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A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing
Applied Soft Computing 13 (2013) 2906–2912
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A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing Ali R. Yildiz ∗ Bursa Technical University, Department of Mechanical Engineering, Bursa, Turkey
a r t i c l e
i n f o
Article history: Received 15 February 2012 Received in revised form 8 April 2012 Accepted 12 April 2012 Available online 24 April 2012 Keywords: Artificial bee colony Manufacturing Milling Structural optimization Hybrid approach
a b s t r a c t The purpose of this paper is to develop a novel hybrid optimization method (HRABC) based on artificial bee colony algorithm and Taguchi method. The proposed approach is applied to a structural design optimization of a vehicle component and a multi-tool milling optimization problem. A comparison of state-of-the-art optimization techniques for the design and manufacturing optimization problems is presented. The results have demonstrated the superiority of the HRABC over the other techniques like differential evolution algorithm, harmony search algorithm, particle swarm optimization algorithm, artificial immune algorithm, ant colony algorithm, hybrid robust genetic algorithm, scatter search algorithm, genetic algorithm in terms of convergence speed and efficiency by measuring the number of function evaluations required. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Designing and manufacturing new products possessing desired property are important in industry. With the advent of ever faster computing platforms, the computer aided design and optimization tools are becoming more attractive due to its great contribution to cost, material and time savings in the procedures of the engineering design. The application of these tools allows a more rapid design process and more detailed design studies. Over the past decades, a number of optimization algorithms have been used extensively in structural and manufacturing optimization tasks. The early works on the topics mostly use various mathematical techniques. These methods may not be used efficiently in finding global optimum solutions. As an alternative to traditional techniques, population-based optimization approaches, such as, genetic algorithm, particle swarm optimization algorithm, artificial immune algorithm, cuckoo search algorithm and artificial bee colony algorithm have been developed by mimicking natural phenomena and widely applied in various fields of science [1–12]. Artificial bee colony algorithm (ABC) is one of the most recently introduced swarm-based algorithms based on the intelligent foraging behaviour of honey bee swarm [13]. The ABC has been found to be successful in a wide variety of optimization tasks [13–15].
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On the other hand, researchers are paying more and more interest on hybrid algorithms to solve optimization problems. The hybrid algorithms have shown outstanding reliability and efficiency in application to the engineering optimization problems [16–20]. The main goal of the present research is to develop a robust optimization approach based on artificial bee colony algorithm and Taguchi method to solve design and manufacturing optimization problems. In the new hybrid approach, S/N values are calculated and ANOVA (analysis of variance) table for the objective function is formed using S/N ratios respectively. According to results of ANOVA table, appropriate interval levels of design variables are found and then, initial population of artificial bee colony algorithm is defined according to these interval levels. Then optimum results of the problem are obtained using artificial bee colony algorithm. Since the ABC has been found to be successful in a wide variety of optimization tasks, it is used in this paper. The developed new hybrid optimization approach entitled hybrid robust artificial bee colony algorithm (HRABC) is applied to optimum design of a vehicle part taken from automotive industry and to optimization of the machining parameters in multi-tool milling operations. The results of the HRABC for each case study show that the proposed optimization method converges rapidly to the global optimum solution and provides reliable and accurate solutions. The remaining contents of the paper are organized as follows. Literature review is given in Section 2. The standard ABC and Taguchi method are presented in Section 3. In Section 4, the proposed approach is used for optimization of a vehicle component.
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The results are discussed in Section 5. An application of the HRABC to optimization of the machining parameters in multi-tool milling operations is given in the Appendix A. 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 [1–23]. For instance, a novel approach for multi-component topology optimization of continuum structures using a multi-objective genetic algorithm is developed by Yildiz and Saitou [2]. The developed approach is applied to multi-component topology optimization of a vehicle floor frame. Artificial bee colony (ABC) algorithm originally developed by Karaboga [13] is inspired with social behavior, such as bird flocking, fish schooling, which is used successfully in the solution of optimization problems. The ABC algorithm has been used in many areas of optimization studies. The use of the ABC algorithms in the optimum solution of problems resulted better solutions compared to classical methods [13–15,25–32]. The robustness issues have been used to solve optimization problems by researchers [33–37]. Robinson et al. [38] presents a review paper which focuses largely on the work done since 1992 and a historical perspective of parameter design is also given. Kunjur and Krishnamurty [39] presented a robust optimization approach that integrates 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 [36,40]. Hybrid methods are also used to enhance the performance of evolutionary algorithm. For instance, the artificial immune algorithm is hybridized with hill climbing local search algorithm by Yildiz [16] and applied to multi-objective disc brake and manufacturing optimization problems from literature. In another paper [18], the particle swarm optimization approach is hybridized with receptor editing property of immune system. The proposed approach is used to solve optimization problems in design and manufacturing areas. Tsai et al. [41] 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. It is known that the ABC algorithm is an efficient approach at exploring the solution space, but it does not guarantee the global optimum as other evolutionary methods. The introduction of hybrid methods comes from the need to tackle more and more complex real-world problems. Some of the hybrid approaches in literature have been made on hybrid ABC [28,42,43]. 3. Hybrid bee colony optimization algorithm for structural optimization Structural design optimization has always been a very interesting and creative segment in a large variety of engineering designs. Structures, of course, should be designed such that they can resist applied forces (stress constraints), and do not exceed certain deformations (displacement constraints). Moreover, structures should
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be economical. Theoretically, the best design is the one that satisfies the stress and displacement constraints, and results in the least cost of construction. Although there are many factors that may affect the design cost, the first and most important one is the amount of material used to design the structures. Therefore, minimizing the weight of the structure is usually the goal of structural optimization. In this paper, a new hybrid optimization approach (HRABC) is developed to solve structural design and manufacturing optimization problems. In the proposed optimization 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 population, then they apply throughout bee colony algorithm for finding optimal solutions. 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 bee colony optimization algorithm and Taguchi’s method are given and, finally, the proposed hybrid approach is explained.
3.1. Bee colony algorithm Artificial bee colony algorithm (ABC) developed by Karaboga [13] and further developed by Karaboga and Basturk [14,24–27] is a nature inspired algorithm based on the intelligent foraging behavior of honey bee swarm. The ABC algorithm describes the foraging behavior, learning, memorizing and information sharing characteristics of honeybees. A basic model of foraging behavior of honeybee swarms consists of two essential components and define two leading modes of the behavior. The colony of artificial bees consists of three groups of bees: employed bees, onlookers and scouts. The colony of the artificial bees is divided into two groups, first half of the colony consists of the employed artificial bees and the second half includes the onlooker bees. Scout bees are the employed bees whose food source has been abandoned. In ABC algorithm, the position of a food source represents a possible solution to the optimization problem (value of design variables) and the nectar amount of a food source corresponds to the quality of the associated solution (fitness value). At the first step, the ABC generates a randomly distributed initial population Pinitial of N solutions, where N denotes the size of population. Each solution xi (i = 1, 3, . . ., N) is a S-dimensional vector where S is the number of optimization parameters (design variables). After initialization, the population of the solutions is subjected to repeated cycles, C = 1, 2, . . ., G, of the search processes of the employed bees, the onlooker bees and scout bees. An employed bee generates a modification in the solution in her memory depending on the local information. If the objective function value (fitness) of the new solution is better than that of the previous one, the bee memorizes the new position and forgets the old one. Otherwise, she keeps the position of the previous one in her memory. After all employed bees complete the search process; they share the nectar information of the food sources and their position information with the onlooker bees on the dance area. An onlooker bee evaluates the fitness information taken from all employed bees and chooses a food source with a probability related to its fitness value. An onlooker bee also produces a new solution and it memorizes the new position if its fitness value is better than the previous position. An artificial onlooker bee chooses a food source depending on the probability value associated with that food source, Pi , calculated by the following expression: Pi =
Fi
Nb
F n=1 n
(1)
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where Fi is the fitness value of the solution i which is proportional to the nectar amount of the food source in the position i and Nb is the number of food sources which is equal to the number of employed bees. In order to produce a candidate food position from the old one in memory, the ABC uses the following expression:
vij = xij + rij (xij − xkj )
(2)
where k ∈ {1, 2, . . ., N} and j ∈ {1, 2, . . ., D} are randomly chosen indexes. Although k is determined randomly, it has to be different from i . rij is a random number between (−1, 1). It controls the production of neighbour food sources around xij and represents the comparison of two food positions visually by a bee. As can be seen from (2), as the difference between the parameters of the xij and xkj decreases, the perturbation on the position xij gets decreased too. Thus, as the search approaches to the optimum solution in the search space, the step length is adaptively reduced. If the position of the food source cannot be improved for some predetermined number of cycles then that food source is abandoned. The abandoned food source is replaced with a new food source by the scouts. In ABC, this is simulated by producing a position randomly and replacing it with the abandoned one. The value of predetermined number of cycles is an important control parameter of the ABC algorithm, which is called ‘limit’ for abandonment. The value of limit is generally considered as N × S. Assume that the abandoned source is xi and j ∈ {1, 2, . . ., D}, then the scout discovers a new food source to be replaced with xi . This operation can be expressed as, j
j
j
j
xi = xmin + rand(0, 1)(xmax − xmin )
(3)
So from the above explanation it is clear that the control parameters used in ABC are number of food sources which is equal to number of employed bees, number of onlooker and scout bee, the value of ‘limit’ and the maximum cycle number. The effect of these control parameters on the convergence and fitness value of the objective function is discussed in the later section of the present work. Implementation steps of the ABC are summarized below: Step 1: Initialize swarm with randomly generated N food sources and evaluate them. Step 2: Find a candidate food source for each employed bee according to Eq. (2) and evaluate the candidate food source and select the better one as the new food source. Step 3: Calculate the probability values of food sources based on their fitness values. Step 4: Select a food source for each onlooker bee according to Eq. (1) by roulette wheel selection and generates a candidate solution according to Eq. (2). Step 5: Select the better one as the new food source and select a better one as the new food source. Step 6: Memorize the position of the best food source found so far. Step 7: Find the abandoned food sources and produce new positions for exhausted food sources according to Eq. (3). Step 8: Repeat the procedure from step 2 till the termination criterion is met. At termination, position of the food source and its nectar amount are the optimum values of the design variables and objective function for the considered problem for single run of the algorithm.
3.2. Taguchi method The Taguchi’s method is a universal approach, which is widely used in robust design [44]. There are three stages to achieve Taguchi’s objective: (1) concept design, (2) robust parameter design, and (3) tolerance design. The robust parameter design is used to determine the levels of factors and to minimize the sensitivity of noise. That is, a parameter setting should be determined with the intention that the product response has minimum variation while its mean is close to the desired target. Taguchi’s method is based on statistical and sensitivity analysis for determining the optimal setting of parameters to achieve robust performance. The responses at each setting of parameters were treated as a measure that would be indicative of not only the mean of some quality characteristic, but also the variance of that characteristic. The mean and the variance would be combined into a single performance measure known as the signal-to-noise (S/N) ratio. Taguchi classifies robust parameter design problems into different categories depending on the goal of the problem and for each category as follows: Smaller the better For these kind of problems, the target value of y, that is, quality variable, is zero. In this situation, S/N ratio is defined as follows:
S/N ratio = −10 log
yi2
(4)
n
Larger the better In this situation, the target value of y, that is, quality variable, is infinite and S/N ratio is defined as follows:
S/N ratio = −10 log
1/yi2
n
(5)
Nominal the best For these kind of problems, the certain target value is given for y value. In this situation S/N ratio is defined as follows:
S/N ratio = −10 log
y2
s2
(6)
Taguchi’s method uses an orthogonal array and analysis of mean to analyse the effects of parameters based on statistical analysis of experiments. To compare performances of parameters, the statistical test known as the analysis of variance (ANOVA) is used. Further details and technical merits about robust parameter design can be found in the [44]. Finally, optimum results are obtained by applying artificial bee colony algorithm. Stage 1: Determine efficient solution space for design variables using Taguchi’s method Stage 2: Apply the artificial bee colony algorithm to find the optimum solutions. 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 objective, 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 for the objective function. According to results of ANOVA, appropriate levels of design parameters
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Fig. 1. Design variables.
are found and then, initial swarm is defined according to the levels. Finally, optimum results of the optimization problem are obtained by applying the artificial bee colony algorithm process in two steps as follows • • • •
define initial population set, use artificial bee colony operators to create the next generation, evaluate objective function and constraints, repeat the loop until the optimum solutions are found.
In this paper, a new hybrid approach is proposed to improve the performance of the artificial bee colony algorithm. The hybridization of the artificial bee colony algorithm with Taguchi method is resulted in a solution, which leads to better parameter values for optimal structural design problems. The algorithm of proposed hybrid approach can be outlined as follows Program: HRABC BEGIN Begin Set the input parameters Select suitable orthogonal array Select suitable S/N ratio type While (not termination condition for experiments) do Compute S/N ratios Compute objective function values Conduct matrix of experiments End While (not termination condition for parameters) do Compute contributions Generate analysis of variance (ANOVA table) End Define RPD intervals for parameters Begin Generate randomly initial population of solutions Repeat Place the employed bees on their food sources Calculate the probability values Place the onlooker bees on the food sources Send the scouts to the search area for discovering new food sources Memorize the best food source found so far Until a termination is satisfied, and output the best food source found so far End END.
4. Structural design optimization using improved hybrid bee colony algorithm The hybrid approach proposed in Section 3 is applied to solve a structural design optimization problem taken from automotive industry for the optimal design of a vehicle component in this
section. The objective function is the volume of the part which is to be designed for minimum volume subject to strength constraints. The objective function value of the component is provided by ANSYS [45] during optimization loop. In this research, structural optimization is performed using the proposed approach. In the first stage, the experiments are designed to evaluate the effects of four design variables related to the objective function. The four design variables x1 , x2 , x3 and x4 are selected as shown in Fig. 1. 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 the objective as given in Table 1. Smaller the better characteristic is applied to compute S/N ratios based on objective as smaller the better for the volume. 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 [44]. The ANOVA table for the objective function is formed using S/N ratios as shown in Table 2. The most effective parameter is x1 with 62.4% contribution for the objective (see Table 2). Level 4 is considered for x1 since the smaller the better characteristic is employed for the first 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. 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. The results of HRABC are given in Table 3. It can be seen that a volume of 16008.1 mm3 with 298 MPa is obtained. It is clearly seen that structural design optimization performance is improved compared to traditional CAD, genetic algorithm, artificial immune algorithm, harmony search algorithm, particle swarm optimization algorithm, differential evolution algorithm and artificial bee colony algorithm. 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 the ABC solution. The structural layout results of the present hybrid approach using the ABC and robust design based on Taguchi’s method for the vehicle part is given in Fig. 2. The results of the HRABC for the optimal design of the vehicle part are given in Table 3. It is seen that structural design optimization performance is improved compared to other approaches. Using the HRABC, the vehicle component is designed with minimum volume (weight) without exceeding allowable stress. The HRABC provides the lightest design with minimum function
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Table 1 Experimental results and S/N ratios for volume. Exp. No.
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
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
F (volume)
S/N (volume) −87.49 −87.01 −86.45 −85.80 −86.47 −86.14 −86.62 −86.04 −85.59 −85.55 −86.12 −86.19 −85.53 −85.83 −84.78 −85.11
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
Table 2 Results of the analysis of variance for volume.
X1 X2 X3 X4
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 6.797
Error Total
DOF
M
F
Cont. (%)
3 3 3 3
1.4148 0.1469 0.0970 0.6051
539.9 56.09 37.03 230.9
62.43 6.473 4.281 26.70
3 15
0.0026
Table 3 Comparison of the optimization results for the side door bracket design.
Initial design Ansys Genetic algorithm (GA) Artificial immune algorithm (AIA) Harmony search algorithm (HS) Particle swarm optimization algorithm (PSO) Differential evolution algorithm (DE) Artificial bee colony algorithm (ABC) Hybrid robust artificial bee colony algorithm (HRABC)
Volume (cm3 )
Stress (MPa)
Function Evaluations
20727.2 18763.9 17663.7 17514.1 17495.9 17455.6 17312.8 17288.9 16008.1
256 277 293 294 296 298 293 290 298
– 30000 5000 4600 4250 3800 3100 2500 1000
Table 4 Comparison of the results for multi-tool milling operation. Method
Cu – Unit cost
Tu – Unit time
Pr – Profit rate
Handbook [46] Method of feasible direction [47] Genetic algorithm [19] Ant colony algortihm [48] Hybrid particle swarm (PSRE) [20] Immune algorithm Hybrid immune algorithm [19] Proposed approach (HRABC)
$18.36 $11.35 $11.11 $10.20 $10.90 $11.08 $10.91 $10.90
9.40 min 5.48 min 5.22 min 5.43 min 5.052 min 5.07 min 5.04 min 5.00 min
0.71/min 2.49/min 2.65/min 2.72/min 2.79/min 2.75/min 2.79/min 2.82/min
evaluation number as genetic algorithm, artificial immune algorithm, harmony search algorithm, particle swarm optimization algorithm, differential evolution algorithm and artificial bee colony algorithm.
5. Conclusions
Fig. 2. The optimal structural layout.
This paper describes a new optimization approach (HRABC) based on artificial bee colony algorithm and Taguchi method for solving structural design and manufacturing optimization problems. The Taguchi method is used to define robust initial population to achieve better initialize the artificial bee colony algorithm. The solution space of the artificial bee colony algorithm is refined based
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on the effect of the various design variables on the objective function. The HRABC is applied to the optimization of a vehicle component taken from automotive industry and the multi-tool milling problem. Results obtained from the HRABC for structural design and multi-tool milling problems have been compared with those obtained differential evolution algorithm, harmony search algorithm, particle swarm optimization algorithm, artificial immune algorithm, hybrid robust genetic algorithm, scatter search algorithm, ant colony algorithm and genetic algorithm. The results and comparisons in Tables 2 and 3 demonstrate the effectiveness and robustness of the HRABC in solving optimization problems in the different area of the engineering problems. The HRABC can not only improve the solution quality but also reduce the computational effort. The studies clearly indicate that the HRABC outperforms the state-of-the-art evolutionary algorithms for the problems solved in this article. The future work is to apply the HRABC for solving other engineering optimization problems. Appendix A. Case study of computational machining optimization Since the achievement of a machining operation is affected by depth of cut, feed rate and cutting speed, only these parameters are considered in practice. The proposed approach is applied to the mathematical model of Tolouei-Rad and Bidhendi [46] to show effectiveness of the new method in solving real-world optimization problems in the manufacturing industry. In the optimization of machine parameters for milling operations, the purpose is to maximize the total profit rate. The maximization of the total profit rate is carried out according to the two objective functions, which are unit production time and unit production cost. The unit cost is the sum of material cost, set up cost, machining cost and tool changing cost. In order to maximize the profit rate, allowable range of cutting speed and feed rate are imposed restriction by constraints. The constraints taken into consideration in this paper are maximum machine power, surface finish requirement and maximum cutting force permitted by the rigidity of the tool.The comparison of the results obtained by the proposed approach, against other techniques such as immune algorithm, GA, the feasible direction method and handbook recommendations, is given in Table 4. Function evaluation numbers are 20,000 and 15,000 to find optimal solutions for GA and immune algorithm, respectively. The HRABC also improves the convergence rate by computing the best value and maintaining the less function evaluations 3500. It can be seen that better results for the best computed solutions are achieved for the milling optimization problem compared to immune algorithm, genetic algorithm and feasible direction method. References [1] A.R. Yildiz, A new design optimization framework based on immune algorithm and Taguchi method, Computers in Industry 60 (8) (2009) 613–620. [2] A.R. Yildiz, K. Saitou, Topology synthesis of multi-component structural assemblies in continuum domains, Journal of Mechanical Design 133 (2011) 011008-1–011008-9. [3] S. Bureerat, J. Limtragool, Performance enhancement of evolutionary search for structural topology optimisation, Finite Elements in Analysis and Design 42 (6) (2006) 547–566. [4] K.Y. Chan, Modeling of a liquid epoxy molding process using a particle swarm optimization-based fuzzy regression approach, IEEE Transactions on Industrial Informatics 7 (1) (2011) 148–158. [5] I. Durgun, A.R. Yildiz, Structural design optimization of vehicle components using Cuckoo search algorithm, Materials Testing 54 (3) (2012) 185–188. [6] K.Y. Chan, C.K. Kwong, H. Jiang, M.E. Aydin, T.C. Fogarty, A new orthogonal array based crossover, with analysis of gene interactions, for evolutionary algorithms and its application to car door design, Expert Systems with Applications 37 (5) (2010) 3853–3862.
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Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing Ali R. Yildiz ∗ Bursa Technical University, Department of Mechanical Engineering, Bursa, Turkey
a r t i c l e
i n f o
Article history: Received 15 February 2012 Received in revised form 8 April 2012 Accepted 12 April 2012 Available online 24 April 2012 Keywords: Artificial bee colony Manufacturing Milling Structural optimization Hybrid approach
a b s t r a c t The purpose of this paper is to develop a novel hybrid optimization method (HRABC) based on artificial bee colony algorithm and Taguchi method. The proposed approach is applied to a structural design optimization of a vehicle component and a multi-tool milling optimization problem. A comparison of state-of-the-art optimization techniques for the design and manufacturing optimization problems is presented. The results have demonstrated the superiority of the HRABC over the other techniques like differential evolution algorithm, harmony search algorithm, particle swarm optimization algorithm, artificial immune algorithm, ant colony algorithm, hybrid robust genetic algorithm, scatter search algorithm, genetic algorithm in terms of convergence speed and efficiency by measuring the number of function evaluations required. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Designing and manufacturing new products possessing desired property are important in industry. With the advent of ever faster computing platforms, the computer aided design and optimization tools are becoming more attractive due to its great contribution to cost, material and time savings in the procedures of the engineering design. The application of these tools allows a more rapid design process and more detailed design studies. Over the past decades, a number of optimization algorithms have been used extensively in structural and manufacturing optimization tasks. The early works on the topics mostly use various mathematical techniques. These methods may not be used efficiently in finding global optimum solutions. As an alternative to traditional techniques, population-based optimization approaches, such as, genetic algorithm, particle swarm optimization algorithm, artificial immune algorithm, cuckoo search algorithm and artificial bee colony algorithm have been developed by mimicking natural phenomena and widely applied in various fields of science [1–12]. Artificial bee colony algorithm (ABC) is one of the most recently introduced swarm-based algorithms based on the intelligent foraging behaviour of honey bee swarm [13]. The ABC has been found to be successful in a wide variety of optimization tasks [13–15].
∗ Tel.: +1 734 5658028. E-mail address: [email protected] 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.04.013
On the other hand, researchers are paying more and more interest on hybrid algorithms to solve optimization problems. The hybrid algorithms have shown outstanding reliability and efficiency in application to the engineering optimization problems [16–20]. The main goal of the present research is to develop a robust optimization approach based on artificial bee colony algorithm and Taguchi method to solve design and manufacturing optimization problems. In the new hybrid approach, S/N values are calculated and ANOVA (analysis of variance) table for the objective function is formed using S/N ratios respectively. According to results of ANOVA table, appropriate interval levels of design variables are found and then, initial population of artificial bee colony algorithm is defined according to these interval levels. Then optimum results of the problem are obtained using artificial bee colony algorithm. Since the ABC has been found to be successful in a wide variety of optimization tasks, it is used in this paper. The developed new hybrid optimization approach entitled hybrid robust artificial bee colony algorithm (HRABC) is applied to optimum design of a vehicle part taken from automotive industry and to optimization of the machining parameters in multi-tool milling operations. The results of the HRABC for each case study show that the proposed optimization method converges rapidly to the global optimum solution and provides reliable and accurate solutions. The remaining contents of the paper are organized as follows. Literature review is given in Section 2. The standard ABC and Taguchi method are presented in Section 3. In Section 4, the proposed approach is used for optimization of a vehicle component.
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The results are discussed in Section 5. An application of the HRABC to optimization of the machining parameters in multi-tool milling operations is given in the Appendix A. 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 [1–23]. For instance, a novel approach for multi-component topology optimization of continuum structures using a multi-objective genetic algorithm is developed by Yildiz and Saitou [2]. The developed approach is applied to multi-component topology optimization of a vehicle floor frame. Artificial bee colony (ABC) algorithm originally developed by Karaboga [13] is inspired with social behavior, such as bird flocking, fish schooling, which is used successfully in the solution of optimization problems. The ABC algorithm has been used in many areas of optimization studies. The use of the ABC algorithms in the optimum solution of problems resulted better solutions compared to classical methods [13–15,25–32]. The robustness issues have been used to solve optimization problems by researchers [33–37]. Robinson et al. [38] presents a review paper which focuses largely on the work done since 1992 and a historical perspective of parameter design is also given. Kunjur and Krishnamurty [39] presented a robust optimization approach that integrates 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 [36,40]. Hybrid methods are also used to enhance the performance of evolutionary algorithm. For instance, the artificial immune algorithm is hybridized with hill climbing local search algorithm by Yildiz [16] and applied to multi-objective disc brake and manufacturing optimization problems from literature. In another paper [18], the particle swarm optimization approach is hybridized with receptor editing property of immune system. The proposed approach is used to solve optimization problems in design and manufacturing areas. Tsai et al. [41] 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. It is known that the ABC algorithm is an efficient approach at exploring the solution space, but it does not guarantee the global optimum as other evolutionary methods. The introduction of hybrid methods comes from the need to tackle more and more complex real-world problems. Some of the hybrid approaches in literature have been made on hybrid ABC [28,42,43]. 3. Hybrid bee colony optimization algorithm for structural optimization Structural design optimization has always been a very interesting and creative segment in a large variety of engineering designs. Structures, of course, should be designed such that they can resist applied forces (stress constraints), and do not exceed certain deformations (displacement constraints). Moreover, structures should
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be economical. Theoretically, the best design is the one that satisfies the stress and displacement constraints, and results in the least cost of construction. Although there are many factors that may affect the design cost, the first and most important one is the amount of material used to design the structures. Therefore, minimizing the weight of the structure is usually the goal of structural optimization. In this paper, a new hybrid optimization approach (HRABC) is developed to solve structural design and manufacturing optimization problems. In the proposed optimization 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 population, then they apply throughout bee colony algorithm for finding optimal solutions. 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 bee colony optimization algorithm and Taguchi’s method are given and, finally, the proposed hybrid approach is explained.
3.1. Bee colony algorithm Artificial bee colony algorithm (ABC) developed by Karaboga [13] and further developed by Karaboga and Basturk [14,24–27] is a nature inspired algorithm based on the intelligent foraging behavior of honey bee swarm. The ABC algorithm describes the foraging behavior, learning, memorizing and information sharing characteristics of honeybees. A basic model of foraging behavior of honeybee swarms consists of two essential components and define two leading modes of the behavior. The colony of artificial bees consists of three groups of bees: employed bees, onlookers and scouts. The colony of the artificial bees is divided into two groups, first half of the colony consists of the employed artificial bees and the second half includes the onlooker bees. Scout bees are the employed bees whose food source has been abandoned. In ABC algorithm, the position of a food source represents a possible solution to the optimization problem (value of design variables) and the nectar amount of a food source corresponds to the quality of the associated solution (fitness value). At the first step, the ABC generates a randomly distributed initial population Pinitial of N solutions, where N denotes the size of population. Each solution xi (i = 1, 3, . . ., N) is a S-dimensional vector where S is the number of optimization parameters (design variables). After initialization, the population of the solutions is subjected to repeated cycles, C = 1, 2, . . ., G, of the search processes of the employed bees, the onlooker bees and scout bees. An employed bee generates a modification in the solution in her memory depending on the local information. If the objective function value (fitness) of the new solution is better than that of the previous one, the bee memorizes the new position and forgets the old one. Otherwise, she keeps the position of the previous one in her memory. After all employed bees complete the search process; they share the nectar information of the food sources and their position information with the onlooker bees on the dance area. An onlooker bee evaluates the fitness information taken from all employed bees and chooses a food source with a probability related to its fitness value. An onlooker bee also produces a new solution and it memorizes the new position if its fitness value is better than the previous position. An artificial onlooker bee chooses a food source depending on the probability value associated with that food source, Pi , calculated by the following expression: Pi =
Fi
Nb
F n=1 n
(1)
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where Fi is the fitness value of the solution i which is proportional to the nectar amount of the food source in the position i and Nb is the number of food sources which is equal to the number of employed bees. In order to produce a candidate food position from the old one in memory, the ABC uses the following expression:
vij = xij + rij (xij − xkj )
(2)
where k ∈ {1, 2, . . ., N} and j ∈ {1, 2, . . ., D} are randomly chosen indexes. Although k is determined randomly, it has to be different from i . rij is a random number between (−1, 1). It controls the production of neighbour food sources around xij and represents the comparison of two food positions visually by a bee. As can be seen from (2), as the difference between the parameters of the xij and xkj decreases, the perturbation on the position xij gets decreased too. Thus, as the search approaches to the optimum solution in the search space, the step length is adaptively reduced. If the position of the food source cannot be improved for some predetermined number of cycles then that food source is abandoned. The abandoned food source is replaced with a new food source by the scouts. In ABC, this is simulated by producing a position randomly and replacing it with the abandoned one. The value of predetermined number of cycles is an important control parameter of the ABC algorithm, which is called ‘limit’ for abandonment. The value of limit is generally considered as N × S. Assume that the abandoned source is xi and j ∈ {1, 2, . . ., D}, then the scout discovers a new food source to be replaced with xi . This operation can be expressed as, j
j
j
j
xi = xmin + rand(0, 1)(xmax − xmin )
(3)
So from the above explanation it is clear that the control parameters used in ABC are number of food sources which is equal to number of employed bees, number of onlooker and scout bee, the value of ‘limit’ and the maximum cycle number. The effect of these control parameters on the convergence and fitness value of the objective function is discussed in the later section of the present work. Implementation steps of the ABC are summarized below: Step 1: Initialize swarm with randomly generated N food sources and evaluate them. Step 2: Find a candidate food source for each employed bee according to Eq. (2) and evaluate the candidate food source and select the better one as the new food source. Step 3: Calculate the probability values of food sources based on their fitness values. Step 4: Select a food source for each onlooker bee according to Eq. (1) by roulette wheel selection and generates a candidate solution according to Eq. (2). Step 5: Select the better one as the new food source and select a better one as the new food source. Step 6: Memorize the position of the best food source found so far. Step 7: Find the abandoned food sources and produce new positions for exhausted food sources according to Eq. (3). Step 8: Repeat the procedure from step 2 till the termination criterion is met. At termination, position of the food source and its nectar amount are the optimum values of the design variables and objective function for the considered problem for single run of the algorithm.
3.2. Taguchi method The Taguchi’s method is a universal approach, which is widely used in robust design [44]. There are three stages to achieve Taguchi’s objective: (1) concept design, (2) robust parameter design, and (3) tolerance design. The robust parameter design is used to determine the levels of factors and to minimize the sensitivity of noise. That is, a parameter setting should be determined with the intention that the product response has minimum variation while its mean is close to the desired target. Taguchi’s method is based on statistical and sensitivity analysis for determining the optimal setting of parameters to achieve robust performance. The responses at each setting of parameters were treated as a measure that would be indicative of not only the mean of some quality characteristic, but also the variance of that characteristic. The mean and the variance would be combined into a single performance measure known as the signal-to-noise (S/N) ratio. Taguchi classifies robust parameter design problems into different categories depending on the goal of the problem and for each category as follows: Smaller the better For these kind of problems, the target value of y, that is, quality variable, is zero. In this situation, S/N ratio is defined as follows:
S/N ratio = −10 log
yi2
(4)
n
Larger the better In this situation, the target value of y, that is, quality variable, is infinite and S/N ratio is defined as follows:
S/N ratio = −10 log
1/yi2
n
(5)
Nominal the best For these kind of problems, the certain target value is given for y value. In this situation S/N ratio is defined as follows:
S/N ratio = −10 log
y2
s2
(6)
Taguchi’s method uses an orthogonal array and analysis of mean to analyse the effects of parameters based on statistical analysis of experiments. To compare performances of parameters, the statistical test known as the analysis of variance (ANOVA) is used. Further details and technical merits about robust parameter design can be found in the [44]. Finally, optimum results are obtained by applying artificial bee colony algorithm. Stage 1: Determine efficient solution space for design variables using Taguchi’s method Stage 2: Apply the artificial bee colony algorithm to find the optimum solutions. 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 objective, 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 for the objective function. According to results of ANOVA, appropriate levels of design parameters
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Fig. 1. Design variables.
are found and then, initial swarm is defined according to the levels. Finally, optimum results of the optimization problem are obtained by applying the artificial bee colony algorithm process in two steps as follows • • • •
define initial population set, use artificial bee colony operators to create the next generation, evaluate objective function and constraints, repeat the loop until the optimum solutions are found.
In this paper, a new hybrid approach is proposed to improve the performance of the artificial bee colony algorithm. The hybridization of the artificial bee colony algorithm with Taguchi method is resulted in a solution, which leads to better parameter values for optimal structural design problems. The algorithm of proposed hybrid approach can be outlined as follows Program: HRABC BEGIN Begin Set the input parameters Select suitable orthogonal array Select suitable S/N ratio type While (not termination condition for experiments) do Compute S/N ratios Compute objective function values Conduct matrix of experiments End While (not termination condition for parameters) do Compute contributions Generate analysis of variance (ANOVA table) End Define RPD intervals for parameters Begin Generate randomly initial population of solutions Repeat Place the employed bees on their food sources Calculate the probability values Place the onlooker bees on the food sources Send the scouts to the search area for discovering new food sources Memorize the best food source found so far Until a termination is satisfied, and output the best food source found so far End END.
4. Structural design optimization using improved hybrid bee colony algorithm The hybrid approach proposed in Section 3 is applied to solve a structural design optimization problem taken from automotive industry for the optimal design of a vehicle component in this
section. The objective function is the volume of the part which is to be designed for minimum volume subject to strength constraints. The objective function value of the component is provided by ANSYS [45] during optimization loop. In this research, structural optimization is performed using the proposed approach. In the first stage, the experiments are designed to evaluate the effects of four design variables related to the objective function. The four design variables x1 , x2 , x3 and x4 are selected as shown in Fig. 1. 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 the objective as given in Table 1. Smaller the better characteristic is applied to compute S/N ratios based on objective as smaller the better for the volume. 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 [44]. The ANOVA table for the objective function is formed using S/N ratios as shown in Table 2. The most effective parameter is x1 with 62.4% contribution for the objective (see Table 2). Level 4 is considered for x1 since the smaller the better characteristic is employed for the first 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. 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. The results of HRABC are given in Table 3. It can be seen that a volume of 16008.1 mm3 with 298 MPa is obtained. It is clearly seen that structural design optimization performance is improved compared to traditional CAD, genetic algorithm, artificial immune algorithm, harmony search algorithm, particle swarm optimization algorithm, differential evolution algorithm and artificial bee colony algorithm. 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 the ABC solution. The structural layout results of the present hybrid approach using the ABC and robust design based on Taguchi’s method for the vehicle part is given in Fig. 2. The results of the HRABC for the optimal design of the vehicle part are given in Table 3. It is seen that structural design optimization performance is improved compared to other approaches. Using the HRABC, the vehicle component is designed with minimum volume (weight) without exceeding allowable stress. The HRABC provides the lightest design with minimum function
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Table 1 Experimental results and S/N ratios for volume. Exp. No.
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
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
F (volume)
S/N (volume) −87.49 −87.01 −86.45 −85.80 −86.47 −86.14 −86.62 −86.04 −85.59 −85.55 −86.12 −86.19 −85.53 −85.83 −84.78 −85.11
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
Table 2 Results of the analysis of variance for volume.
X1 X2 X3 X4
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 6.797
Error Total
DOF
M
F
Cont. (%)
3 3 3 3
1.4148 0.1469 0.0970 0.6051
539.9 56.09 37.03 230.9
62.43 6.473 4.281 26.70
3 15
0.0026
Table 3 Comparison of the optimization results for the side door bracket design.
Initial design Ansys Genetic algorithm (GA) Artificial immune algorithm (AIA) Harmony search algorithm (HS) Particle swarm optimization algorithm (PSO) Differential evolution algorithm (DE) Artificial bee colony algorithm (ABC) Hybrid robust artificial bee colony algorithm (HRABC)
Volume (cm3 )
Stress (MPa)
Function Evaluations
20727.2 18763.9 17663.7 17514.1 17495.9 17455.6 17312.8 17288.9 16008.1
256 277 293 294 296 298 293 290 298
– 30000 5000 4600 4250 3800 3100 2500 1000
Table 4 Comparison of the results for multi-tool milling operation. Method
Cu – Unit cost
Tu – Unit time
Pr – Profit rate
Handbook [46] Method of feasible direction [47] Genetic algorithm [19] Ant colony algortihm [48] Hybrid particle swarm (PSRE) [20] Immune algorithm Hybrid immune algorithm [19] Proposed approach (HRABC)
$18.36 $11.35 $11.11 $10.20 $10.90 $11.08 $10.91 $10.90
9.40 min 5.48 min 5.22 min 5.43 min 5.052 min 5.07 min 5.04 min 5.00 min
0.71/min 2.49/min 2.65/min 2.72/min 2.79/min 2.75/min 2.79/min 2.82/min
evaluation number as genetic algorithm, artificial immune algorithm, harmony search algorithm, particle swarm optimization algorithm, differential evolution algorithm and artificial bee colony algorithm.
5. Conclusions
Fig. 2. The optimal structural layout.
This paper describes a new optimization approach (HRABC) based on artificial bee colony algorithm and Taguchi method for solving structural design and manufacturing optimization problems. The Taguchi method is used to define robust initial population to achieve better initialize the artificial bee colony algorithm. The solution space of the artificial bee colony algorithm is refined based
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on the effect of the various design variables on the objective function. The HRABC is applied to the optimization of a vehicle component taken from automotive industry and the multi-tool milling problem. Results obtained from the HRABC for structural design and multi-tool milling problems have been compared with those obtained differential evolution algorithm, harmony search algorithm, particle swarm optimization algorithm, artificial immune algorithm, hybrid robust genetic algorithm, scatter search algorithm, ant colony algorithm and genetic algorithm. The results and comparisons in Tables 2 and 3 demonstrate the effectiveness and robustness of the HRABC in solving optimization problems in the different area of the engineering problems. The HRABC can not only improve the solution quality but also reduce the computational effort. The studies clearly indicate that the HRABC outperforms the state-of-the-art evolutionary algorithms for the problems solved in this article. The future work is to apply the HRABC for solving other engineering optimization problems. Appendix A. Case study of computational machining optimization Since the achievement of a machining operation is affected by depth of cut, feed rate and cutting speed, only these parameters are considered in practice. The proposed approach is applied to the mathematical model of Tolouei-Rad and Bidhendi [46] to show effectiveness of the new method in solving real-world optimization problems in the manufacturing industry. In the optimization of machine parameters for milling operations, the purpose is to maximize the total profit rate. The maximization of the total profit rate is carried out according to the two objective functions, which are unit production time and unit production cost. The unit cost is the sum of material cost, set up cost, machining cost and tool changing cost. In order to maximize the profit rate, allowable range of cutting speed and feed rate are imposed restriction by constraints. The constraints taken into consideration in this paper are maximum machine power, surface finish requirement and maximum cutting force permitted by the rigidity of the tool.The comparison of the results obtained by the proposed approach, against other techniques such as immune algorithm, GA, the feasible direction method and handbook recommendations, is given in Table 4. Function evaluation numbers are 20,000 and 15,000 to find optimal solutions for GA and immune algorithm, respectively. The HRABC also improves the convergence rate by computing the best value and maintaining the less function evaluations 3500. It can be seen that better results for the best computed solutions are achieved for the milling optimization problem compared to immune algorithm, genetic algorithm and feasible direction method. References [1] A.R. Yildiz, A new design optimization framework based on immune algorithm and Taguchi method, Computers in Industry 60 (8) (2009) 613–620. [2] A.R. Yildiz, K. Saitou, Topology synthesis of multi-component structural assemblies in continuum domains, Journal of Mechanical Design 133 (2011) 011008-1–011008-9. [3] S. Bureerat, J. Limtragool, Performance enhancement of evolutionary search for structural topology optimisation, Finite Elements in Analysis and Design 42 (6) (2006) 547–566. [4] K.Y. Chan, Modeling of a liquid epoxy molding process using a particle swarm optimization-based fuzzy regression approach, IEEE Transactions on Industrial Informatics 7 (1) (2011) 148–158. [5] I. Durgun, A.R. Yildiz, Structural design optimization of vehicle components using Cuckoo search algorithm, Materials Testing 54 (3) (2012) 185–188. [6] K.Y. Chan, C.K. Kwong, H. Jiang, M.E. Aydin, T.C. Fogarty, A new orthogonal array based crossover, with analysis of gene interactions, for evolutionary algorithms and its application to car door design, Expert Systems with Applications 37 (5) (2010) 3853–3862.
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