Metamodel Multi-objective Optimization Tool for Mechatronic System Design

Adel Younis and Zuomin Dong Department of Mechanical Engineering,

University of Victoria Victoria, BC, Canada

E-mail: zdong@uvic.ca

Abstract — The use of approximation models in multiobjective optimization problems that involve expensive analysis and simulation processes such as multi-physics modeling and simulation, finite element analysis (FEA) and computational fluid dynamics (CFD) has become more popular and more attractive, especially for the optimization of complex mechatronics systems. Approximation models have been found as a promising tool for multiobjective optimization problems due to their capability for providing accurate modeling results with much less computations for intensive computation problems. Many present global optimization search techniques involve fitness evaluations that are expensive to perform, even worse for problems with multiple objective black-box functions evaluations. In this work, a new adaptive multiobjective optimization approach based metamodeling (AMOP) techniques is introduced. The approach can identify the Pareto front for multiobjective optimization problems efficiently with high accuracy. The computation cost associated with identifying the Pareto front for expensive black-box functions is reduced. The new search method was tested using benchmark test problems and mechatronics device design examples.

Keywords — multiobjective optimization; approximation models; response surface function, radial basis function; Kriging

I. INTRODUCTION Real world engineering design problems are usually

characterized by the presence of many conflicting objectives. Multiobjective optimization problems (MOOP) can be found in various fields when optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.

Multiobjective optimization simultaneously handles two or more conflicting design objectives subject to certain constraints. There is no crisp definition for the optimum as a single objective or goal. The decision is left to the decision maker to decide what the best is. Several conflicting and non-commensurate objective (criterion) functions have to be optimized over a feasible set determined by constraint functions. Due to the conflicting nature of the criteria, a unique optimal solution that meets all criteria does not exist. Based on the commonly used Pareto concept of optimality, one has to deal with a rather large or infinite number of “optimal” solutions. Two different optimal solutions are characterized by the fact that each of them is better in one criterion but worse in another. The fact that improvement of

one criterion results in a loss in another is known as the trade-off between the solutions. Mechatronics applications involve considerations from several design aspects, naturally leading to multiobjective design optimization and needed tradeoff. The complexity of these problems also leads to computation intensive global optimization.

II. RELATED WORK Efficient, effective, and robust techniques are highly in

demand to solve these complex multiobjective optimization problems. Many algorithms were introduced and more are being proposed to explore the design space of multiobjective optimization problems. Most of the available algorithms are based on evolutionary algorithms such as GA, PSO, and others. Few are those that are proposed for black box functions. These are the ones that are proposed to solve multiobjective optimization problems involving expensive analysis and simulation processes such as multi-physics modeling and simulation, finite element analysis (FEA) and computational fluid dynamics (CFD). The optimization of computationally expensive black box functions need more attention and should be addressed because of the challenges these functions have brought to the optimization community. Approximation models are very promising tool that might help with addressing these challenges. Approximation models help simulate and predict the expensive model (function) with less computation cost.

MOOP with expensive high fidelity computational models such as complicated simulations and detailed analyses, the standard multiobjective optimization methods are hard to perform directly because of their high computational cost of objective and constraint function evaluations. The use of approximation models becomes more attractive in these problems.

Two scenarios exist for solving multiobjective optimization problems with black box functions. The first scenario is by directly approximating the Pareto optimal [1], [2], [3] and this can be seen in many evolutionary algorithms (EAs), these methods are usually computationally expensive because a massive number of non-Pareto set points have to be evaluated. The other scenario is using approximate models in which each single objective function is evaluated. It is obvious that the accuracy of the Pareto optimal frontier will depend on the approximation models. Li et al. [4] used a hyper-ellipse

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surrogate to approximate the Pareto optimal frontier for bi-criteria convex optimization problems. If the approximation is not sufficiently accurate, then the Pareto optimal frontier obtained using the surrogate approximation will not be a good approximation of the actual Pareto optimal frontier. Wilson et al. [2] used the surrogate approximations (response surface and Kriging models) for computationally expensive analyses to explore the multiobjective design space and identify Pareto optimal points, the Pareto set, using the approximation models. Their approach minimizes the loss of the accuracy by validating the approximations before the optimization procedure. Proos et al. [5] developed a new algorithm incorporating multiple criterion optimizations into evolutionary structural optimization (ESO) using the weighting and global criterion method. Yang et al. [3] proposed a strategy named it adaptive approximation in multiobjective optimization (AAMO) for multiobjective optimization problems that need expensive computer simulations. They employed MOGA and approximation models and specifically they used Kriging metamodels. MOGA is used to improve the accuracy of the approximation models. Yang et al. [3] also proposed the first framework as claimed managing approximation models in MOO. In the framework, a GA based method is employed with a sequentially updated approximation model. In their approach they update the approximation model in the optimization process and not before the optimization procedure as in [2]. The fidelity of the identified frontier solutions, however, is still built upon the accuracy of the approximation model. The work done in [3] also suffers from the problems of the GA-based MOO algorithm, i.e., the algorithm has difficulty in finding frontier points near the extreme points (the minimum obtained by considering only one objective function).

Shan and Wang [6] proposed a new method named it Pareto Set Pursuing (PSP) in which a sampling guidance functions based on approximation models were developed. They claimed that PSP demonstrates considerable promises in efficiency, accuracy and robustness. Nain and Deb [7] combined an artificial neural network approximation with the NSGA-II to enable the use of GAs on computationally expensive problems and to achieve a computationally effective search and optimization procedure. Their objective is to enable the use of Gas on computationally expensive problems while retaining their basic robust search capabilities. Knowles [8] proposes an extended efficient global optimization algorithm (ParEGO) for MOO problems using the design and analysis of computer experiments (DACE) models which are sequentially updated in the iteration procedure. Kim and Chung [9] developed a multiobjective design optimization framework by combining GAs and Kriging approximation techniques and have tested it on wing platform design problem and demonstrated that the results obtained from the developed approach were efficient and applicable. Liu et al. [10] suggested a novel multiobjective optimization method based on an approximation model management technique. Response surface approximations were used to predict the Pareto optimal sets and design of experiments DOE were used to

generate the sample points. It is an iterative method in which the approximations gets updated in every single iteration by using of the trust region move limits updating strategy, which guarantees the accuracy of the approximations in the interest region of the design space where the actual Pareto optimal solutions may exist. The presented method focused on obtaining the actual Pareto optimal which makes it less dependent on the accuracy of the approximation models. Lim et al. investigated the use of surrogate models in evolutionary search. Recently, Jang et al. [11] employs adaptive approximation framework to resolve the computation burden of the full stochastic fatigue analysis in the optimization.

The algorithm proposed in this work is an adaptive approach inspired by the work done in [6]. This algorithm adaptively selects the right approximate model from three offered approximate models based on a selection criterion which will be explained in this work later. The algorithm adaptively fits the right metamodel to identify the Pareto frontier for the multiobjective problem. Three metamodels are used in this work, QPF, RBF and Kriging. Classical sampling techniques suffer from the curse of dimensionality that is as the number of design variables increases the sampling points needed to explore the design space increases exponentially. Because of that, Latin Hypercube Designs (LHD) is used in this work. LHD explores the design space away better than classical sampling designs. These contribute largely towards a good reduction in the computation cost and have improved and increased the chances of identifying the Pareto frontier for MOO problems.

III. STATEMENT OF THE MULTIOBJECTIVE OPTIMIZATION PROBLEM

Multiobjective optimization can be defined as finding the vector

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which will satisfy the m inequality constraints

����� � ������ � ���� �� (2)

and the p equality constraints

����� � ������ � ���� � �

(3)

and optimize the vector function

������ � �������� ������ � ������ � (4)

where � � ���� �� �� � is the vector of decision variables. In other words, we wish to determine from among the set � of all numbers which satisfy Eqs. (1) and (2) the particular set ���� ��� � ���� which yields the optimum values of all the objective functions.

A. Pareto frontier and Pareto efficiency

The Pareto frontier or Pareto set, named after economist Vilfredo Pareto is the set of all Pareto efficient outcomes. A vector of �� is Pareto optimal if there exists no feasible vector � that would decrease some objective function without causing a simultaneous increase in at least one objective function.

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Mathematically, the Pareto optimality can be defined as follows: A vector of ����is a Pareto optimum if and only if, for any �and �,

����� � ����������� � �� ��� ���� ��������!��� � �!����

(5)

In multiobjective optimization (MOO), the main task is to identify Pareto set points so that a Pareto frontier can be identified. It is hard to claim that these points are in the Pareto set or Pareto frontier unless there is a convincing way to judge whether these points are in the Pareto set or not. That could be decided if a fitness function is adapted.

B. Fitness function

The fitness function used in this work is defined in Eq. (6). This function firstly appeared in [12]. The fitness function is defined as:

"! � #� $ �%� &��'(�)�! $ �)�� � �)�!

$ �)�� � � �)*! $ �)*� �+,-.

(6)

where, "!�� denotes the fitness value of the ith design; �)�! is scaled kth objective function value of the ith design, / ��� ��; and l is called the frontier exponent. In this work, l is taken as a constant 1. The max in Eq. (6) is over all other designs j�i in the set and the min is over all the objectives. The objectives �)�� �)� � �)*��in Eq.(6) are scaled to a range [0, 1]. For example for���)�! ,

���)�! �0%1���! $ 0%1���*!�0%1���*23 $ 0%1���*!�

(7)

where, 0%1���!�denotes un-scaled value of the first objective for the ith design; 0%1���*!��denotes the maximum un-scaled value of the first objective among all designs; and rawf1,min denotes the minimum���)�! �un-scaled value of the first objective among all designs. In case that an objective function is a constant, the scaled objective function value is taken as 1 in this work.

IV. METAMODELING TECHNIQUES USED In this work, three metamodeling techniques were used,

Quadratic polynomial function (QPF), radial basis function (RBF) and Kriging metamodel. These metamodels were selected because of their efficiency in imitating the real black box function (expensive function). The metamodel in simple, easy to calculate form, is used to replace the original, black-box computer analysis and simulation model. The introduced metamodel also provides an insight to the optimization problem by visualizing the interactions among design variables, objective functions and constraints. The overall objective is to reduce the computation cost of computationally intensive design simulations and analyses, using inexpensive surrogates of these analyses and simulations. Computational efficiency which refers to the computational effort required to construct the metamodel and to predict response values for a set of new points using the metamodel and transparency which refers the capability for providing information concerning selection of different variables and interactions

among these variables are two major concerns in generating these metamodels [13].

V. THE PROPOSED APPROACH The main objective of the proposed algorithm (AMOP) is

to efficiently identify the Pareto frontier with less computation cost for costly black-box functions.

There are many statistical methods available nowadays that can be used to generate sample points based on a given probability density function (PDF). Wang et al. [14] were inspired by [15] and developed a sampling method called mode pursuing sampling method (MPS) in which they start by construct an approximate model from a few sample points, these points are called expensive points because they are evaluated using the black box function, and then generate a large number of points using the approximate model, these are called cheap points because they are evaluated using the approximate model constructed, sorts the points, and constructs a cumulative function analogous to cumulative density function (CDF) by adding up all the function values before the current point in the sorted point set. A sample is then drawn from the point set according to this cumulative function. This will result in that more new sample points are generated around the current minimum in the region of interest and less in other regions in the design space. Then, the variation of the objective function in MPS is considered as a sampling guidance function. This sampling guidance function is adapted and used in this work.

After generating few sample points in the space of interest. Three metamodels are constructed. These are QPF, RBF and Kriging. The root mean square error (RSME) is measured. The metamodel with less RSME value is selected to be constructed in the next steps. It is known that Kriging always gives less RSME than QPF and RBF. In this approach, a rule was set to guide the metamodel constructing process that is to check RMSE for QPF and RBF first. If RSME is greater than �, set equal to 5 in this work, then construct Kriging. If the opposite is true then construct either QPF or RBF based on RSME.

In this work, the maximum number of iterations is set to be one of the termination criteria besides measuring the difference between frontier points after two consecutive iterations. If the difference is sufficiently small then the algorithms should terminate.

Steps of the proposed approach

The proposed approach steps consist of 9 steps, as shown in flowchart of Fig. 1.

VI. NUMERICAL EXAMPLES AND RESULTS The proposed method is tested with a well-known

multiobjective optimization problem found in the literature. Also a mechatronics device design example is shown.

A. Test problem#1:

The problem was proposed by Kita et al. [16]. It is a two-objective function with a two decision variables.

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reduction in computational cost of the proposed method. Fig. 2 shows the performance space and evaluated points for test problem#1, how they are distributed and how these points are converging to the Optimal Pareto frontier. Fig. 4 show that in all selected 5 runs the algorithm always identifies to the same Pareto frontier.

Tables 1 and 2 shows the results obtained using the proposed algorithms. The number of iteration; the number of function evaluation took the algorithm to identify the Pareto frontier; how many Pareto points were used to construct the Pareto frontier and how long it took to identify the Pareto frontier set pints are presented in Tables 1 and 2.

Table 7 shows a summary of the best results obtained from more than 10 runs with the number of iterations, the number of objective function evaluations, the number of Pareto points in the optimal Pareto frontier, the CPU time took the algorithm to converge to the Optimal Pareto frontier and the metamodel used in that run to achieve that solution for all test problems. It is clear that RBF which is loyal to the sample points and Kriging which interpolates the sample points and produces good accuracy did a good job in identifying the efficient Pareto set with less computation cost and fairly number of Pareto set points. In Table 8, the variation range and the median of the number of iterations, number of function evaluations and Pareto set points are shown.

TABLE 1. TEST RESULTS OF THE PROPOSED METHOD ON TEST PROBLEM #1

Runs # iterations #

Evaluations # Pareto points

Metamodel

Run#1 71 75 39 KRG Run#2 74 80 41 KRG Run#3 65 71 38 KRG Run#4 66 77 38 RBF Run#5 63 69 41 KRG Run#6 71 77 42 KRG

TABLE 2. TEST RESULTS OF THE PROPOSED METHOD ON TEST PROBLEM #2

Runs # iterations

# Evaluations # Pareto points Metamodel

Run#1 328 409 10 QPF Run#2 328 409 10 QPF Run#3 328 409 10 RBF Run#4 328 409 10 QPF Run#5 328 409 10 KRG

Because of the way LHD generates the sample points in the

region of interest, more than 5 runs have been carried out for each test problem. These results represent the best result obtained from those different runs. The number of iterations, total number of evaluated points, number of converged frontier points, CPU time needed to identify the Pareto frontier and the metamodel used were recorded for 5 different runs.

TABLE 3. BEST TEST RESULTS OF THE PROPOSED APPROACH

Test Problem

Num of Iterations

Num of Evaluations

Pareto Set Points

Metamodel Used

Problem1 66 77 38 RBF

Problem2 328 409 10 RBF

FIGURE 2. PERFORMANCE SPACE & EVALUATED POINTS FOR PROBLEM #1

FIGURE 3. PARETO SET POINTS FOR TEST PROBLEM #1

FIGURE 4. EVALUATED POINTS OF FIVE RUNS FOR TEST PROBLEM #1

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