Monte Carlo method is a computer-based simulation or experiment method. It approximately simulates and solves mathematical or physical stochastic problem with statistical random sampling. Compared with traditional algebraic method, due to their reliance on repeated computation of random or pseudo-random numbers, Monte Carlo method can apply Normal distribution, Exponential distribution, Weibull distribution etc. to model phenomena with significant uncertainty in inputs when it is unfeasible or impossible to compute an exact result with a deterministic algorithm, and does not need to know parameter’s distribution type and probability parameter [1] -[5] . In new product design, usually design parameter is a random variable that follows a probability distribution. Design parameter is a key impact factor to design robustness, and distribution of these design parameters determines greatly robustness of output response. It is not reliable to optimize this product design problem using traditional deterministic method. How to find the optimum value of these design parameters is an important problem to improve the design robustness of new product. Here, we propose a robustness optimization method of new product based on Monte Carlo simulation.

Procedures are as followed [6] -[10] :

1) To analyze the problem existing in new product design, and define the relationship between design variables and response output as a response surface model;

2) To analyze design variables’ distribution types and define design variables’ distribution characteristics, such as mean and standard deviation, etc.;

3) To sample from populations of random design variables X, and get the sample;

4) To bring sample into position error model between response surface model and standard value, , then get a r position error sample and compose an experiment

, (is response output,);

5) To check if it meets, is allowed error maximum.

6) To repeat step 2) to 5) by k times, and calculate the number of in all independent samples,

, here is the number of,;

7) To calculate reliability ratio. If (is acceptable reliability), then process or product is robust, otherwise we need to modify the precision value of random design variables to enhance reliability ratio and robustness of the manufacture process, then the manufacture process is optimized.

Considering design of a pressure container, according to mechanics of materials, the pressure container’s axial

stress is, hoop stress is, here is internal pressure of container; is wall

thickness of container; is internal radius of container; is half height of container. Material of container is 15 MnV. Through observation of experiment, we defined the design variables and their distribution characteristics as followed: internal work pressure of container follows lognormal distribution,

the tolerance of is (10, 20), the material strength limit follows lognormal distribution,

the tolerance of is (340, 420), wall thickness of container follows normal distribution,

the tolerance of is (1.5, 4.5), internal radius of container R follows normal distribution,

the tolerance of R is (50, 100), half height of container H follows normal distribution, the to-

lerance of H is (130, 210). Our objective is to maximize container’s volume under 95% failure probability (con-

fidence level) of strength and container size falling Interval.

As known, our objective function is volume maximum of the pressure container, that is

Constraints are

Firstly we established response surface model of response variables concerned pressure container and constraints condition, and run Monte Carlo experiment 3000 times using Crystal Ball 7. According to Anderson-Darling testing, Chi-Square testing and K-S testing [11] -[13] , we can fit the probability distribution type of all four response variables as Normal distribution, Gamma distribution, Gamma distribution and Beta distribution, see Table 1.

Then we get probability distribution and cumulative probability distribution of response variable and constraints as showed in Figure 1.

When random changes exist in design variables, traditional determined optimization method cannot guarantee robustness of product and process design to the extent. While Monte Carlo method can be used to precision control and optimization in product and process design, which can avoid increasing cost due to duplicate experi- ments and excessive design precision, as well as low pass percentage due to deficient design precision. In robust analysis and design, Monte Carlo method also can study on change of response model of product and process brought by modification of design variables, get probability distribution and statistical parameters’ values of response variables, further improve design robustness of product and process and realize robustness design and optimization of product and process. It proved that analysis and optimization of response surface model based on Monte Carlo method was a good robustness design method, and can markedly improve robustness, precision and pass percentage of product and process.

It was supported by National Natural Science Foundation of China (71102047, 71302016, 71302017) and “The Fundamental Research Funds for the Central Universities” (NKZXB1202). Many thanks are also given to anonymous reviewers.