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A new design optimization framework based on immune algorithm and Taguchi's method
Computers in Industry 60 (2009) 613–620
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Computers in Industry journal homepage: www.elsevier.com/locate/compind
A new design optimization framework based on immune algorithm and Taguchi’s method Ali Rıza Yıldız Uludag University, Mechanical Engineering Department, 16059, Bursa, Turkey
A R T I C L E I N F O
A B S T R A C T
Article history: Available online 1 July 2009
This paper describes an innovative optimization approach that offers significant improvements in performance over existing methods to solve shape optimization problems. The new approach is based on two-stages which are (1) Taguchi’s robust design approach to find appropriate interval levels of design parameters (2) Immune algorithm to generate optimal solutions using refined intervals from the previous stage. A benchmark test problem is first used to illustrate the effectiveness and efficiency of the approach. Finally, it is applied to the shape design optimization of a vehicle component to illustrate how the present approach can be applied for solving shape design optimization problems. The results show that the proposed approach not only can find optimal but also can obtain both better and more robust results than the existing algorithm reported recently in the literature. ß 2009 Elsevier B.V. All rights reserved.
Keywords: Shape optimization Immune algorithm Taguchi’s method Robust design
1. Introduction The optimal design of structures includes sizing, shape and topology optimization. In the last 30 years, there has been extensive research focused on shape optimization due to its great contribution to cost, material and time savings in the procedures of the engineering design. The purpose of shape optimization is to determine the optimal shape of a continuum medium to maximize or minimize a given criterion (often called an objective function), such as minimize the weight of the body, maximize the stiffness of the structure or remove the stress concentrations, subjected to the stress or displacement constraint conditions. Computer-aided optimization has been commonly used to obtain more economical designs since 1970s. Numerous algorithms have been developed to solve shape design optimization problems in the last four decades. The early works on the topic mostly use various mathematical techniques. These methods are not only time consuming in solving complex nature problems but also they may not be used efficiently in finding global or near global optimum solutions. In the past few decades, a number of innovative approaches, such as tabu search, genetic algorithm, simulated annealing, particle swarm optimization algorithm, ant colony algorithm and immune algorithm have been developed and widely applied in various fields of science [1–10]. Fast convergence speed and robustness in finding the global minimum are not easily achieved at the same time. Fast convergence requires a minimum number of calculations, increas-
E-mail address: [email protected]. 0166-3615/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.compind.2009.05.016
ing the probability of missing important points; on the other hand, the evaluation of more points for finding the global minimum decreases the convergence speed. This leads to the question: ‘how to obtain both fast convergence speed and global search capability at the same time.’ There have been a number of attempts to answer this question, while hybrid algorithms have shown outstanding reliability and efficiency in application to the engineering optimization problems [11–15]. Therefore, the researchers are paying great attention on hybrid approaches to answer this question, particularly to avoid premature convergence towards a local minimum and to reach the global optimum results. There is an increasing interest to apply the new approaches and to further improve the performance of optimization techniques for the solution of shape design optimization problems. Although some improvements regarding shape design optimization issues are achieved, the complexity of design problems presents shortcomings. The main goal of present research is to further develop and strengthen the immune algorithm which is a computational intelligence paradigm inspired by the biological immune system to generate real world design solutions. A new hybrid approach based on robustness issues is used to help better initialize immune algorithm search. It has been aimed to reach optimum designs by using Taguchi’s robust parameter design approach coupled with immune algorithm. In this new hybrid approach, S/N values are calculated and ANOVA (analysis of variance) table for each of the objectives are formed using S/N ratios respectively. According to results of ANOVA table, appropriate interval levels of design parameters are found and then, initial antibody population of immune algorithm is defined according to these interval levels.
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Then, optimum results of design optimization problem are obtained using immune algorithm. The hybrid approach is evaluated using a well-known benchmark problem and compared with other optimization methods in the literature. Finally, the developed new hybrid optimization approach is applied to a vehicle part design optimization problem taken from automotive industry to demonstrate the application of the present approach to real world design problems. The results show that the proposed optimization method converges rapidly to the global optimum solution and provides reliable and accurate solutions for even the most complicated of optimization problems. 2. Literature review Recently, new approaches in the area of optimization research are presented to further improve the solution of optimization problems with complex nature. Over the past few years, the studies on evolutionary algorithms have shown that these methods can be efficiently used to eliminate most of the difficulties of classical methods. Evolutionary algorithms are widely used to solve engineering optimization problems with complex nature. Various research works are carried out to enhance the performance of evolutionary algorithm [16–22]. Immune algorithm inspired from human immune system is one of the modern heuristic algorithms, which is used successfully in the solution of optimization problems. The use of immune algorithms in the optimum solution of problems resulted better solutions compared to classical methods [22–30]. The immune algorithm was used for multi-modal topology optimization [23] and optimal design of truss structure by Luh and Chueh [24]. Coello and Cortes [25] hybridized immune algorithm with genetic algorithm and applied to optimization problems taken from the evolutionary optimization literature and compared their results with other evolutionary optimization approaches which are representative of the state-of-the-art in evolutionary optimization literature. Some researchers have used the robustness issues to solve multiobjective optimization problems [31–34]. Robinson et al. [35] presents a review paper that focuses largely on the work done since 1992 and a historical perspective of parameter design is also given. Kunjur and Krishnamurty [36] presented a robust multiple criteria optimization approach that integrates multi-objective optimization concepts with statistical robust design techniques. Although Taguchi’s methods have been successfully applied to processes in the design and manufacturing, they are also criticized for their efficiency [37,38]. Hybrid methods are also used to enhance the performance of evolutionary algorithm. Yildiz et al. [15] developed a hybrid robust genetic algorithm (HRGA) based on Taguchi’s method and genetic algorithm. After the approach was validated by multi-objective welded beam design problems, it was applied to shape design optimization problem of an automobile component from industry. Yildiz [26] hybridized immune algorithm with hill climbing local search algorithm and applied to design and manufacturing optimization problems from literature. Tsai et al. [39] proposed a hybrid algorithm which the Taguchi’s method is inserted between crossover and mutation operations of a genetic algorithm. Taguchi method is incorporated in the crossover operations to select the better genes to achieve crossover, and consequently, enhance the performance of genetic algorithm. In this research, a test problem is used to evaluate and observe the performance of the proposed approach. The results are compared with the solutions of different methods given in Ref. [25]. Finally, the proposed hybrid approach is applied to the design optimization of a vehicle component to illustrate how the present approach can be applied for solving multi-objective shape design optimization problems.
3. Hybrid optimization approach for shape optimization In this paper, a new hybrid optimisation approach, named HTIA, is developed to solve shape optimisation problems. In the proposed optimisation approach, the refinement of the population space is introduced by Taguchi’s method. The bounds selected on the design variables are first used for the initial antibody population, then they apply throughout immune algorithm for finding optimal design paramaters. The aim is to overcome the limitations caused by larger population regarding computational cost and quality of solutions for global optimization. First, some brief explanations about multiobjective optimization, immune system and Taguchi’s method are given and, finally, the proposed hybrid approach is explained. Optimization problems that have more than one objective function are usually called as multi-objective optimization problems. A multi-objective optimization problem has a number of objective functions which are to be minimized or maximized. Most real-world engineering optimization problems are multiobjective in nature, since they usually have several objectives and constraints that must be satisfied at the same time. A general mathematical model of constrained multi-objective optimization problem can be defined as follows [15,40–44]: Objective function: Minimize=maximize Subject to
g j ðxÞ 0;
hk ðxÞ ðLÞ
xi
f m ðxÞ;
m ¼ 1; 2; . . . ; M;
j ¼ 1; 2; . . . ; J;
¼ 0; k ¼ 1; 2; . . . ; K; ðUÞ xi xi i ¼ 1; 2; . . . ; n;
(1) (2)
(3)
A solution x is a vector of n decision variables: x = (x1, x2,. . .,xn)T. ðLÞ Each decision variable xi to take a value between a lower xi and an ðUÞ upper xi bound. These bounds define decision variable space. In this mathematical model gj(x) and hk(x) define equality and inequality constraint function respectively. M objective functions f(x) = (f1(x), f1(x),. . ., fM(x))T can be either minimized or maximized. The problem is to find optimum variables that satisfy the constraints given by (2) and (3). Immune algorithm inspired from human immune system is an optimization procedure. It begins its search with a random set of solutions. Then, cloning, mutation and receptor editing operations are applied to find better populations of solutions. A larger population makes the algorithm more likely to locate a good making string, but also increases the time taken by the algorithm. The problem with larger population is to tend evolutionary algorithms to converge and stick around certain solutions; therefore, there is a need to define the efficient range of population intervals to achieve better Pareto optimal sets in shorter times. This shortcoming is eliminated by introducing Taguchi’s based initial antibody population. 3.1. Immune system The human immune system has a vital role to protect our body health from infectious foreign organisms called antigens including viruses, bacteria and other parasites. The immune system needs to distinguish all cells (or molecules) within our body and categorize these cells as self or non-self. While the disease causing external cells are named as non-self its own harmless cells are named as self. There are several types of immune cells in immune system. Lymphocytes are the main type of immune cells and are classified as ‘‘T’’ and ‘‘B’’ cells. B-cells are capable of recognizing antigens free in the blood stream whereas T-cells can recognize via other accessory cells [45]. The immune algorithm used in this paper is based on the clonal selection and affinity maturation principles of
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Fig. 2. Schematic representation of shape-space for antigen-binding sites [47].
Fig. 1. The clonal selection principle [27].
the immune system. Clonal selection mechanism is used to explain how an immune response is given when a non-self antigenic pattern is recognized by a B-cell. The principle is shown in Fig. 1. Both B-cells and T-cells have receptor molecules on their surfaces. The B-cells are called as antibody and recognize disease causing pathogens. When antigens and receptor molecules have complementary shapes, they can bind together [46]. When an antigen is recognized by immune cell receptors, an immune response is obtained through the production of antibody from the bone marrow. After these antibodies combine with antigens, the antigen stimulates the B-cell to proliferate and mature into terminal (non-dividing) antibody secreting cells, called plasma cells. The proliferation in the immune system is achieved by cell divisions (mitosis). The proliferation (mitosis) generates a clone of cells that are the children of a single cell. De Castro and Von Zuben [27] developed the clonal selection algorithm (CSA) based on clonal selection to solve engineering problems. In their paper, they presented the overview of this mechanism from the points of view of immunology and engineering problems. Affinity maturation is the all mutation processes and selection that guarantees the survival of the variant offspring that better recognizes the antigen. The affinity defines the degree of binding of the cell receptor with the antigen. The higher the affinity the stronger the binding and thus the better the immune recognition and response. The two main mechanisms of affinity maturation are hypermutation and receptor editing [27,46]. Random changes (mutations) happen in the variable region genes of antibody molecules and cause structurally different cells. Rarely, one such change will lead to an increase in the affinity of the antibody. Since the lymphocytes of the immune system are somatic cells, the mutation that occurs during affinity maturation is named as somatic mutation. Somatic mutation helps the immune system to keep the high affinity cells and to ensure large mutations for the low affinity ones in order to get better affinity cells. The last stage of artificial immune system is receptor editing. After cloning and mutation processes are completed, a percentage of antibodies in the antibody population are replaced with antibodies created randomly. This process is named as receptor editing. Receptor editing mechanism introduces diversity and
helps to escape from local optima on an affinity landscape and leads to possible new candidates [47]. In Fig. 2, a simplified landscape is illustrated. In the figure, the x-axis show all possible antigen-binding sites, with the most similar adjacent to each other. The y-axis shows the antigen–antibody affinity. When it is taken a particular antibody (A) selected during the primary immune response, point mutations permit the immune system to explore local regions around A by making small steps towards an antibody with higher affinity. As a result, the antibodies might become stuck at a local optimum (A0 ) [27,47]. Receptor editing allows an antibody to take large steps through the landscape. Although this will land the antibody in a local optimum (B), occasionally the leap will lead to an antibody on the side of a higher hill (C), where the climbing region have more chance to reach to the global optimum. From this point (C), the antibodies can reach to the top of the hill (C0 ). In conclusion, receptor editing may rescue immune responses stuck on unsatisfactory local optima. In receptor editing mechanism, a percentage of the antibodies in the antibody population are replaced with randomly created antibodies. This mechanism allows finding new antibodies that correspond to new search regions in the all search space. Exploring new search regions may help the algorithm to escape from local optimal. 3.2. Taguchi method Taguchi method is a method that chooses the most suitable combination of the levels of controllable factors by using S/N table and orthogonal arrays against the factors that form the variation and are uncontrollable in product and process. Hence, it tries to reduce the variation in product and process to least. Taguchi uses statistical performance measure which is known as S/N ratio that takes both medium and variation into consideration. Most of the shape design optimization problems in automotive industry are usually multi-objective, often conflicting, and they have uncontrollable variations in their design parameters with complex nature. There is a need to overcome the shortcomings due to the traditional optimization methods and also to further improve the strength of recent approaches to achieve better results for the real world design problems. Therefore, in this research, a hybrid approach is developed to solve shape design optimization problems. The architecture of the proposed approach is given in Fig. 3. Taguchi’s robust parameter design is introduced to help to define robust initial population levels of design parameters and to reduce the effects of noise factors to achieve better initialize immune algorithm search. The problem with larger population may tend immune algorithm to converge and stick around certain
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Fig. 3. Hybrid immune algorithm based multi-objective design optimization approach.
solutions which may not be the best ones. This is handled with the help of robust parameter levels which are embedded into immune algorithm process as being initial population intervals. In other words, the design space is restricted and refined based on the effect of the various design variables on objective functions. The purpose of the ANOVA tables is to help differentiate the robust designs from the non-robust ones. Finally, optimum results of multi-objective problem are obtained by applying immune algorithm through cloning, mutation and receptor editing operations. The present approach is considered in two stages as follows: Stage 1 Determine efficient solution space for shape optimization variables using Taguchi’s method. Stage 2 Apply immune algorithm to find shape design variables. In the first stage, Taguchi’s robust parameter design procedure is used to find the levels of variables for efficient search space as follows:
identify the objectives, constraints and design parameters, determine the settings of the design parameter levels, conduct the experiments using orthogonal array, compute S/N ratios and ANOVA analysis, find the optimal settings of design parameters.
The main issue of experimental analysis is the ANOVA analysis which is formed using S/N ratios respectively for each of the objectives. According to results of ANOVA, appropriate levels of design parameters are found and then, initial search population of immune algorithm process is defined according to levels. Finally, optimum results of the optimization problem are obtained by applying immune algorithm process through cloning, mutation and receptor editing operations in two steps as follows:
define initial antibody set, use immune operators to create the next generation, evaluate objective function and constraints, repeat the loop until the optimum solutions are found.
immune algorithm is resulted in a solution, which leads to better parameter values for shape design optimization problems. The algorithm of proposed hybrid approach can be outlined as follows: BEGIN Step 1: Taguchi’s method Begin 1.a Choose convenient orthogonal array from Taguchi’s standard orthogonal arrays 1.b Define levels and intervals 1.c For i: = 1 to NOE (number of experiments) do begin Compute objective function values end; 1.d Choose convenient S/N ratio type (smaller the best or larger the best or nominal the best) based on minimization or maximization of objective functions 1.e For i: = 1 to NOE do begin Compute S/N ratios end; 1.f Constitute Anova table for objective functions using S/N ratios 1.g Determine optimum levels and intervals using percentage contribution to performance using Anova table Use these levels and intervals for forming initial population end; ‘Generate optimal solution set using immune algorithm and computed robust initial population space’ Begin Input Use Initial population found in previous part of the program as input to immune algorithm Step 2: Artificial Immune Algorithm 2.a Initial antibody population
In this paper, a new hybrid approach is proposed to improve the performance of the immune algorithm. Our argument behind the proposed approach is that the strength of one algorithm can be used to improve the performance of another approach in the optimization process. The combination of Taguchi’s method and
For i: = 1 to the number of antibodies do begin Generate randomly initial antibody population End; While (not termination condition) do begin
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Table 1 Statistical results of different methods for test problem. Design variables
Best
Mean
Worst
Std dev
Function evaluations
HTIA Coello and Cortes [25] Hamida and Schoenauer [48] Koziel and Michalewicz [49]
6961.814 6961.7608 6961.81 6952.100
6961.279 6961.273 6961.81 6342.6
6960.257 6960.607 NA 5473.9
0.116 0.3598 NA NA
100,000 150,000 1,500,000 1,400,000
Calculate affinity values of the antibodies 2.b Cloning Generate copies of antibodies using affinity values of the antibodies 2.c Mutation For i: = 1 to the number of the antibodies do begin Apply mutation to the antibodies End; 2.d Receptor editing For i: = 1 to (0.25* number of the antibodies) do begin Apply receptor editing to randomly selected antibodies end; end; END. 4. Evaluation of the proposed approach using test problem In order to evaluate the performance of the proposed hybrid approach, a single objective benchmark problem commonly used in the optimization literature is successfully solved by the proposed algorithm. The results of the benchmark problem are compared with those of other methods that are representative of the state-of-the-art in the optimization literature. After it is shown that the proposed approach is successful to optimize the optimization problems, it is applied to a case study from automobile industry. The test problem is a minimization problem for a single objective function with two variables and two inequality constraints. This problem is employed by several authors to evaluate the performances of their approaches [25,48,49]. It is defined as follows: Minimize : Subject to :
f ðxÞ ¼ ðx1 10Þ3 þ ðx2 20Þ3 g 1 ðxÞ ¼ ðx1 5Þ2 þ ðx2 5Þ2 100 0 g 2 ðxÞ ¼ ðx1 6Þ2 ðx2 5Þ2 þ 82:81 0
where 13 x1 100 and 0 x2 100. The global optimum is located at x* = (14.095, 0.84296), where f(x*) = 6961.81388. Both constraints are active at the optimum. The parameters used by the proposed hybrid approach for optimization process are the following:
The results of the proposed hybrid approach for test problem are compared against those provided by Coello and Cortes [25]. HTIA gives better solutions for this problem compared with those given in Table 1 as far as the number of function evaluations, the best solution computed, and the statistical analysis results are taken into account together. The statistical values of HTIA show the robustness of the solutions with the standard deviation of 0.116. The standard deviation is very low, which indicates that HTIA is among the most robust approach in finding an optimum solution. The HTIA requires only 100,000 function evaluations to find the best-known solution of 6961.814. Coello and Cortes [25] found the best solution 6961.7608 for 150,000 function evaluations with standard deviation of 0.3598. The use of the HTIA improves the convergence rate by computing the best value 6961.814 with respect to very low standard deviation 0.116 and maintaining the less function evaluations 100,000. 5. Shape optimization using improved hybrid immune algorithm The hybrid approach proposed in the Section 3 is applied to solve the shape design optimization problem in this section. The second example is taken from automotive industry for the optimal design of a vehicle component. The objective functions are due to the volume and the frequency of the part which is to be designed for minimum volume and avoiding critical frequency subject to strength constraints. In this research, then shape optimization is performed using present approach. In the first stage, the experiments are designed to evaluate the effects of four design variables related to objective functions. The four shape design variables x1, x2, x3 and x4 are selected as shown in Fig. 4. The feasible range of design variables without shape distortions is considered as 6 < x1 < 30, 21 < x2 < 27, 8 < x3 < 14 and 28 < x4 < 46. Matrix experiments are designed using L16 orthogonal arrays and S/N ratios are conducted for each objective as given in Table 2. Smaller the better and larger the better characteristics are applied to compute S/N ratios based on each objective as smaller the better for volume and frequency. The details about orthogonal array, S/N ratios, ANOVA analysis, and how they are computed and used for experimental evaluations are given in the reference of Phadke [50].
(a) antibody size: 100; (b) maximum number of generations: 1000; (c) number of objective function evaluations: 100,000. The efficiency of each approach can be measured by comparing the number of function evaluations, which is equal to the antibody size (population size) and multiplied by the number of generations as each solution is evaluated once in every generation. In the evolutionary optimization approaches, the aim is to find the optimum results with minimum number of function evaluations. In this paper, the HTIA is run 30 times independently.
Fig. 4. Design variables.
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Table 2 Experimental results and S/N ratios for volume and frequency. Ex no
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
F1 (volume)
F2 (frequency)
S/N1 volume
S/N2 frequency
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6 6 6 6 14 14 14 14 22 22 22 22 30 30 30 30
21 23 25 27 21 23 25 27 21 23 25 27 21 23 25 27
8 10 12 14 10 8 12 14 12 14 8 10 14 12 10 8
28 34 40 46 40 46 28 34 46 40 34 28 34 28 46 40
23,712 22,432 21,032 19,512 21,062 20,282 21,452 20,052 19,042 18,962 20,242 20,402 18,912 19,572 17,352 18,012
8.1066 7.7716 7.5002 7.2879 8.4748 8.2572 7.8605 7.5219 9.1969 8.7593 8.4097 8.1266 10.082 9.7315 9.6848 9.1402
87.4994 87.0174 86.4576 85.8060 86.4700 86.1422 86.6294 86.0432 85.5943 85.5577 86.1251 86.1935 85.5347 85.8327 84.7870 85.1112
18.1768 17.8102 17.5015 17.2520 18.5626 18.3367 17.9090 17.5266 19.2728 18.8494 18.4956 18.1982 20.0709 19.7636 19.7218 19.2191
Table 3 Results of the analysis of variance for volume.
X1 X2 X3 X4 Error
Level 1
Level 2
Level 3
Level 4
S
86.70 86.27 86.22 86.54
86.32 86.14 86.12 86.18
85.87 86.00 86.13 85.90
85.32 85.79 85.74 85.58
4.244 0.440 0.291 1.815 0.007
3 3 3 3 3
6.797
15
Total
DOF
M
F
Cont. (%)
1.4148 0.1469 0.0970 0.6051 0.0026
539.9 56.09 37.03 230.9
62.43 6.473 4.281 26.70
Table 4 Results of the analysis of variance for frequency.
X1 X2 X3 X4 Error
Level 1
Level 2
Level 3
Level 4
17.69 19.02 18.56 18.51
18.08 18.69 18.57 18.48
18.70 18.41 18.61 18.53
19.69 18.05 18.42 18.65
Total
S
DOF 9.189 1.973 0.009 0.064 0.006
11.24
ANOVA for each objective is formed using S/N ratios for the first and second objective functions as shown in Tables 3 and 4. The most effective parameter is x1 with 62.4% contribution for the first objective (see Table 3). Level 4 is considered for x1 since the smaller the better characteristic is employed for the first objective. The most effective parameter is x1 with 81.7% contribution for the second objective (see Table 4). Level 4 is considered for x1 since the larger the better characteristic is employed for the second objective. Therefore, level 4 can be selected for x1 as 30 mm. However, the levels for x2, x3, and x4 cannot be defined since contributions are weak and there are some conflicts in selecting them. For example, x2 has 6% contribution for the first objective (level 4) and 17.5% contribution for the second objective (level 1); thus, x2 must be taken between levels 1 and 4 as 21 < x2 < 27. In case of other two variables, ANOVA results with contributions less than 1 as 0.08 and 0.57, this cannot be used to select the
3 3 3 3 3
M
F
Cont. (%)
3.06300 0.65799 0.00324 0.02160 0.00203
1512.49 324.91 1.60 10.67
81.745 17.55 0.008 0.569
15
levels. Thus, ANOVA of stress constraint is considered to define the levels of x3 and x4 (Table 4). Then, the variables are considered as x1 = 30, 21 < x2 < 27, 8 < x3 < 14, and 28 < x4 < 46. The parameter levels are taken as x1 = 30, 21 < x2 < 27, 8 < x3 < 14, and 28 < x4 < 46, which are obtained from the ANOVA analysis. Immune algorithm begins its search with a set of solutions, with its population range defined by Taguchi’s method. The results of HTIA are given in Table 5. It can be seen that a volume of 16,416 mm3 and frequency of 9.36 Hz with 296 MPa is obtained. Then, the problem is solved using immune algorithm approach. Immune algorithm begins its search with a set of solutions, with its population range defined by Taguchi’s method. Each antibody has an affinity value suitable with the structure of the problem. Cloning, mutation and receptor editing process are applied to the whole antibody population through the immune algorithm process. Then, HTIA works by repeatedly modifying a population
Table 5 Comparison of the optimization results for the side door bracket design.
Initial design CAD optimum design Immune algorithm HTIA
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
Volume (cm3)
Frequency (Hz)
Stress (MPa)
18 22 27.5 30
24 27 26.6 26.85
11 13.1 12.7 13.52
32 36.7 43.1 46.2
20,727.2 18,763.9 17,288.92 16,416.41
8.3 8.1 8.9 9.36
256 277 290 296
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Fig. 5. The optimal structural layout.
of artificial structures through the application of immune operators to compute optimal set of design variables. It is clearly seen that shape design optimization performance is improved compared to traditional CAD and immune algorithm solutions. ANSYS is used for CAD optimization process. The intervals of design variables are considered as 6 < x1 < 30, 21 < x2 < 27, 8 < x3 < 14, and 28 < x4 < 46 for multi-objective immune algorithm solution. The structural layout results of the present hybrid approach using immune algorithm and robust design based on Taguchi’s method for the vehicle part are given in Fig. 5. The results of present hybrid approach for the optimal design of the vehicle part are given in Table 5. It is seen that shape design optimization performance is improved compared to other approaches. 6. Conclusions This research describes a new optimization approach based on immune algorithm and Taguchi’s robust design approach for solving shape design optimization problems. Taguchi’s robust design approach is introduced to help to define robust initial population levels of design parameters to achieve better initialize immune algorithm search. The design solution space of immune algorithm is refined based on the effect of the various design variables on objective functions. The validity and efficiency of HTIA are evaluated and illustrated with a test problem taken from literature. Then, it is applied to a vehicle component taken from automotive industry. It is seen that better results can be achieved with present hybrid optimization approach. Therefore, HTIA is a suitable optimization technique for the solution of shape design optimization problems. Since HTIA is a generalized optimization method, it can also be applied to other optimization problems. Further research can be carried out to improve the HTIA by integrating the key issues from the fields of multi-objective optimization, evolutionary algorithms and local search techniques. Now, as further research, the author consider hybridizing of HTIA by local search techniques like hill-climbing to extend its improvements for solving multi-objective design optimization problems. References [1] A.R. Yildiz, K. Saitou, Topology Synthesis of Multi-Component Structural Assembly in Continuum Domain DETC 2008, New York, USA, 3–5 August, 2008. [2] N. Lyu, K. Saitou, Topology optimization of multicomponent structures via decomposition-based assembly synthesis, Transactions of ASME Journal of Mechanical Design 127 (2005) 170–183. [3] P.C. Fourie, A.A. Groenwold, The particle swarm optimization algorithm in size and shape optimization, Structural and Multidisciplinary Optimization 23 (2002) 259–267. [4] N. Kaya, Machining fixture locating and clamping position optimization using genetic algorithms, Computers in Industry 57 (2006) 112–120.
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[37] G. Box, Signal-to-noise ratios, performance criteria, and transformations, Technometrics 1 (30) (1988) 1–17. [38] V.N. Nair, Taguchi’s parameter design: a panel discussion, Technometrics 34 (1992) 127–161. [39] J.T. Tsai, T.K. Liu, J.H. Chou, Hybrid Taguchi genetic algorithm for global numerical optimization, IEEE Transactıons on Evolutıonary Computation 84 (2004) 365– 377. [40] S.S. Rao, Game Theory approach for multiobjective structural optimization, Computers and Structures 26 (1) (1987) 119–127. [41] F.Y. Cheng, X.S. Li, Generalized method for multiobjective engineering optimization, Engineering Optimization 31 (1999) 641–661. [42] K.M. Liew, P.K. Tan, T. Ray, Leader identification and leader selection: its effect on a swarms performance for multi-objective design optimization problems, Structural and Multidisciplinary Optimization 28 (2004) 156–169. [43] D.E. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning, Addison-Wesley Publishing, 1989. [44] K. Deb, Multiobjective Optimization Using Evolutionary Algorithms, John Wiley & Sons Ltd., England, 2001. [45] L.N. De Castro, F.J. Von Zuben, Artificial immune systems, Part 1, Basic theory and applications, Technical Report, TR-DCA 01/99, 1999. [46] L.N. De Castro, F.J. Von Zuben, Learning and optimization using the clonal selection principle, IEEE Transactions on Evolutionary Computation 6 (2002) 239–251. [47] A.J.T. George, D. Gray, Receptor editing during affinity maturation, Immunology Today 20 (1999) 196. [48] S.B. Hamida, M. Schoenauer, ASCHEA: new results using adaptive segregational constraint handling, in: Proc Congr Evolutionary Computation, vol. 1, 2002, 884– 889.
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Ali Rıza Yildiz is a Lecturer at Uludag University. He received his PhD in Mechanical Engineering from Uludag University, Turkey in 2006. He worked on ‘‘Topology Synthesis of Multi-Component Structural Assembly in Continuum Domain’’ as a Post Doctoral Researcher between 2006 and 2008 at the Discrete Design Optimization Laboratory, University of Michigan, Ann Arbor, USA. His major research interests includes multi-objective shape optimization, topology optimization and hybrid optimization algorithms. He has published more than 25 journal articles and conference papers. Dr. Yildiz served on the technical program committee of international conferences including, ASME 34th Design Automation Conference (IDETC 2008), World Congress on Engineering and Computer Science (WCECS 2008), ASME 33rd Design Automation Conference (IDETC 2007) and more. He is an Editorial Board Member of several journals, including ‘‘Expert Systems: The Journal of Knowledge Engineering’’ and ‘‘International Journal of Mathematical Modelling and Numerical Optimisation’’ and is a reviewer for a variety of additional journals.
Contents lists available at ScienceDirect
Computers in Industry journal homepage: www.elsevier.com/locate/compind
A new design optimization framework based on immune algorithm and Taguchi’s method Ali Rıza Yıldız Uludag University, Mechanical Engineering Department, 16059, Bursa, Turkey
A R T I C L E I N F O
A B S T R A C T
Article history: Available online 1 July 2009
This paper describes an innovative optimization approach that offers significant improvements in performance over existing methods to solve shape optimization problems. The new approach is based on two-stages which are (1) Taguchi’s robust design approach to find appropriate interval levels of design parameters (2) Immune algorithm to generate optimal solutions using refined intervals from the previous stage. A benchmark test problem is first used to illustrate the effectiveness and efficiency of the approach. Finally, it is applied to the shape design optimization of a vehicle component to illustrate how the present approach can be applied for solving shape design optimization problems. The results show that the proposed approach not only can find optimal but also can obtain both better and more robust results than the existing algorithm reported recently in the literature. ß 2009 Elsevier B.V. All rights reserved.
Keywords: Shape optimization Immune algorithm Taguchi’s method Robust design
1. Introduction The optimal design of structures includes sizing, shape and topology optimization. In the last 30 years, there has been extensive research focused on shape optimization due to its great contribution to cost, material and time savings in the procedures of the engineering design. The purpose of shape optimization is to determine the optimal shape of a continuum medium to maximize or minimize a given criterion (often called an objective function), such as minimize the weight of the body, maximize the stiffness of the structure or remove the stress concentrations, subjected to the stress or displacement constraint conditions. Computer-aided optimization has been commonly used to obtain more economical designs since 1970s. Numerous algorithms have been developed to solve shape design optimization problems in the last four decades. The early works on the topic mostly use various mathematical techniques. These methods are not only time consuming in solving complex nature problems but also they may not be used efficiently in finding global or near global optimum solutions. In the past few decades, a number of innovative approaches, such as tabu search, genetic algorithm, simulated annealing, particle swarm optimization algorithm, ant colony algorithm and immune algorithm have been developed and widely applied in various fields of science [1–10]. Fast convergence speed and robustness in finding the global minimum are not easily achieved at the same time. Fast convergence requires a minimum number of calculations, increas-
E-mail address: [email protected]. 0166-3615/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.compind.2009.05.016
ing the probability of missing important points; on the other hand, the evaluation of more points for finding the global minimum decreases the convergence speed. This leads to the question: ‘how to obtain both fast convergence speed and global search capability at the same time.’ There have been a number of attempts to answer this question, while hybrid algorithms have shown outstanding reliability and efficiency in application to the engineering optimization problems [11–15]. Therefore, the researchers are paying great attention on hybrid approaches to answer this question, particularly to avoid premature convergence towards a local minimum and to reach the global optimum results. There is an increasing interest to apply the new approaches and to further improve the performance of optimization techniques for the solution of shape design optimization problems. Although some improvements regarding shape design optimization issues are achieved, the complexity of design problems presents shortcomings. The main goal of present research is to further develop and strengthen the immune algorithm which is a computational intelligence paradigm inspired by the biological immune system to generate real world design solutions. A new hybrid approach based on robustness issues is used to help better initialize immune algorithm search. It has been aimed to reach optimum designs by using Taguchi’s robust parameter design approach coupled with immune algorithm. In this new hybrid approach, S/N values are calculated and ANOVA (analysis of variance) table for each of the objectives are formed using S/N ratios respectively. According to results of ANOVA table, appropriate interval levels of design parameters are found and then, initial antibody population of immune algorithm is defined according to these interval levels.
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Then, optimum results of design optimization problem are obtained using immune algorithm. The hybrid approach is evaluated using a well-known benchmark problem and compared with other optimization methods in the literature. Finally, the developed new hybrid optimization approach is applied to a vehicle part design optimization problem taken from automotive industry to demonstrate the application of the present approach to real world design problems. The results show that the proposed optimization method converges rapidly to the global optimum solution and provides reliable and accurate solutions for even the most complicated of optimization problems. 2. Literature review Recently, new approaches in the area of optimization research are presented to further improve the solution of optimization problems with complex nature. Over the past few years, the studies on evolutionary algorithms have shown that these methods can be efficiently used to eliminate most of the difficulties of classical methods. Evolutionary algorithms are widely used to solve engineering optimization problems with complex nature. Various research works are carried out to enhance the performance of evolutionary algorithm [16–22]. Immune algorithm inspired from human immune system is one of the modern heuristic algorithms, which is used successfully in the solution of optimization problems. The use of immune algorithms in the optimum solution of problems resulted better solutions compared to classical methods [22–30]. The immune algorithm was used for multi-modal topology optimization [23] and optimal design of truss structure by Luh and Chueh [24]. Coello and Cortes [25] hybridized immune algorithm with genetic algorithm and applied to optimization problems taken from the evolutionary optimization literature and compared their results with other evolutionary optimization approaches which are representative of the state-of-the-art in evolutionary optimization literature. Some researchers have used the robustness issues to solve multiobjective optimization problems [31–34]. Robinson et al. [35] presents a review paper that focuses largely on the work done since 1992 and a historical perspective of parameter design is also given. Kunjur and Krishnamurty [36] presented a robust multiple criteria optimization approach that integrates multi-objective optimization concepts with statistical robust design techniques. Although Taguchi’s methods have been successfully applied to processes in the design and manufacturing, they are also criticized for their efficiency [37,38]. Hybrid methods are also used to enhance the performance of evolutionary algorithm. Yildiz et al. [15] developed a hybrid robust genetic algorithm (HRGA) based on Taguchi’s method and genetic algorithm. After the approach was validated by multi-objective welded beam design problems, it was applied to shape design optimization problem of an automobile component from industry. Yildiz [26] hybridized immune algorithm with hill climbing local search algorithm and applied to design and manufacturing optimization problems from literature. Tsai et al. [39] proposed a hybrid algorithm which the Taguchi’s method is inserted between crossover and mutation operations of a genetic algorithm. Taguchi method is incorporated in the crossover operations to select the better genes to achieve crossover, and consequently, enhance the performance of genetic algorithm. In this research, a test problem is used to evaluate and observe the performance of the proposed approach. The results are compared with the solutions of different methods given in Ref. [25]. Finally, the proposed hybrid approach is applied to the design optimization of a vehicle component to illustrate how the present approach can be applied for solving multi-objective shape design optimization problems.
3. Hybrid optimization approach for shape optimization In this paper, a new hybrid optimisation approach, named HTIA, is developed to solve shape optimisation problems. In the proposed optimisation approach, the refinement of the population space is introduced by Taguchi’s method. The bounds selected on the design variables are first used for the initial antibody population, then they apply throughout immune algorithm for finding optimal design paramaters. The aim is to overcome the limitations caused by larger population regarding computational cost and quality of solutions for global optimization. First, some brief explanations about multiobjective optimization, immune system and Taguchi’s method are given and, finally, the proposed hybrid approach is explained. Optimization problems that have more than one objective function are usually called as multi-objective optimization problems. A multi-objective optimization problem has a number of objective functions which are to be minimized or maximized. Most real-world engineering optimization problems are multiobjective in nature, since they usually have several objectives and constraints that must be satisfied at the same time. A general mathematical model of constrained multi-objective optimization problem can be defined as follows [15,40–44]: Objective function: Minimize=maximize Subject to
g j ðxÞ 0;
hk ðxÞ ðLÞ
xi
f m ðxÞ;
m ¼ 1; 2; . . . ; M;
j ¼ 1; 2; . . . ; J;
¼ 0; k ¼ 1; 2; . . . ; K; ðUÞ xi xi i ¼ 1; 2; . . . ; n;
(1) (2)
(3)
A solution x is a vector of n decision variables: x = (x1, x2,. . .,xn)T. ðLÞ Each decision variable xi to take a value between a lower xi and an ðUÞ upper xi bound. These bounds define decision variable space. In this mathematical model gj(x) and hk(x) define equality and inequality constraint function respectively. M objective functions f(x) = (f1(x), f1(x),. . ., fM(x))T can be either minimized or maximized. The problem is to find optimum variables that satisfy the constraints given by (2) and (3). Immune algorithm inspired from human immune system is an optimization procedure. It begins its search with a random set of solutions. Then, cloning, mutation and receptor editing operations are applied to find better populations of solutions. A larger population makes the algorithm more likely to locate a good making string, but also increases the time taken by the algorithm. The problem with larger population is to tend evolutionary algorithms to converge and stick around certain solutions; therefore, there is a need to define the efficient range of population intervals to achieve better Pareto optimal sets in shorter times. This shortcoming is eliminated by introducing Taguchi’s based initial antibody population. 3.1. Immune system The human immune system has a vital role to protect our body health from infectious foreign organisms called antigens including viruses, bacteria and other parasites. The immune system needs to distinguish all cells (or molecules) within our body and categorize these cells as self or non-self. While the disease causing external cells are named as non-self its own harmless cells are named as self. There are several types of immune cells in immune system. Lymphocytes are the main type of immune cells and are classified as ‘‘T’’ and ‘‘B’’ cells. B-cells are capable of recognizing antigens free in the blood stream whereas T-cells can recognize via other accessory cells [45]. The immune algorithm used in this paper is based on the clonal selection and affinity maturation principles of
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Fig. 2. Schematic representation of shape-space for antigen-binding sites [47].
Fig. 1. The clonal selection principle [27].
the immune system. Clonal selection mechanism is used to explain how an immune response is given when a non-self antigenic pattern is recognized by a B-cell. The principle is shown in Fig. 1. Both B-cells and T-cells have receptor molecules on their surfaces. The B-cells are called as antibody and recognize disease causing pathogens. When antigens and receptor molecules have complementary shapes, they can bind together [46]. When an antigen is recognized by immune cell receptors, an immune response is obtained through the production of antibody from the bone marrow. After these antibodies combine with antigens, the antigen stimulates the B-cell to proliferate and mature into terminal (non-dividing) antibody secreting cells, called plasma cells. The proliferation in the immune system is achieved by cell divisions (mitosis). The proliferation (mitosis) generates a clone of cells that are the children of a single cell. De Castro and Von Zuben [27] developed the clonal selection algorithm (CSA) based on clonal selection to solve engineering problems. In their paper, they presented the overview of this mechanism from the points of view of immunology and engineering problems. Affinity maturation is the all mutation processes and selection that guarantees the survival of the variant offspring that better recognizes the antigen. The affinity defines the degree of binding of the cell receptor with the antigen. The higher the affinity the stronger the binding and thus the better the immune recognition and response. The two main mechanisms of affinity maturation are hypermutation and receptor editing [27,46]. Random changes (mutations) happen in the variable region genes of antibody molecules and cause structurally different cells. Rarely, one such change will lead to an increase in the affinity of the antibody. Since the lymphocytes of the immune system are somatic cells, the mutation that occurs during affinity maturation is named as somatic mutation. Somatic mutation helps the immune system to keep the high affinity cells and to ensure large mutations for the low affinity ones in order to get better affinity cells. The last stage of artificial immune system is receptor editing. After cloning and mutation processes are completed, a percentage of antibodies in the antibody population are replaced with antibodies created randomly. This process is named as receptor editing. Receptor editing mechanism introduces diversity and
helps to escape from local optima on an affinity landscape and leads to possible new candidates [47]. In Fig. 2, a simplified landscape is illustrated. In the figure, the x-axis show all possible antigen-binding sites, with the most similar adjacent to each other. The y-axis shows the antigen–antibody affinity. When it is taken a particular antibody (A) selected during the primary immune response, point mutations permit the immune system to explore local regions around A by making small steps towards an antibody with higher affinity. As a result, the antibodies might become stuck at a local optimum (A0 ) [27,47]. Receptor editing allows an antibody to take large steps through the landscape. Although this will land the antibody in a local optimum (B), occasionally the leap will lead to an antibody on the side of a higher hill (C), where the climbing region have more chance to reach to the global optimum. From this point (C), the antibodies can reach to the top of the hill (C0 ). In conclusion, receptor editing may rescue immune responses stuck on unsatisfactory local optima. In receptor editing mechanism, a percentage of the antibodies in the antibody population are replaced with randomly created antibodies. This mechanism allows finding new antibodies that correspond to new search regions in the all search space. Exploring new search regions may help the algorithm to escape from local optimal. 3.2. Taguchi method Taguchi method is a method that chooses the most suitable combination of the levels of controllable factors by using S/N table and orthogonal arrays against the factors that form the variation and are uncontrollable in product and process. Hence, it tries to reduce the variation in product and process to least. Taguchi uses statistical performance measure which is known as S/N ratio that takes both medium and variation into consideration. Most of the shape design optimization problems in automotive industry are usually multi-objective, often conflicting, and they have uncontrollable variations in their design parameters with complex nature. There is a need to overcome the shortcomings due to the traditional optimization methods and also to further improve the strength of recent approaches to achieve better results for the real world design problems. Therefore, in this research, a hybrid approach is developed to solve shape design optimization problems. The architecture of the proposed approach is given in Fig. 3. Taguchi’s robust parameter design is introduced to help to define robust initial population levels of design parameters and to reduce the effects of noise factors to achieve better initialize immune algorithm search. The problem with larger population may tend immune algorithm to converge and stick around certain
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Fig. 3. Hybrid immune algorithm based multi-objective design optimization approach.
solutions which may not be the best ones. This is handled with the help of robust parameter levels which are embedded into immune algorithm process as being initial population intervals. In other words, the design space is restricted and refined based on the effect of the various design variables on objective functions. The purpose of the ANOVA tables is to help differentiate the robust designs from the non-robust ones. Finally, optimum results of multi-objective problem are obtained by applying immune algorithm through cloning, mutation and receptor editing operations. The present approach is considered in two stages as follows: Stage 1 Determine efficient solution space for shape optimization variables using Taguchi’s method. Stage 2 Apply immune algorithm to find shape design variables. In the first stage, Taguchi’s robust parameter design procedure is used to find the levels of variables for efficient search space as follows:
identify the objectives, constraints and design parameters, determine the settings of the design parameter levels, conduct the experiments using orthogonal array, compute S/N ratios and ANOVA analysis, find the optimal settings of design parameters.
The main issue of experimental analysis is the ANOVA analysis which is formed using S/N ratios respectively for each of the objectives. According to results of ANOVA, appropriate levels of design parameters are found and then, initial search population of immune algorithm process is defined according to levels. Finally, optimum results of the optimization problem are obtained by applying immune algorithm process through cloning, mutation and receptor editing operations in two steps as follows:
define initial antibody set, use immune operators to create the next generation, evaluate objective function and constraints, repeat the loop until the optimum solutions are found.
immune algorithm is resulted in a solution, which leads to better parameter values for shape design optimization problems. The algorithm of proposed hybrid approach can be outlined as follows: BEGIN Step 1: Taguchi’s method Begin 1.a Choose convenient orthogonal array from Taguchi’s standard orthogonal arrays 1.b Define levels and intervals 1.c For i: = 1 to NOE (number of experiments) do begin Compute objective function values end; 1.d Choose convenient S/N ratio type (smaller the best or larger the best or nominal the best) based on minimization or maximization of objective functions 1.e For i: = 1 to NOE do begin Compute S/N ratios end; 1.f Constitute Anova table for objective functions using S/N ratios 1.g Determine optimum levels and intervals using percentage contribution to performance using Anova table Use these levels and intervals for forming initial population end; ‘Generate optimal solution set using immune algorithm and computed robust initial population space’ Begin Input Use Initial population found in previous part of the program as input to immune algorithm Step 2: Artificial Immune Algorithm 2.a Initial antibody population
In this paper, a new hybrid approach is proposed to improve the performance of the immune algorithm. Our argument behind the proposed approach is that the strength of one algorithm can be used to improve the performance of another approach in the optimization process. The combination of Taguchi’s method and
For i: = 1 to the number of antibodies do begin Generate randomly initial antibody population End; While (not termination condition) do begin
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Table 1 Statistical results of different methods for test problem. Design variables
Best
Mean
Worst
Std dev
Function evaluations
HTIA Coello and Cortes [25] Hamida and Schoenauer [48] Koziel and Michalewicz [49]
6961.814 6961.7608 6961.81 6952.100
6961.279 6961.273 6961.81 6342.6
6960.257 6960.607 NA 5473.9
0.116 0.3598 NA NA
100,000 150,000 1,500,000 1,400,000
Calculate affinity values of the antibodies 2.b Cloning Generate copies of antibodies using affinity values of the antibodies 2.c Mutation For i: = 1 to the number of the antibodies do begin Apply mutation to the antibodies End; 2.d Receptor editing For i: = 1 to (0.25* number of the antibodies) do begin Apply receptor editing to randomly selected antibodies end; end; END. 4. Evaluation of the proposed approach using test problem In order to evaluate the performance of the proposed hybrid approach, a single objective benchmark problem commonly used in the optimization literature is successfully solved by the proposed algorithm. The results of the benchmark problem are compared with those of other methods that are representative of the state-of-the-art in the optimization literature. After it is shown that the proposed approach is successful to optimize the optimization problems, it is applied to a case study from automobile industry. The test problem is a minimization problem for a single objective function with two variables and two inequality constraints. This problem is employed by several authors to evaluate the performances of their approaches [25,48,49]. It is defined as follows: Minimize : Subject to :
f ðxÞ ¼ ðx1 10Þ3 þ ðx2 20Þ3 g 1 ðxÞ ¼ ðx1 5Þ2 þ ðx2 5Þ2 100 0 g 2 ðxÞ ¼ ðx1 6Þ2 ðx2 5Þ2 þ 82:81 0
where 13 x1 100 and 0 x2 100. The global optimum is located at x* = (14.095, 0.84296), where f(x*) = 6961.81388. Both constraints are active at the optimum. The parameters used by the proposed hybrid approach for optimization process are the following:
The results of the proposed hybrid approach for test problem are compared against those provided by Coello and Cortes [25]. HTIA gives better solutions for this problem compared with those given in Table 1 as far as the number of function evaluations, the best solution computed, and the statistical analysis results are taken into account together. The statistical values of HTIA show the robustness of the solutions with the standard deviation of 0.116. The standard deviation is very low, which indicates that HTIA is among the most robust approach in finding an optimum solution. The HTIA requires only 100,000 function evaluations to find the best-known solution of 6961.814. Coello and Cortes [25] found the best solution 6961.7608 for 150,000 function evaluations with standard deviation of 0.3598. The use of the HTIA improves the convergence rate by computing the best value 6961.814 with respect to very low standard deviation 0.116 and maintaining the less function evaluations 100,000. 5. Shape optimization using improved hybrid immune algorithm The hybrid approach proposed in the Section 3 is applied to solve the shape design optimization problem in this section. The second example is taken from automotive industry for the optimal design of a vehicle component. The objective functions are due to the volume and the frequency of the part which is to be designed for minimum volume and avoiding critical frequency subject to strength constraints. In this research, then shape optimization is performed using present approach. In the first stage, the experiments are designed to evaluate the effects of four design variables related to objective functions. The four shape design variables x1, x2, x3 and x4 are selected as shown in Fig. 4. The feasible range of design variables without shape distortions is considered as 6 < x1 < 30, 21 < x2 < 27, 8 < x3 < 14 and 28 < x4 < 46. Matrix experiments are designed using L16 orthogonal arrays and S/N ratios are conducted for each objective as given in Table 2. Smaller the better and larger the better characteristics are applied to compute S/N ratios based on each objective as smaller the better for volume and frequency. The details about orthogonal array, S/N ratios, ANOVA analysis, and how they are computed and used for experimental evaluations are given in the reference of Phadke [50].
(a) antibody size: 100; (b) maximum number of generations: 1000; (c) number of objective function evaluations: 100,000. The efficiency of each approach can be measured by comparing the number of function evaluations, which is equal to the antibody size (population size) and multiplied by the number of generations as each solution is evaluated once in every generation. In the evolutionary optimization approaches, the aim is to find the optimum results with minimum number of function evaluations. In this paper, the HTIA is run 30 times independently.
Fig. 4. Design variables.
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Table 2 Experimental results and S/N ratios for volume and frequency. Ex no
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
F1 (volume)
F2 (frequency)
S/N1 volume
S/N2 frequency
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6 6 6 6 14 14 14 14 22 22 22 22 30 30 30 30
21 23 25 27 21 23 25 27 21 23 25 27 21 23 25 27
8 10 12 14 10 8 12 14 12 14 8 10 14 12 10 8
28 34 40 46 40 46 28 34 46 40 34 28 34 28 46 40
23,712 22,432 21,032 19,512 21,062 20,282 21,452 20,052 19,042 18,962 20,242 20,402 18,912 19,572 17,352 18,012
8.1066 7.7716 7.5002 7.2879 8.4748 8.2572 7.8605 7.5219 9.1969 8.7593 8.4097 8.1266 10.082 9.7315 9.6848 9.1402
87.4994 87.0174 86.4576 85.8060 86.4700 86.1422 86.6294 86.0432 85.5943 85.5577 86.1251 86.1935 85.5347 85.8327 84.7870 85.1112
18.1768 17.8102 17.5015 17.2520 18.5626 18.3367 17.9090 17.5266 19.2728 18.8494 18.4956 18.1982 20.0709 19.7636 19.7218 19.2191
Table 3 Results of the analysis of variance for volume.
X1 X2 X3 X4 Error
Level 1
Level 2
Level 3
Level 4
S
86.70 86.27 86.22 86.54
86.32 86.14 86.12 86.18
85.87 86.00 86.13 85.90
85.32 85.79 85.74 85.58
4.244 0.440 0.291 1.815 0.007
3 3 3 3 3
6.797
15
Total
DOF
M
F
Cont. (%)
1.4148 0.1469 0.0970 0.6051 0.0026
539.9 56.09 37.03 230.9
62.43 6.473 4.281 26.70
Table 4 Results of the analysis of variance for frequency.
X1 X2 X3 X4 Error
Level 1
Level 2
Level 3
Level 4
17.69 19.02 18.56 18.51
18.08 18.69 18.57 18.48
18.70 18.41 18.61 18.53
19.69 18.05 18.42 18.65
Total
S
DOF 9.189 1.973 0.009 0.064 0.006
11.24
ANOVA for each objective is formed using S/N ratios for the first and second objective functions as shown in Tables 3 and 4. The most effective parameter is x1 with 62.4% contribution for the first objective (see Table 3). Level 4 is considered for x1 since the smaller the better characteristic is employed for the first objective. The most effective parameter is x1 with 81.7% contribution for the second objective (see Table 4). Level 4 is considered for x1 since the larger the better characteristic is employed for the second objective. Therefore, level 4 can be selected for x1 as 30 mm. However, the levels for x2, x3, and x4 cannot be defined since contributions are weak and there are some conflicts in selecting them. For example, x2 has 6% contribution for the first objective (level 4) and 17.5% contribution for the second objective (level 1); thus, x2 must be taken between levels 1 and 4 as 21 < x2 < 27. In case of other two variables, ANOVA results with contributions less than 1 as 0.08 and 0.57, this cannot be used to select the
3 3 3 3 3
M
F
Cont. (%)
3.06300 0.65799 0.00324 0.02160 0.00203
1512.49 324.91 1.60 10.67
81.745 17.55 0.008 0.569
15
levels. Thus, ANOVA of stress constraint is considered to define the levels of x3 and x4 (Table 4). Then, the variables are considered as x1 = 30, 21 < x2 < 27, 8 < x3 < 14, and 28 < x4 < 46. The parameter levels are taken as x1 = 30, 21 < x2 < 27, 8 < x3 < 14, and 28 < x4 < 46, which are obtained from the ANOVA analysis. Immune algorithm begins its search with a set of solutions, with its population range defined by Taguchi’s method. The results of HTIA are given in Table 5. It can be seen that a volume of 16,416 mm3 and frequency of 9.36 Hz with 296 MPa is obtained. Then, the problem is solved using immune algorithm approach. Immune algorithm begins its search with a set of solutions, with its population range defined by Taguchi’s method. Each antibody has an affinity value suitable with the structure of the problem. Cloning, mutation and receptor editing process are applied to the whole antibody population through the immune algorithm process. Then, HTIA works by repeatedly modifying a population
Table 5 Comparison of the optimization results for the side door bracket design.
Initial design CAD optimum design Immune algorithm HTIA
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
Volume (cm3)
Frequency (Hz)
Stress (MPa)
18 22 27.5 30
24 27 26.6 26.85
11 13.1 12.7 13.52
32 36.7 43.1 46.2
20,727.2 18,763.9 17,288.92 16,416.41
8.3 8.1 8.9 9.36
256 277 290 296
A.R. Yıldız / Computers in Industry 60 (2009) 613–620
Fig. 5. The optimal structural layout.
of artificial structures through the application of immune operators to compute optimal set of design variables. It is clearly seen that shape design optimization performance is improved compared to traditional CAD and immune algorithm solutions. ANSYS is used for CAD optimization process. The intervals of design variables are considered as 6 < x1 < 30, 21 < x2 < 27, 8 < x3 < 14, and 28 < x4 < 46 for multi-objective immune algorithm solution. The structural layout results of the present hybrid approach using immune algorithm and robust design based on Taguchi’s method for the vehicle part are given in Fig. 5. The results of present hybrid approach for the optimal design of the vehicle part are given in Table 5. It is seen that shape design optimization performance is improved compared to other approaches. 6. Conclusions This research describes a new optimization approach based on immune algorithm and Taguchi’s robust design approach for solving shape design optimization problems. Taguchi’s robust design approach is introduced to help to define robust initial population levels of design parameters to achieve better initialize immune algorithm search. The design solution space of immune algorithm is refined based on the effect of the various design variables on objective functions. The validity and efficiency of HTIA are evaluated and illustrated with a test problem taken from literature. Then, it is applied to a vehicle component taken from automotive industry. It is seen that better results can be achieved with present hybrid optimization approach. Therefore, HTIA is a suitable optimization technique for the solution of shape design optimization problems. Since HTIA is a generalized optimization method, it can also be applied to other optimization problems. Further research can be carried out to improve the HTIA by integrating the key issues from the fields of multi-objective optimization, evolutionary algorithms and local search techniques. Now, as further research, the author consider hybridizing of HTIA by local search techniques like hill-climbing to extend its improvements for solving multi-objective design optimization problems. References [1] A.R. Yildiz, K. Saitou, Topology Synthesis of Multi-Component Structural Assembly in Continuum Domain DETC 2008, New York, USA, 3–5 August, 2008. [2] N. Lyu, K. Saitou, Topology optimization of multicomponent structures via decomposition-based assembly synthesis, Transactions of ASME Journal of Mechanical Design 127 (2005) 170–183. [3] P.C. Fourie, A.A. Groenwold, The particle swarm optimization algorithm in size and shape optimization, Structural and Multidisciplinary Optimization 23 (2002) 259–267. [4] N. Kaya, Machining fixture locating and clamping position optimization using genetic algorithms, Computers in Industry 57 (2006) 112–120.
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Ali Rıza Yildiz is a Lecturer at Uludag University. He received his PhD in Mechanical Engineering from Uludag University, Turkey in 2006. He worked on ‘‘Topology Synthesis of Multi-Component Structural Assembly in Continuum Domain’’ as a Post Doctoral Researcher between 2006 and 2008 at the Discrete Design Optimization Laboratory, University of Michigan, Ann Arbor, USA. His major research interests includes multi-objective shape optimization, topology optimization and hybrid optimization algorithms. He has published more than 25 journal articles and conference papers. Dr. Yildiz served on the technical program committee of international conferences including, ASME 34th Design Automation Conference (IDETC 2008), World Congress on Engineering and Computer Science (WCECS 2008), ASME 33rd Design Automation Conference (IDETC 2007) and more. He is an Editorial Board Member of several journals, including ‘‘Expert Systems: The Journal of Knowledge Engineering’’ and ‘‘International Journal of Mathematical Modelling and Numerical Optimisation’’ and is a reviewer for a variety of additional journals.