1. Introduction
Presently, computer-aided optimization is one of the most important computer-aided design tools for product development and improvement. In recent years, the optimization design for expensive function was carried out using a combination of global optimization techniques. Due to high-cost computation function, a surrogate model and an optimization technique are combined to increase the efficiency of the optimization algorithm. The optimization with a surrogate model is popular and used in the various fields such as, mechanical engineering [
1,
2], chemical engineering [
3,
4], aerospace engineering [
5,
6], etc. One of the popular methods that combined the surrogate model and the optimization technique to solved an expensive function is an efficient global optimization (EGO) [
7] because it can reduce the computational cost for an optimization process. The EGO process starts with generating an initial set of sampling points. The objective of this process is to increase the diversity of the sampling points. The most popular method for generating the initial sampling is the Latin Hypercube Sampling (LHS) [
8]. Most of researchers try to increase the efficiency of this process [
9,
10,
11]. Then, the surrogate model is used to estimate the actual function value. The original EGO uses the Kriging method, which the accuracy of this function directly affects the efficiency of the optimization method. Finally, the optimization method is used to find the optimal point of the surrogate model and that point is used to evaluate the next additional sampling point to increase the accuracy of the surrogate model and find the better optimal solution. Therefore, the important process of the EGO is to find additional points. The original EGO is designed for finding single additional sampling points per one iteration. However, the time required for completing a single simulation is often one day or more, thus, finding numerous sampling points for the original EGO will take a longer time. Much research work has solved this problem by using the parallel computing technique [
12]. For example, Ginsbourger et al. [
13,
14] extended the original EGO to parallel computing by applying the multivariate expected improvement (q-EI) and implementing it via Monte Carlo simulation. Then, Frazier [
15] proposed a q-EI stochastic gradient approach for calculation the q-EI. Masahiro M. [
16] proposed a method for adding various points in combination with maximization of EI by using a multi-additional sampling (MAs) method.
The weak point of the multiple additional sampling EI is it requires more experiments or the expensive data than the single additional sampling [
17] because it can use less data to predict a function with the surrogate model. The target to increase the efficiency of the multiple additional sampling points by EGO is to increase the accuracy of the surrogate model. Fortunately, in real-world design problems, it is possible to select a variety of physical computation models to tackle a design problem. For example, an efficiency of a helicopter rotor blade [
18] could be calculated by using the blade element momentum method (BEMT) or using Navier–Stokes equations. However, the advantage of the BEMT is that it can calculate the efficiency within a few minutes but it has lower efficiency than calculation via the Navier–Stokes equation which takes significantly longer computing time. The efficiency of an airfoil could be calculated using a potential flow equation [
19] and the Navier–Stokes equation [
20]. For the structural analysis, the Finite Element Method (FEM) and a novel numerical implementation of the asymptotic homogenization method (NIAH) [
21] could be used to predict the buckling analysis. With this advantage, many researchers try to combine the data of several fidelity levels to increase the accuracy of the surrogate model for the EGO process. Forrester, Sóbester and Keane [
22] proposed the co-Kriging method for the multi-fidelity optimization. Huang [
23] used the co-Kriging method as the multi-fidelity surrogate model to design the aircraft. Brooks [
24] also used the co-Kriging method to design the transonic compressor rotor. Sun [
25] used the radial basis function (RBF) and the response surface hybrid surrogate model to find the optimal shape of the honeycomb structure. Liu [
26] proposed the Gaussian/RBF surrogate model for multi-fidelity optimization. In addition, Ariyarit [
18,
27] proposed the RBF/Kriging surrogate model applied with the EGO to find the optimal shape of the helicopter rotor blade and the airfoil. However, most of the proposed multi-fidelity surrogate models are designed for single-additional sampling points.
In this study, a multi-additional multi-fidelity EGO, which could improve the efficiency in searching for an optimum point, is proposed and investigated by implementing it to symmetry and asymmetry mathematical test problems. The co-Kriging and RBF/Kriging hybrid surrogate model are selected to show which surrogate model is most suitable for the proposed method. The results of the proposed method are compared with the original multi-additional sampling EGO.
2. Efficient Global Optimization
The EGO [
7] procedure, shown in
Figure 1, starts with the generated samples for low fidelity and high-fidelity using the design of experiment technique (DoE) [
28]. In this research, the LHS [
8] is selected for this process, because it could maintain the diversity of the data and can fix the number of experiments by a user. The next process is to construct the surrogate model, in which the Kriging method is selected for a single-fidelity optimization process and co-Kriging and RBF/Kriging hybrid surrogate model are selected for a multi-fidelity optimization process. The schematic diagram of the single-fidelity and multi-fidelity surrogate models is shown in
Figure 2. After the surrogate model is constructed, an additional sample point for an optimization method can be found by maximizing the EI [
7] via genetic algorithms [
29].
The EI(
) at point
can be expressed as
where
is the probability density function representing uncertainty about
, which predicts a function value from a surrogate model. Additional sampling points based on using the EI is repeated until an objective function converges.
2.1. Kriging Surrogate Model
An ordinary Kriging model [
30] can predict the unknown function
as
where
and
denote a global model and a local model, respectively. The global model (
) is expressed as
where
is a matrix denoting the correlation between the sample points, and
is a vector containing the evaluation value of each sampling point. For the Kriging surrogate model,
denotes the constant global model. The local model (
) is expressed as
where
is a vector written in terms of
is a vector of sampling points. The correlation between
and
is a function of the distance between
and
. In the Kriging surrogate model, the local derivation at an unknown point
is expressed using stochastic processes. Several design points are generated as sampling points and then a surrogate model is constricted using a Gaussian random function as the correlation function for the estimation of the trend using the stochastic process.
2.2. Co-Kriging Surrogate Model
A Co-Kriging surrogate model [
22] is the extension of a Kriging surrogate model by combining the low-fidelity data and high fidelity data. A co-Kriging method can be predicted an unknown function
by Equation (
3). However, the different point of these method is the co-Kriging model estimates the global model
and the local model
by combining data of the low-fidelity and the high-fidelity data by using the scaling parameter (
) of the different model (
), which is defined as:
where
is a vector that contains the high-fidelity evaluation value of each high-fidelity sample points, and
is the estimated value via the Kriging surrogate model of each high-fidelity sample points.
2.3. Rbf/Kriging Multi-Fidelity Surrogate Model
A RBF/Kriging multi-fidelity surrogate model [
18] is the combination surrogate model between a Radial Basis Function (RBF) [
31] and the Kriging surrogate model. In this surrogate model, the RBF is used to predicted the local model (
) of the Kriging surrogate model in Equation (
3). The different point of the RBF/Kriging hybrid surrogate model and co-Kriging surrogate model is the hybrid surrogate model uses the low-fidelity data to predict the global model (
), but the co-Kriging method uses the low-fidelity data to predict the local model (
).The hybrid surrogate model can predict the unknown function
as
In this study,
is calculated using the data of low-fidelity data and the
at an unknown design point (
) can be expressed as
where
N is the number of sampling design solutions,
denotes the weight coefficient of the
ith design sampling point, and
represents a positive coefficient.
3. Multiple Additional Sampling for Multi-Fidelity Optimization
The original multi-fidelity EGO can be obtained one additional sample in each iteration where the process of the original multi-fidelity EGO is illustrated in
Figure 1b. A designer can only employ computational resources for one evaluation in this case, whereas they can use a parallel environment in the initial sampling. Here, available computational resources are not employed during the additional sampling stage, as illustrated in
Figure 3a.
Here, multi-additional sampling (MAs) for multi-fidelity optimization is proposed. In EGO with MAs, a sub-iteration is additionally included, as illustrated in
Figure 3b. In this sub-iteration, an additional sampling solution (obtained by an iterative approach) is included to construct an updated multi-fidelity surrogate model by using the predicted point
as a temporal function value. Another additional sampling solution is obtained through EI maximization.
A schematic illustration of additional sampling in the proposed MAs method is illustrated in
Figure 4. This process begins by having
from using EI maximization on the initial surrogate model. The predicted value
is also computed, and then the model is then updated temporally using (
). As the EI value around
should not be greater, the next additional point (
) can be obtained. This process is repeated until an arbitrary number of additional sampling solutions are obtained. Thereafter, the exact values
are calculated using a parallel evaluation environment for
by using the expensive function. The set of these additional sampling points is finally added to the data set for the improvement of the model.
In this sub-iteration, only the values predicted by the surrogate model are required. Therefore, multiple additional samples can be rapidly obtained. In the main iteration, the exact value for these additional sampling solutions can be computed using a parallel evaluation. As a result, a design using this EGO method can be completed more quickly compared to the original EGO. The number of sub-iterations can be found using the number of parallel evaluations, where the value of EI or the number of experiments could be the termination condition of the EGO process.
4. Investigation Problems
Here, the proposed method is investigated by solving test functions. The effect of multi-fidelity surrogate model, which are co-Kriging, RBF/Kriging surrogate model, for the multi-sampling EGO process results are compared with results given by the single-fidelity multi-sampling EGO to demonstrate the efficiency of these methods.
In this study, four test functions [
18,
32,
33] are used.
denotes a high-fidelity function and
denoted a low-fidelity function.
The six-hump camel-back function (SC) is defined as follows:
The Himmelblau function (HIM) is defined as follows:
The Rosenbrock function (ROS) is defined as follows:
The Colville function (COL) is defined as follows:
In each investigation, 10 initial high-fidelity points and 200 low-fidelity point were acquired using LHS. The number of additional sampling points was set as 12 for all test functions and 2–4 multi-additional sampling were selected to test the effect of multi-additional sampling compared with the results of single-additional sampling for the single-fidelity and multi-fidelity EGO. The function of the low-fidelity data is slightly different from the high-fidelity data because in the real world computation there are some errors between these functions. However, the characteristic of the low-fidelity and high-fidelity function is they, to some extent, have similar function landscapes. To simulate the situation, the low fidelity function is always set as its high fidelity counterpart multiplied by a factor. This implies that the high fidelity function is always more accurate.
6. Conclusions
This study investigated the effect of multi-fidelity technique to the multi-additional EGO process. The RBF/Kriging hybrid multi-fidelity EGO and CO-Kriging multi-fidelity EGO are selected to test the effect of this framework. The MAs technique could help the EGO process to run the parallel computing that can reduce the computation time. However, the weak point of the MAs technique combined with the EGO results in the accuracy of the surrogate model being reduced, which leads to the use of resources to find the optimum solution. Consequently, the multi-fidelity technique is proposed to combineEGO with MAs to increase the efficiency of the EGO process by increasing the accuracy of the surrogate model.
To examine the multi-fidelity EGO framework, four test functions are solved, and the results are compared with those obtained by the ordinary Kriging EGO. The optimum results for the test functions show that the RBF/Kriging hybrid multi-fidelity EGO has the lowest effect when using the RBF/Kriging hybrid multi-additional sampling framework because the RBF/Kriging hybrid multi-fidelity surrogate model can maintain the accuracy of the surrogate model when it runs with higher number of MAs. For some of the number of MAs test functions the Co-Kriging could receive the best optimal solution when solving with the SAS but it cannot obtain a good solution when solving via multi-additional sampling framework. The ordinary EGO has a high-effect when using the multi-additional sampling framework. The optimal solution of the ordinary EGO gets worse results when solving the optimal solution via higher number of multi-additional sampling points. This results suggest that the hybrid RBF/Kriging multi-fidelity EGO is the best choice for the MAs framework because it can maintain the accuracy of the surrogate model that affects the efficiency of the optimization process.