Invasive weed optimization algorithm for optimization no-idle flow ...

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Author's personal copy Neurocomputing 137 (2014) 285–292

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Invasive weed optimization algorithm for optimization no-idle flow shop scheduling problem Yongquan Zhou a,b, Huan Chen a, Guo Zhou c a

College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Nanning 530006, China c Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100081, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 8 February 2013 Received in revised form 4 April 2013 Accepted 14 May 2013 Available online 19 February 2014

In this paper, an invasive weed optimization (IWO) scheduling algorithm is presented for optimization no-idle flow-shop scheduling problem (NFSP) with the criterion to minimize the maximum completion time (makespan). Firstly, a simple approach is put forward to calculate the makespan of job sequence. Secondly, the most position value (MPV) method is used to code the weed individuals so that fitness values can be calculated. Then, use the global exploration capacity of IWO to select the best fitness value and its corresponding processing sequence of job by evaluating the fitness of individuals. The results of 12 different scale NFSP benchmarks compared with other algorithms show that NFSP can be effectively solved by IWO with stronger robustness. & 2014 Elsevier B.V. All rights reserved.

Keywords: No-idle flow shop scheduling problem Invasive weed optimization Makespan Most position value method Global exploration capacity Robustness

1. Introduction Production scheduling plays a basic and key role in achieving advanced manufacturing and in improving the production efficiency. Under the premise of rapid market change, good production scheduling program can greatly improve production efficiency and resource utilization ratio to maintain enterprises with strong competitiveness. In actual production scheduling, a machine, once opened, is not allowed to interrupt cause costs and benefits. This type of production scheduling problem is known as no-idle flow shop scheduling problem. For example, the process of pottery drum drying furnace needs a large amount of natural gas in using. Due to the thermal inertia of the furnace, open and stop it all will take a few days, so once the device is opened, free time is not allowed. Another example is a glass fiber production process, the molding device brushed molding of the glass melt, once the equipment is shut down, the production of glass fiber will come to naught. Baraz and Mosheiov have proved that no-idle flow shop scheduling problem with 3 machines is NP-hard problem [6], so high-efficiency scheduling program is very important. At present, approaches for solving no-idle flow shop scheduling problem

E-mail addresses: [email protected] (Y. Zhou), [email protected] (H. Chen), [email protected] (G. Zhou). http://dx.doi.org/10.1016/j.neucom.2013.05.063 0925-2312 & 2014 Elsevier B.V. All rights reserved.

conclude exact methods, constructed heuristic algorithms and intelligent optimization algorithms. Exact methods, such as a branch and bound method used by Baptiste and Lee [1] and Saadani et al. [2], often require a lot of computing and storage space. Constructed heuristic algorithms, such as the Johnson method [3], the NEH [4] method, the KK method [5] and the IG method [6], can quickly construct scheduling solution, while the quality is often unsatisfactory. In recent years, with the rapid development of intelligent optimization algorithms, intelligent optimization algorithms, such as GA [7], TS [8], SA [9], ACO [10], DE [11], PSO [12], HS [13], ANN [30,31], PSO [32], etc., are also applied to solve the NFSP. An invasive weed optimization (IWO) algorithm [14] was proposed by Mehrabian and Lucas in 2006, which was inspired from a common phenomenon in agriculture: colonization of invasive weeds. The algorithm has a simple structure, less parameters, strong robustness, easy to understand and easy to program features. At present, IWO as a new optimization method has been successfully applied to the training feed-forward neural networks [15], constrained optimization of combustion at a coal-fired utility boiler [16], PID controller design [17], energy efficient trajectory planning [18], multi-user detection for MC-CDMA [19], tuning of auto-disturbance rejection controller [20], cooperative multiple task assignment of UAVS [21], design of non-uniform circular antenna arrays [22], design of encoding sequences for DNA [23], the piezoelectric actuator placement [24], image clustering [25], constrained

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engineering design [26] and pattern recognition [33,34], multimodal function optimization [35–38], permutation flow-shop scheduling problem [39] and circle packing problem [40]. In this paper, the IWO algorithm is used as a solution for solving NFSP. Firstly, weed individuals in IWO are encoded to construct mapping between weed individuals and scheduling solution. Then, the global exploration capacity of IWO is used to select the best fitness value and its corresponding processing sequence of jobs by evaluating the fitness of individuals. The simulation results show that the IWO can effectively solve the NFSP, showing a stronger robustness.

m3

t j ,3

m2 m1

t j ,2

t j ,1

Fig. 2. Calculate Dðπ 2 ; 2; 3Þ when Dðπ 1 ; 2; 3Þ 4t j2 ;2 .

t j ,3

m3

2. Description of NFSP

m1

Dðπ 1 ; j; j þ1Þ ¼ t j1 ;j þ 1 ; j ¼ 1; 2…; m  1 Dðπ 2 ; j; j þ1Þ ¼ max fDðπ 1 ; j; j þ1Þ t j2 ;j ; 0g þt j2 ;j þ 1 ; j ¼ 1; 2; …; m  1 Dðπ i ; j; j þ 1Þ ¼ max fDðπ i  1 ; j; j þ 1Þ t ji ;j ; 0g þ t ji ;j þ 1 i ¼ 2; 3; …; n; j ¼ 1; 2; …; m  1 m1

n

j¼1

i¼1

C max ðπ n Þ ¼ ∑ Dðπ n ; j; jþ 1Þ þ ∑ t ji ;1 π n ¼ argfC max ðπ n Þg- min ; 8 π A Π where n is the number of all jobs and m is the number of all machines, times of job i processed on machine j is denoted as t i;j (1 ri r n; 1 r jr m), assuming preparation time for each job is zero or is included in the processing time t i;j , π n ¼ ðj1 ; j2 ; …; jn Þ is a scheduling solution, all scheduling solutions are denoted as Π, π i ¼ fj1 ; j2 ; …; ji gis a partial solution, Dðπ i ; j; j þ1Þ denotes the minimum difference between the completion of processing the last job ji on machine j and jþ 1 restricted by the no-idle constraint. C max ðπ n Þ is the makespan. According to the above description, the computing complexities of calculating the minimum difference Dðπ i ; j; j þ 1Þ and the makespan C max ðπ n Þ are Oðm2 Þ and Oðnm2 Þ respectively. In order to facilitate understanding, the process of calculating Dðπ 2 ; 2; 3Þ with 2 jobs and 3 machines is shown in Figs. 1–3. 3. Invasive weed optimization algorithm for solving NFSP 3.1. Invasive weed optimization algorithm In the basic IWO, weeds represent the feasible solutions of problems and population is the set of all weeds. A finite number

m3

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t j ,2 t j ,1 D(π , 2,3) Fig. 1. Calculate Dðπ 1 ; 2; 3Þ.

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Firstly, the general characterizations of NFSP are as follows [12]: there are n jobs and m machines, all jobs are processed in the same sequence to minimize a given objective function, every job is processed in one machine only once and each machine can only process one job at a time, machines must process jobs without any interruption from the start of processing the first job to the completion of processing the last job, and all jobs are processed in an identical processing order on all machines. Secondly, the mathematical model of NFSP is given as follows:

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t j ,1

t j ,3

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t j ,1 t

Fig. 3. Calculate Dðπ 2 ; 2; 3Þ when Dðπ 1 ; 2; 3Þ r t j2 ;2 .

of weeds are being dispread over the search area. Every weed produces new weeds depending on its fitness. The generated weeds are randomly distributed over the search space by normally distributed random numbers with a mean equal to zero. This process continues until maximum number of weeds is reached. Only the weeds with better fitness can survive and produce seed, others are being eliminated. The process continues until maximum iterations are reached or hopefully the weed with best fitness is closest to optimal solution. The process is addressed in detail as follows: Step 1. Initialize a population A population of initial solutions is being dispread over the D dimensional search space with random positions. Step 2. Reproduction The higher the weed’s fitness, the more seeds it produces. The formula of weeds producing seeds is weedn ¼

f  f min ðsmax  smin Þ þsmin f max  f min

ð1Þ

where f is the current weed’s fitness. f max and f min respectively represent the maximum and the least fitness of the current population. smax and smin respectively represent the maximum and the least value of a weed. Step 3. Spatial dispersal The generated seeds are randomly distributed over the D dimensional search space by normally distributed random numbers with a mean equal to zero, but with a varying variance. This ensures that seeds will be randomly distributed so that they abide near to the parent plant. However, standard deviation (s) of the random function will be reduced from a previously defined initial value (sinit ) to a final value (sfinal ) in every generation. In simulations, a nonlinear alteration has shown satisfactory performance, given as follows

scur ¼

ðiter max  iterÞn ðsinit  sfinal Þ þ sfinal ðiter max Þn

ð2Þ

where iter max is the maximum number of iterations, scur is the standard deviation at the present time step and n is the nonlinear modulation index. Generally, n is set to 3. Step 4. Competitive exclusion After passing some iteration, the number of weeds in a colony will reach its maximum (P_MAX) by fast reproduction. At this time, each weed is allowed to produce seeds. The produced seeds are then allowed to spread over the search area. When all seeds have found their position in the search area, they are ranked together with their parents (as a colony of weeds). Next, weeds with lower fitness are eliminated to reach the maximum

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allowable population in a colony. In this way, weeds and seeds are ranked together and the ones with better fitness survive and are allowed to replicate. The population control mechanism is also applied to their offspring to the end of a given run, realizing competitive exclusion.

3.2. Individual coding scheme The IWO algorithm originally solves the continuous space optimization problems, so individuals must be encoded appropriately to solve scheduling problems. In this paper, the most position value method (MPV) has been used. [29]. The MPV rule is applied to enable the continuous Differential Evolution (DE) algorithm to be used in all kinds of sequencing problems, mutant individual is constructed by the optimal target individual, and trial individual is obtained through crossover of target and mutant individual. We used to transform weed individuals from encoded by float numbers into jobs sequence. Then weed individuals’ fitness can be calculated by their corresponding sequence. Suppose that x ¼ ðx1 ; x2 ; …; xn Þ is a weed individual, π ¼ ðj1 ; j2 ; …; jn Þ represents the transformed corresponding jobs sequence, where j1 is the column number of the maximum float in x; j2 is the column number of the second largest float in x, and so on. There may exist such a situation that many positions have the same float value; in this case, the left position is given priority. For example, x1 and x2 have the same float values, so the column number of x1 will be firstly saved in jobs sequence π, and then x2. Suppose x ¼ ½0:06; 2:99; 1:86; 3:73; 1:86; 0:67, the following mapping is shown in Table 1. 3.3. Pseudo-code of solving NFSP by IWO algorithm

Pseudo Code Begin Initialize(pop), set parameters; iteration_number_cur ¼1; While iteration_number_cur o iteration_max job_sequence_weed ¼ MPV_sort(pop); For i¼1:N fitness(i) ¼-F(job_sequence_weed); End For BestFitness ¼max(fitness); WorstFitness ¼min(fitness);

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stepLength¼ (iteration_max -iteration_number_cu)^n *(stepLength_ini-stepLength_final)/ (iteration_max)^n þstepLength_final; For i¼1:N num ¼(seed_max-seed_min)*(fitness(i)-WorstFitness)/ (BestFitness-WorstFitness) þseed_min; seed ¼ normrnd(0,stepLength,num); End For job_sequence_seed ¼MPV_sort(seed); If total_num(weed, seed) 4 P_SIZE pop¼ selectBetter(weed, seed, P_SIZE); Else pop¼ join(weed, seed); End If End While Output: select the best job_sequence(job_sequence_weed or job_sequence_seed) -F(job_sequence); End

4. Simulation experiments and results analysis 4.1. Test platform and parameters settings The experimental program testing platform as: Processor: CPU Intel Core i3-370, Frequency: 2.40 GHz, Memory: 4 GB, Operating system: Windows 7, Run software: Matlab7.6. Parameters settings is shown in Table 2. These parameters are all tested effectively by many experiments. In order to test the performance of IWO for solving NFSP, benchmarks designed by Taillard [27] are employed, which are composed of 12 sub-sets of given problems with the size from 20 jobs and 5 machines to 500 jobs and 20 machines, and each subset consists of 10 instances. 4.2. Comparison of IWO with exact method NEH

Table 1 Locations of individual and its corresponding jobs sequence. Locations

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2

3

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6

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0.06 4

2.99 2

1.86 3

3.73 5

1.86 6

0.67 1

Because the results of the same algorithm implemented by a different langue in a different computer may differ, we reimplemented the NEH. Performance improvement percent (PIP) is defined as |IWO  NEH|/NEH  100%. With regard to the length of paper, only the first instance of 12 different scale benchmarks is tested, and the results are shown in Table 3. Iteration graphs of Ta061–Ta111 are shown in Figs. 4–9. Table 4 shows the scheduling program obtained by IWO. It can be seen from Table 3 that the IWO algorithm obtained better makespan than the NEH algorithm for every instance of 12 different scale benchmarks. The highest PIP is 19.773%, and the average PIP is 12.363%. From Figs. 4–9, it can be seen that the IWO algorithm can effectively jump out of the local optimum and explore unceasingly until about 400 iterations. Table 4 shows the scheduling program of 12 instances. What should be noted is that more than one scheduling program can be obtained by IWO in terms of determinate makespan.

Table 2 Parameters settings. Parameter meaning

Variable

Value

Parameter meaning

Variable

Value

Lower bound of individual Upper bound of individual The initial number of population The maximum number of population The maximum iteration number

X_ min X_ max G_SIZE P_MAX iter_ max

 200 200 10 15 500

The The The The The

step_ini step_f inal seed_ max seed_ min n

100 0.001 15 1 4

initial variance final variance maximum number of seed generated minimum number of seed generated nonlinear modulation index

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Table 3 Comparison of IWO and NEH. Scale nm

Ta001 20  5 Ta011 20  10 Ta021 20  20 Ta031 50  5 Ta041 50  10 Ta051 50  20 Ta061 100  5 Ta071 100  10 Ta081 100  20 Ta091 200  10 Ta101 200  20 Ta111 500  20 Average PIP:12.363%

NEH Makespan

IWO Makespan

PIP

1486 2366 3991 3099 4171 6560 6150 7839 1,1723 1,3134 1,7911 3,6898

1389 2207 3226 3020 3465 5475 5839 6815 9405 1,1783 1,5217 3,0730

6.527% 6.720% 19.168% 2.549% 16.926% 16.539% 5.056% 13.062% 19.773% 10.286% 15.041% 16.716%

x 104

1.08 1.06 1.04 makespan

Problem

1.1

1.02 1 0.98 0.96 0.94

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Fig. 6. Iteration graph of Ta081.

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Fig. 4. Iteration graph of Ta061.

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Fig. 7. Iteration graph of Ta091.

7600 1.85

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Fig. 5. Iteration graph of Ta071. 1.5

Figs. 10 and 11 are respectively the Gantt charts of Ta001 and Ta011. The best scheduling solution can be obtained by the Gantt chart. For example, for Ta001, the best scheduling sequence is

0

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iteration

Fig. 8. Iteration graph of Ta101.

400

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x 10

3.7

12,11,15,3,6,19,8,13,16,14,9,17,5,7,1,18,2,4,20,10, the best Makespan is 1389.

4

4.3. Comparison of IWO with heuristic methods IG and KK

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makespan

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Fig. 9. Iteration graph of Ta111.

400

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IG [6] (improved greedy) is a heuristic method put forward by Baraz and Mosheiov in 2007 for solving NFSP. Their numerical tests showed that the IG heuristic method performed better. KK is a heuristic method with time complexity Oðmn2 Þ. Their numerical tests also showed that the KK algorithm performed better than NEH. In this section, each instance of 12 sub-sets conducts 5 replications, and each replication is compared to the solution produced by NEH. The average relative percent error (ARE) and average standard deviation (ASD) of the ARE and average time (AT) of each scale benchmark are calculated as the statistics for the performance measures. Results of IG and KK in the literature [12] are compared with those of IWO. Compared results are shown in Table 5. It can be seen from Table 5 that ARE obtained by IWO is better than both IG and KK. That is to say the makespan obtained by IWO is smaller than those by IG and KK; what is’ more the ASD of IWO is the smallest in three algorithms. In addition, the AT of each scale

Table 4 Scheduling program obtained by IWO. Problem

Scheduling program

Ta001 Ta011 Ta021 Ta031 Ta041 Ta051 Ta061

12,11,15,3,6,19,8,13,16,14,9,17,5,7,1,18,2,4,20,10 1389 17,4,2,9,10,11,8,5,16,12,20,7,6,18,1,13,19,14,15,3 2207 16,1,5,7,3,17,8,4,15,10,9,11,2,13,12,19,6,20,18,14 3226 31,34,49,25,30,9,48,4,45,46,13,2,26,8,12,38,23,42,32,11,6,22,14,41,1,29,44,27,17,28,19,35,39,47,15,33,18,10,36,43,21,7,20,5,24,16,37,40,50,3 3020 36,22,44,25,1,31,32,29,20,49,27,19,13,15,2,14,42,12,6,18,48,47,37,34,23,46,38,17,43,3,11,41,5,45,4,33,28,10,21,40,8,16,50,30,26,7,9,35,39,24 3465 43,31,39,12,24,6,15,10,29,47,38,7,36,9,32,48,42,17,22,50,44,3,33,19,28,18,16, 46,23,35,40,14,25,30,49,8,45,27,4,1,2,20,26,37,34,11,21,13,5,41 5475 10,87,75,58,49,2,27,5,74,40,22,16,44,31,11,93,52,82,20,100,84,64,71,14,25, 1,65,96,33,94,46,15,29,42,80,62,35,39,43,72,24,19,34,66,55,13,21,8,90,68, 5839 54,85,45,3,81,59,61,76,86,88,50,32,70,73,38,78,57,67,9,53,12,51,36,92,69, 7,83,37,56,79,47,91,95,41,26,4,28,18,97,98,63,30,48,60,6,23,77,99,89,17 93,51,21,34,6,25,39,38, 3,29,68,84,67,1,47,28,19,17,18,72,13,37,26,96, 7, 32,58,100,43,15,33,20,81,80,85,40,10,57,79,64,71,50,63,76,22,14,98,69,60,54, 6815 66,253,89,99,24, 9,56,4,61,23,92,31,74,42,73,35,78,41,87,75,86,11,49,62, 90,83,48,82,94,30,27,77,46,55,12,88,8,59,36,52,5,44,70,95,97,16,65,91,45 37,98, 5,21,72,25,89,81,79,45,24,40,91,36,47, 4,14, 7,42,27,38,58,80,23,28, 67, 90,48,55,33,74,44,95,64, 9,22,61,19,11,18,76,99,35,31,92,73,94,68,20,84, 9405 85,10, 83,17,65,71,70,15,60,86,96,43,34,75,32, 1,16,82, 2,39,53,93,30,62,66, 41,46,26, 3,78,100,50,69,63,51,54,12, 8,77,88,56,97,87, 6,29,49,57,13,52,59 1,1783 76,53,15,75,134,131,59,135,200,68,190,71,136,28,87,41,48,155,37,160,92,113,78,16,97,18,10,176,172, 83,8,98,164,108,110,142,120,1,196,145,138,105,79,181,167,128,36,159,63,74,177,191,144,64,115,137,14,88,81,198,189,111,174,32, 101,4,153,66,125,96,197,51,195,126,38,43,152,129,150,35,194,31,49,33,47,106,1163,20,57,94,44,175, 69,173,157,171,117,80,6,93,143,46,112,162,193,11,67,5,56,122,13,147,184,104,2,109,42,165,21,12,14 9,84,30,24,70,182,188,34,23,121,192,17,54,146,156,62,187,9,100,86,22,7,154,60,52,166,45,199,95,10 3,141,118,114,102,158,90,163,148,186,89,99,91,61,19,58,124,25,77,133,119,73,26,123, 139,185,107,170,161,82,85,29,65,50,130,169,151,183,180,179,140,72,127,178,39,27,132, 55, 40,168 1,5217 83,183,65,174,71,109,48,2,177,192,30,57,69,14,5,88,82,17,125,120,147,189, 139,196,123,135,97,140,89,37,79,87,151,194,25,56,22,130,4,146,27,108,169, 46,63,133,152,67,180,166,95,105,100,1,159,55,81,52,42,182,122,23,195,77,90,153,70,9,40,198,54,137, 35,6,199,176,181,115,190,41,13,38,132,102,31,110,33,154,111,98,51,61,21,113,145,158,142,68,138,128,175,84,200,162,10,143,134, 161,24,184,127,18,75,104,78,160,155,185,117,28,16,58,164,12,197,74,126,141,173,80,76,170,50,149,129,44,60,39,47,92,136,188,163,34,150,101,148,94,20, 103,66,64,168,32,121,86,131,19,72,107,7,45,62,8,186,144,11,187,119,15,156, 193,167,157,171,178,53,3,73,118,106,93,43,172,59,96,49,29,91,191,116,112,26,114,165,36,179, 99, 85,124 3,0730 15,327,13,433,264,279,361,37,12,59,86,190,100,30,293,47,14,421,201,54,488, 89,8,68,155,119,447,181,231,346,172,99,440,43,411,10,381,44,153,472,395, 183,297,366,39,177,479,149,338,430,349,240,375,315,116,166,148,388,79,51, 176,88,255,102,341,200,161,62,258,300,340,105,163,266,154,131,457,170,408,274,464,259,139,291,207, 76,329,322,81,286,487,396,74,110,298,299,394,22, 167,120,369,157,323,345,306,452,130,92,456,257,19,444,371,85,49,492,463, 310,50,194,476,28,143,278,478,336,384,324,403,429,280,26,101,7,358,477,268,458,485,281,179,87,419, 57,121,4,267,158,374,449,146,184,171,316,75,412, 118,174,398,222,446,18,245,71,242,484,495,434,91,228,391,31,60,332,217,218,32,344,138,150,107,254, 152,401,343,409,454,25,202,337,189,308,147,342,459,219,234,156,132,480,468,354,61,442,422,353,93,247,221,198,55,489,114,238, 52,23,387,208,423,382,104,352,453,236,98,117,372,275,233,443,16,326,491, 216,53,392,490,348,3,141,97,441,203,45,380,406,450,465,17,335,5,249,134, 427,78,431,271,289,368,225,95,331,58,351,24,40,164,112,230,191,428,496,113,33,416,312,36,20,90,49 9,334,321,229,402,304,84,339,284,140,211,292,436, 175,48,437,379,360,282,35,432,307,470,77,96,333,21,438,383,239,215,34,197,122,178,439,404,426,42, 251,363,295,461,206,277,142,214,241,196,253,424, 204,256,347,192,169,500,173,135,69,186,212,497,250,27,407,2,448,364,469, 471,288,405,359,357,168,11,193,276,435,320,63,260,144,303,270,220,325,386,309,6,224,66,350,80,82, 373,451,455,187,397,482,151,127,252,460,70,29,73, 486,377,285,160,106,389,393,199,108,296,356,318,235,137,109,305,38,367, 410,126,65,269,145,64,133,425,319,246,129,128,473,243,483,244,378,9,385, 290,399,248,209,415,462,180,362,474,103,272,355,72,376,330,466,365,111, 418,317,493,213,162,185,227,283,223,125,263,414,445,262,165,311,400,287, 136,370,313,195,123,210,83,46,188,498,273,294,1,420,182,94,481,301,328,494,413,56,314,265,390,232,226,159,205,124, 41,237,261,417,467,475,302,115, 67

Ta071 Ta081 Ta091

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Table 5 Comparison of IWO, and IG and KK.

benchmark obtained by IWO is smaller than those by IG and KK. The most position value of AT IWO used is only 68.8 s. So, we can say that IWO can effectively solve NFSP with better robustness.

Scale

IG

KK

IWO

nm

ARE

ARE

ARE

ASD

AT

20  5 20  10 20  20 50  5 50  10 50  20 100  5 100  10 100  20 200  10 200  20 500  20 Average

9.19 8.37 5.40 11.53 12.34 12.44 16.40 13.98 14.78 17.18 16.38 18.68 13.055

0.87 2.00 1.29  0.03  1.79  0.55  0.24 0.08  2.57  0.55  2.23  1.65  0.447

 8.507  12.946  14.135  7.201  15.218  17.311  4.914  12.074  17.004  9.113  15.058  12.474  12.162

0.028 0.038 0.025 0.035 0.029 0.027 0.012 0.021 0.029 0.018 0.016 0.029 0.025

0.9 s 1.2 s 1.6 s 1.3 s 2.6 s 4.1 s 3.0 s 4.2 s 8.8 s 9.2 s 31.9 s 68.8 s 11.466 s

4.4. Comparison of IWO with intelligence algorithm PSOvns and HDPSO In [28], a PSO algorithm called PSOvns with encoding scheme based on random key representation and variable neighborhood search (VNS) is proposed for flow shop problem. Their numerical results show that PSOvns performance is better than VNS and GA. In [12], a hybrid discrete PSO (HDPSO) is put forward to solving NFSP. The numerical results show that HDPSO is better than other heuristic methods and PSO algorithms. In this section, results obtained by IWO with the same criterion in Section 4.2 are compared with that in [12] of PSOvns and HDPSO. Compared results are shown in Table 6.

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Table 6 Comparison of IWO, and PSOvns and HDPSO. Scale

PSOvns

HDPSO

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ARE

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AT

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AT

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AT

20  5 20  10 20  20 50  5 50  10 50  20 100  5 100  10 100  20 200  10 200  20 500  20 average

 3.01  7.16  6.81  0.75  5.75  6.65  0.72  1.40  4.66  1.41  3.23  1.85  3.62

0.04 0.15 0.27 0.00 0.22 0.42 0.03 0.11 0.33 0.12 0.31 0.15 0.18

2.0 s 2.0 s 2.0 s 5.0 s 5.0 s 5.0 s 10.0 s 10.0 s 10.0 s 20.0 s 20.0 s 50.0 s 11.75

 3.03  7.28  7.04  0.75  5.88  7.27  0.73  1.55  5.36  1.54  3.71  2.16  3.86

0.00 0.02 0.03 0.00 0.10 0.21 0.00 0.03 0.08 0.02 0.13 0.10 0.06

2.0 s 2.0 s 2.0 s 5.0 s 5.0 s 5.0 s 10.0 s 10.0 s 10.0 s 20.0 s 20.0 s 50.0 s 11.75

 8.507  12.946  14.135  7.201  15.218  17.311  4.914  12.074  17.004  9.113  15.058  12.474  12.162

0.028 0.038 0.025 0.035 0.029 0.027 0.012 0.021 0.029 0.018 0.016 0.029 0.025

0.9 s 1.2 s 1.6 s 1.3 s 2.6 s 4.1 s 3.0 s 4.2 s 8.8 s 9.2 s 31.9 s 68.8 s 11.466 s

It can be seen from Table 6 that ARE obtained by IWO is better than those by PSOvns and HDPSO. That is to say the makespan obtained by IWO is smaller than those by IG and KK. The reason is that the global exploration of IWO based on normal distribution is better than two other algorithms. Early in the algorithm, it can search solution with a large range, and as the algorithm proceeds, more good solutions are found among the solutions found. In addition, the ASD of some scale benchmarks obtained by IWO is bigger than PSOvns and HDPSO, but the average of ASD is smaller than those of PSOvns and HDPSO. That is to say, the robustness of IWO, which has nothing to do with platform (computer), is better than PSOvns and HDPSO. In terms of AT, 10 scale benchmarks out of 12 used by IWO are shorter than PSOvns and HDPSO and the average AT is the smallest in 3 algorithms, while the average AT of Ta101 and Ta111 is larger than two other algorithms. This also says that the AT has nothing to do with computer. If our computer is much faster than the other authors’, AT of all the benchmarks should be shorter. So, we can conclude again that the IWO can effectively solve NFSP with better robustness.

5. Conclusions In this paper, an Invasive Weed Optimization (IWO) scheduling algorithm is presented for solving no-idle flow-shop scheduling problem (NFSP). Firstly, a simple approach is put forward to calculate the makespan of job sequence. Secondly, the most position value (MPV) method is used to code the weed individuals so that fitness values can be calculated. Finally, different algorithms are used to compare with IWO and the results show that IWO can effectively solve NFSP with better robustness. As the IWO algorithm has proved much better than the others in terms of both final accuracy and convergence we expect that it will be an attractive alternative tool for different types of optimization problems.

Acknowledgments This work is supported by National Science Foundation of China (61165015), Key Project of Guangxi Science Foundation (2012GXNSFDA053028), Key Project of Guangxi High School Science Foundation (20121ZD008) and funded by open research fund program of Key Lab of Intelligent Perception and Image Understanding of Ministry of Education of China under Grant (IPIU01201100).

IWO

References [1] P. Baptiste, K.H.A Lee. branch and bound algorithm for the F|no-idle|Cmax, in: Proceedings of the International Conference on Industrial Engineering and Production Management, Lyon, 1997, pp. 429–438. [2] N.E.H. Saadani, P. Baptiste, M. Moalla, The simple F2//Cmax with forbidden tasks in first or last position: a problem more complex than it seems, Eur. J. Oper. Res. 161 (1) (2005) 21–31. [3] Johnson’s rule. 〈http://www.en.wikipedia.org/wiki/Johnson’s_Rule〉, 2012 (accessed 20.4.12). [4] M. Nawaz, E.E. Enscore, I. Ham, A heuristic algorithm for the m-machine, n-job flow shop sequencing problem, Omega 11 (1) (1983) 91–95. [5] P.J. Kalczynski, J. Kamburowski, A heuristic for minimizing the makespan in no-idle permutation flow shop, Comput. Ind. Eng. 49 (1) (2005) 146–154. [6] D. Baraz, G. Mosheiov, A note on a greedy heuristic for the flow shop makespan minimization with no machine idle-time, Eur. J. Oper. Res. 184 (2) (2008) 810–813. [7] B.B. Li, L. Wang, A hybrid quantum-inspired genetic algorithm for multiobjective flow shop scheduling, IEEE Trans. Syst., Man, Cybern., Part B 37 (3) (2007) 576–591. [8] W.J. Ren, Q.K. Pan, H.Y. Han, Tabu search algorithm for no-idle flow shop scheduling problems, Comput. Eng. Des. 31 (23) (2010) 5071–5074. [9] L. Wang, Intelligence optimization algorithm with applications, Tsinghua University Press, Beijing, 2001. [10] F.R. Zhang, J.H. Duan, R.B. Pang, et al., Ant colony heuristic algorithm for noidle flow shop scheduling problem, Appl. Res. Comput. 28 (3) (2011) 859–861. [11] Q.K. Pan, L. Wang, A novel differential evolution algorithm for no-idle permutation flow shop scheduling problems, Eur. J. Ind. Eng. 2 (3) (2008) 279–297. [12] Q.K. Pan, L. Wang, No-idle permutation flow shop scheduling based on a hybrid discrete particle swarm optimization algorithm, Int. J. Adv. Manuf. Technol. 39 (7–8) (2008) 796–807. [13] L. Wu, Q.K. Pan, H.Y. Sang, et al., Harmony search algorithms for no-idle flow shop scheduling problems, Comput. Integr. Manuf. Syst. 15 (10) (2009) 1960–1967. [14] A.R. Mehrabian, C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecol. Inform. 1 (4) (2006) 355–366. [15] Ritwik Giri, Aritra Chowdhury, Arnob Ghosh, et al., A modified invasive weed optimization algorithm for training of feed-forward neural networks, in: Proceedings of the IEEE International Conference on Systems Man and Cybernetics (SMC), Istanbul, IEEE, 2010, pp. 3166–3173. [16] Huan Zhao, Pei-hong Wang, Xianyong Peng, et al. Constrained optimization of combustion at a coal-fired utility boiler using hybrid particle swarm optimization with invasive weed, in: Proceedings of the International Conference on Energy and Environment Technology (ICEET), Washington, DC, IEEE Computer Society, 2009, pp. 564–567. [17] D. Kundu, K. Suresh, S. Ghosh, et al. Designing fractional-order PIλDμ controller using a modified invasive weed optimization algorithm, in: Proceedings of the 2009 World Congress on Nature & Biologically Inspired Computing, 2009, pp. 1315-1320. [18] A. Sengupta,, T. Chakraborti,, A. Konar,. Energy efficient trajectory planning by a robot arm using invasive weed optimization technique, in: Proceedings of the Third World Congress on Nature and Biologically Inspired Computing, 2011, pp. 311-316. [19] Ho-Lung Hung, Chien-Chi Chao, Chia-Hsin Cheng. Invasive weed optimization method based blind multi-user detection for MC-CDMA interference suppression over multipath fading channel, in: Proceedings of the IEEE International Conference on Systems Man and Cybernetics (SMC), 2010, pp. 2145–2150. [20] Zhihua Chen, Shuo Wang, Zhonghua Deng. Tuning of auto-disturbance rejection controller based on the invasive weed optimization, in: Proceedings

Author's personal copy 292

[21]

[22]

[23]

[24]

[25]

[26]

[27] [28]

[29]

[30]

[31] [32]

[33] [34]

[35]

[36] [37]

[38]

Y. Zhou et al. / Neurocomputing 137 (2014) 285–292

of the Sixth International Conference on Bio-Inspired Computing: Theories and Applications, 2011, pp. 314–318. Mohsen Ramezani Ghalenoei, Hossein Hajimirsadeghi, Caro Lucas, Discrete invasive weed optimization algorithm: application to cooperative multiple task assignment of UAVS, in: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 Held Jointly with the 2009 28th Chinese Control Conference, 2009, pp. 1695–1670. Gourab Ghosh Roy, Swagatam Das, Prithwish Chakraborty, Design of nonuniform circular antenna arrays using a modified invasive weed optimization algorithm, IEEE Trans. Antennas Propag. 59 (1) (2011) 110–118. X. Zhang, Y. Wang, G. Cui, et al., Application of a novel IWO to the design of encoding sequences for DNA computing, Comput. Math. Appl. 57 (11/12) (2009) 2001–2008. A.R. Mehrabian, A. Yousefi-Koma, A novel technique for optimal placement of piezoelectric actuators on smart structure, J. Frankl. Inst. 348 (1) (2011) 12–23. Su Shoubao, Fang Jie, Wang Jiwen, et al., Image clustering method based on invasive weed colonization, J. South China Univ. Technol. (Nat. Sci. Ed.) 36 (5) (2008) 95–105. Su Shoubao, Wang Jiwen, Zhang Ling, et al., An invasive weed optimization algorithm for constrained engineering design problems, J. Univ. Sci. Technol. China 39 (8) (2009) 885–893. E. Taillard, Benchmarks for basic scheduling problems, Eur. J. Oper. Res. 64 (2) (1993) 278–285. M.F. Tasgetiren, Y.C. Liang, M. Sevkli, et al., A particle swarm optimization algorithm for makespan and total flow time minimization in permutation flow shop sequencing problem, Eur. J. Oper. Res. 177 (3) (2007) 1930–1947. H.Y. Sang, Q.K. Pan, L. Wu, et al., Effective hybrid differential evolution algorithms for lot-streaming flow shop scheduling problem, Comput. Eng. Appl. 46 (21) (2010) 47–50. D.S Huang, Ji-Xiang Du, A constructive hybrid structure optimization methodology for radial basis probabilistic neural networks, IEEE Trans. Neural Netw. 19 (12) (2008) 2099–2115. D.S. Huang, Radial basis probabilistic neural networks: model and application, Int. J. Pattern Recognit. Artif. Intell. 13 (7) (1999) 1083–1101. Fei Han, Qing-Hua Ling, D.S. Huang, An improved approximation approach incorporating particle swarm optimization and a priori information into neural networks, Neural Comput. Appl. 19 (2) (2010) 255–261. Li Bo, D.S. Huang, Locally linear discriminant embedding: an efficient method for face recognition, Pattern Recognit. 41 (12) (2008) 3813–3821. Bo Li, D.S. Huang, Chao Wang, Kun-Hong Liu, Feature extraction using constrained maximum variance mapping, Pattern Recognit. 41 (11) (2008) 3287–3294. Fei Han, Qing-Hua Ling, D.S. Huang, Modified constrained learning algorithms incorporating additional functional constraints into neural networks, Inf. Sci. 178 (3) (2008) 907–919. D.S. Huang, Mi. Jian-Xun, A new constrained independent component analysis method, IEEE Trans. Neural Netw. 18 (5) (2007) 1532–1535. Ji-Xiang Du, D.S. Huang, Xiao-Feng Wang, Xiao Gu, Shape recognition based on neural networks trained by differential evolution algorithm, Neurocomputing 70 (4–6) (2007) 896–903. Jun Zhang, D.S. Huang, Tat-Ming Lok, Michael R. Lyu, A novel adaptive sequential niche technique for multimodal function optimization, Neurocomputing 69 (16–18) (2006) 2396–2401.

[39] Huan Chen, Yongquan Zhou, Sucai He, Xinxin Ouyang, Peigang Guo, Invasive weed optimization algorithm for solving permutation flow-shop scheduling problem, J. Comput. Theor. Nanosci. 10 (3) (2013) 708–713. [40] Yongquan Zhou, Qifang Luo, Huan Chen, Jinzhao Wu, Using differential evolution invasive weed optimization for solving circle packing problem, Adv. Sci. Lett. 19 (6) (2013) 1807–1810.

Yongquan Zhou, Ph.D and Prof, received the MS degree in computer science from Lanzhou University, Lanzhou, China, in 1993 and the PhD degree in computation intelligence from the Xiandian University, Xi’an, China, in 2006. He is currently a professor at Guangxi University for Nationalities. His research interests include computation intelligence, neural networks, and intelligence information processing, etc. He has published 1 book, and more than 150 research papers in journals.

Huan Chen, M.S., received the BS degree from Henan University of Science and Technology, Zhengzhou, China, in 2010. His current research interest is in computation intelligence, swarm intelligence algorithm.

Guo Zhou, Ph.D., received the BS degree in network engineering from Beijing University of Posts and Telecommunications, Beijing, China, in 2010. His current research interest is in computer graphics and vision, machine learning and web development.