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In this study, three factors such as die temperature, liquid fraction and compression holding time are selected as the independent variables. Hardness average measured at eight
locations of the piston and standard deviation of the hardness data are resolved as the dependent variables. The mathematical model of this system can be expressed as follows
η1=f ξ1; ξ2; ξ3 e T (1)
η2=g ξ1; ξ2; ξ3 e T (2)
Here, η1 is average hardness and η2 is standard deviation of hardness.
3.1.2 Central composite design and analysis of variation
Central composite design is suitable for the second order regression model and has significantly fewer experiments than the 33 factorial designs. At the same time, it has advantageous characteristics for sequential experiment from which once the first parts of the experiment, 23 factorial designs, are performed for first-order regression analysis and simultaneously the analysis is determined to be inappropriate. Then, for new experimental design, the additional experimental conditions, such as the axial points and the center points, make it possible to move into the
second order regression schedules. Another advantage could be found from the fact that it allows variation analysis because it contains the 23 factorial designs. Table 1 shows the experimental conditions obtained from the methodology of central composite design.
3.1.3 Second-order regression analysis
To approximate the trend of nonlinearity of the system, the quadratic equation is adopted for regression models. The quadratic equations of k independent variables are expressed as follows.
Each coefficient can be calculated from the matrix equations.
3.2 Optimization of the parameters by genetic algorithm and neural network
With two response surface equations related to the hardness average and the standard deviation of the hardness values, optimization of the process parameters could be regarded as a multi-objective problem in which optimization of the
process parameters could be achieved both by maximizing the hardness average and minimizing the standard deviation of the hardness values simultaneously. This paper
proposes the NN decision maker and the genetic algorithm as a solution technique for the multi-objective problem. The superiority of the NN decision maker attributes to the fact that it could reflect supervisors’ decisions over the whole range of the problem. Genetic algorithms have little restriction on the form of the objective functions and, therefore, NN decision maker could be splendidly mounted into the optimization. At the same time, its excellent ability of searching for the global optimal solution cannot be overlooked.
3.2.1 Fitness evaluation of dual-objective problem
Fitness functions in genetic algorithms are similar to objective functions in optimization problems. The fitness score is a potentially transformed rating used by a genetic algorithm to determine the fitness of inpiduals for mating. The fitness score is typically obtained by linear scaling of the objective score, which is the output of objective functions. The optimization problem of this study, as mentioned before, is not a single objective problem but a multi-criteria optimization problem that can be achieved both by maximizing the hardness average and
minimizing the standard deviation of the hardness values simultaneously. In multi-objective optimization the notion of optimality is not at all obvious.A weight factor strategy is generally used and regarded as a simple and convenient solution technique of multicriteria optimization. It uses values of the linear combination between the objective functions and their weight factors determined by the ranking of dominance on the
goal. However, despite its simplicity in application, the mechanics of its operation are somewhat unclear and even the optimality of final solutions are often distracted from the supervisor’s purpose.This paper introduces a new methodology, neural network(NN) decision maker, which could reflect effectively a supervisor’s intention on the whole range of the optimization problem. Firstly, a decision making pattern is created in a tabular form whose data is evaluated according to the supervisor’s decision on the optimality of the dualobjective function. Secondly, the data of the decision making pattern is trained into a multi-layer neural network by back-propagation training skill. For this reason, NN decision maker can be regarded to yield a reasonable evaluation of optimality based on the generated decision pattern.Multi-layer neural networks are known to have an aspect of the nonlinear interpolation function, since their