Enhanced parallel cat swarm optimization based on the Taguchi method

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Enhanced parallel cat swarm optimization based on the Taguchi method

Pei-Wei Tsai

a

, Jeng-Shyang Pan

a,b

, Shyi-Ming Chen

c,d,⇑

, Bin-Yih Liao

a

Department of Electronic Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, ROC b

Innovative Information Industry Research Center, Harbin Institute of Technology, Shenzhen Graduate School, Nanshan, Shenzhen City, Guangdong Province, China c

Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC d

Graduate Institute of Educational Measurement and Statistics, National Taichung University of Education, Taichung, Taiwan, ROC

a r t i c l e

i n f o

Keywords:

Cat swarm optimization

Enhanced parallel cat swarm optimization Parallel cat swarm optimization Taguchi method

a b s t r a c t

In this paper, we present an enhanced parallel cat swarm optimization (EPCSO) method for solving numerical optimization problems. The parallel cat swarm optimization (PCSO) method is an optimiza-tion algorithm designed to solve numerical optimizaoptimiza-tion problems under the condioptimiza-tions of a small population size and a few iteration numbers. The Taguchi method is widely used in the industry for optimizing the product and the process conditions. By adopting the Taguchi method into the tracing mode process of the PCSO method, we propose the EPCSO method with better accuracy and less compu-tational time. In this paper, five test functions are used to evaluate the accuracy of the proposed EPCSO method. The experimental results show that the proposed EPCSO method gets higher accuracies than the existing PSO-based methods and requires less computational time than the PCSO method. We also apply the proposed method to solve the aircraft schedule recovery problem. The experimental results show that the proposed EPCSO method can provide the optimum recovered aircraft schedule in a very short time. The proposed EPCSO method gets the same recovery schedule having the same total delay time, the same delayed flight numbers and the same number of long delay flights as theLiu, Chen, and Chou method (2009). The optimal solutions can be found by the proposed EPCSO method in a very short time.

1. Introduction

In recent years, some artiﬁcial intelligence (AI) methods have been presented to solve optimization problems. Chu, Tsai, and

Pan (2006) and Chu and Tsai (2007)presented the cat swarm

opti-mization (CSO) method for solving optiopti-mization problems by retaining the natural behaviors of cats.Tsai, Pan, Chen, Liao, and

Hao (2008) presented a parallel cat swarm optimization (PCSO)

method based on the framework of parallelizing the structure of the CSO method. Genetic algorithms (GA) have successfully been used in the internet service (Elliott & Krzymien, 2009) and imped-ance measurements (Janeiro & Ramos, 2009); particle swarm opti-mization (PSO) techniques have successfully been used to design antennas (Wu et al., 2009) and to construct parameters in neural network systems (Lin, Chen, & Lin, 2009); ant colony optimization (ACO) techniques have successfully been used to solve the travel-ing salesman problem (TSP) (Dorigo & Gambardella, 1997) and the routing problem of networks (Pinto, Nägele, Dejori, Runkler,

& Sousa, 2009); artiﬁcial bee colony (ABC) techniques have

successfully been used to solve the lot-streaming ﬂow shop sched-uling problem (Pan, Tasgetiren, Suganthan, & Chua, 2011); cat swarm optimization (CSO) techniques have successfully been used to adjust the parameters of the SVM (Lin et al., 2009). Moreover, in the industry, the Taguchi method (Taguchi, Chowdhury, & Taguchi, 2000) has successfully been used for optimizing the product-line design and the process conditions. Tsai, Liu, and Chou (2004) successfully adopted the Taguchi method into the GA’s crossover process and presented the hybrid Taguchi-genetic algorithm (HTGA).

Although the parallel cat swarm optimization (PCSO) method presented inTsai et al. (2008)has the ability to ﬁnd the near best solution under more strict conditions, its computational speed is not efﬁcient. It is obvious that to reduce the computational time of the PCSO method and to keep high accuracy results with a small population size simultaneously are the desired goals of the PCSO method. Therefore, in this paper, we propose the en-hanced parallel cat swarm optimization (EPCSO) method by adopting the orthogonal array of the Taguchi method into the tracing mode process of the PCSO method. The proposed EPCSO method can successfully been used for solving optimization problems.

⇑ Corresponding author. Tel.: +886 2 27376417; fax: +886 2 27301081. E-mail address:[email protected](S.-M. Chen).

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2. Parallel cat swarm optimization

Tsai et al. (2008)have proposed the parallel cat swarm

optimi-zation (PCSO) method for solving optimioptimi-zation problems. The basic idea of the PCSO method utilizes the major structure of the cat swarm optimization (CSO) method proposed byChu et al. (2006). The CSO method has two modes, i.e., the seeking mode and the tracing mode, for simulating the behaviors of cats to move the indi-viduals in the solution space. By adjusting the parameter MR, the ratio of individuals moved by the seeking process and the tracing process can be controlled, where MR 2 [0, 1]. Some methods for splitting a population into several sub-populations to construct a parallel structure have been presented, such as the island-model genetic algorithm (Whitley, Rana, & Heckendorn, 1999), the parallel genetic algorithm (Abramson & Abela, 1992), the ant colony system with communication strategies (Chu, Roddick, &

Pan, 2004) and the parallel particle swarm optimization algorithm

with communication strategies (Chang, Chu, Roddick, & Pan, 2005). Each of the sub-populations evolves independently and shares the information they have occasionally. It results in the reducing of the population size for each sub-population and the beneﬁt of cooper-ation is achieved.

In the PCSO method (Tsai et al., 2008), the individuals are sep-arated into a predeﬁned number of groups in the initial process to construct the virtual parallel space for the individuals. If we let the predeﬁned number of groups be equal to 1, then the PCSO method becomes the CSO method (Chu et al., 2006) due to the fact that there is only one group. The individuals in the same group provide a local near best solution for their group in every generation, and the global near best solution found so far can be discovered by comparing the local near best solutions collected from the parallel groups. The individuals in a group can only access the near best solution discovered by their own group, but when the process of information exchanging is applied, the parallel groups can receive a near best solution from another randomly picked group. The dif-ference between the PCSO method and the CSO method is de-scribed as follows. At the beginning of the PCSO method, N individuals are created and then they are separated into G groups. The calculation of the PCSO method in the tracing mode is different from that of the CSO method and there exists an information exchanging process. The process of the PCSO method is reviewed as follows (Tsai et al., 2008):

Step 1: Create N cats, randomly sprinkle the cats into the M-dimensional solution space within the constrain ranges of the initial value and randomly collect them into G groups. Generate the velocities for each dimension. Set the motion flags of the cats to make them move into the tracing mode or the seeking mode according to the user predefined value of MR, where MR 2 [0, 1]. The cats moved by the seeking mode process present higher exploitative capacity. Contrarily, the cats gain a higher explorative abil-ity when they are moved by the tracing mode process. Hence, MR affects the ratio of artificial agents to work on the exploitation and exploration.

Step 2: Evaluate the fitness values of the cats, respectively, by tak-ing the coordinates into the fitness function, which repre-sents the benchmark and the characteristics of the problem to be solved. After calculating the fitness values of the cats, respectively, record the coordinate xbest and

the ﬁtness value of the cat which has the best ﬁtness value found so far.

Step 3: Move the cats by taking the operations in the seeking mode by Eqs.(1)–(4)or the parallel tracing mode by Eqs.

(5) and (6)according to the statuses of the motion ﬂags.

Step 4: Reset the motion ﬂags of all cats and separate them into statuses that indicating the seeking or the tracing by re-pick [N (1 MR)] cats to move in the seeking mode and (N MR) cats to move in the parallel tracing mode. Step 5: Check whether the number of iterations reaches a

prede-ﬁned iteration number. If the condition is satisﬁed, apply the information exchanging process.

Step 6: Check whether the process satisﬁes the termination condi-tion. If the process is terminated, output the coordinate which represents the found best solution and Stop. Other-wise, go to Step 2.

2.1. The seeking mode process

In the seeking mode, the cat moves slowly and conservatively. It observes the environment before it moves. Four essential factors are defined in the seeking mode, i.e., Seeking Memory Pool (SMP), Seeking Range of the selected Dimension (SRD), Counts of Dimension to Change (CDC) and Self-Position Considering (SPC). SMP is used to define the size of the seeking memory for each cat to indicate the points sought by the cat. The cat will pick a point from the memory pool according to the rules described later; SRD declares the mutative ratio for the selected dimensions; CDC discloses how many dimensions will be varied. In the seeking mode, if a dimension is selected to mutate, the difference between the new value and the old one cannot be out of the range defined by SRD; SPC is a Boolean variable, which decides whether the point in which the cat is already standing will be one of the candidates to move to. No matter the value of SPC is true or false, the value of SMP will not be influenced. These factors are all playing important roles in the seeking mode. The process of the seeking mode is re-viewed as follows (Tsai et al., 2008):

Step 1: Generate j copies of catk, where j = SMP. If the value of SPC

is true, let j = SMP 1 and return the present position as one of the candidates.

Step 2: According to CDC, plus/minus SRD percents of the current value randomly and replace the old one for all copies according to Eqs.(1)–(3): M ¼ Modify [ ð1 ModifyÞ; ð1Þ jModifyj ¼ CDC M; ð2Þ xjd¼ xjd; d R Modify; ð1 þ rand SRDÞ xjd; d 2 Modify; d¼1;2;...;M

8

j; ð3Þ

where Modify denotes the elements of the selected dimensions, whose values will be modiﬁed, and rand is a random variable in the range [0, 1].

Step 3: Calculate the ﬁtness values of all candidate points, respectively.

Step 4: If all the ﬁtness values are not exactly equal, calculate the selecting probability Piof each candidate point, shown as

follows:

Pi¼

1; if FSmax¼ FSmin;

jFSiFSbj

FSmaxFSmin; where 0 < i < j; otherwise:

(

ð4Þ

If the goal of the ﬁtness function is to ﬁnd the minimum solution, then let FSb= FSmax. Otherwise, let FSb= FSmin, where FSmaxdenotes

the largest FS in the candidates and FSmindenotes the smallest one.

Step 5: Randomly pick the point to move from the candidate points and replace the position of catk.

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2.2. The parallel tracing mode process

The parallel tracing mode process is reviewed as follows (Tsai et al., 2008):

Step 1: Update the velocities for every dimension

v

k,d(t) for the

catkat the current iteration, shown as follows:

v

k;dðtÞ ¼

v

k;dðt 1Þ þ r1 c1 ½xlbest;dðt 1Þ xk;dðt 1Þ;

where d ¼ 1; 2; . . . ; M ð5Þ

where t denotes the iteration number, xlbest;dðt 1Þ denotes the

posi-tion of the cat which has the best ﬁtness value at the previous iter-ation in the group that catkbelongs to and M denotes the dimension

of the solution space.

Step 2: Check whether the velocities are in the range of maximum velocity. The new velocity is bounded to the maximum velocity in case the new velocity is over-range.

Step 3: Update the position of catkaccording to Eq.(6):

xk;dðtÞ ¼ xk;dðt 1Þ þ

v

k;dðtÞ: ð6Þ

2.3. The information exchanging process

The information exchange process forces the sub-populations to exchange their information and achieves somehow the coopera-tion. It deﬁned a parameter ECH to control the exchanging of the information between sub-populations. The information exchang-ing process is applied once per ECH iterations. The process of infor-mation exchanging consists of the following four steps (Tsai et al.,

2008):

Step 1: Pick up a group of sub-populations sequentially and sort the individuals in this group according to their ﬁtness values.

Step 2: Randomly select a local best solution from an unrepeat-able group.

Step 3: The individual whose ﬁtness value is the worst in the group is replaced by the selected local best solution. Step 4: Repeatedly perform Step 1 to Step 3 G times to let every

group receives a local best solution from the others. 3. The Taguchi method

The Taguchi method (Taguchi et al., 2000) is an important tool for robust design. It is widely used for optimizing the product-line design and the process conditions due to the fact that it can pro-vide high quality products with low development costs (Tsai

et al., 2004). One of the major tools in the Taguchi method is called

the orthogonal array, which is adopted by the proposed EPCSO method. In the EPCSO method, only the two-level orthogonal array is used to take part in the process. In the following, we brieﬂy review the concept of two-levels orthogonal arrays fromTaguchi

et al. (2000). An array is called orthogonal due to that each column

indicates a value of a considered factor, and the factors listed in the orthogonal array can be evaluated independently. Every row in an orthogonal array represents a set of parameters for one run of the experiment. The two-levels orthogonal array can be described as follows:

Lnð2n1Þ; ð7Þ

where n = 2k_{, k is a positive integer which is greater than 1, and}

n 1 denotes the number of columns in the two-levels orthogonal array. For example, assume that we have two sets of solutions with 7 parameters in our design and assume that we want to ﬁnd the

optimal combination of their values. Then, the L8(27) orthogonal

array is shown inTable 1.

The elements in the orthogonal array shown inTable 1indicate which parameter value should be taken into the experiment on a specific considered factor. The values ‘‘0’’ and ‘‘1’’ in the elements of the orthogonal array shown inTable 1indicate which factor’s value should be used in a run of the experiment, where the value ‘‘0’’ inTable 1means that the factor’s value should be taken from the first set of the solution, and the value ‘‘1’’ inTable 1means that the factor’s value should be taken from the other set of the solu-tion. Therefore, fromTable 1, we can see that the 5th run of the experiment takes the values of the factors A, B, E and F from the first solution set and takes the values of the factors C, D and G from the second solution set to compose the run of the experiment. The orthogonal array provided by the Taguchi method (2000) can reduce the runs of the experiment efficiently. The case given in

Table 1requires 27_{runs of the experiment to explore the full}

com-bination, whereas the Taguchi method in this example reduces the experiment to 8 runs only.

4. The proposed enhanced parallel cat swarm optimization (EPCSO) method

InTsai et al. (2008), we have proposed the PCSO method by

organizing the artiﬁcial agents into a predeﬁned number of groups. When the agent moves in the process of the parallel tracing mode, it collects the local best information found so far by its own group and use the information to update its velocity. Although it presents higher accuracies and faster convergence than the CSO method

(Chu et al., 2006) under the conditions of a few number of artiﬁcial

agents and a short limited computational time, its computational time could be further improved in case that the number of artiﬁcial agents increases due to the fact that the number of artiﬁcial agents moving in the process of parallel seeking mode is much more than those move in the tracing mode in the original design. In this pa-per, in order to overcome the drawbacks of the CSO method (2006) and the PCSO method (2008), we adopt the Taguchi meth-od’s orthogonal array into the process of the parallel tracing mode process to propose the enhanced parallel cat swarm optimization (EPCSO) method.

The orthogonal array in the Taguchi method is quite useful for reducing the run of the experiment. When the artiﬁcial agent moves in the new designed parallel tracing mode, it accesses not only the local near best solution in its own group, but also utilizes the information comes from the global near best solution overall groups. The operation of the enhanced parallel tracing mode is de-scribed as follows:

Step 1: Generate two sets of candidate velocities, which are denoted by the symbols cv1,dand cv2,d, respectively, where cv1;dðtÞ ¼vk;dðt 1Þ þ r1 c1 ½xgbest;dðt 1Þ xk;dðt 1Þ; where d ¼ 1; 2; . . . ; M;ð8Þ cv2;dðtÞ ¼vk;dðt 1Þ þ r1 c1 ½xlbest;dðt 1Þ xk;dðt 1Þ; where d ¼ 1; 2; . . . ; M;ð9Þ

Table 1

The L8(27) orthogonal array.

Experiment number Considered factors

A B C D E F G 1 0 0 1 0 1 1 0 2 1 0 0 0 0 1 1 3 0 1 0 0 1 0 1 4 1 1 1 0 0 0 0 5 0 0 1 1 0 0 1 6 1 0 0 1 1 0 0 7 0 1 0 1 0 1 0 8 1 1 1 1 1 1 1

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M denotes the dimension of the solution space, xgbest denotes the

global near best solution found so far, xlbest denotes the local near

best solution of the group, l represents the group that xkbelongs

to, r1is a random value in the range of [0, 1], and c1is a constant.

Use the candidate velocities and the Taguchi orthogonal array to create a series of velocity sets, shown as follows:

v

sets;dðtÞ ¼

c

v

1;dðtÞ; if the element in the orthogonal array is \0";

c

v

2;dðtÞ; otherwise;

ð10Þ where d = 1, 2, . . . , M, M denotes the dimension of the solution space, and s is the index of the velocity set.

Step 2: Take one velocity set to update the original velocity

v

k,d(t)

of the artiﬁcial agent each time, shown as follows:

v

k;dðtÞ ¼

v

max;if ½

v

k;dðt 1Þ þ

v

sets;dðtÞ

exceeds the maximum velocity;

v

k;dðt 1Þ þ

v

sets;dðtÞ; otherwise; 8 > < > : ð11Þ

update the position xk,d(t) of the artiﬁcial agent, shown as follows:

xk;dðtÞ ¼ xk;dðt 1Þ þ

v

k;dðtÞ ð12Þ and calculate its ﬁtness value for later use. Accumulate the ﬁtness values contributed by the column factors and take the most adap-tive factors to compose the latest velocity.

Step 3: Move the artiﬁcial agent with the latest velocity by Eq. (12)to update its position.

Let us consider the orthogonal array shown inTable 1. Assume that the ﬁtness function is shown as follows:

f ðXÞ ¼X M d¼1 1 xd ð13Þ

and assume that the goal is to minimize the fitness value. The cur-rent position X of the artificial agent and the velocity of the artificial agent in the previous run are assumed to be the values shown in Table 2. First, the candidate velocity sets cv1,dand cv2,dare

gener-ated, as shown in Table 3. Therefore, eight solution sets will be created and the dimension M of the solution space for this example is 7. Then, the eight solution sets are composed with the elements of cv1,dand cv2,dbased on the values shown inTable 1. The

experimen-tal results and the accumulated ﬁtness values are shown inTable 4. The ﬁnal velocity of the agent in this run of the experiment is shown inTable 5.

The proposed EPCSO method has the same operation as the CSO method (Chu et al., 2006) for the process of the seeking mode. The flowchart of the proposed EPCSO method is shown inFig. 1. The PCSO method (Tsai et al., 2008) only limits the artificial agents, who move in the parallel tracing mode, to access the local near best solution of which group they belong. Although this constrain divides the solution space into several parallel spaces, the efficiency of improving the accuracy and the searching speed of the PCSO method is not promoted. In this paper, in order to improve the efficiency of the PCSO method, we further adopt the Taguchi orthogonal array into process of the parallel tracing mode of the proposed EPCSO method. The enhanced parallel tracing mode utilizes the orthogonal array to run the test samples

constructed by the updated velocities with the distances between the local near best solution to the artificial agent and the global near best solution to the artificial agent. It makes the enhanced parallel tracing mode returns the searching result more stable and the searching efficiency is improved.

In the proposed EPCSO method, the number G of groups is de-ﬁned as follows:

G ¼ 2njn 2 N0 ð14Þ

and the total population size N is deﬁned as follows:

N ¼ 2m Gjm 2 N0; ð15Þ

where N0_{is a set of natural number including zero.}

InFig. 1, MR is a user deﬁned value in the range of [0, 1]. If MR is equal to 0, then it implies that all the cats will move into the seek-ing mode; if it is equal to 1, then it implies that all the cats will move into the tracing mode. In our experiment, MR is set to 0.1 to let 10% cats to move into the tracing mode and to let 90% cats to move into the seeking mode.

5. Experimental results

In order to evaluate the accuracy of the proposed EPCSO meth-od, a series of experiments are taken with ﬁve familiar benchmark functions, shown as follows:

f1ðXÞ ¼ XD d¼1 x2 d Initial Range : ½100; 100 D ; ð16Þ f2ðXÞ ¼ XD d¼1 x2 d 4000 YD d¼1 cos xdffiffiffi d p þ 1 Initial Range : ½0; 600 D ; ð17Þ f3ðXÞ ¼ 20exp 0:2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 D PD d¼1 x2 d s ! exp 1 D PD d¼1 cosð2pxdÞ þ20 þ expð1Þ 8 > < > : 9 > = > ; Initial Range : ½32; 32D ; ð18Þ f4ðXÞ ¼ XD d¼1 x2 d 10cosð2

p

xdÞ þ 10 Initial Range : ½5; 5 D ; ð19Þ f5ðXÞ ¼ XD d¼1 Xd b¼1 xb !2 Initial Range : ½100; 100 D : ð20Þ

All the benchmark functions are evaluated according to the condi-tions shown inChen and Li (2007) with 2000 iterations and re-peated 25 runs. The number of dimensions of the solution space are set to 30 and 100, respectively, for evaluating the performance in different dimensional situations. Because the parallelism struc-ture is constructed by 4 parallel groups, the size of the swarm for the proposed EPCSO method and the PCSO method (Tsai et al., 2008) are set to 16. The parameters setting for PSO-type algorithms can refer to (Chen & Li, 2007) and the parameters setting of the pro-posed EPCSO method and the PCSO method (Tsai et al., 2008) are shown inTable 6.

Table 2

The coordinate and the original velocity of the artiﬁcial agent. Current position Considered factors

A B C D E F G

X 0 0 0 0 0 0 0

vk,d(t 1) 1 1 1 1 1 1 1

Table 3

The candidate velocity sets.

Candidate velocity Considered factors

A B C D E F G

cv1 2 1 3 2 1 0 1

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e = 825), the valid candidate positions, in this case, are the location of 823 and 893. After the seeking operation, the selected element and the elements locate after it are swapped with the calculated location. The tracing operation is also brought to the location index to insure every movement still retains the consistency between the destination of the former ﬂight and the origin of the current ﬂight.

L ¼ flij^p0e;j¼ p0i;j1g; 1 6 i 6 M; 1 6 j 6 N; 1 6 li6M: ð28Þ In our experiment, the population size of the proposed EPCSO meth-od is set to 16 and the populations are divided into four groups. The maximum iteration is set to 2000 and the experiment is repeated with 25 runs. The experiment is run on an Intel Core 2 2.4G CPU with the GCC compiler for C with a Freebsd 7.1-Release operating system. The experimental results show that the optimal recovery schedules can be found in every run by delay 24 flights with totally delay 595 min, and only 5 flights will be delayed relatively longer (more than half hour). To fully run the 2000 iterations costs less than 4.6 s and the optimum solutions can be found with 24 itera-tions in average.Fig. 6shows four of the recovered schedules found by the proposed EPCSO method, where the darker zone is the dis-turbance period of the closure of the airport ‘‘TSA’’. The proposed EPCSO method gets different recovered schedules with stable fitness values at final in all 25 runs. For all 25 runs, the standard deviation of the final fitness values is 4.547 1013_{and the total}

delay time is stabilized on 595 min.Fig. 5shows the recovery re-sults delay 24 ﬂights with totally 595 min delay, and 5 ﬂights are delayed more than 30 minutes. The standard deviation of the statis-tical total delay time for every recovery schedule is 14.

7. Conclusions

In this paper, we have presented the enhanced parallel cat swarm optimization (EPCSO) method for solving optimization

problems. In this paper, five test functions are used to evaluate the accuracy of the proposed EPCSO method. The experimental re-sults show that the proposed EPCSO method gets higher accuracies with less computational time than the existing methods. We also have applied the proposed method to solve the aircraft schedule recovery problem. Aircraft schedule recovery contains many essen-tialities and should be solved within a short period of time due to the disrupted aircraft schedule implies departure time delay for many flights. The experimental results show that the proposed EPCSO method can provide the optimum recovered aircraft sche-dule in a very short time. The proposed EPCSO method gets the same recovery schedule having the same total delay time, the same delayed flight numbers and the same number of long delay flights as theLiu et al.’s method (2009). The optimal solutions can be found by the proposed EPCSO method in a very short time, i.e., 24 iterations in average.

Acknowledgement

The authors thank Mrs. Szu-Ping Hao, Department of Mechani-cal Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, for her help during this work.

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