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Relevance vector machine and fuzzy system based multi-objective dynamic design optimization: A case study
Expert Systems with Applications 37 (2010) 3598–3604
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Relevance vector machine and fuzzy system based multi-objective dynamic design optimization: A case study Xuemei Liu a, Xiao-Hui Zhang a,*, Jin Yuan b,a a b
School of Mechanical and Electronic Engineering, Shandong Agricultural University, 271018 Tai’an, China School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Keywords: Multi-objective dynamic design optimization Relevance vector machine (RVM) Multi-objective genetic algorithm (MOGA) Fuzzy system Glass stacking machine
a b s t r a c t To improve the original design flaws of overturning assembly of glass stacking machine taken as a case study, a multi-objective optimization approach integrated relevance vector machines (RVM), multiobjective genetic algorithms (MOGA) and fuzzy system are presented for the optimal dynamic design problem. Firstly, the multi-objectives of the overturning assembly are constructed by the use of dynamic structure optimization design theory. The motion simulation and finite element analysis of overturning assembly are utilized for sampling scheme given by uniform design to collect the train dataset. The dataset could describe the non-linear behaviors of dynamic and static characteristics of variety of mechanical structures, which is identified by RVMs. Sequentially, RVM- based meta-model as fitness function is combined with MOGA to obtain the Pareto optimal set. Finally, a fuzzy inference system is established as decision-making support to obtain the optimum preference solution. Therefore, the modified physical prototype with the round solution proofed feasibility and efficiency of this approach. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, dynamic design methods, as an important aspect of modern design, are becoming increasingly common in the field of structural design problems, which can be effective in improving the dynamics performance and stability of equipment, reducing the costs of production. The objective function such as eigenvalue or structural response is a complex implicit non-linear function in the analysis of the dynamics. The huge computation burden is often caused by dynamic design analysis, simulation and optimization. Wang and Shan (2007) described the meta-modeling technology in the optimization of engineering design. In the physical process analysis and simulation, in order to obtain a comparable level of accuracy as physical testing data and not too expensive computational consumption, the meta-modeling has been widely used in various disciplines, as well as in the field of engineering optimization. The meta-modeling is an approximation model used in the non-linear process modeling, no more than to approximate the non-linear finite element model. The statistical learning theory has been popularly developed in non-linear system regression based on input and output data with noise (Chan, Chan, & Cheung, 2001). Relevant vector machine (Tipping, 2001) (RVM) is the general Bayesian learning framework of kernel method for obtaining * Corresponding author. E-mail address: [email protected] (X.-H. Zhang). 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.10.032
state-of-the art sparse solutions to regression and classification tasks, which lead to significant reduction in the expense of computational complexity of the decision function and memory consumption, thereby making it more suitable for meta-modeling applications. RVM in many applications (Yuan, Wang, Yu, & Fang, 2007; Widodo et al., 2009) have produced very good results. In dynamic design problem, a number of conflicting objectives should trade off. As a result, the dynamic design problem is in fact a highly non-linear, complex solution space-based multi-objective optimization problem, which has the complexity of a general multi-objective optimization but also has the inherent difficulty of dynamic design. Glass stacking machine could have automatic unstacking glass sheets from stock stands and could have been transmitting the glass sheet to the glass-cutting table. There are some flaws in original glass stacking machine due to the application of traditional design methods: the 2 mm thickness of big glass sheets overturned by the stacking machine will be fragmentized, no sufficient accurate positioning, and unwieldy and unreasonable structure. As the glass overturning and transmission is a dynamic process, the dynamic properties are essential to the design. That is, it must be optimized with dynamic design methods to achieve smooth overturning process. In this paper, in order to improve the product performance, a hybrid approach that integrated relevant vector machine, genetic algorithms and fuzzy inference system is presented to the multiobjective optimization problems of dynamic design. The key
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Fig. 1. Components of glass stacking machine.
component of glass stacking machine - overturning assembly taken as a case study - instantiates the efficiency and feasibility of this approach.
Since the main structure bears dynamic load in the whole process and the dynamic nature of the system is very important, the optimization of the overturning assembly should be used for dynamic design methods.
2. Glass stacking machine Glass stacking machine (shown in Fig. 1) is composed of frame, overturning assembly, conveying system, roller feeding system, hydraulic system and pneumatic system components. The overturning assembly mainly consists of overturning arm, overturning cylinder, lifting cylinder and sucker. According to the function decomposition, the overturning assembly should meet the following features: 1. Could overturn the largest glass sheets of the original size of 6:0 m 3:3 m with thickness of 2—19 mm. 2. Overturned range in 0 —105 . 3. After loading the sheets, turn the glass sheets from 105°to horizontal position in 15 s. 4. Stacking process meet security and smooth requirements.1
3. Multi-objective optimization of dynamic design of overturning assembly Relevance vector machine (Tipping, 2001) is simply a specialization of a spares Bayesian model which utilizes the same datadependent kernel basis. The key feature of RVM is that the inferred predictors are good in generalization performance, as well as exceedingly sparse in that they contain relatively few ‘‘relevance vectors”, which means shorter consuming prediction time. Therefore, a well-trained RVM is used as a fitness function in GA-based optimization for saving the fitness value computation and guiding the search direction for further optimum solutions. The optimization architecture refers to Yuan et al. (2007). For this self-contained paper, RVM for regression is introduced concisely here. 3.1. Relevance vector machine
The original designed overturning assembly (the quality of overturning arm is 670.2 kg; the natural frequency of overturning arm is 14.81 Hz; using a 3 5 vacuum sucker matrix) is shown in Fig. 1. Moreover, an on–off valve designed to control the overturning cylinder output the overturning torque. In the physical prototype testing, stacking the 6:0 m 3:3 m 19 mm glass sheets, the front braced plate of hinge axis of overturning cylinder and the sleeve strengthen plate near the overturning cylinder on the main beam axle appear with obvious deformation and cracking phenomenon. As a result, the original design of the structure cannot meet the strength condition of overturning assembly. Meanwhile, the original design of stacking machine resulted in debris phenomenon stacking 2 mm glass sheets. From the above analysis, the key to stacking and unstacking process is an optimized design of overturning assembly, and especially, for the brittle and easily broken characteristics of the glass sheets, focusing on the dynamic system features analysis of overturning the 6:0 m 3:3 m 2 mm specifications of glass sheets. 1 As the original glass plates in stock stands are separated by papers or desiccants, the coefficient of friction is negligible, so it is assumed that the separation process between glass plates and stock stands is safe and stable.
Given a multiple-input–single-out dataset of N input vectors N D ¼ fX i ; yi gi¼1 , the input vector X i belongs Rd ,and the corresponding scalar-valued outputs yi are assumed iid (independent, identically distributed) observations. In the engineering view, some observations could be assumed to contain mean-zero Gaussian noise with variance r2 : pðen jr2 Þ ¼ Nð0; r2 Þ.
y ¼ f ðX; WÞ þ e
ð1Þ
where W ¼ ½x1 ; . . . ; xM T is the weight vector, after the Bayesian learning process, few weight vector is non-zero which named ^ at a value sparse feature. In the kernel model, regression estimate y X is given by:
^ ¼ f ðX; WÞ ¼ y
N X
xi KðX; X i Þ ¼ UW
ð2Þ
i¼0
And U is the N M design matrix, wherein its element is unm ¼ KðX m ; X n Þ. In fact, the sparse Bayesian learning framework has the ability to utilize arbitrary basis functions, such as Gaussian kernel, splines kernel, symmlet wavelet kernel and so on. (Schmolck & Everson, 2007). This paper utilizes the common Gaussian basis function /i ðX; X i Þ ¼ expfkX X i k2 =c2 g to the regression of the
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non-linear dynamic characteristics of overturning assembly. To the parameter c, especially for an insufficient dataset, using cross validation to find the best parameter values is crucial. To control the complexity of model and avoid overfitting, a zero-mean Gaussian prior probability distribution is defined over every xi with variance r1 i , the likelihood of W is written as:
pðW jaÞ ¼ ð2pÞM=2
M Y
a1=2 m expf
m¼1
am x2m 2
g
ð3Þ
where hyperparameters vector a ¼ ½a0 ; a1 ; a2 ; . . . aN T , controls how far from zero each weight is allowed to deviate. To control the model sparseness, RVM defined a hierarchical prior over a : pðaÞ and the inverse noise variance r2 : pðr2 Þ is specified uninformative hyperpriors: Gamma distributions. The posterior distribution over W is inferred by Bayes’ posterior inference as Gaussian: pðWjt; a; r2 Þ Nðl; RÞ. Where the posterior mean l and covariance R are as follows: 2
T
l ¼ r RU t R ¼ ðr2 UT U þ AÞ1
ð4Þ ð5Þ
where A ¼ diagða0 ; a1 ; . . . aN Þ. For a given unseen data X , the predicted distribution of corresponding output y : pðy jyÞ Nðl ; d2 Þ:
l ¼ yðX ; lÞ r2 ¼ r2MP þ /ðX ÞT R/ðX Þ
ð6Þ ð7Þ
The predictive mean l is the predictor of the model output with the unseen data X and the posterior mean weights l. The predictive variance r2 is the sum of variance associated with both the noise process and the uncertainty of the weight estimates. 3.2. Objective functions of dynamic optimization design of overturning assembly Different mechanical structures have different dynamic performance. The dynamic optimization design of overturning assembly firstly decides the structure dynamics performance as the dynamic design criteria for optimization, according to the criteria, and then determines the optimal design of the design variables, the objective function and constraints. The common criteria of dynamic design are the respond, modal characteristics, physical parameters and synthesis criteria. The structure of the technical and economic indicators is integrated as synthesis criteria. In this paper, the synthesis criteria are chose to the overturning assembly dynamic design as optimization criteria. According to the synthesis criteria, the 10 factors of sizes, shown in Fig. 2, impacted to the structure and layout of overturning arm are selected as the design variables X of the dynamic optimize design. These 10 design variables and their design range D are shown in Table 1. Moreover, X ¼ ½x1 ; x2 ; . . . ; x10 2 D is taken as the constraints of the multi-objective optimization. 3.2.1. The constrain of strength condition According to the strength of the synthesis criteria, for unstacking a variety of the glass sheets, the strength of the overturning assembly must be under the limit condition of the material strength. The minimum safety factor of the overturning assembly is sðXÞ, and carbon structural steel Q235A is selected as the material of the overturning arm. In accordance with the provisions of the yield stress, the safety factor is usually taken from 1.5 to 2.2, and then the dynamic design optimization of strength condition should be:
sðXÞ P 1:5
ð8Þ
3.2.2. The objective of response condition The unstacking process divides into two stages: acceleration and deceleration process.2 The hydraulic cylinder drives the overturning assembly rotating around the rotation axis of lifting cylinder connector. In the unstacking process, the overturning assembly and the sucker compose a system with two-degrees-of-freedom (TDOF), and its dynamic model is shown in Fig. 3. Where m1 is the quality of overturning arm; m2 is the quality of glass sheet and sucker assembly; k1 is the stiffness of overturning arm; k2 is the stiffness of glass sheet and sucker; c2 is viscous damping coefficient of glass sheet and sucker; P is the exciting force. According to the system of motion equations, the amplitudes of overturning arm and glass sheet can be respectively computed by
P A1 ¼ k1
A2 ¼
P k1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 k2 Þ2 þ ð2f2 akÞ2 ½ð1 k2 Þða2 k2 Þ lk2 a2 2 þ ð2f2 akÞ2 ð1 k2 lk2 Þ2 ð9Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 a þ ð2f2 akÞ ½ð1 k2 Þða2 k2 Þ lk2 a2 2 þ ð2f2 akÞ2 ð1 k2 lk2 Þ2 ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi wherepx 1 ¼ ffi k1 =m1 is the natural frequency of overturning arm; ffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ¼ k2 =m2 is the natural frequency of glass sheet and sucker. a ¼ x2 =x1 is the ratio of two natural frequency, while k ¼ x=x1 is the ratio of exciting frequency and natural frequency of overturning arm. l ¼ m2 =m1 is the ratio of quality. f2 is the ratio of viscous damping of glass sheet and sucker. Because of the exciting force PðtÞ on the system is proportional to the angular acceleration aðtÞ of unstacking process, thus the type of aðtÞ determines the type of dynamic features. In this paper, the angular acceleration aðtÞ and velocity is choosen as shown in Fig. 4, therefore, such TDOF system works with variational acceleration from 0 s to 7.5 s, and variational deceleration from 7.5 s to 15 s in unstacking process, thus no step variation of angular velocity. The overturning motion could be driven by the output force of overturning cylinder controlled by the proportional valve. Taking into account the responses of the synthesis criteria, the optimization design is to find the best structure of overturning assembly that minimize the vibration of unstacking the designed largest 2 mm glass sheets to ensure the safety and smooth of the overturning process. As a result, the maximum response of unstacking largest glass sheets is the objective function, namely, the minimum amplitude of the 6:0 m 3:3 m 2 mm type glass sheet is first optimization objective:
min AðXÞ ¼ min A1 ðx1 ; x2 ; . . . ; x10 Þ
ð11Þ
where A1 ðXÞ is the amplitude of glass sheets with 6:0 m 3:3 m 2 mm specification computed by (10). 3.2.3. The objective of quality Minimizing the quality of overturning arm could reduce material consumption, decrease cost and improve the speed of loading glass sheets, so as to improve the performance of the entire overturning assembly. Therefore, the minimum quality of overturning arm is set as the second objective:
min MðXÞ ¼ min Mðx1 ; x2 ; . . . ; x10 Þ
ð12Þ
3.2.4. The objective of natural frequency In mechanical design, to enhance the anti-vibration ability of mechanical structure is one of the important ways to improve the characteristics of machine, such as high-quality, high-speed, high 2 For the facilitation in the analysis, both the stages of unstacking process are assumed to be 7.5 s.
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Fig. 2. Design variables scheme.
Table 1 Design variables.
a
Length of overturning arm x1
Length of main beam x2
Length of main beam x3 a
Lateral plate of overturning arm a x4
1:8 6 x1 6 2:6 m
4:0 6 x2 6 5:5 m
6 6 x3 6 12:5 mm
0 6 x4 6 64 mm
0 6 x5 6 0:1 m
Lateral plate of overturning arm c x6 0 6 x6 6 0:3 m
Thickness of lateral plate of overturning arm x7 5 6 x7 6 17 mm
Braced plate of overturning cylinder a x8 0 6 x8 6 64 mm
Braced plate of overturning cylinder b x9 0 6 x9 6 160 mm
Thickness of braced plate of overturning cylinder x10 5 6 x10 6 17 mm
Lateral plate of overturning arm b x5
The thickness of the main beam is selected by GB6728-86.
rity of loading process, so as to increase the speed of glass stacking machine. The stiffness of TDOF system kðXÞ is computed by:
kðXÞ ¼
k1 ðXÞk2 k2 ¼ k1 ðXÞ þ k2 1 þ k2 =k1 ðXÞ
ð13Þ
where the stiffness of overturning arm is k1 ðXÞ ¼ x2 ðXÞ . . . MðXÞ, and xðXÞ denotes the natural frequency of overturning arm. Because k2 was assumed as invariable, obviously, it is necessary to improve the system stiffness kðXÞ by increasing the stiffness k1 ðXÞ of overturning arm. Furthermore, under the condition of fixed quality MðXÞ of overturning arms, the stiffness k1 ðXÞ can be measured by its natural frequency xðXÞ, the greater xðXÞ cause the greater kðXÞ. Namely, maximizing the system stiffness is equivalent to minimizing the natural frequency of overturning arm. As a result, the third objective is minimizing the natural frequency of overturning arm:
max xðXÞ ¼ min½xðXÞ
ð14Þ
To sum up, the dynamic multi-objective optimization design formulates:
min FðXÞ ¼ min½AðXÞ; MðXÞ; xðXÞT Fig. 3. The dynamic model of two-degrees-of-freedom system composed by the overturning assembly and sucker.
efficiency, low noise and low cost. In the unstacking process, the TDOF system composed of overturning arm and the sucker may vibrate due to different working conditions, therefore, a consideration with increasing of the anti-vibration of such system, mainly by improving the stiffness of the structure, could complete the stationa-
s:t: sðXÞ X
P 1:5
ð15Þ
¼ ½x1 ; x2 ; . . . ; x10 2 D
3.3. Uniform design sampling scheme The uniform design (UD) is a statistical design of experiment method proposed by Fang and Wang (1994), which produces high
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Fig. 4. Overturning angular acceleration and velocity.
Table 2 Uniform design sampling scheme of overturning assembly. No.
Input (design variables)
Output (objective function)
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
MðXÞ
AðXÞ
xðXÞ
sðXÞ
1 2 ... 50
2000 2000 ... 1800
4000 4500 ... 5500
12.5 6 ... 9
0 16 ... 0
0 25 ... 0
0 75 ... 225
5 8 ... 8
0 16 ... 32
0 40 ... 80
5 8 ... 8
488713.67 412964.26 ... 511321.09
0.8276 2.1642 ... 1.1442
22.6840 13.3100 ... 20.5340
1.4110 0.9347 ... 2.0260
Fig. 5. The safety factor distribution of overturning assembly.
representative samples in the domain and allows the largest possible amount of levels for each factor among all experimental designs. In this paper, UD is adopted for the sampling scheme of dynamic design, denoted by U 50 510 , which means 10 factors with 5 levels for each of the factors in the problem space and 50 presented as the number of experiment. The sampling scheme of dynamic design utilizes full-featured motion simulation software to obtain the force responses of overturning assembly with different structure parameters in the kinematic and dynamic simulation process of unstacking variable type glass sheets. Sequentially, the strength analysis of the overturning arms and modal analysis of the TDOF system are studied by the rapid finite element analysis software in order to obtain and calculate the responses AðXÞ; xðXÞ of dynamic design and the performance constraint sðXÞ, the objective function MðXÞ obtained from three-dimensional CAD software.
The sampling dataset (partial samples shown in Table 2) is utilized for RVM training dataset. Fig. 5 shows the safety factor sðXÞ distribution of the overturning arm with structure parameters designed in first experimental scheme, as unstacking 19mm specification glass sheet.
3.4. Multi-objective genetic algorithm of dynamic optimization design of overturning assembly Due to good generalization abilities, RVM could be used to model and approximate the map from the design variables to the performance of overturning assembly. So, the model based on RVMs could be used in the fitness evaluation of GAs rather than the real non-linear system (Yao, 1999). The optimization scheme refers to Yuan et al. (2007).
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Fig. 6. Pareto front of multi-objective optimization of dynamic design of overturning assembly.
Fig. 7. Mapping relationship between Pareto front and evaluation function.
Table 3 Fuzzy rules in linguistic form for selection of Pareto optimal set. IF MðXÞlight, AðXÞ small, xðXÞ high, sðXÞ large, THEN Pareto front good; IF MðXÞ medium, AðXÞ not large, xðXÞ not low, sðXÞ not small, THEN Pareto front normal; IF MðXÞ not weight, AðXÞ medium, xðXÞ not low, sðXÞ not small, THEN Pareto front normal; IF MðXÞ not weight£AðXÞ not large £xðXÞ medium£sðXÞ not small£THEN Pareto front normal; IF MðXÞ weight, THEN Pareto front bad; IF AðXÞ large, THEN Pareto front bad; IF xðXÞ low, THEN Pareto front bad; IF sðXÞ small, THEN Pareto front bad.
Yao and Xu (2006) pointed out about multi-objective optimization algorithm that NSGA-II (Deb, Agrawal, Pratap, & Meyarivan, 2002), compared with other multi-objective optimization algorithm, shows a better performance in a multiply test function set and adapts more objectives. In this paper, NSGA-II is selected as a optimization tool for solve the multi-objective dynamic design. Chromosome is encoded by corresponding design parameters in real ½x1 x2 . . . x10 . The Pareto front of multi-objective optimization searched by NSGA-II is shown in Fig. 6. 3.5. Fuzzy inference system for the selection of Pareto optimal set Pareto optimal set cannot be directly applied to structure design. In this paper, a fuzzy inference system is used to help decision makers to give the most satisfaction of a non-dominate solution of the problem as the ultimate solution. The rules of the fuzzy infer-
Table 4 Optimum results comparison between quadratic programming and this approach. Optimization approach
MðX Þ ðkgÞ
AðX Þ ðmmÞ
xðX ÞðHzÞ
sðX Þ
Proposed method Quadratic programming method
462 440.8
0.80 0.66
21.05 21.96
2.13 1.93
ence system follow experts in the field and built on the basis of experience and prior knowledge (see Table 3). The variable language of the fuzzy inference system defined by MATLAB toolbox is expressed by Gaussian membership function. From the input–output mapping relationship shown in Fig. 7, we can see that the smaller mass of the overturning arm, the smaller the amplitude of 2 mm large glass sheet, the higher natural frequency and the larger minimum safety factor of overturning arm, the larger evaluation function of Pareto optimal solution. 4. Discussion The difference of test dataset between the prediction based on RVM and the finite element analysis is less than 3.56%, which meet the precision of engineering requirements. Furthermore, the computation time of prediction by RVM is hundred times faster than that of finite element analysis. To apply the ideal point approach, the dynamic multi-objective optimization functions could be changed to the quadratic programming model, thus the optimum results compared with this approach are shown in Table 4, which shows better solution with this approach.
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Table 5 Comparison before and after optimization. Result comparison
M FEM ðkgÞ
AFEM ðmmÞ
xFEM ðHzÞ
Before optimization After optimization
670.2 461.8
1.99 0.87
14.81 22.77
The ultimate solution X is rounded to modify the structure parameters of the overturning assembly. The strength analysis is done for overturning arm with these round structure parameters, and the modal analysis of system composed of overturning assembly and glass sheet is done, thus those results are shown in Table 5. After the optimization, the mass of overturning arm is decreased 31.1% than that of original design, the amplitude of 2 mm big glass sheet improvement design over the reduce of 55.9%, and the natural frequency of overturning arm is increased by 53.7% than that of original design, the strength of overturning assembly meet engineering requirements. The loading process of overturning assembly designed with above optimum structure parameters is safer, smooth and its cost is more economic, so as to enhance the loading speed and improve the positioning accuracy. Applying the optimum designed parameters to the physical prototype of the glass stacking machine, test results showed that the overturning structure meet the requirements of strength, and its structure size decreases. To improve the on–off valve for the proportional valve to control the output force of overturning cylinder, the vibration control improved the stability of the loading process. Therefore, the loading of 2 mm big glass sheet is no more debris. 5. Conclusion In this paper, a hybrid intelligent approach that integrated RVM, GA and fuzzy inference system is proposed for the dynamic design of multi-objective optimization. To modify the original design flaw, the dynamic design- based three objectives and a performance constraint are given out for dynamic optimization design of overturning assembly as a case study. First, uniform design sampling strategy is adopted to acquire the samples dataset that reflects the dynamic characteristics of mechanical structure. Thus, the dy-
namic characteristics are identified by the sparse RVM-learning process. Secondly, NSGA-II algorithm is utilized to search Pareto optimal solution set from the design variables space expressed by the chromosomes, and the fitness value is calculated by the RVM predictions. Finally, through a fuzzy inference system defined by fuzzy rules, the decision making determines the best design parameters combinations. Experiments of the case study have proved improved dynamic performance and economic performance; therefore, the proposed method applied to glass stacking machine is feasible and efficient. Acknowledgements This work is supported by the Science-Tech Tackle Key Project of Shandong Province of China (Grant No. 2007GG10009001) and also supported by National Science Foundation for Post-doctoral Scientists of China (Grant No.20090450700). The authors thank to Tao Wang for her help on the experimental work. References Chan, W. C., Chan, C. W., Cheung, K., et al. (2001). On the modeling of nonlinear dynamic systems using support vector neural networks. Engineering Application of Artificial Intelligence, 14, 105–113. Deb, K., Agrawal, S., Pratap, A., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197. Fang, K., & Wang, Y. (1994). Number-theoretic methods in statistics. Chapman and Hall. Schmolck, A., & Everson, R. (2007). Smooth relevance vector machine: a smoothness prior extension of the RVM. Machine Learning, 68(2), 107–135. Tipping, M. (2001). Sparse Bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1, 211–244. Wang, G. G., & Shan, S. (2007). Review of meta-modeling techniques in support of engineering design optimization. ASME Transactions Journal of Mechanical Design, 129(4), 370–380. Widodo, A., Kim, Eric Y., Son, J.-D., Yang, B.-S., Tan, A. C. C., Gu, D.-S., et al. (2009). Fault diagnosis of low speed bearing based on relevance vector machine and support vector machine. Expert Systems with Applications, 363, 7252–7261. Yao, X. (1999). Evolving artificial neural networks. Proceedings of the IEEE, 87(9), 1423–1447. Yao, X., & Xu, Y. (2006). Recent advances in evolutionary computation. Journal of Computer Science and Technology, 21(1), 1–18. Yuan, J., Wang, K. S., Yu, T., & Fang, M. L. (2007). Integrating relevance vector machines and genetic algorithms for optimization of seed-separating process. Engineering Applications of Artificial Intelligence, 20(7), 970–979.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Relevance vector machine and fuzzy system based multi-objective dynamic design optimization: A case study Xuemei Liu a, Xiao-Hui Zhang a,*, Jin Yuan b,a a b
School of Mechanical and Electronic Engineering, Shandong Agricultural University, 271018 Tai’an, China School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Keywords: Multi-objective dynamic design optimization Relevance vector machine (RVM) Multi-objective genetic algorithm (MOGA) Fuzzy system Glass stacking machine
a b s t r a c t To improve the original design flaws of overturning assembly of glass stacking machine taken as a case study, a multi-objective optimization approach integrated relevance vector machines (RVM), multiobjective genetic algorithms (MOGA) and fuzzy system are presented for the optimal dynamic design problem. Firstly, the multi-objectives of the overturning assembly are constructed by the use of dynamic structure optimization design theory. The motion simulation and finite element analysis of overturning assembly are utilized for sampling scheme given by uniform design to collect the train dataset. The dataset could describe the non-linear behaviors of dynamic and static characteristics of variety of mechanical structures, which is identified by RVMs. Sequentially, RVM- based meta-model as fitness function is combined with MOGA to obtain the Pareto optimal set. Finally, a fuzzy inference system is established as decision-making support to obtain the optimum preference solution. Therefore, the modified physical prototype with the round solution proofed feasibility and efficiency of this approach. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, dynamic design methods, as an important aspect of modern design, are becoming increasingly common in the field of structural design problems, which can be effective in improving the dynamics performance and stability of equipment, reducing the costs of production. The objective function such as eigenvalue or structural response is a complex implicit non-linear function in the analysis of the dynamics. The huge computation burden is often caused by dynamic design analysis, simulation and optimization. Wang and Shan (2007) described the meta-modeling technology in the optimization of engineering design. In the physical process analysis and simulation, in order to obtain a comparable level of accuracy as physical testing data and not too expensive computational consumption, the meta-modeling has been widely used in various disciplines, as well as in the field of engineering optimization. The meta-modeling is an approximation model used in the non-linear process modeling, no more than to approximate the non-linear finite element model. The statistical learning theory has been popularly developed in non-linear system regression based on input and output data with noise (Chan, Chan, & Cheung, 2001). Relevant vector machine (Tipping, 2001) (RVM) is the general Bayesian learning framework of kernel method for obtaining * Corresponding author. E-mail address: [email protected] (X.-H. Zhang). 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.10.032
state-of-the art sparse solutions to regression and classification tasks, which lead to significant reduction in the expense of computational complexity of the decision function and memory consumption, thereby making it more suitable for meta-modeling applications. RVM in many applications (Yuan, Wang, Yu, & Fang, 2007; Widodo et al., 2009) have produced very good results. In dynamic design problem, a number of conflicting objectives should trade off. As a result, the dynamic design problem is in fact a highly non-linear, complex solution space-based multi-objective optimization problem, which has the complexity of a general multi-objective optimization but also has the inherent difficulty of dynamic design. Glass stacking machine could have automatic unstacking glass sheets from stock stands and could have been transmitting the glass sheet to the glass-cutting table. There are some flaws in original glass stacking machine due to the application of traditional design methods: the 2 mm thickness of big glass sheets overturned by the stacking machine will be fragmentized, no sufficient accurate positioning, and unwieldy and unreasonable structure. As the glass overturning and transmission is a dynamic process, the dynamic properties are essential to the design. That is, it must be optimized with dynamic design methods to achieve smooth overturning process. In this paper, in order to improve the product performance, a hybrid approach that integrated relevant vector machine, genetic algorithms and fuzzy inference system is presented to the multiobjective optimization problems of dynamic design. The key
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Fig. 1. Components of glass stacking machine.
component of glass stacking machine - overturning assembly taken as a case study - instantiates the efficiency and feasibility of this approach.
Since the main structure bears dynamic load in the whole process and the dynamic nature of the system is very important, the optimization of the overturning assembly should be used for dynamic design methods.
2. Glass stacking machine Glass stacking machine (shown in Fig. 1) is composed of frame, overturning assembly, conveying system, roller feeding system, hydraulic system and pneumatic system components. The overturning assembly mainly consists of overturning arm, overturning cylinder, lifting cylinder and sucker. According to the function decomposition, the overturning assembly should meet the following features: 1. Could overturn the largest glass sheets of the original size of 6:0 m 3:3 m with thickness of 2—19 mm. 2. Overturned range in 0 —105 . 3. After loading the sheets, turn the glass sheets from 105°to horizontal position in 15 s. 4. Stacking process meet security and smooth requirements.1
3. Multi-objective optimization of dynamic design of overturning assembly Relevance vector machine (Tipping, 2001) is simply a specialization of a spares Bayesian model which utilizes the same datadependent kernel basis. The key feature of RVM is that the inferred predictors are good in generalization performance, as well as exceedingly sparse in that they contain relatively few ‘‘relevance vectors”, which means shorter consuming prediction time. Therefore, a well-trained RVM is used as a fitness function in GA-based optimization for saving the fitness value computation and guiding the search direction for further optimum solutions. The optimization architecture refers to Yuan et al. (2007). For this self-contained paper, RVM for regression is introduced concisely here. 3.1. Relevance vector machine
The original designed overturning assembly (the quality of overturning arm is 670.2 kg; the natural frequency of overturning arm is 14.81 Hz; using a 3 5 vacuum sucker matrix) is shown in Fig. 1. Moreover, an on–off valve designed to control the overturning cylinder output the overturning torque. In the physical prototype testing, stacking the 6:0 m 3:3 m 19 mm glass sheets, the front braced plate of hinge axis of overturning cylinder and the sleeve strengthen plate near the overturning cylinder on the main beam axle appear with obvious deformation and cracking phenomenon. As a result, the original design of the structure cannot meet the strength condition of overturning assembly. Meanwhile, the original design of stacking machine resulted in debris phenomenon stacking 2 mm glass sheets. From the above analysis, the key to stacking and unstacking process is an optimized design of overturning assembly, and especially, for the brittle and easily broken characteristics of the glass sheets, focusing on the dynamic system features analysis of overturning the 6:0 m 3:3 m 2 mm specifications of glass sheets. 1 As the original glass plates in stock stands are separated by papers or desiccants, the coefficient of friction is negligible, so it is assumed that the separation process between glass plates and stock stands is safe and stable.
Given a multiple-input–single-out dataset of N input vectors N D ¼ fX i ; yi gi¼1 , the input vector X i belongs Rd ,and the corresponding scalar-valued outputs yi are assumed iid (independent, identically distributed) observations. In the engineering view, some observations could be assumed to contain mean-zero Gaussian noise with variance r2 : pðen jr2 Þ ¼ Nð0; r2 Þ.
y ¼ f ðX; WÞ þ e
ð1Þ
where W ¼ ½x1 ; . . . ; xM T is the weight vector, after the Bayesian learning process, few weight vector is non-zero which named ^ at a value sparse feature. In the kernel model, regression estimate y X is given by:
^ ¼ f ðX; WÞ ¼ y
N X
xi KðX; X i Þ ¼ UW
ð2Þ
i¼0
And U is the N M design matrix, wherein its element is unm ¼ KðX m ; X n Þ. In fact, the sparse Bayesian learning framework has the ability to utilize arbitrary basis functions, such as Gaussian kernel, splines kernel, symmlet wavelet kernel and so on. (Schmolck & Everson, 2007). This paper utilizes the common Gaussian basis function /i ðX; X i Þ ¼ expfkX X i k2 =c2 g to the regression of the
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non-linear dynamic characteristics of overturning assembly. To the parameter c, especially for an insufficient dataset, using cross validation to find the best parameter values is crucial. To control the complexity of model and avoid overfitting, a zero-mean Gaussian prior probability distribution is defined over every xi with variance r1 i , the likelihood of W is written as:
pðW jaÞ ¼ ð2pÞM=2
M Y
a1=2 m expf
m¼1
am x2m 2
g
ð3Þ
where hyperparameters vector a ¼ ½a0 ; a1 ; a2 ; . . . aN T , controls how far from zero each weight is allowed to deviate. To control the model sparseness, RVM defined a hierarchical prior over a : pðaÞ and the inverse noise variance r2 : pðr2 Þ is specified uninformative hyperpriors: Gamma distributions. The posterior distribution over W is inferred by Bayes’ posterior inference as Gaussian: pðWjt; a; r2 Þ Nðl; RÞ. Where the posterior mean l and covariance R are as follows: 2
T
l ¼ r RU t R ¼ ðr2 UT U þ AÞ1
ð4Þ ð5Þ
where A ¼ diagða0 ; a1 ; . . . aN Þ. For a given unseen data X , the predicted distribution of corresponding output y : pðy jyÞ Nðl ; d2 Þ:
l ¼ yðX ; lÞ r2 ¼ r2MP þ /ðX ÞT R/ðX Þ
ð6Þ ð7Þ
The predictive mean l is the predictor of the model output with the unseen data X and the posterior mean weights l. The predictive variance r2 is the sum of variance associated with both the noise process and the uncertainty of the weight estimates. 3.2. Objective functions of dynamic optimization design of overturning assembly Different mechanical structures have different dynamic performance. The dynamic optimization design of overturning assembly firstly decides the structure dynamics performance as the dynamic design criteria for optimization, according to the criteria, and then determines the optimal design of the design variables, the objective function and constraints. The common criteria of dynamic design are the respond, modal characteristics, physical parameters and synthesis criteria. The structure of the technical and economic indicators is integrated as synthesis criteria. In this paper, the synthesis criteria are chose to the overturning assembly dynamic design as optimization criteria. According to the synthesis criteria, the 10 factors of sizes, shown in Fig. 2, impacted to the structure and layout of overturning arm are selected as the design variables X of the dynamic optimize design. These 10 design variables and their design range D are shown in Table 1. Moreover, X ¼ ½x1 ; x2 ; . . . ; x10 2 D is taken as the constraints of the multi-objective optimization. 3.2.1. The constrain of strength condition According to the strength of the synthesis criteria, for unstacking a variety of the glass sheets, the strength of the overturning assembly must be under the limit condition of the material strength. The minimum safety factor of the overturning assembly is sðXÞ, and carbon structural steel Q235A is selected as the material of the overturning arm. In accordance with the provisions of the yield stress, the safety factor is usually taken from 1.5 to 2.2, and then the dynamic design optimization of strength condition should be:
sðXÞ P 1:5
ð8Þ
3.2.2. The objective of response condition The unstacking process divides into two stages: acceleration and deceleration process.2 The hydraulic cylinder drives the overturning assembly rotating around the rotation axis of lifting cylinder connector. In the unstacking process, the overturning assembly and the sucker compose a system with two-degrees-of-freedom (TDOF), and its dynamic model is shown in Fig. 3. Where m1 is the quality of overturning arm; m2 is the quality of glass sheet and sucker assembly; k1 is the stiffness of overturning arm; k2 is the stiffness of glass sheet and sucker; c2 is viscous damping coefficient of glass sheet and sucker; P is the exciting force. According to the system of motion equations, the amplitudes of overturning arm and glass sheet can be respectively computed by
P A1 ¼ k1
A2 ¼
P k1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 k2 Þ2 þ ð2f2 akÞ2 ½ð1 k2 Þða2 k2 Þ lk2 a2 2 þ ð2f2 akÞ2 ð1 k2 lk2 Þ2 ð9Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 a þ ð2f2 akÞ ½ð1 k2 Þða2 k2 Þ lk2 a2 2 þ ð2f2 akÞ2 ð1 k2 lk2 Þ2 ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi wherepx 1 ¼ ffi k1 =m1 is the natural frequency of overturning arm; ffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ¼ k2 =m2 is the natural frequency of glass sheet and sucker. a ¼ x2 =x1 is the ratio of two natural frequency, while k ¼ x=x1 is the ratio of exciting frequency and natural frequency of overturning arm. l ¼ m2 =m1 is the ratio of quality. f2 is the ratio of viscous damping of glass sheet and sucker. Because of the exciting force PðtÞ on the system is proportional to the angular acceleration aðtÞ of unstacking process, thus the type of aðtÞ determines the type of dynamic features. In this paper, the angular acceleration aðtÞ and velocity is choosen as shown in Fig. 4, therefore, such TDOF system works with variational acceleration from 0 s to 7.5 s, and variational deceleration from 7.5 s to 15 s in unstacking process, thus no step variation of angular velocity. The overturning motion could be driven by the output force of overturning cylinder controlled by the proportional valve. Taking into account the responses of the synthesis criteria, the optimization design is to find the best structure of overturning assembly that minimize the vibration of unstacking the designed largest 2 mm glass sheets to ensure the safety and smooth of the overturning process. As a result, the maximum response of unstacking largest glass sheets is the objective function, namely, the minimum amplitude of the 6:0 m 3:3 m 2 mm type glass sheet is first optimization objective:
min AðXÞ ¼ min A1 ðx1 ; x2 ; . . . ; x10 Þ
ð11Þ
where A1 ðXÞ is the amplitude of glass sheets with 6:0 m 3:3 m 2 mm specification computed by (10). 3.2.3. The objective of quality Minimizing the quality of overturning arm could reduce material consumption, decrease cost and improve the speed of loading glass sheets, so as to improve the performance of the entire overturning assembly. Therefore, the minimum quality of overturning arm is set as the second objective:
min MðXÞ ¼ min Mðx1 ; x2 ; . . . ; x10 Þ
ð12Þ
3.2.4. The objective of natural frequency In mechanical design, to enhance the anti-vibration ability of mechanical structure is one of the important ways to improve the characteristics of machine, such as high-quality, high-speed, high 2 For the facilitation in the analysis, both the stages of unstacking process are assumed to be 7.5 s.
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Fig. 2. Design variables scheme.
Table 1 Design variables.
a
Length of overturning arm x1
Length of main beam x2
Length of main beam x3 a
Lateral plate of overturning arm a x4
1:8 6 x1 6 2:6 m
4:0 6 x2 6 5:5 m
6 6 x3 6 12:5 mm
0 6 x4 6 64 mm
0 6 x5 6 0:1 m
Lateral plate of overturning arm c x6 0 6 x6 6 0:3 m
Thickness of lateral plate of overturning arm x7 5 6 x7 6 17 mm
Braced plate of overturning cylinder a x8 0 6 x8 6 64 mm
Braced plate of overturning cylinder b x9 0 6 x9 6 160 mm
Thickness of braced plate of overturning cylinder x10 5 6 x10 6 17 mm
Lateral plate of overturning arm b x5
The thickness of the main beam is selected by GB6728-86.
rity of loading process, so as to increase the speed of glass stacking machine. The stiffness of TDOF system kðXÞ is computed by:
kðXÞ ¼
k1 ðXÞk2 k2 ¼ k1 ðXÞ þ k2 1 þ k2 =k1 ðXÞ
ð13Þ
where the stiffness of overturning arm is k1 ðXÞ ¼ x2 ðXÞ . . . MðXÞ, and xðXÞ denotes the natural frequency of overturning arm. Because k2 was assumed as invariable, obviously, it is necessary to improve the system stiffness kðXÞ by increasing the stiffness k1 ðXÞ of overturning arm. Furthermore, under the condition of fixed quality MðXÞ of overturning arms, the stiffness k1 ðXÞ can be measured by its natural frequency xðXÞ, the greater xðXÞ cause the greater kðXÞ. Namely, maximizing the system stiffness is equivalent to minimizing the natural frequency of overturning arm. As a result, the third objective is minimizing the natural frequency of overturning arm:
max xðXÞ ¼ min½xðXÞ
ð14Þ
To sum up, the dynamic multi-objective optimization design formulates:
min FðXÞ ¼ min½AðXÞ; MðXÞ; xðXÞT Fig. 3. The dynamic model of two-degrees-of-freedom system composed by the overturning assembly and sucker.
efficiency, low noise and low cost. In the unstacking process, the TDOF system composed of overturning arm and the sucker may vibrate due to different working conditions, therefore, a consideration with increasing of the anti-vibration of such system, mainly by improving the stiffness of the structure, could complete the stationa-
s:t: sðXÞ X
P 1:5
ð15Þ
¼ ½x1 ; x2 ; . . . ; x10 2 D
3.3. Uniform design sampling scheme The uniform design (UD) is a statistical design of experiment method proposed by Fang and Wang (1994), which produces high
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Fig. 4. Overturning angular acceleration and velocity.
Table 2 Uniform design sampling scheme of overturning assembly. No.
Input (design variables)
Output (objective function)
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
MðXÞ
AðXÞ
xðXÞ
sðXÞ
1 2 ... 50
2000 2000 ... 1800
4000 4500 ... 5500
12.5 6 ... 9
0 16 ... 0
0 25 ... 0
0 75 ... 225
5 8 ... 8
0 16 ... 32
0 40 ... 80
5 8 ... 8
488713.67 412964.26 ... 511321.09
0.8276 2.1642 ... 1.1442
22.6840 13.3100 ... 20.5340
1.4110 0.9347 ... 2.0260
Fig. 5. The safety factor distribution of overturning assembly.
representative samples in the domain and allows the largest possible amount of levels for each factor among all experimental designs. In this paper, UD is adopted for the sampling scheme of dynamic design, denoted by U 50 510 , which means 10 factors with 5 levels for each of the factors in the problem space and 50 presented as the number of experiment. The sampling scheme of dynamic design utilizes full-featured motion simulation software to obtain the force responses of overturning assembly with different structure parameters in the kinematic and dynamic simulation process of unstacking variable type glass sheets. Sequentially, the strength analysis of the overturning arms and modal analysis of the TDOF system are studied by the rapid finite element analysis software in order to obtain and calculate the responses AðXÞ; xðXÞ of dynamic design and the performance constraint sðXÞ, the objective function MðXÞ obtained from three-dimensional CAD software.
The sampling dataset (partial samples shown in Table 2) is utilized for RVM training dataset. Fig. 5 shows the safety factor sðXÞ distribution of the overturning arm with structure parameters designed in first experimental scheme, as unstacking 19mm specification glass sheet.
3.4. Multi-objective genetic algorithm of dynamic optimization design of overturning assembly Due to good generalization abilities, RVM could be used to model and approximate the map from the design variables to the performance of overturning assembly. So, the model based on RVMs could be used in the fitness evaluation of GAs rather than the real non-linear system (Yao, 1999). The optimization scheme refers to Yuan et al. (2007).
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Fig. 6. Pareto front of multi-objective optimization of dynamic design of overturning assembly.
Fig. 7. Mapping relationship between Pareto front and evaluation function.
Table 3 Fuzzy rules in linguistic form for selection of Pareto optimal set. IF MðXÞlight, AðXÞ small, xðXÞ high, sðXÞ large, THEN Pareto front good; IF MðXÞ medium, AðXÞ not large, xðXÞ not low, sðXÞ not small, THEN Pareto front normal; IF MðXÞ not weight, AðXÞ medium, xðXÞ not low, sðXÞ not small, THEN Pareto front normal; IF MðXÞ not weight£AðXÞ not large £xðXÞ medium£sðXÞ not small£THEN Pareto front normal; IF MðXÞ weight, THEN Pareto front bad; IF AðXÞ large, THEN Pareto front bad; IF xðXÞ low, THEN Pareto front bad; IF sðXÞ small, THEN Pareto front bad.
Yao and Xu (2006) pointed out about multi-objective optimization algorithm that NSGA-II (Deb, Agrawal, Pratap, & Meyarivan, 2002), compared with other multi-objective optimization algorithm, shows a better performance in a multiply test function set and adapts more objectives. In this paper, NSGA-II is selected as a optimization tool for solve the multi-objective dynamic design. Chromosome is encoded by corresponding design parameters in real ½x1 x2 . . . x10 . The Pareto front of multi-objective optimization searched by NSGA-II is shown in Fig. 6. 3.5. Fuzzy inference system for the selection of Pareto optimal set Pareto optimal set cannot be directly applied to structure design. In this paper, a fuzzy inference system is used to help decision makers to give the most satisfaction of a non-dominate solution of the problem as the ultimate solution. The rules of the fuzzy infer-
Table 4 Optimum results comparison between quadratic programming and this approach. Optimization approach
MðX Þ ðkgÞ
AðX Þ ðmmÞ
xðX ÞðHzÞ
sðX Þ
Proposed method Quadratic programming method
462 440.8
0.80 0.66
21.05 21.96
2.13 1.93
ence system follow experts in the field and built on the basis of experience and prior knowledge (see Table 3). The variable language of the fuzzy inference system defined by MATLAB toolbox is expressed by Gaussian membership function. From the input–output mapping relationship shown in Fig. 7, we can see that the smaller mass of the overturning arm, the smaller the amplitude of 2 mm large glass sheet, the higher natural frequency and the larger minimum safety factor of overturning arm, the larger evaluation function of Pareto optimal solution. 4. Discussion The difference of test dataset between the prediction based on RVM and the finite element analysis is less than 3.56%, which meet the precision of engineering requirements. Furthermore, the computation time of prediction by RVM is hundred times faster than that of finite element analysis. To apply the ideal point approach, the dynamic multi-objective optimization functions could be changed to the quadratic programming model, thus the optimum results compared with this approach are shown in Table 4, which shows better solution with this approach.
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Table 5 Comparison before and after optimization. Result comparison
M FEM ðkgÞ
AFEM ðmmÞ
xFEM ðHzÞ
Before optimization After optimization
670.2 461.8
1.99 0.87
14.81 22.77
The ultimate solution X is rounded to modify the structure parameters of the overturning assembly. The strength analysis is done for overturning arm with these round structure parameters, and the modal analysis of system composed of overturning assembly and glass sheet is done, thus those results are shown in Table 5. After the optimization, the mass of overturning arm is decreased 31.1% than that of original design, the amplitude of 2 mm big glass sheet improvement design over the reduce of 55.9%, and the natural frequency of overturning arm is increased by 53.7% than that of original design, the strength of overturning assembly meet engineering requirements. The loading process of overturning assembly designed with above optimum structure parameters is safer, smooth and its cost is more economic, so as to enhance the loading speed and improve the positioning accuracy. Applying the optimum designed parameters to the physical prototype of the glass stacking machine, test results showed that the overturning structure meet the requirements of strength, and its structure size decreases. To improve the on–off valve for the proportional valve to control the output force of overturning cylinder, the vibration control improved the stability of the loading process. Therefore, the loading of 2 mm big glass sheet is no more debris. 5. Conclusion In this paper, a hybrid intelligent approach that integrated RVM, GA and fuzzy inference system is proposed for the dynamic design of multi-objective optimization. To modify the original design flaw, the dynamic design- based three objectives and a performance constraint are given out for dynamic optimization design of overturning assembly as a case study. First, uniform design sampling strategy is adopted to acquire the samples dataset that reflects the dynamic characteristics of mechanical structure. Thus, the dy-
namic characteristics are identified by the sparse RVM-learning process. Secondly, NSGA-II algorithm is utilized to search Pareto optimal solution set from the design variables space expressed by the chromosomes, and the fitness value is calculated by the RVM predictions. Finally, through a fuzzy inference system defined by fuzzy rules, the decision making determines the best design parameters combinations. Experiments of the case study have proved improved dynamic performance and economic performance; therefore, the proposed method applied to glass stacking machine is feasible and efficient. Acknowledgements This work is supported by the Science-Tech Tackle Key Project of Shandong Province of China (Grant No. 2007GG10009001) and also supported by National Science Foundation for Post-doctoral Scientists of China (Grant No.20090450700). The authors thank to Tao Wang for her help on the experimental work. References Chan, W. C., Chan, C. W., Cheung, K., et al. (2001). On the modeling of nonlinear dynamic systems using support vector neural networks. Engineering Application of Artificial Intelligence, 14, 105–113. Deb, K., Agrawal, S., Pratap, A., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197. Fang, K., & Wang, Y. (1994). Number-theoretic methods in statistics. Chapman and Hall. Schmolck, A., & Everson, R. (2007). Smooth relevance vector machine: a smoothness prior extension of the RVM. Machine Learning, 68(2), 107–135. Tipping, M. (2001). Sparse Bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1, 211–244. Wang, G. G., & Shan, S. (2007). Review of meta-modeling techniques in support of engineering design optimization. ASME Transactions Journal of Mechanical Design, 129(4), 370–380. Widodo, A., Kim, Eric Y., Son, J.-D., Yang, B.-S., Tan, A. C. C., Gu, D.-S., et al. (2009). Fault diagnosis of low speed bearing based on relevance vector machine and support vector machine. Expert Systems with Applications, 363, 7252–7261. Yao, X. (1999). Evolving artificial neural networks. Proceedings of the IEEE, 87(9), 1423–1447. Yao, X., & Xu, Y. (2006). Recent advances in evolutionary computation. Journal of Computer Science and Technology, 21(1), 1–18. Yuan, J., Wang, K. S., Yu, T., & Fang, M. L. (2007). Integrating relevance vector machines and genetic algorithms for optimization of seed-separating process. Engineering Applications of Artificial Intelligence, 20(7), 970–979.