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Reliability Engineering and System Safety 135 (2015) 1–8

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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

Multiobjective robust design optimization of fatigue life for a truck cab Jianguang Fang a,c, Yunkai Gao a,n, Guangyong Sun b,nn, Chengmin Xu a, Qing Li c a b c

School of Automotive Studies, Tongji University, Shanghai 201804, China State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha 410082, China School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia

art ic l e i nf o

a b s t r a c t

Article history: Received 6 May 2013 Received in revised form 12 September 2014 Accepted 7 October 2014 Available online 6 November 2014

Structural optimization for vehicle fatigue durability signifies an exciting topic of research to improve its long-term safety and performance with minimum cost. Nevertheless, majority of the existing studies has been dealing with deterministic optimization and has not involved uncertainties, which could lead to an unstable or even useless design in practice. In order to simultaneously enhance the performance and robustness of the fatigue life for a truck cab, a multiobjective optimization is proposed in this study. After validating the simulation model, different dual surrogate modeling (DSM) methods are attempted to overcome the limitation of classical dual response surface (DRS) method; and subsequently the most accurate model, namely dual Kriging (DKRG) in this case, is selected through a comparative study. Then, the multiobjective particle swarm optimization (MOPSO) algorithm is adopted to perform the optimization. Compared with traditional single objective optimization strategies which yield only one specific optimum, MOPSO allows producing a set of non-dominated solutions over the entire Pareto space for a non-convex problem, which provides designers with more insightful information. Finally, a multi-criteria decision making (MCDM) model, which integrates the techniques of order preference by similarity to ideal solution (TOPSIS) with grey relation analysis (GRA), is implemented to find a best compromise optimum from the Pareto set. The selected optimum demonstrated not only to improve the fatigue life of the truck cab, but also to enable the design less sensitive to presence of uncertainties. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Robust design optimization Multiobjective particle swarm optimization Fatigue design Dual surrogate model TOPSIS Grey relational analysis

1. Introduction Fatigue life indicates one of the key factors in determining the long-term performance, safety and durability, which has drawn significant attention in automotive engineering. As an effective computational tool, structural optimization techniques have played an increasingly important role in vehicular design for improving fatigue performance. In this regard, many topics, ranging from key components to automotive body, have been explored. For example, Buciumeanu et al. [1] designed suspension component by taking into account fatigue requirements. Kang et al. [2] optimized fatigue criteria to ensure the durability of a lower control arm. Mrzyglod and Zielinski [3] conducted a parametric optimization for a suspension arm with multiaxial highcycle fatigue criteria. Song et al. [4] adopted the surrogate models to optimize a control arm with consideration of strength and durability performance. Kim et al. [5] conducted the structural

n

Corresponding author. Tel./fax: þ 86 21 6958 9845. Corresponding author. Tel.: þ 86 731 8881 1445; fax: þ 86 731 8882 2051. E-mail addresses: fangjg87@gmail.com (Y. Gao), sgy800@126.com (G. Sun).

nn

http://dx.doi.org/10.1016/j.ress.2014.10.007 0951-8320/& 2014 Elsevier Ltd. All rights reserved.

optimization of an outer tie rod for a passenger car by considering the durability criterion. Hsu and Hsu [6] minimized the weight of aluminum disc wheels under constraints of fatigue life through a sequential neural network approximation method. Lee and Jung [7] developed metamodel for optimizing a connecting rod subjected to a certain fatigue life. Schafer and Finke [8] carried out a shape optimization of steel wheels by using design of experiment (DoE) and finite element method (FEM), aiming to reduce fatigue failure and increase durability. Kaya et al. [9] re-designed a failed vehicle component subjected to cyclic loading using topology and shape optimization approach. Saoudi et al. [10] optimized fatigue life and weight of automobile aluminum part under random road excitation. Ping et al. [11] performed fatigue analysis of the body for a sport utility vehicle and performed topological optimization of the spot weld location in the critical region by using homogenization method to improve its durability. Adl and Panahi [12] conducted a multiobjective optimization for a passenger car body, where artificial neural network (ANN) was used to estimate the functional performance attributes in terms of the weight and fatigue life. Nevertheless, most of these abovementioned studies have not involved uncertainties and therefore been of considerable

2

J. Fang et al. / Reliability Engineering and System Safety 135 (2015) 1–8

limitation to practical applications. In real world, uncertainties caused by manufacturing and operation are inevitable and should be taken into account in analysis and design. Traditional constrained deterministic optimization that often pushes the design to boundary and cannot accommodate any uncertainties for achieving an expected performance in real life [13–18]. On the other hand, traditional optimization with uncertainties likely leads to a large scatter, which may not only cause significant fluctuations from the desired performance, but also add to life-cycle costs, including inspection, repair and other maintenance expenses. Thus, the robust design optimization (RDO), which enables to reduce the scatter of the structural performance and improve structural safety in a probabilistic manner of risk assessment without eliminating the source of variability, has drawn increasing attention for solving real-world fatigue problems recently. For example, McDonald and Heller [19] incorporated robust design strategies to develop an iterative 2D finite-element-based optimization procedure, which was used to determine precise shape of a hole in a plate to maximize its fatigue life. Li et al. [20] presented robust optimization of structural fatigue life based upon stochastic finite element method, aiming to reduce the scatter of fatigue life. Ozturk [21] proposed an efficient method for fatigue based shape optimization to obtain the robust design of an oil sump shape with consideration of the variation of the clamping forces. To the authors' best knowledge, however, there have been very limited reports available on RDO for structural fatigue in automotive engineering to date [22]. It remains unclear whether and how the design optimization could maintain or improve targeted fatigue life and design stability even if uncertainties are presented. Furthermore, based upon the previous studies [13,14], it appears that the performance and robustness of the optimum tend to conflict with each other and often a tradeoff needs to be made appropriately in the design [23,24]. The tradeoff can be addressed by defining such a decision-making problem as a multiobjective optimization (MOO) problem [25]. However, there are limited reports available to explore such a tradeoff. In this study; we propose to perform the fatigue RDO of a truck cab by integrating dual surrogate modeling techniques with multiobjective particle swarm optimization (MOPSO) algorithm, where both the mean value and standard deviation of the fatigue life are regarded as the design objectives. For a decision maker, the ultimate goal in practical applications is often to find the best compromise solution from the Pareto set. It characterizes a multi-criteria decision making (MCDM) problem, which actually is commonly encountered in various real-life applications [26–28]. In this paper, we will combine the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) with Grey relational analysis (GRA) to rank the priority of the Pareto solutions, thereby determining the best possible compromise from the multiple objectives.

2. Methodology 2.1. Robust design optimization (RDO)

Fig. 1. Illustration of robust design optimization vs deterministic design optimization.

the parametric perturbation and often cannot be recommended as a design in practice, even though it has a better mean value (lower in the objective function) than Solution 2. By taking into account the uncertainties, the variability of structural performance is often assessed by the standard deviation. From optimization perspective, one can summarize several typical formulations to address robust design problems as follows: Design 1: Minimize the mean value of the objective while constraining the standard deviation to a predefined level fσ0 [29] 8 > < min f μ ðxÞ s:t: f σ ðxÞ r f σ 0 ð1Þ > : x rxrx L

U

where f μ ðxÞ is the mean value of the objective function, f σ ðxÞ and f σ 0 are the standard deviation of the objective function and its corresponding upper-limit constraint, respectively. Design 2: Minimize the standard deviation of the objective while constraining the mean value under an upper-limit 8 > < min f σ ðxÞ s:t: f μ ðxÞ r f μ0 ð2Þ > : x rxrx L

U

Design 3: Formulate a single robustness cost function by weighing the mean value and standard deviation of the objective function as 8 < min αf μ ðnxÞ þ ð1 αÞf σ ðnxÞ; 0 r α r 1 fσ fμ ð3Þ : s:t: x r x r x L

U

where α denotes the weighting factor, and f μ and f σ are the mean and standard deviation values, respectively. Design 4 (the proposed method): Optimize both mean and standard deviation functions in a concurrent manner. Usually, the performance and robustness could conflict with each other (i.e., the more the emphasis on robustness, the worse the objective mean value, and vice versa). Therefore, the robust design problem is formulated in the framework of multiobjective optimization as follows: n

Unlike deterministic optimization that deals with nominal variables and parameters without considering uncertainties, a robust optimization develops a solution that can be least sensitive to variations of the nominal design. As shown in Fig. 1, let the x-axis represent the uncertain parameters, including random design variables and other noise factors, and the vertical axis represent an objective function f(x) to be minimized. Of these two optimal Solutions 1 and 2 pointed in this figure, Solution 2 (x2) is considered more robust as a variation of 7 Δx in design variables does not alter the objective function too much and maintains the solution within the feasible region when the design variables and/or noise factors are perturbed. On contrary, Solution 1 (x1) appears highly sensitive to

(

min s:t:

ðf μ ðxÞ; f σ ðxÞÞ xL r x r xU

n

ð4Þ


2.2. Dual surrogate model (DSM) In the abovementioned robust optimization problems, the mean and standard deviation functions need to be first defined, respectively. As a traditional method, the dual response surface (DRS) method adopts polynomial response surfaces (PRS) to model both mean and variance functions, which has been widely adopted in robust designs in literature [18,30–33], e.g. quadratic polynomials given as k

k

k

i¼1

i¼1

ioj

k

k

k

i¼1

i¼1

ioj

Y^ μ ðxÞ ¼ b0 þ ∑ bi xi þ ∑ bii x2i þ ∑ ∑ bij xi xj

Y^ σ ðxÞ ¼ c0 þ ∑ ci xi þ ∑ cii x2i þ ∑ ∑ cij xi xj

ð5Þ

ð6Þ

The specific steps to fit the dual PRS models can be found in literature [13,14]. The PRS model is relatively easy to be constructed and rather simple to derive the sensitivity, but it might be difficult to accommodate high nonlinearity in some cases [34]. To tackle this problem, other surrogate models, such as radial basis function (RBF) [35] and Kriging (KRG) models [36], may be more effective. In this study, we generalize the DRS to dual surrogate model (DSM) to formulate both the mean and standard deviation of the objective by using different surrogate algorithms, as in Eqs. (7) and (8) for the dual Kriging (DKRG) model, Y^ μ ðxÞ ¼ β^ μ þ rμ T ðxÞR μ 1 ðyμ f μ β^ μ Þ

ð7Þ

Y^ σ ðxÞ ¼ β^ σ þ rσ T ðxÞR σ 1 ðyσ f σ β^ σ Þ

ð8Þ

and Eqs. (9) and (10) for the dual radial basis function (DRBF) model, respectively. m

ns

j¼1

i¼1

m

ns

j¼1

i¼1

Y^ μ ðxÞ ¼ ∑ cμj pμj ðxÞ þ ∑ λμi ϕμ ðr μ ðx; xi Þ Y^ σ ðxÞ ¼ ∑ cσ j pσ j ðxÞ þ ∑ λσ i ϕσ ðr σ ðx; xi Þ

ð9Þ

ð10Þ

2.3. Multiobjective particle swarm optimization (MOPSO) algorithm Real-life engineering problems are typically characterized by a number of quality and/or performance indices, while some of which could conflict with each other as aforementioned. In order to provide designers with more insightful data for justification, we formulate the fatigue based optimization as a multiobjective optimization (MOO) problem in this study, where its fatigue performance and corresponding robustness indicator are regarded as two design objectives. To solve such a MOO problem, a multiobjective particle swarm optimization (MOPSO) algorithm is adopted herein. The particle swarm optimization (PSO) algorithm [37] is a relatively new heuristic algorithm inspired by the choreography of a bird flock. As an extended version to PSO, multiobjective particle swarm optimization (MOPSO) algorithm is featured by its fast convergence and well-distributed Pareto frontier compared with other heuristic multiobjective optimization algorithms such as NSGA-II [38,39], PEAS, microGA [40–42], etc. It is noted that MOPSO has been employed successfully in various practical problems [43–48] and more details about this algorithm can be consulted from Ref. [41]. 2.4. Decision model for MCDM Often a multiobjective optimization needs to cope with some conflicting objectives, typically forming a Pareto frontier from MOPSO.

3

In order to rank the solutions in Pareto frontier and choose the best possible compromise, TOPSIS induced by Wang and Yoon [49] and Grey relational analysis (GRA) introduced by Deng [50] are integrated in this paper. The process of this hybrid method to determine the best compromise solution is presented as follows: Step 1. Input S and w (i.e., S forms the Pareto frontier), where the component sij is the jth objective value at the ith alternative Pareto point, component wj is the weight of the jth objective, and weight vector w must satisfy ∑2j ¼ 1 wj ¼ 1. In this study, the weights are determined using the entropy method [51]. _ Step 2. Normalize S to be S according to Eq. (11). _ s ij ¼

max sij sij 8j

max sij min sij 8j

;

for i ¼ 1; 2; :::; τ

and

j ¼ 1; 2:

ð11Þ

8j

Step 3. Determine the ideal solution s þ and the negative ideal solution s using Eqs. (12) and (13), respectively, s i1 ; max_ s i2 Þ ð12Þ s þ ¼ ðmax_ 8i

8i

s ¼ ðmin_ s i1 ; min_ s i2 Þ 8i

ð13Þ

8i

Step 4. Calculate the grey relation coefficient of each alternative s ij Þ and the negative ideal solution γ ðsj ;_ s ij Þ to the ideal γ ðsj þ ;_ by taking s þ and s as the referential sequence and each alternative to be the comparative sequence min minsj þ _ s ij þ ζ max maxsj þ _ s ij 8 i 8 j 8 i 8 j _ γ ðsj þ ; s ij Þ ¼ ð14Þ sj þ _ s ij þ ζ max maxsj þ _ s ij 8i

8j

min minsj _ s ij þ ζ max maxsj _ s ij 8 i 8 j 8 i 8 j γ ðsj ;_s ij Þ ¼ sj _ s ij þ ζ max maxsj _ s ij 8i

ð15Þ

8j

where ζ is the distinguished coefficient (0 r ζ r1). In this study, ζ ¼0.5 [52]. Step 5. Determine the grade of grey relation of each alternative to s þ and s by using Eqs. (16) and (17) 2

γ ðs þ ;_s i Þ ¼ ∑ wj γ ðsj þ ;_s ij Þ

ð16Þ

j¼1 2

γ ðs ;_s i Þ ¼ ∑ wj γ ðsj ;_s ij Þ

ð17Þ

j¼1

Step 6. Find the relative closeness Ci of the distance that an alternative is close to the ideal solution, which is defined in Eq. (18) γ ðs þ ;_s i Þ Ci ¼ ð18Þ γ ðs ;_s Þ i

Step 7. Rank the priority of alternatives in a descending order of Ci and choose the best possible compromise solution.

It can be noted that the difference of the above integrated method from conventional TOPSIS lies in its introduction of the grey relation coefficient (i.e., γ ðsj þ ;_ s ij Þ and γ ðsj ;_ s ij Þ) of Grey relation model to replace the general distance. Meanwhile, the conventional grey relation is revised in order to reflect the impact of decision-making theory. As a result, this method is considered to be able to acquire a satisfactory compromise solution for a MCDM problem [53].

4


3. Fatigue optimization for a truck cab 3.1. Numerical model and experimental validation Fig. 2a displays the finite element analysis (FEA) model of a truck cab. For its durability prediction, the approach of a direct transient response is adopted. MSC.Nastran is used herein to obtain the stress and strain histories for each element of interest by using load time histories as input. The cab is subjected to a torsional cyclic load at the rear body mounts when the front body mounts are fixed. As shown in Fig. 2a, F1 and F2 form the torsional moment, and its amplitude and frequency can be consulted in our testing specification [54]. The strain response histories extracted from FEA are directly used in MSC.Fatigue in order to determine the durability performance. Strain life method of durability prediction can be used in transient analysis. The strain–life (ε–N) curve is expressed according to the classical Coffin–Manson equation as follows [55,56]:

Δε 2

¼

Δεe Δεp 2

þ

2

¼

σ f' E

ð2N f Þb þ εf'ð2Nf Þc

ð19Þ

where Δε=2, Δεe =2 and Δεp =2 are total, elastic and plastic strain amplitudes, respectively, and σ f', b, εf' and c are fatigue strength, fatigue strength exponent, fatigue ductility coefficient and fatigue ductility exponent, respectively. Commonly, these parameters are estimated in industrial applications when no detailed cyclic material data are available. For metallic materials, such as steel, aluminum, and titanium alloys, Baumel et al. [57] proposed a uniform material law, which has proved effective to obtain reasonably good fatigue prediction [58,59]. Following Bäumel–Seeger's method, the parameters can be obtained because only ultimate tensile strength (σb) and elastic modulus (E) of material are required in the calculation. In order to validate the simulation model, an experiment is conducted in this study as illustrated in Fig. 2b. It is found from Fig. 3 that the numerical simulation and experimental test of the baseline model have the same fatigue failure location, i.e., in the upper region of the front left pillar. And it can be seen that the fatigue life is also comparable between the numerical simulation and experimental test as summarized in Table 1.

functions of fatigue life, and Fnμ and Fnσ are their ideal optimums. 8 > < min F μ ðxÞ s:t: F σ ðxÞ rF σ 0 ¼ 0:050 ð20Þ Design 1 : > : x rxrx L

U

8 > < min F σ ðxÞ s:t: F μ ðxÞ Z F μ0 ¼ 4:70 Design 2 : > : x rxrx L U

Design 3 :

8 < min : s:t: (

Design 4 :

min s:t:

α

F μ ðxÞ þ ð1 F nμ

ð21Þ

αÞF σFðxÞ; n

σ

0rαr1

ð22Þ

xL r x r xU F μ ðxÞ; F σ ðxÞÞ

ð23Þ

xL r x r xU

Fig. 4 and Table 2 present three thickness design variables and their dimensional ranges to be optimized in this study. With regard to uncertainties, Grujicic et al. [60] pointed out the importance of

Fig. 3. Correlation between experimental and numerical fatigue failures. (a) Simulation and (b) Experiment.

Table 1 Comparison between simulation and physical tests (Log of fatigue life).

3.2. Definition of optimization problem As mentioned above, the possible formulations of a robust design optimization can be mathematically presented in Eqs. (12)–(15), where Fμ(x) and Fσ(x) are the mean value and standard deviation

Simulation

Experiment

4.46

4.60

Fig. 2. Fatigue analysis. (a) Simulation and (b) Experiment.


5

considering variations in material properties to predict the fatigue performance of vehicle components. Thus, we would like to restrict our attention on uncertainties induced by the material properties in this RDO problem. Specifically, the ultimate tensile strength (σb), elastic modulus (E) and density (ρ), which can be affected by rolling process [14], are chosen as the noise factors to take into account the uncertainties. Their fluctuations are in the ranges of E¼ [200 GPa, 220 GPa], ρ ¼[7700 kg/m3, 7900 kg/m3], and σb ¼ [300 MPa, 340 MPa], respectively, which are from the statistical data in the ASM International [61]. 3.3. Optimization problem process The entire design procedure is described in the flowchart seen in Fig. 5. In the cross product array, the noise factors (i.e., material uncertainties) are arranged in outer array as in Table 3, which is sampled by orthogonal array, while the control factors (i.e., thickness design variables) are arranged in the inner array, which is generated by Optimal Latin Hypercube Sampling (OLHS) approach [13,14,62–64]. The results of cross product array are summarized in Table 4, where the first 30 designs are the training points and the last five are the assessment points. In the surrogate approach, the established model should be sufficiently accurate to replace the high-fidelity full simulation in

Fig. 4. Illustration of design variables.

Table 3 Outer array for the noise factors.

Table 2 Variable ranges. Design variables

x1 (mm) x2 (mm) x3 (mm)

Varying ranges Lower bounds

Upper bounds

0.7 0.7 0.7

2.0 1.5 1.0

No.

E (GPa)

ρ (kg/m3)

σb (MPa)

1 2 3 4 5

200 200 220 220 210

7700 7900 7700 7900 7800

300 340 340 300 320

Fig. 5. Flowchart of optimization process.

6


the optimization. In this study, three metrics, namely R-square, relative average absolute error (RAAE), and relative maximum absolute error (RMAE) [34,35,65] are used to assess the accuracy of surrogate models through the assessment point given in Table 4 as, R2 ¼ 1

∑qi¼ 1 ðyi y^ i Þ2

ð24Þ

∑qi¼ 1 ðyi yÞ2

RAAE ¼

∑qi¼ 1 jyi y^ i j

RMAE ¼

ð25Þ

∑qi¼ 1 jyi yj

max fjyi y^ 1 j; …; jyi y^ q jg

ð26Þ

∑qi¼ 1 jyi yj=q

where yi denotes the exact function value for assessment point i, y^ i is the corresponding surrogate value, y stands for the mean of yi , and q is the number of the assessment points. It can be seen that a larger value of R-square and a smaller value of RAAE are preferred, which indicate a higher surrogate accuracy of overall performance in the design space. On the other hand, a larger value of RMAE indicates lower surrogate accuracy in one region of the design space even though a very good global assessment can be represented by the R-square and RAAE. In optimization, we usually focus more on the global behavior and thus more emphasis is placed on the first two metrics. For the fatigue criteria, the accuracy assessments of different DSMs are listed in Table 5. Clearly, the DKRG model exhibits the best global and local accuracy for both the mean and standard Table 4 Results of cross product array. No.

x1

x2

x3

Fμ

Fσ

Training points 1 1.69 2 1.96 3 1.91 4 1.60 5 1.10 6 0.97 7 1.87 8 0.74 9 1.37 10 1.82 11 1.15 12 0.92 13 1.33 14 1.46 15 1.06 16 0.79 17 1.73 18 0.70 19 1.19 20 1.42 21 0.88 22 1.51 23 0.83 24 1.64 25 2.00 26 1.55 27 1.78 28 1.28 29 1.24 30 1.01

0.76 1.31 1.11 1.22 0.98 0.73 1.17 1.25 1.14 0.81 1.33 1.09 1.36 1.00 1.20 0.87 1.03 1.06 0.78 1.50 0.92 0.70 1.42 1.39 0.89 0.95 1.44 0.84 1.28 1.47

0.95 0.82 0.97 0.91 0.83 0.88 0.73 0.80 0.79 0.77 0.87 0.72 0.99 1.00 0.96 0.78 0.86 0.90 0.74 0.84 0.98 0.81 0.93 0.75 0.89 0.71 0.94 0.92 0.70 0.76

4.12 4.75 4.74 4.53 4.11 3.72 4.63 3.42 4.31 4.31 4.12 3.78 4.23 4.38 4.04 3.46 4.56 3.32 3.99 4.35 3.78 3.94 3.70 4.54 4.57 4.41 4.69 4.09 4.23 3.99

0.0618 0.0974 0.0964 0.0888 0.0947 0.0512 0.0921 0.0644 0.0919 0.0680 0.0956 0.0759 0.0849 0.0842 0.0853 0.0710 0.0898 0.0629 0.0787 0.0899 0.0655 0.0642 0.0734 0.0935 0.0736 0.0919 0.0952 0.0744 0.1006 0.0589

Assessment 31 32 33 34 35

1.30 0.70 1.50 1.10 0.90

1.00 0.93 0.85 0.70 0.78

4.85 3.85 4.02 4.50 3.24

0.1036 0.0660 0.0581 0.0905 0.0636

points 2.00 1.35 1.03 1.68 0.70

Table 5 Accuracy assessment of different dual surrogate models. DSM

Response

R2

RAAE

RMAE

DPRS

Fμ(x) Fσ(x)

0.9536 0.7761

0.2006 0.4212

0.2821 0.6727

DKRG

Fμ(x) Fσ(x)

0.9980 0.9530

0.0374 0.1775

0.0737 0.3808

DRBF

Fμ(x) Fσ(x)

0.9963 0.6299

0.0586 0.4276

0.0742 1.2603

deviation, and thus is chosen for the subsequent robust design optimization. 3.4. Optimization results The results of single objective optimization are summarized in Table 6, where objective weight α as in Eq. (22) is assigned five values evenly distributed in [0, 1] for Design 3. For Design 1 (Eq. (20)) and Design 2 (Eq. (21)), the design achieve the optimum by pushing the design onto the bound of each constraint (i.e., Fσ(x)¼0.050 and Fμ(x)¼4.72, respectively). Note that the constraints Fσ0 and Fμ0 in Eqs. (20) and (21) can actively affect the optimums. In practical application, it is however difficult to define a constraint prior to optimization. For Design 3, when the weighting factor α ¼0 or 1, the optimization problems become to minimize the standard deviation or maximize the mean value of the fatigue life, respectively. That is to say that at these two cases, the optimizations solve for Fnσ and Fnμ in Eq. (14), respectively. Furthermore, when α increases from 0 to 1 evenly, the optimums obtained do not distribute evenly in the solution space. Specifically, the optimum almost remains unchanged when α changes from 0 to 0.50, whilst it changes noticeably when α increases from 0.50 to 1. Fig. 6 plots the Pareto frontier of MOPSO obtained from Design 4, together with the results of the single objective optimizations and the baseline design. Obviously, MOPSO generates a welldistributed Pareto frontier over the entire design space, and each point represents a non-dominated solution. It can be seen that the optimal values of Fμ(x) and Fσ(x) strongly conflict with each other, indicating that there is no any other point in the Pareto frontier that allows minimizing these two objectives concurrently without compromising one from another. It is noted that while Designs 1 and 2 yield only one single point in the Pareto frontier, indicating one special Pareto solution, Design 3 can produce a number of solutions by changing the value of the weighting factor α. Nevertheless, some solutions from Design 3 (e.g., the solutions from α ¼0 to 0.50) may distribute in one small region rather than spread over the Pareto space uniformly. Moreover, as a type of weighted cost function, Design 3 is effective only if the Pareto frontier is convex when it is used to generate a Pareto frontier [66]. Obviously, Fig. 6 indicates that the convexity could be problematic in some region of the design space. Therefore, the adoption of MOPSO appears essential in this case. 3.5. MCDM result After acquiring the Pareto set from MOPSO, the decision maker often needs to determine a compromise solution for accomplishing the assignment. By using the TOPSIS based on GRA, we rank these 100 Pareto solutions and select the best compromise, which is listed in Table 7 together with the baseline design. It can also be seen that the simulation results agree well with the DSM results, and more importantly, the optimal solution (signified with star in Fig. 6) selected by integrating TOPSIS with GRA not only improves


7

Table 6 Robust design results with single objective optimization. Description

Design 1

Design 2

Design 3 α¼ 0

α ¼0.25

α ¼ 0.50

α¼ 0.75

α¼ 1

Design variables

x1 x2 x3

0.97 0.73 1.00

1.80 1.21 1.00

0.96 0.70 1.00

0.96 0.70 1.00

0.96 0.70 1.00

2.00 0.86 1.00

2.00 1.50 1.00

Fatigue life

Fμ(x) Fσ(x)

3.74 0.050

4.72 0.083

3.68 0.0486

3.68 0.0486

3.67 0.0487

4.56 0.0703

4.90 0.100

surface (DRS) method, a more generalized dual surrogate method (DSM) was introduced in this study, in which different DSMs, namely dual polynomial response surface (DPRS), dual Kriging (DKRG) and dual radial basis function (DRBF) models were employed to fit the mean value and standard deviation of the fatigue life. Following the surrogate modeling, MOPSO were conducted for attaining a nondeterministic optimization solution that provides the decision-makers with insightful design information. Besides, through the comparative studies with traditional single objective or weighted-based multiobjective optimization strategies, it was found that particle swarm optimization (PSO) based multiobjective optimization can produce a series of non-dominated solutions over the Pareto space; and more importantly, MOPSO can tackle the non-convex optimization problem, which may not be properly addressed by the weighted-based method. Finally, the optimum selected by combining TOPSIS with Grey relational analysis proved to be able to provide a proper compromise between performance and robustness of fatigue life. Fig. 6. Pareto frontier of fatigue mean and standard deviation.

Acknowledgment Table 7 Comparisons of the baseline and the best compromise designs. Description

Baseline

Optimized (MCDM) DSM

Design variables

Fatigue life

x1 (mm) x2 (mm) x3 (mm) Fμ(x) Fσ(x)

1.5 1.5 1.5

1.87 0.82 1.00

4.45 0.0755

4.57 0.0707

Simulation

This work was supported from the National 973 Project of China (2011CB711205), the National Natural Science Foundation of China (11202072), and the Doctoral Fund of Ministry of Education of China (20120161120005). The first author is a recipient of the doctoral scholarships from both China Scholarship Council (CSC) and the University of Sydney. References

4.54 0.069

the performance and robustness of the fatigue life simultaneously compared with the baseline, but also provides a proper compromise between performance and robustness compared with other Pareto points. Note that in real-life engineering, design problems could be more complex, where more design variables and uncertainty types might be involved. In this study, while only three design variables and their material uncertainties were considered, the proposed multiobjective optimization method combined with TOPSIS-based GRA procedure can be potentially extended to such more complicated problems.

4. Concluding remarks To simultaneously enhance the robustness and performance of the fatigue life involving the material uncertainties, the structural design of a truck cab is formulated as a robust optimization problem by integrating the multiobjective particle swarm optimization (MOPSO) algorithm. To overcome the limitation of the classical dual response

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