基于满意度函数的多响应曲面稳健优化
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摘要
本文研究具有多个质量特性的产品或过程稳健参数优化问题,目的在于获得对某些不确定性扰动不敏感的稳健最优解。以此为目标,本文使用响应曲面法建立各个响应变量与设计因子之间的经验模型,然后借助满意度函数法对多响应问题进行处理。本文重点考虑两种误差扰动:因子的制造容差和响应曲面模型的预测误差,并使用稳健对等方法定义满意度函数的稳健性指标。具体研究内容包括:
首先,分别引入遗传算法和模拟退火这两种智能算法对总体满意度函数进行极大化寻优,并使用模式搜索算法对返回的解进行进一步细探。算例表明,与模式搜索算法相比,智能算法更适合处理复杂函数优化问题;与单一智能优化算法相比,混合算法则能够提高解的收敛精度。
其次,针对因子的容差扰动定义满意度函数的稳健对等式,并使用遗传+模式搜索混合算法对稳健最优解进行搜索。算例表明,该方法能够成功获得稳健可行域中的解,这样的解对因子的制造误差不敏感。
再次,分析预测响应的波动特性对满意度函数的影响,并借助蒙特卡罗方法模拟出满意度函数的分布形状并研究其统计规律。算例表明,传统满意度函数法所获得的全局最优解可能具有太高的概率风险,而局部最优解所处的可行域往往对预测响应的波动更加稳健,这有助于对稳健最优解的进一步探索。
最后,将响应曲面模型的预测误差考虑到满意度函数法的优化模型中,使用稳健对等方法定义满意度函数的稳健性指标,并借助遗传+模式搜索混合算法对该稳健性指标进行极大化寻优。算例表明,该方法能够成功返回稳健可行域中的解,并且大大减小满意度的波动范围,使其对模型的预测误差抗干扰。
尽管本文所使用的算例来自化工和半导体行业,但本文所给出的多响应稳健优化方法不局限于这些行业,而是对不同领域中的稳健设计与优化问题均具有一定的普适性。
This study copes with robust parameter optimization of product or process involving multiple independent quality characteristics. The purpose is to acquire robust optimal solutions which are insensitive to some pre-defined uncertainties. In this thesis, the response surface model is established to relate each response and design factors, and then the desirability function method is utilized to compromise all the responses. Using robust counterpart approach to measure robustness, we mainly deal with two types of uncertainty, i.e. tolerances bands on design factors and prediction errors of response surface models. The topics discussed in this paper are outlined as following.
Firstly, we develop two intelligent algorithms, i.e. the genetic algorithm (GA) and the simulated annealing (SA) algorithm to find the maximum of overall desirability function. The pattern search (PS) algorithm is utilized to refine solutions found by GA and SA. Computational examples reveal that the intelligent algorithms have outperformed a single PS algorithm when the optimization problem is complex, but the PS algorithm can enchance the convergence precision of GA and SA. Thus, we propose to use the hybrid algorithm in this study rather than a single algorithm alone.
Secondly, we present the robust counterpart for the desirability function method when the variations on input variables are considered. The GA post-hybridized with PS algorithm is employed to search for the robust optimum. The numerical example demonstrates that the proposed method can successfully find solutions lying in the robust feasible region. The so-obtained solutions are of more practical meanings since they are robust against production tolerances or manufacturing imprecision.
Thirdly, we investigate the impact of (poor) model predictions on the desirability function. The Monte Carlo approach is used to simulate the distribution of desirability function and to give a statistical analysis of the simulated results. The case example shows that the traditional global optimum is more likely to have a high probabilistic risk, while the local optimum usually implys a new feasible operating region, which is helpful to guide for robust optimum solutions.
Finally, we embed the uncertainty information of model predictions into the standard desirability function method. The GA combined with PS algorithm is used to find the robust optimum. The illustrated example indicates that the presented approach is very useful in indentifying solutions lying in the robust feasible region. It can also alleviate the impact of prediction errors a great deal on the desirability function. The found solutions are of more practical values because they are insensitive to model prediction errors.
Although the examples illustrated in this thesis are motivated from the chemical or semiconductor industry, the procedures are quite general and are not restricted to these fields.
首先,分别引入遗传算法和模拟退火这两种智能算法对总体满意度函数进行极大化寻优,并使用模式搜索算法对返回的解进行进一步细探。算例表明,与模式搜索算法相比,智能算法更适合处理复杂函数优化问题;与单一智能优化算法相比,混合算法则能够提高解的收敛精度。
其次,针对因子的容差扰动定义满意度函数的稳健对等式,并使用遗传+模式搜索混合算法对稳健最优解进行搜索。算例表明,该方法能够成功获得稳健可行域中的解,这样的解对因子的制造误差不敏感。
再次,分析预测响应的波动特性对满意度函数的影响,并借助蒙特卡罗方法模拟出满意度函数的分布形状并研究其统计规律。算例表明,传统满意度函数法所获得的全局最优解可能具有太高的概率风险,而局部最优解所处的可行域往往对预测响应的波动更加稳健,这有助于对稳健最优解的进一步探索。
最后,将响应曲面模型的预测误差考虑到满意度函数法的优化模型中,使用稳健对等方法定义满意度函数的稳健性指标,并借助遗传+模式搜索混合算法对该稳健性指标进行极大化寻优。算例表明,该方法能够成功返回稳健可行域中的解,并且大大减小满意度的波动范围,使其对模型的预测误差抗干扰。
尽管本文所使用的算例来自化工和半导体行业,但本文所给出的多响应稳健优化方法不局限于这些行业,而是对不同领域中的稳健设计与优化问题均具有一定的普适性。
This study copes with robust parameter optimization of product or process involving multiple independent quality characteristics. The purpose is to acquire robust optimal solutions which are insensitive to some pre-defined uncertainties. In this thesis, the response surface model is established to relate each response and design factors, and then the desirability function method is utilized to compromise all the responses. Using robust counterpart approach to measure robustness, we mainly deal with two types of uncertainty, i.e. tolerances bands on design factors and prediction errors of response surface models. The topics discussed in this paper are outlined as following.
Firstly, we develop two intelligent algorithms, i.e. the genetic algorithm (GA) and the simulated annealing (SA) algorithm to find the maximum of overall desirability function. The pattern search (PS) algorithm is utilized to refine solutions found by GA and SA. Computational examples reveal that the intelligent algorithms have outperformed a single PS algorithm when the optimization problem is complex, but the PS algorithm can enchance the convergence precision of GA and SA. Thus, we propose to use the hybrid algorithm in this study rather than a single algorithm alone.
Secondly, we present the robust counterpart for the desirability function method when the variations on input variables are considered. The GA post-hybridized with PS algorithm is employed to search for the robust optimum. The numerical example demonstrates that the proposed method can successfully find solutions lying in the robust feasible region. The so-obtained solutions are of more practical meanings since they are robust against production tolerances or manufacturing imprecision.
Thirdly, we investigate the impact of (poor) model predictions on the desirability function. The Monte Carlo approach is used to simulate the distribution of desirability function and to give a statistical analysis of the simulated results. The case example shows that the traditional global optimum is more likely to have a high probabilistic risk, while the local optimum usually implys a new feasible operating region, which is helpful to guide for robust optimum solutions.
Finally, we embed the uncertainty information of model predictions into the standard desirability function method. The GA combined with PS algorithm is used to find the robust optimum. The illustrated example indicates that the presented approach is very useful in indentifying solutions lying in the robust feasible region. It can also alleviate the impact of prediction errors a great deal on the desirability function. The found solutions are of more practical values because they are insensitive to model prediction errors.
Although the examples illustrated in this thesis are motivated from the chemical or semiconductor industry, the procedures are quite general and are not restricted to these fields.
引文
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