficient condition of convergence for fuzzy sequences is provided. Finally, ...The notions of fuzzy (super) pramart are introduced. Then the completeness and separability of metric space are discussed. A necessary and sufficient condition of convergence for fuzzy sequences is provided. Finally, the graph Kuratowski-Mosco convergence and D-convergence of fuzzy (super) pramart and quasi-martingale are studied.展开更多
文摘In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d<sup>∞</sup><sub>H</sub> . The results are based on the result of single-valued random variables obtained by Taylor [1] and set-valued random variables obtained by Li Guan [2].
文摘In this paper, we shall represent a strong law of large numbers (SLLN) for weighted sums of set- valued random variables in the sense of the Hausdorff metric dH, based on the result of single-valued random variable obtained by Taylor [1].
文摘The character and an algorithm about DRVIP（ discrete random variable with interval probability） and the secured kind DRVFP （discrete random variable with crisp event-fuzzy probability） are researched. Using the fuzzy resolution theorem, the solving mathematical expectation of a DRVFP can be translated into solving mathematical expectation of a series of RVIP. It is obvious that solving mathematical expectation of a DRVIP is a typical linear programming problem. A very functional calculating formula for solving mathematical expectation of DRVIP was obtained by using the Dantzig＇s simplex method. The example indicates that the result obtained by using the functional calculating formula fits together completely with the result obtained by using the linear programming method, but the process using the formula deduced is simpler.
文摘Mathematic description about the second kind fuzzy random variable namely the random variable with crisp event-fuzzy probability was studied. Based on the interval probability and using the fuzzy resolution theorem, the feasible condition about a probability fuzzy number set was given,go a step further the definition and characters of random variable with fuzzy probability (RVFP) and the fuzzy distribution function and fuzzy probability distribution sequence of the RVFP were put forward. The fuzzy probability resolution theorem with the closing operation of fuzzy probability was given and proved. The definition and characters of mathematical expectation and variance of the RVFP were studied also. All mathematic description about the RVFP has the closing operation for fuzzy probability,as a result, the foundation of perfecting fuzzy probability operation method is laid.
文摘The random variable with fuzzy probability caused by fuzziness of probability density function was studied. The basic concepts/definitions and calculating methods of the interval/fuzzy probability density function, the random variable with fuzzy density function (RVFDF) and its distribution function, mathematical expectation and variance are given and same theorems related to the RVFDF are proved.
基金Project 60374022 supported by the National Natural Science Foundation of China.
文摘By constructing a mcan-square performance index in the case of fuzzy random variable, the optimal estimation theorem for unknown fuzzy state using the fuzzy observation data are given. The state and output of linear discrete-time dynamic fuzzy system with Gaussian noise are Gaussian fuzzy random variable sequences. An approach to fuzzy Kalman filtering is discussed. Fuzzy Kalman filtering contains two parts： a real-valued non-random recurrence equation and the standard Kalman filtering.
基金supported by Aeronautical Science Foundation of China (No. 20100251006)Technological Foundation Project of China (No. J132012C001)
文摘For random vibration of airborne platform, the accurate evaluation is a key indicator to ensure normal operation of airborne equipment in flight. However, only limited power spectral density（PSD） data can be obtained at the stage of flight test. Thus, those conventional evaluation methods cannot be employed when the distribution characteristics and priori information are unknown. In this paper, the fuzzy norm method（FNM） is proposed which combines the advantages of fuzzy theory and norm theory. The proposed method can deeply dig system information from limited data, which probability distribution is not taken into account. Firstly, the FNM is employed to evaluate variable interval and expanded uncertainty from limited PSD data, and the performance of FNM is demonstrated by confidence level, reliability and computing accuracy of expanded uncertainty. In addition, the optimal fuzzy parameters are discussed to meet the requirements of aviation standards and metrological practice. Finally, computer simulation is used to prove the adaptability of FNM. Compared with statistical methods, FNM has superiority for evaluating expanded uncertainty from limited data. The results show that the reliability of calculation and evaluation is superior to 95%.
文摘In this paper, we propose a novel approach for Fuzzy random-valued Optimization. The main idea behind our approach consists of taking advantage of interplays between fuzzy random variables and random sets in a way to get an equivalent stochastic program. This helps avoiding pitfalls due to severe oversimplification of the reality. We consider a numerical example that shows the efficiency of the proposed method.
基金Sponsored by the National Natural Science Foundation of China (70471063)
文摘A new prioritization method in the analytic hierarchy process （AHP）, which improves the group fuzzy preference programming （GFPP） method, is proposed. The fuzzy random theory is applied in the new prioritization method. By modifying the principle of decision making implied in the GFPP method, the improved group fuzzy preference programming （IGFPP） method is formulated as a fuzzy linear programming problem to maximize the average degree of the group satisfaction with all possible group priority vectors. The IGFPP method inherits the advantages of the GFPP method, and solves the weighting trouble existed in the GFPP method. Numerical tests indicate that the IGFPP method performs more effectively than the GFPP method in the case of very contradictive comparison judgments from decision makers.
基金supported by the National Natural Science Foundation of China (Grant No. NSFC 50875213)
文摘For structure system with fuzzy input variables as well as random ones, a new importance measure system is presented for evaluating the effects of the two kinds of input variables on the output response. Based on the fact that the fuzziness of the output response is determined by that of the input variable, the presented measure system defines the importance measures which evaluate the effect of the fuzzy input variable. And for the random input variable, the importance measure system analyzes its effect from two aspects, i.e. its effect on the central distribution position and that on the fuzzy degree of the membership function of the output response. Taking the effects of the two kinds of input variables on the first moment and second one of the output response into account, the definitions of the importance measures of the input variables are given and their engineering significations are demonstrated. Combining with the advantages of the point estimates of Zhao and Ono, a solution of the proposed importance measures is provided. Several examples show that the proposed measure system is comprehensive and reasonable, and the proposed solution can improve computational efficiency considerably with acceptable precision.
文摘Tuned mass dampers (TMD) are well known as one of the most widely adopted devices in vibration control passive strategies. In the past few decades,many methods have been developed to find the optimal parameters of a TMD installed on a structure and subjected to a random base excitation process,but most of them are usually based on an implicit assumption that all of the structural parameters are deterministic. However,in many real cases this simplification is unacceptable,so robust optimal design criteria becomes aviable alternative to better support engineers in the design process. In Robust Design Optimization (RDO) approaches,indeed the solution must be able to not only minimize the performance but also to limitits variation induced by uncertainty. Most of the currently available RDO methods are based on a probabilistic description of the model uncertainty,even if in many cases they are not able to explicitly include the influence of all the possible sources of uncertainties. Therefore,in this study,a fuzzy version of the robust TMD design optimization problem is proposed. The consistency of the fuzzy approach is studied with respect to the available non-probabilistic formulations reported in the literature and an application to an example of a robust design of a linear TMD subjected to base random vibrations in the presence of fuzzy uncertainties. The results show that the proposed fuzzy-based approach is able to give a set of optimal solutions both in terms of structural efficiency and sensitivity to mechanical and environmental uncertainties.
基金Supported by the Key Grant Project of Chinese Ministry of Education(309018)National Natural Science Foundation of China(70973104 and 11171304)the Zhejiang Natural Science Foundation of China(Y6110023)
文摘The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.
文摘There are some mathematical models(see Example2.4)and analogous results in standard martingale theorywhich can not be described by the usual fuzzy martingaletheory because of the lack of corresponding semi-orderin the fuzzy number space(E^n,D).In this paper,asuitable semi-order in the fuzzy number space(E^n,D)and the semi-order fuzzy supermartingale and submar-tingale are introduced,the charaterlstics of semi-ordersupermartingales and submartingales,as well as theDood’s stopping theorem for them(the bounded stoppingtimes theorem and the general stopping times theoremfor a class of closable semi-order fuzzy supermartin-gales and submartingales)are established.
基金Supported by National Basic Research Programof China (973Program, No.2007CB814901)Research Funds for Doctorial Programs of Higher Education (No.20060255006)Anhui Natural Science Foundation of University (No. KJ2008B143)
文摘Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are extended. In order to prove results of this paper, the notion of fuzzy martingale difference sequences is also introduced.
基金the Key Project of the Ministry of Education of China (205073)Research Fund for Doctorial Program of Higher Education (No.20060255006)
文摘The notions of fuzzy (super) pramart are introduced. Then the completeness and separability of metric space are discussed. A necessary and sufficient condition of convergence for fuzzy sequences is provided. Finally, the graph Kuratowski-Mosco convergence and D-convergence of fuzzy (super) pramart and quasi-martingale are studied.
