Regression-based multiobjective optimization approaches for the autoclave composite parts placement problem: a mixed integer nonlinear programming model and an evolutionary algorithm

Abstract

The autoclave curing process is an important step in composite parts production, in which a batch of parts is cured in an autoclave simultaneously by systematically changing the inner temperature and pressure. Typically, curing cycles have three main phases: heating, dwell, and cooling. During the heating phase, parts are heated until they reach a curing temperature. Due to factors such as positions of parts inside the autoclave and batch composition, it is often not possible for the parts to reach the curing temperature simultaneously. Parts that reach the curing temperature earlier than the others are overcured, which negatively affects their quality. Moreover, shorter curing cycles are preferred due to the savings in cost, energy, and time. This study addresses these two considerations with a unified approach that integrates two decision support methods: regression and optimization. In the first stage, we determine the factors affecting the time to reach the curing temperature and relate them using multiple linear regression models. In the second stage, we utilize the regression models of the first stage and determine efficient placements of the parts in the autoclave considering two objectives: minimization of the duration of the heating phase and the maximum time delay between parts in reaching the curing temperature. The former corresponds to increasing productivity and the latter corresponds to minimizing quality losses. We propose a biobjective mixed integer nonlinear programming model together with its equivalent linear model to generate the efficient frontier. Additionally, to obtain solutions faster, a multiobjective evolutionary algorithm and its mechanisms that address the problem are proposed. The approach is applied on real cases in a composite factory. The estimates of the regression models are significantly close to the realizations, and considerable gains in both objectives are observed with the optimization tools.

Appendix: Verification of the regression models

The following assumptions for the regression models are checked:

• Linear relation between the predictors and the response variable: The scatter plots and Spearman’s correlation coefficients are used to evaluate the relationship. The correlation coefficients take values in the range between 0.2 and 0.8, and 30% of the relations have a coefficient of 0.7. Since values close to 1.0 indicate a strong association, we conclude that predictor-response variable pairs have considerable associations.

• Normally-distributed residuals: For all regression models, the residuals form an approximate straight line in the normal probability plots and considered to be approximately normal distributed.

• No multicollinearity: The variance inflation factors of all predictor variables are between 1 and 10, from which we conclude that they are not correlated with each other.

• Homoscedasticity: No patterns are observed from the residual plots.

Next, the goodness-of-fit of the models is checked using the standard error of the regression, $S$ and the coefficient of multiple determination, adjusted-R². Smaller values of $S$ indicate smaller deviations between the data and the fitted values. Larger values of adjusted-R² indicate that the model explains a larger variation percentage of the response variable. In the study, the $S$ values range between 3.8 and 8.2, with an average of 5.92. The adjusted-R² values range between 37.9 and 91.0, with an average of 76.0. In general, the models of the areas that are closer to the fan fit the data better. The reason for this can be other external factors affecting the heating duration in those areas, which we could not account for in the regression models. Overall, the fit of the models is satisfactory.