Methods for optimization of nonlinear problems with discrete variables: A review

Meta-Models and Genetic Algorithm Application to Approximate Optimization with Discrete Variables for Fire Resistance Design of A60 Class Bulkhead Penetration Piece

A60 class bulkhead penetration piece is a fire-resistance apparatus installed on bulkhead compartments to protect lives and to prevent flame diffusion in case of fire accident in ships and offshore plants. In this study, approximate optimization with discrete variables was carried out for the fire-resistance design of an A60 class bulkhead penetration piece (A60 BPP) using various meta-models and multi-island genetic algorithms. Transient heat transfer analysis was carried out to evaluate the fire-resistance design of the A60 class bulkhead penetration piece, and we verified the results of the analysis via a fire test. The design of the experiment’s method was applied to generate the meta-models to be used for the approximate optimization, and the verified results of the transient heat transfer analysis were integrated with the design of the experiment’s method. The meta-models used in the approximate optimization were response surface model, Kriging, and radial basis function-based neural network. In the approximate optimization, the bulkhead penetration piece length, diameter, material type, and insulation density were applied to discrete design variables, and constraints that were considered include temperature, productivity, and cost. The approximate optimum design problem based on the meta-model was formulated such that the discrete design variables were determined by minimizing the weight of the A60 class bulkhead penetration piece subject to the limit values of constraints. In the context of approximate accuracy, the solution results from the approximate optimization were compared to actual analysis results. It was concluded that the radial basis function-based neural network, among the meta-models used in the approximate optimization, showed the most accurate optimum design results for the fire-resistance design of the A60 class bulkhead penetration piece.

Flexible piecewise linear model for investigating dose-response relationship in meta-analysis: Methodology, examples, and comparison

OBJECTIVES

Dose-response meta-analysis (DRMA) is widely employed in establishing the potential dose-response relationship between continuous exposures and disease outcomes. However, there is no valid DRMA method readily for discrete exposures, especially when the possible dose-response trend not likely to be linear. We proposed a piecewise linear DRMA model as a solution to this issue.

METHODS

We illustrated the methodology of piecewise linear model in both one-stage DRMA approach and two-stage DRMA approach. The method by testing the equality of slopes of each piecewise was employed to judge if there is "piecewise effect" against a simple linear trend. We then used sleep (continuous exposure) and parity (discrete exposure) data as examples to illustrate how to apply the model in DRMA using the Stata code attached. We also empirically compared the slopes of piecewise linear model with simple linear as well as restricted cubic spline model.

RESULTS

Both one-stage and two-stage piecewise linear DRMA model fitted well in our examples, and the results were similar. Obvious "piecewise effects" were detected in both the two samples by the method we used. In our example, the new model showed a better fitting effect and practical, reliable results compared to the simple linear model, while similar results for to restricted cubic spline model.

CONCLUSION

Piecewise linear function is a valid and straightforward method for DRMA and can be used for discrete exposures, especially when the simple linear function is under fitted. It represents a superior model to linear model in DRMA and may be an alternative model to the nonlinear model.

A Modified Jaya Algorithm for Mixed-Variable Optimization Problems

Mixed-variable optimization problems consist of the continuous, integer, and discrete variables generally used in various engineering optimization problems. These variables increase the computational cost and complexity of optimization problems due to the handling of variables. Moreover, there are few optimization algorithms that give a globally optimal solution for non-differential and non-convex objective functions. Initially, the Jaya algorithm has been developed for continuous variable optimization problems. In this paper, the Jaya algorithm is further extended for solving mixed-variable optimization problems. In the proposed algorithm, continuous variables remain in the continuous domain while continuous domains of discrete and integer variables are converted into discrete and integer domains applying bound constraint of the middle point of corresponding two consecutive values of discrete and integer variables. The effectiveness of the proposed algorithm is evaluated through examples of mixed-variable optimization problems taken from previous research works, and optimum solutions are validated with other mixed-variable optimization algorithms. The proposed algorithm is also applied to two-plane balancing of the unbalanced rigid threshing rotor, using the number of balance masses on plane 1 and plane 2. It is found that the proposed algorithm is computationally more efficient and easier to use than other mixed optimization techniques.