General Response Surface Reliability Analysis for Fuzzy-Random Uncertainty both in Basic Variables and in State Variables

A general response surface（RS） method is presented for reliability analysis of complex structure/mechanism with fuzzy-random uncertainty both in basic variables and in failure state variables. On the basis of equivalent transformation from fuzzy basic variable to random basic variable, the fuzziness and randomness in the basic variables are considered simultaneously in the presented general RS method. Once the fuzzy basic variables are transformed into the random basic variables, the conventional RS method is employed to establish the general RS for the complex structure/mechanism with implicit limit state equation by finite element numerical simulation. Furthermore, the general failure probability is defined according to the probability formula for fuzzy-random event by taking the fuzziness and randomness in the failure-safety state into consideration, and an appropriate fuzzy operator is adopted to calculate the general failure probability for the complex structure/mechanism with multiple implicit failure modes. Finally, a general reliability analysis of an elastic linkage mechanism is introduced to illustrate the present method.

DYNAMIC CHARACTERISTIC ANALYSIS OF FUZZY-STOCHASTIC TRUSS STRUCTURES BASED ON FUZZY FACTOR METHOD AND RANDOM FACTOR METHOD

A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented. Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously, the structural stiffness and mass matrices axe constructed based on the fuzzy factor method and random factor method; from the Rayleigh＇s quotient of structural vibration, the structural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic; the fuzzy numeric characteristics of dynamic characteristic axe then derived by using the random variable＇s moment function method and algebra synthesis method. Two examples axe used to illustrate the validity and rationality of the method given. The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameters on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.

Random Fuzzy Chance-constrained Programming Based on Adaptive Chaos Quantum Honey Bee Algorithm and Robustness Analysis

This paper proposes an adaptive chaos quantum honey bee algorithm （CQHBA） for solving chance-constrained program- ming in random fuzzy environment based on random fuzzy simulations. Random fuzzy simulation is designed to estimate the chance of a random fuzzy event and the optimistic value to a random fuzzy variable. In CQHBA, each bee carries a group of quantum bits representing a solution. Chaos optimization searches space around the selected best-so-far food source. In the marriage process, random interferential discrete quantum crossover is done between selected drones and the queen. Gaussian quantum mutation is used to keep the diversity of whole population. New methods of computing quantum rotation angles are designed based on grads. A proof of con- vergence for CQHBA is developed and a theoretical analysis of the computational overhead for the algorithm is presented. Numerical examples are presented to demonstrate its superiority in robustness and stability, efficiency of computational complexity, success rate, and accuracy of solution quality. CQHBA is manifested to be highly robust under various conditions and capable of handling most random fuzzy programmings with any parameter settings, variable initializations, system tolerance and confidence level, perturbations, and noises.

MATHEMATICAL EXPECTATION ABOUT DISCRETE RANDOM VARIABLE WITH INTERVAL PROBABILITY OR FUZZY PROBABILITY

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.

RANDOM VARIABLE WITH FUZZY PROBABILITY

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.

Integrated evaluation system under randomness and fuzziness for groundwater contamination risk assessment in a little town, Central China

An integrated evaluation system under randomness and fuzziness was developed in this work to systematically assess the risk of groundwater contamination in a little town, Central China. In this system, randomness of the parameters and the fuzziness of the risk were considered simultaneously, and the exceeding standard probability of contamination and human health risk due to the contamination were integrated. The contamination risk was defined as a combination of "vulnerability" and "hazard". To calculate the value of "vulnerability", pollutant concentration was simulated by MODFLOW with random input variables and a new modified health risk assessment(MRA) model was established to analyze the level of "hazard". The limit concentration based on environmental-guideline and health risk due to manganese were systematically examined to obtain the general risk levels through a fuzzy rule base. The "vulnerability" and "hazard" were divided into five categories of "high", "medium-high", "medium", "low-medium" and "low", respectively. Then, "vulnerability" and "hazard" were firstly combined by integrated evaluation. Compared with the other two scenarios under deterministic methods, the risk obtained in the proposed system is higher. This research illustrated that ignoring of uncertainties in evaluation process might underestimate the risk level.

A NEW METHOD FOR CONSTRUCTING FUZZY EVIDENCE BASED ON THE NON-CONSONANT RANDOM SET

Natural-language information is often mathematically expressed by fuzzy sets. With the random set theory as a bridge, this kind of information can be transformed into fuzzy evidence in Dempster-Shafer (DS) theory. Then Dempster's combination rule or other combination rules of evi- dence can be used perfectly for fusing natural-language and other information. However, this traditional transformation involves the use of α -cutsets to construct the focal elements which have to be repre- sented as consonant sets. This construction is very inflexible and unreasonable in some practical ap- plications. In this paper, with the desire to overcome this limitation, a method for constructing more general non-consonant focal elements is proposed based on the random set theory. Some examples are given to show the generality and the efficiency of this new method. Finally, we validate that non-consonant constructions provide less degrees of total uncertainty than that of the consonant case in these examples by using the evaluation criterion of total uncertainty.

On Fuzzy Random-Valued Optimization

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.

RANDOM-FUZZY SAFETY ANALYSIS FOR AN AERO ENGINE TURBINE DISK

A numerical simulation method is presented for the random-fuzzy safety analysis of an aero engine disk. Based on the equivalent transformation from a fuzzy variable to a random variable, the equivalent random Probability Density Functions(PDFs) are got from their corresponding Fuzzy Possibility Distributions(FPDs) for the fuzzy variables. In that case the perfect numerical simulation method for the random uncertainty is employed to solve the fuzzy uncertainty. For the complex structure such as the aero engine disk with implicit relationship between the input basic variable and the response variable, the equivalent PDFs of the input basic variables are delivered simultaneously to the response variable by an empirical PDF, which is simulated by Finite Element Method(FEM). Then, in view of the fuzzy application requirement occurring in engineering usually, the reliability definition and calculation are discussed for the aero engine disk with multiple fuzzy failure modes. On the other hand, through the inverse transformation of the fuzzy variable to the random variable, the FPDs of the response variables can be calculated from their empirical PDFs as well.

INTERVAL ANALYSIS OF FUZZY-RANDOM HEAT CONDUCTION IN COMPOSITE TUBES

During the analysis of stability heat conduction in the composite tubes, firstly, when the temperature boundary conditions are the random conditions, equations of the mean values and variances of the random thermal function are transformed. Secondly, when the heat conduct parameters are the fuzzy numbers and the temperature boundary conditions are the random numbers, interval equations of the heat conduction are presented. Thirdly, by comparison of the interval results, the result in the interval analysis is larger than that in the confidence interval. Moreover the error expecting equation is presented. Finally, with upper （lower） approximation in rough set theory, a new method of the interval analysis to deal with the stability heat conduction is presented.

Improved Group Fuzzy Preference Programming Method Based on Fuzzy Random Theory

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.

RANDOM-FUZZY MODEL FOR THE DUCTILE/BRITTLE TRANSITION

Based on a large experimental sample, the influence of stress triaxiality and temperature have been studied on the growth of micro voids and the ductile/brittle transition (DBT) of 40Cr steel. Ductile and brittle fractures have been investigated simultaneously. The experiments show that the ductile fracture is controlled by the micro void mechanism,and the critical void growth ratio can be used as the criterion of ductile fracture. The brittle fracture is modeled by an embedded penny crack, and a micro stress intensity with a characteristic length can be used as the criterion of the brittle fracture. The DBT is the result of the competition of the two mechanisms. Which exists in the fracture of all specimens simultaneously. The distributions of model parameters were measured experimentally. A random model and a random-fuzzy model for DBT were presented. The comparison between the calculated and experimental results shows that the random-fuzzy model can model the DBT satisfactorily.

MATHEMATIC DESCRIPTION ABOUT RANDOM VARIABLE WITH FUZZY DENSITY FUNCTION (RVFDF)

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.

Interactive Fuzzy Programming for Random Fuzzy Two-Level Integer Programming Problems through Fractile Criteria with Possibility

This paper considers two-level integer programming problems involving random fuzzy variables with cooperative behavior of the decision makers. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile criteria to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy two-level integer programming problems are reduced to deterministic ones. Through the introduction of genetic algorithms with double strings for nonlinear integer programming problems, interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.

Representation Theorems for Fuzzy Random Sets and Fuzzy Stochastic Processes

The fuzzy static and dynamic random phenomena in an abstract separable Banach space is discussed in this paper. The representation theorems for fuzzy set valued random sets, fuzzy random elements and fuzzy set valued stochastic processes are obtained.

Stochastic Optimal Estimation with Fuzzy Random Variables and Fuzzy Kalman Filtering

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.

Modeling Multisource-heterogeneous Information Based on Random Set and Fuzzy Set Theory

This paper presents a new idea, named as modeling multisensor-heterogeneous information, to incorporate the fuzzy logic methodologies with mulitsensor-multitarget system under the framework of random set theory. Firstly, based on strong random set and weak random set, the unified form to describe both data (unambiguous information) and fuzzy evidence (uncertain information) is introduced. Secondly, according to signatures of fuzzy evidence, two Bayesian-markov nonlinear measurement models are proposed to fuse effectively data and fuzzy evidence. Thirdly, by use of "the models-based signature-matching scheme", the operation of the statistics of fuzzy evidence defined as random set can be translated into that of the membership functions of relative point state variables. These works are the basis to construct qualitative measurement models and to fuse data and fuzzy evidence.

Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in Gα Space

In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d∞H . 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].

Multisensor Fuzzy Stochastic Fusion Based on Genetic Algorithms

To establish a parallel fusion approach of processing high dimensional information, the model and criterion of multisensor fuzzy stochastic data fusion were presented. In order to design genetic algorithm fusion, the fusion parameter coding, initial population and fitness function establishing, and fuzzy logic controller designing for genetic operations and probability choosing were completed. The discussion on the highly dimensional fusion was given. For a moving target with the division of 1 64 (velocity) and 1 75 (acceleration), the precision of fusion is 0 94 and 0 98 respectively. The fusion approach can improve the reliability and decision precision effectively.