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Monte Carlo method can analyze, solve and optimize many mathematical or physical problems through generating a large number of statistical random samples to simulating stochastic events. It also can be used to remarkably improve design quality of new product. In new product design process, setting distribution characteristics of the design variables is vital to product quality and production robustness. Firstly, response surface model between output characteristics and design variables in new product design is proposed, and the distribution characteristics of design variables and response output are analyzed; then position error model of response output and standard value and allowed error maximum is presented; and then the differences of position error model and allowed error maximum are count, and reliability ratio is built and calculated, and design robustness of the new product is increased by adjusting the precision value of random design variables in Monte Carlo experiments. Finally, a case is brought forward to verify the validity of the method.

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 [

Procedures are as followed [

1) To analyze the problem existing in new product design, and define the relationship between design variables

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

5) To check if it meets

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

7) To calculate reliability ratio

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

stress is

thickness of container;

the tolerance of

the tolerance of

the tolerance of

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

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

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

We can know from _{1} is_{2} is_{3} is

wave range of constraints both g_{1} and g_{2} go beyond their allowed range. According to requirements, current response output value is not robust and need to improve product design level.

We can know from Sensitivity Analysis of Response Variables and Constraints in

According to relationships between design variables and response output variable, and contributions to variance view from

After Modification of design variables precision, we analyze robustness of response variable and constraints

. Analysis of response variables’ distribution characteristics

Variables | Mean | Median | Standard deviation | Skewness | Kurtosis | Ceff. of variability | Distribution type |
---|---|---|---|---|---|---|---|

f | 8568233.77 | 8568233.77 | 534933.47 | 0.00 | 2.97 | 0.0624 | Normal |

g_{1} | 1522.92 | 1511.21 | 208.31 | 0.3379 | 3.17 | 0.1368 | Gamma |

g_{2} | -0.22 | −3.93 | 47.08 | 0.4748 | 3.34 | −216.04 | Gamma |

g_{3} | 2.50 | 2.50 | 0.13 | 0.00 | 2.97 | 0.0519 | Beta |

Pressure container chart

Probability distribution and cumulative probability distribution of response variable and constraints

Sensitivity analysis of response variables and constraints

again, and get the probability that product design met constraints g_{1} is_{2} is_{3} is

tainer is robust enough, and pass percentage of pressure container has been enhanced greatly. And the maximum volume of container is

Now we further optimize the pressure container to enhance the design robustness using OptQuest optimizer. Here we create the OptQuest model and run simulation experiments 1500 times according to distribution characteristics of design variables and constraints condition [

ity that product design met constraints g_{1} are_{2} was_{3} is

showed, so design robustness of pressure container has been further improved than that of last time.

Probability distribution and cumulative probability distribution of response variable and constraints after modification of parameter precision

Probability distribution and cumulative probability distribution of response variable and constraints after optimization after OptQuest optimization

. Optimum design settings of pressure container with OptQuest

Status | Time Remaining: 00 Simulation:1245 | ||||||||
---|---|---|---|---|---|---|---|---|---|

Simulation | Maximize objective mean | Requirement g_{1} | Requirement g_{2} | Requirement g_{3} | |||||

1 | 4.5380E+06 | 100.00 | 98.8506 | 100.00 | 15 | 385 | 3 | 75 | 170 |

4 | 4.5390E+06 | 100.00 | 98.7106 | 100.00 | 19.6990 | 414.026 | 1.65588 | 79.4806 | 131.772 |

12 | 4.5416E+06 | 100.00 | 99.4676 | 100.00 | 16.6921 | 410.204 | 1.64555 | 78.5969 | 143.661 |

27 | 4.5426E+06 | 100.00 | 99.4423 | 100.00 | 19.6574 | 412.131 | 1.50000 | 73.8756 | 130 |

Best:38 | 4.5532E+06 | 100.00 | 99.1782 | 100.00 | 17.9259 | 395.177 | 3.49114 | 69.6651 | 173.633 |

Optimum design performance gragh of pressure container with OptQuest

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.