2。2。2。 Metamodel-based optimization method

Metamodel-based optimization methods is an approach that objective functions are frequently approximated into the explicit form of low order polynomials with acceptable accuracy。 Once the metamodel mathematically renders the process with minimum error, the optimization problem is easy to tackle by applying appropriate optimization techniques。 Metamod- el-based optimization methods are widely used compared to direct optimization。 The common metamodels are response surface methodology (RSM), artificial neural network (ANN), radial basis function (RBF), Kriging, and hybrid model。 The re- view of metamodeling technique for computer-based engineering design and optimization can be found in the survey of Simpson et al。 [3] and the work of Wang and Shan [4]。 This optimization method has some benefits such as being easy to connect to simulation program, to render a view of entire design space as well as computational efficiency as claimed by Papalambros [5], Wang and Shan [4], and Park and Dang   [6]。

2。3。 Review of process parameter optimization in injection   molding

Direct optimization method is not often used in injection molding。 This method requires a complex integration between simulation tool and optimization code。 There are several authors who have used this approach。 Lam et al。 [7] proposed a GA/ gradient hybrid approach for injection molding conditions optimization。 GA method optimization approach requires a huge number of function evaluations or an enormous number of simulation cycles。 Parallel computing can reduce the simulation time when some computers run simultaneously。 Wu et al。 adopted an enhanced genetic algorithm, referred to distributed multi-population genetic algorithm。 Their approach combined an optimization algorithm and commercial Moldflow soft- ware with a dominance-based constraint-handling technique and a master–slave distributed architecture [8]。 Direct optimi- zation method can also be carried out using only gradient-based optimization technique。 This approach sometimes converges quickly when the optimization problem is low   nonlinear。

Metamodel-based optimization methods are widely used in injection molding。 Most of the common metamodeling tech- niques such as RSM, ANN, RBF, and Kriging model are applied。 The application of metamodel-based optimization methods depends on particular cases and on the preferred use of the researchers。 Following are the common optimization methods appeared in the literature in the field of plastic injection molding。

2。3。1。 RSM model

RSM is one of the metamodeling techniques in which the relationship between input and output is often expressed in the form of quadratic polynomial。 Although this is a traditional method, it is widely used by many authors due to its maturity and the ease of use。 Orthogonal array is often used as the design of experiment (DOE) method for this approach。 RSM is used in conjunction with GA optimization algorithms to minimize the warpage, sink-mark or shrinkage [9–13]。 In fact, we can use any optimization techniques to resolve the optimization problem expressed in terms of RSM model。 However, most of the authors used GA because they thought that GA is a global optimization。 GA can avoid being trapped in local extremum。 Other authors used RSM in conjunction with gradient-based optimization techniques, or they applied RSM to predict the effects of process parameters on the quality of molded parts   [11,14–17]。

2。3。2。 Artificial neural network model

ANN that mimics some basic aspects of the functionality of human brain is an emerging approach because ANN is a pow- erful tool for predicting high nonlinear responses via function approximation。 There are many authors who used ANN as a predictor model showing the relationship between process parameters and quality index。 Kwak et al。 [18], Yarlagadda and Teck Khong [19], and Yarlagadda [20] stated that the neural network predictor using learning data extracted by CAE analysis agrees well with the experimental results。 Kenig et al。 [21], Mok and Kwong [22], Chen et al。 [23], and Altan [24] claimed that the neural network model can accurately predict the product quality, and this approach is usable and efficient tool for quality criteria prediction (shrinkage, weight, or tensile strength)。 ANN is considered as a robust model to predict the relationship between process parameters and the quality of molded parts。 The process parameter optimization can be carried out based on this approximate  relation。