a hybrid biogeography-based optimization algorithm for job shop scheduling...

Computers & Industrial Engineering 73 (2014) 96–114

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Computers & Industrial Engineering

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A hybrid biogeography-based optimization algorithm for job shopscheduling problem

http://dx.doi.org/10.1016/j.cie.2014.04.0060360-8352/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 10 82317318.E-mail address: [email protected] (H. Duan).

Xiaohua Wang, Haibin Duan ⇑State Key Laboratory of Virtual Reality Technology and Systems, School of Automation Science and Electrical Engineering, Beihang University (BUAA), Beijing 100191, PR China

a r t i c l e i n f o

Article history:Received 8 July 2013Received in revised form 30 December 2013Accepted 15 April 2014Available online 9 May 2014

Keywords:Job-shop scheduling problem (JSP)Biogeography-based optimization (BBO)Chaos theoryHybrid biogeography-based optimization(HBBO)

a b s t r a c t

In this paper, a hybrid biogeography-based optimization (HBBO) algorithm has been proposed for thejob-shop scheduling problem (JSP). Biogeography-based optimization (BBO) is a new bio-inpired compu-tation method that is based on the science of biogeography. The BBO algorithm searches for the globaloptimum mainly through two main steps: migration and mutation. As JSP is one of the most difficultcombinational optimization problems, the original BBO algorithm cannot handle it very well, especiallyfor instances with larger size. The proposed HBBO algorithm combines the chaos theory and ‘‘searchingaround the optimum’’ strategy with the basic BBO, which makes it converge to global optimum solutionfaster and more stably. Series of comparative experiments with particle swarm optimization (PSO), basicBBO, the CPLEX and 14 other competitive algorithms are conducted, and the results show that our pro-posed HBBO algorithm outperforms the other state-of-the-art algorithms, such as genetic algorithm (GA),simulated annealing (SA), the PSO and the basic BBO.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Knowledge and services have been emphasized in modern man-ufacturing in the present decade. Many enterprises are facing chal-lenges of unprecedented and abrupt changes, such as intenseglobal competition, rapid IT infrastructure changes and develop-ment of advanced management technologies. Due to its impor-tance in the field of manufacturing industries, the job-shopscheduling problem (JSP) attracts considerable attention (Ma,Chu, & Zuo, 2010; Michael et al., 2013; Zhang, 2010).

The JSP is one of the most difficult combinatorial optimizationproblems and it has been well known as an NP-hard problem(Lawler et al., 1982; Xia & Wu, 2005). The JSP can be describedbriefly as follows. There are a number of jobs that need to be pro-cessed on a number of machines. Each job has several operationswhich have to be processed in a predefined sequence on differentmachines and each machine can only process one job at a time(Sels et al., 2012). The goal is to schedule the jobs on the machinesto minimize the completion time needed for processing all jobs.

Because of the practical importance of the JSP, huge researcheffort has been devoted to fields of engineering, computer science,management science and business. Numerical efficient methodshave been proposed, such as guided local search with shifting bot-

tleneck (SB) (Balas & Vazacopoulos, 1998), tabu search (TS)(Azzouz et al., 2012; Brucker & Neyer, 1998; Eshlaghy &Sheibatolhamdy, 2011; Eugeniusz & Czesław, 2005; Nowicki &Smutnicki, 1996; Watson, Christopher Beck, Howe, & DarrellWhitley, 2003) and simulated annealing (SA) (Atabak et al., 2011;Van Laarhoven, Aarts, & Lenstra, 1992). Since that biologicallyinspired techniques have been advanced in the literature (Duan,Shao, Su, & Zhang, 2010; Duan, Xu, & Xing, 2010; Liu, Duan, &Deng, 2012; Sun, Duan, & Shi, 2013; Xu, Duan, & Liu, 2010; Yu &Duan, 2013; Zhang & Duan, 2014), there are also many bio-inspiredoptimization algorithms applied to the JSP, such as genetic algo-rithm (GA) (Azzouz et al., 2012; De Giovanni & Pezzella, 2010;Pezzella, Morganti, & Ciaschetti, 2007; Sun et al., 2010; Zhang,Shao, Lia, & Gao, 2009), particle swarm algorithm (PSO) (Gu,Tang, & Zheng, 2012; Lian, Jiao, & Gu, 2006; Xia, Wu, Zhang, &Yang, 2004), ant colony algorithm (ACO) (Anitha & Karpagam,2013; Azzouz et al., 2012; Chen & Zhang, 2013; Xing & Chena,2010), and artificial bee colony (ABC) algorithm (Deming, 2012;Tasgetiren et al., 2011; Wang, Duan, & Luo, 2013; Wang, Wang,Yu, & Zhang, 2013). In this paper, a hybrid biogeography-basedoptimization (HBBO), which takes advantages of the chaos theory(Xu et al., 2010) and the ‘‘searching around the optimum’’ strategy,is structured to solve the JSP.

Biogeography-based optimization (BBO) (Simon, 2008) is a newalgorithm, which emulates the geographical distribution and themigration of species in an ecosystem. In this method each feasible

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solution is represented by a habitat. The habitat suitability index(HSI) is utilized to measure the goodness of the habitat. In order tofind a solution with the best aspects, the concepts and models of bio-geography are applied. The basic BBO works mainly based on thetwo mechanisms: migration and mutation. With the migrationmechanism, poor solutions can accept a lot of new features, whichmay improve the quality of those solutions. Furthermore, solutionsdo not have the tendency to clump together in similar groups due tothe new type of mutation operation. Elitism operation (Simon et al.,2009), which retains the best solutions in the population from onegeneration to the next, can also make the basic BBO algorithm moreefficient in the both aspects of migration and mutation.

The chaos theory and ‘‘searching around the optimum’’ strategyare integrated into the basic BBO. Therefore, the exploration abilityof the algorithm can be enhanced and the diversity of populationcan be improved. Furthermore, premature converge can beavoided. Thus the potential weakness lying in the basic methodis avoided. We verify experimentally that our proposed HBBOapproach outperforms the basic BBO approach, and is significantlybetter than the PSO and several state-of-art approaches.

The remainder of this paper is organized as follows. In Section 2,the JSP is formulated. In Section 3, the main principle of the basicBBO with elitism is presented. Section 4 describes the chaos theoryand the principle of the ‘‘searching around the optimum’’ strategy.In Section 5, the detailed implementation procedures of the HBBOalgorithm for the JSP are specified. In Section 6, a series of comparativeexperiments are given to verify that the HBBO approach outperformsthe CPLEX, the PSO, the basic BBO and numerical existing algorithms.Our concluding remarks are contained in the final section.

2. The mathematical description of the JSP

Suppose that there are m machines M = {M1, M2, . . ., Mm} and njobs J = {J1, J2, . . ., Jn}, where job i has p operations Oi = {Oi1, Oi2, . . .,Oip}, i = 1, 2, . . ., n. In this paper we make the number of the oper-ations in each job the same, and it is also same to the number of themachines. Thus, ip = m, i = 1, 2, . . ., n and the operation set can bedescribed as O = {r1, r2, . . ., rn�m}. Makespan Cmax is the total timecost in completing operations of all jobs under conventionalassumptions. The conventional assumptions can be described asfollowing (Allaoui & Artiba, 2004; Wang, Duan, et al., 2013;Wang, Wang, et al., 2013).

(1) Each operation of one job must be processed exactly once oneach of m machines.

(2) Each job is routed through the m machines in a given orderto ensure that the j-th operation of job i Oij will be processedonly when the processing of the (j�1)-th operation Oij�1 hasbeen finished.

(3) Every machine can process only one job at any time, andevery operation cannot be interrupted.

(4) The machines cannot be breakdown.(5) The jobs are independent from each other.

Matrix S (m�n) is developed to represent that the sij-th operationof the job i is processed on machine j, s is the element of the matrix,and i is the column and j is the row of the matrix.

S ¼

s11 s12 . . . s1m

s21 s22 . . . s2m

..

. ... . .

. ...

sn1 sn2 . . . snm

266664

377775

Another matrix T (m�n) is developed to represent the time cost tij ofthe j-th operation of the job i, i is the column and j is the row of thematrix.

T ¼

t11 t12 . . . t1m

t21 t22 . . . t2m

..

. ... . .

. ...

tn1 tn2 . . . tnm

266664

377775

The processing of operation rj on the machine i is denoted by Gij.Suppose that the p-th operation on machine i, Gip, is directlyconnected by the q-th operation Giq, the relation can be representedby the operator Gip ? Giq. The completion time of Gip ? Giq can becalculated with Ciq = Cip + tiq + tk, where tiq is the time cost of theoperation rq on the machine i and tk is the space time on machinei during the convert of the two operations. Thus the completiontime of all the jobs can be computed as Cmax ¼ maxallGij

ðCijÞ.Therefore, the aim of the JSP is to make the makespan Cmax

minimized.

3. Biogeography-based optimization

3.1. Principles of the basic BBO algorithm

BBO is a bio-inspired computation algorithm inspired by thegeographical distribution and the migration of species in an eco-system. The problem can be of any area in life (Engineering, Eco-nomics, Medicine, Business, Urban Planning, Sports, etc.) as longas we have a qualitative measure of the suitability of a given solu-tion (Bansal et al., 2013; Jain et al., 2012; Jamuna & Swarup, 2011;Li, Wang, Zhou, & Yin, 2011; Ma & Simon, 2011; Sayed, Saad,Emara, & Abou El-Zahab, 2013; Silva, Coelho, Lebensztajn, &Lebensztajn, 2012; Wang & Duan, 2013). The BBO technique is uti-lized to solve the JSP in this paper.

BBO is developed based on the mathematics models of bioge-ography, which explains how species emigrate and immigratewithin the habitats, how new species arise, and how speciesbecome extinct. A quantitative performance index HSI is used toevaluate if a habitat is suitable for biological survival. Featuresthat correlate with HSI include such factors as rainfall, diversityof vegetation, diversity of topographic features, land area andtemperature (Simon, 2008). The variables that characterize habit-ability are named suitability index variables (SIVs). SIVs are theindependent variables of the habitat, while HSI are the dependentvariable.

A set of habitats is used to present the candidate solutions inthe BBO. The basic BBO works mainly on two mechanisms,migration and mutation. HSI is the evaluation criteria to measureif a solution is good, analogous to fitness in other population-based optimization algorithms. The species migrate from onehabitat to other habitats that have good geographical conditions.Habitats with a high HSI tend to have a larger number of species,more species that emigrate to nearby habitats, and a lower spe-cies immigration rate. The solutions with high HSI tends to sharetheir features with those with low HSI, and poor solutions canaccept a lot of new features from good solutions. Mutation is aprobabilistic operator that randomly modifies habitat SIVs basedon the habitats a priori probability of existence. Suppose that wehave a habitat H, a vector of SIVs, following the migration andmutation steps to reach the optimal solution. In this way, newcandidate habitat is generated from all of the salutation inpopulation.

3.2. The migration strategy

BBO migration is a probabilistic operator that adjusts each solu-tion Hi by sharing features between solutions. In the BBO, themigration strategy is similar to the evolutionary strategy in whichmany parents can contribute to a single offspring (Li et al., 2011).

98 X. Wang, H. Duan / Computers & Industrial Engineering 73 (2014) 96–114

Migration can be expressed as Hi(SIV) Hj(SIV) (Ma & Simon, 2011).Each individual has its own immigration rate k and emigration rate l,which are related with the number of species on the island.When more species inhabit the island, the immigration rate reduceswhile the emigration rate increases (Wang, Duan, et al., 2013;Wang, Wang, et al., 2013). Therefore, the immigration and emigrationrates can be calculated as follows when there are k species in thehabitat.

ls ¼ ESSmax

ks ¼ I 1� SSmax

� �8<: ð1Þ

where I is the maximum possible immigration rate, which occurswhen there are zero species on the island. The largest possible num-ber of species that the habitat can support is Smax, at which pointthe immigration rate becomes zero and the maximum possible emi-gration rate E occurs. The immigration and emigration curves areshown in Fig. 1.

The equilibrium species number S0 is reached when the emigra-tion rate l is equal to the immigration rate k.

Suppose that there are N habitats, Hi is one of them, whoseimmigration rate is ki, and Hj is another habitat whose emigrationrate is lj. The migration process can be presented in Fig. 2.

The probability Ps, which represents that the habitat containsexactly species S, changes from time t to time t + Dt. It is updatedas follows:

Psðt þ DtÞ ¼ PsðtÞð1� ksDt � lsDtÞ þ Ps�1ks�1Dt þ Psþ1lsþ1Dt ð2Þ

Fig. 2. The migration p

Fig. 1. The relationship between the number of species and the migration rates.

3.3. The mutation strategy

As the HSI of a habitat can change suddenly due to apparentlyrandom events, there is mutation of SIVs in the BBO. Solutions withvery high HSI and very low HSI are both equally improbable, whilemedium HSI solutions are relatively probable to mutate. Mutationis used to enhance the diversity of the population, which helps todecrease the chances of getting trapped in local optima. Moreover,elitism (copying some of the fittest individuals to the next genera-tion) is applied to guarantee the survival of the best individual(s)(Simon et al., 2009). Mutation is a probabilistic operator that ran-domly modifies an SIV of a solution based on its a priori probabilityof existence. Species count probabilities PS, computed from ks andls with equation (2), is used to determine mutation rates. Supposea habitat with S species is selected to execute the mutation opera-tion, randomly modify a chosen variable (SIV) based on its associ-ated probability PS. The mutation rate m(s) can be computedaccording to the following function proportional to PS:

mðsÞ ¼ mmax 1� ps

pmax

� �ð3Þ

where mmax is a user-defined parameter, and Pmax is the maximumspecies count probability, while PS is the probability of existence ofS species in the habitat.

Suppose that habitat Hi e SIVR represents a feasible solution tosome problem, the mutation process can be presented in Fig. 3.

4. The HBBO method

The BBO is a flexible, versatile and robust algorithm. However,there are still some weak points of the basic algorithm when usedto solve the JSP, such as requiring a large number of iterations toreach the global optimal solution and the tendency to convergeprematurely. Therefore, the chaos theory and ‘‘searching aroundthe optimum’’ strategy are introduced into the basic BBO to accel-erate the process of looking for the scheduling solution.

4.1. Chaos theory

Chaos theory, which is discovered by a meteorologist namedEdward Lorenz (Lorenz, 1963), studies the behavior of dynamicalsystems that are highly sensitive to initial conditions. It is an effectwhich is popularly referred to as the butterfly effect. The butterflyeffect phenomenon, common to chaos theory, is also known assensitive to initial conditions, which means that each point in sucha system is arbitrarily closely approximated by other points withsignificantly different in future.

Small differences in initial conditions yield widely divergingoutcomes in chaotic systems, rendering long-term prediction

rocess of the BBO.

Fig. 3. The mutation process of the BBO.


impossible in general. Besides the sensitivity to initial conditions,chaotic systems have another two properties. One is that an infi-nite number of unstable periodic orbits embed in the underlyingchaotic set. The other is that the dynamics in the chaotic systemis ergodic, which means that the system ergodically visits a smallneighborhood of every point in each one of the unstable periodicorbits embedded in its temporal evolution.

The criterion for a dynamical system to be classified as a chaoticsystem is described as follows. It must be sensitive to initial condi-tions, and it must be topologically mixing, moreover its periodicorbits must be dense. The basic idea of chaos theory can be under-stood by considering the following well-known one-dimensionallogistic map, which is one of the best studied chaotic systems(Liu et al., 2012; Xu et al., 2010).

xnþ1 ¼ gxnð1� xnÞ ð4Þ

where n is the number of habitats, xn is restricted to the unit inter-val [0, 1] where 1 represents the maximum possible population and0 represents extinction. g is the growth rate of xn. A very small dif-ference in the initial value of xn would give rise to large difference inits long-time behavior, which is the basic characteristic of chaos.Moreover, when the value of g = 4 and xn – 0.25, 0.5, 0.75 will even-tually visit every neighborhood in a subinterval of [0, 1] with thebest periodicity. Hence the logistic map is defined as following.

xnþ1 ¼ 4xnð1� xnÞ ð5Þ

This logistic map develops chaos by the period-doubling bifurcationroute. The track of chaotic variable can travel ergodically over thewhole space of interest in the neighborhood of a periodic orbit.Despite its variation looks like in disorder, the variation of the cha-otic variable has a delicate inherent rule. In the proposed HBBOalgorithm, chaos theory is used to initialize the habitats. In thisway, the ergodicity and irregularity of the chaotic variable canincrease the diversity of the habitats to increase the speed of reach-ing the optimal solution.

4.2. The ‘‘searching around the optimum’’ strategy

As the basic BBO has a tendency to converge prematurely, astrategy of ‘‘searching around the optimum’’ is proposed. In thisstrategy, an effective method that identifies premature stagnationis embedded to the BBO. Thus, once premature stagnation happens,a randomized solution, as a substitute for current optimum, is usedto change the current searching locus. Hence, the habitats can getout of the local optima.

In each generation of the BBO an optimal solution can bereached. When the optimal solution has no change in the processof continuous MAX iterations, we think that the algorithm hasthe potential of stagnation and the habitats have been or will soonget into local optimal solution. MAX is a threshold given by the user

according the specified problem and the experiment results. Agreater MAX indicates that the standard of the premature stagna-tion judgment is more easing.

A counter C is used to record the number of continuous itera-tions that have the same optimal solution. If the current iterationhas the same optimal solution with the last iteration, C = C + 1.Otherwise, C = 0. When C = MAX, we think that the algorithm can-not jump out of the local optima. Then use a random value toreplace a randomly selected dimension of the current optimal solu-tion, which can be described as follows:

Ui ¼ ðui1;ui2; . . . ;uiDÞ ð6Þ

where uik e[mink, maxk], k e[1, D] is randomly chosen, and Ui is theoptimal solution of current iteration.

This strategy can make a fine tuning for the global optimal solu-tion with a random disturbance and help the algorithm jump out ofthe local optimal solution. With this stagnation judgment and therandom disturbance, the invalid iteration can be effectivelyreduced. Thus the convergence speed of the algorithm can beimproved and the accuracy of the optimization results can behigher, which make the algorithm more stable.

5. Implementation of the HBBO for the JSP

5.1. Encoding strategy

Because the JSP is a combinatorial optimization problem, encod-ing is necessary in the HBBO algorithm. Encoding refers to the hab-itats expression, which is a key problem in the application of theHBBO method for the JSP. Think about the combination characteris-tics and complicated constraint of the JSP, the encoding of the HBBOfor the JSP is much more complicated than other schedule problemssuch as the flow-shop scheduling problem. The Lamarkian character,the complexity of the decoding, coding space characteristics, and thestorage requirements all have to be taken into account. Hence,encoding based on the process is used in the HBBO algorithm.

For a JSP with n jobs processed on m machines, the encoding ofhabitats can be described as following.

½h1;h2; . . . ;hi; . . . ;hn�m� ð7Þ

where hi is the sequence number of a job. If hi is the n-th appearanceof the job number, the n-th operation is on processing. This kind ofencoding can guarantee the feasibility of random schedulingresults.

5.2. Decoding strategy

After the completion of the optimization, the scheduling resulthas to be decoded from the habitats. Take a JSP with 3 jobs to beprocessed on 3 machines for example. There are 3�3 operations

Fig. 4. GATT chart of the 3�3 instance.

Fig. 5. Flow chart of our proposed HBBO for the JSP.

Table 1Information of the 6�6 JSP instance.

Job Operationsequence

Machineassigned

Processingtime


Machineassigned

Processingtime

Job1 O11 3 1 Job4 O41 2 5O12 1 3 O42 1 5O13 2 6 O43 3 5O14 4 7 O44 4 3O15 6 3 O45 5 8O16 5 6 O46 6 9




in a complete scheduling, thus there should be 3�3 SIVs in eachhabitats. Suppose that the initial scheduling of the HBBO is[1,2,3,2,2,3,1,3,1], and the operation-machine matrix is given asfollowing:

2 1 32 3 13 2 1

264

375:

The scheduling result of the habitat is presented as following.(1) Execute the first operation of job1 on the machine 2.(2) Execute the first operation of job2 on the machine 2.(3) Execute the first operation of job3 on the machine 3.(4) Execute the second operation of job2 on the machine 3.(5) Execute the third operation of job2 on the machine 1.(6) Execute the second operation of job3 on the machine 2.(7) Execute the second operation of job1 on the machine 1.(8) Execute the third operation of job3 on the machine 1.(9) Execute the third operation of job1 on the machine 3.

In addition, suppose that the time cost matrix is given asfollowing.

10 6 35 7 49 13 8

264

375

0 10 20 30 40 50 60

1

2

3

4

5

6P

roce

ssin

g M

achi

ne

Processing Time

1

15 18

4

18 23

3

26 35

6

35 45

5

49 52

2

52 62

4

0 5

2

5 13

5

13 16

6

16 19

1

19 25

3

3536

5

0 9

3

9 14

1

1415

2

15 20

4

23 28

6

5455

3

14 18

6

19 22

1

25 32

4

32 35

5

5253

2

62 66

2

20 30

5

30 35

4

35 43

3

43 50

6

50 54

1

54 60

3

18 26

6

26 35

2

35 45

5

45 49

1

49 52

4

52 61

Fig. 6. GATT chart for a possible schedule of the 6�6 JSP without optimization.

0 10 20 30 40 50 60

1

2

3

4

5

6

Pro

cess

ing

Mac

hine

Processing Time

4

8 13

3

17 26

1

26 29

6

29 39

5

39 42

2

44 54

6

0 3

4

3 8

2

8 16

5

16 19

1

29 35

3

3536

3

0 5

1

5 6

5

6 15

2

16 21

4

21 26

6

4647

3

5 9

6

9 12

4

26 29

1

35 42

2

54 58

5

5859

5

19 24

2

24 34

4

34 42

6

42 46

3

46 53

1

53 59

3

9 17

6

17 26

5

26 30

2

34 44

1

44 47

4

47 56

Fig. 7. GATT chart for the schedule of the 6�6 JSP optimized by the PSO method.


The makespan time of the whole processing is 41 s.The GATT chart is presented in Fig. 4.

5.3. The procedure of the HBBO for the JSP

Our proposed HBBO combines the basic BBO with the character-istics of random of chaos theory, efficiency and accuracy of the‘‘searching around the optimum’’ strategy. The proposed HBBO

algorithm has better properties with high accuracy and efficiencythan the basic BBO. The main operations involved in this processare given below.

� Step 1: Initialize HBBO parameters.

Map the problem solutions to SIVs and habitats according to theproblem. Initialize the maximum species count Smax, the maximum

0 10 20 30 40 50

1

2

3

4

5

6P

roce

ssin

g M

achi

ne

Processing Time

1

1 4

4

13 18

3

18 27

6

28 38

2

38 48

5

48 51

2

0 8

4

8 13

6

13 16

1

16 22

5

22 25

3

2728

1

0 1

3

1 6

2

8 13

5

13 22

4

22 27

6

4445

3

6 10

6

16 19

1

22 29

4

29 32

2

48 52

5

5253

2

13 23

5

25 30

4

32 40

6

40 44

3

44 51

1

51 57

3

10 18

6

19 28

2

28 38

5

38 42

1

42 45

4

45 54

Fig. 8. GATT chart for the schedule of the 6�6 JSP optimized by the BBO method.

0 10 20 30 40 50

1

2

3

4

5

6

Pro

cess

ing

Mac

hine

Processing Time

1

6 9

4

16 21

3

21 30

6

30 40

2

40 50

5

50 53

2

0 8

6

8 11

4

11 16

5

22 25

1

25 31

3

3132

3

0 5

1

5 6

2

8 13

5

13 22

4

22 27

6

4950

3

5 9

6

11 14

4

27 30

1

31 38

2

50 54

5

5455

2

13 23

5

25 30

4

30 38

3

38 45

6

45 49

1

49 55

3

9 17

6

17 26

2

26 36

5

36 40

1

40 43

4

43 52

Fig. 9. GATT chart for the schedule of the 6�6 JSP optimized by the HBBO method.


migration rates E and I, the maximum mutation rate mmax, theelitism parameter Keep, and the step size used for numericalintegration of probabilities dt .

� Step 2: Initialize habitats.

Use the chaos theory to initialize a set of habitats with eachhabitat corresponding to a potential solution according toequation (5) and the encoding strategy described in Section 5.1.

� Step 3: Calculate k and l.

For each habitat, map the HSI to the number of species S, the immi-gration rate k, and the emigration rate l according to equation (1).

� Step 4: Migrate.

Probabilistically use immigration rate ki and emigration rate li

to modify each non-elite habitat. Re-compute each HSI accordingto Section 5.2.

Fig. 10. Statistics of the CPLEX for the schedule of the 6�6 JSP. (The green line shows the tendency of the objective value’s changing with time. The yellow points representobjective values of possible solutions and the last red point is objective value of the best solution.) (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

0 100 200 300 400 500 600 700 800 900 100054

56

58

60

62

64

66

68

fitne

ss v

alue

evolvement generation

PSOBBOHBBO

Fig. 11. Evolution curves of methods for the 6�6 JSP in comparison.

0 100 200 300 400 500 600 700 800 900 100054

56

58

60

62

64

66

68


fitne

ss v

alue

PSO

Fig. 12. Evolution curves of the PSO.

0 100 200 300 400 500 600 700 800 900 100055

60

65

70BBO


fitne

ss v

alue

Fig. 13. Evolution curves of the basic BBO.

0 100 200 300 400 500 600 700 800 900 100055

60

65

70


fitne

ss v

alue

HBBO

Fig. 14. Evolution curves of the HBBO.


� Step 5: Mutate.

For each habitat, update the probability of its species countaccording to equation (2). Mutate each non-elite habitat basedon its probability according to equation (3), and re-compute eachHSI according to Section 5.2.

� Step 6: Searching around the optimum.

If the algorithm has the potential of stagnation, use the strategyof ‘‘searching around the optimum’’ to randomly replace a dimen-sion of the current optimal solution with a random number to helpthe algorithm jump out of the local optima according to Section 4.2.Re-compute each HSI according to Section 5.2.



Machineassigned

Processingtime


Machineassigned

Processingtime

Job1 O11 3 54 Job6 O61 2 53O12 4 79 O62 3 99O13 1 16 O63 5 60O14 2 66 O64 1 13O15 5 58 O65 4 53






� Step 7: If the stopping criterion is satisfied, stop the iterationsand output the solution, otherwise, go to Step 3 for the nextiteration. This loop can be terminated after a predefined num-ber of iterations, or after an acceptable problem solution hasbeen found.

The detailed flow chart of the proposed HBBO approach for theJSP is shown in Fig. 5.

6. Comparative experimental results

In order to verify the feasibility and effectiveness of the pro-posed algorithm HBBO for solving the JSP, a set of comparativeexperiments are conducted. To prove that our proposed HBBO per-forms better than other algorithms, some competitive intelligent

0 100 200 300 4

1

2

3

4

5

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Fig. 15. GATT chart for a possible schedule

methods should be chosen. The performance of the proposed algo-rithm is compared with that of the CPLEX, the PSO, and the basicBBO algorithm. In addition, the results are also compared with thatof numerical existing methods.

The methods are all coded in Matlab, which run on a 2.4 GHzIntel Core i3 PC with 4.0 GB of RAM under Windows 2007 operat-ing system. As it is generally accepted that the difficulty to solvethe JSP is closely associated with the problem size, the parametersfor different instance should be various. Therefore, the initialparameters of the HBBO method are set as following based onthe tests and practical experience. For smaller instances (i.e.,instance 1 and 2 in experiment 1; FT06 and LA01to LA 05 in exper-iment 2), we set total population size P = 50, number of genera-tions N = 1000, step size used for numerical integration ofprobabilities dt = 1, the maximum migration rates I = 1, E = 1, themaximum mutation probability mmax = 0.001, and the number ofelitisms keep = 3. For larger instances (i.e., instance 3 in experiment1; FT10–FT20, LA06–LA26, LA31 and LA36 in experiment 2; all theinstances in experiment 3) the total population size P = 100, num-ber of generations N = 3000, step size used for numerical integra-tion of probabilities dt = 1, the maximum migration rates I = 1,E = 1, the maximum mutation probability mmax = 0.005. The popu-lation sizes and the numbers of iterations of the PSO and the basicBBO are set to be the same with that in the HBBO in eachexperiment.

6.1. Experiment 1

To test the performance of the HBBO, three JSP instances withdifferent scales are selected in the first experiment. Comparativeexperiments are conducted by using the PSO, and the basic BBOalgorithm and an exact commercial solver IBM ILOG CPLEX Optimi-zation Studio version 12.2. The population sizes and the numbersof iterations of all the three bio-inspired intelligent methods areset to be the same. And the job shop scheduling example in theCPLEX is executed with basic configuration. The GATT charts ofscheduling results conducted by different intelligent algorithmsand the statistics of the CPLEX are presented. In addition the evo-lution curves of each intelligent algorithm are given in comparison.

The processing time and operation sequence for the first 6�6 JSPinstance (Fisher & Thompson, 1963) are given as following in

00 500 600 700g Time

88

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of the 10�5 JSP without optimization.

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Fig. 17. GATT chart for the schedule of the 10�5 JSP optimized by the basic BBO method.


Table 1. Figs. 6–11 present the experimental results of differentmethods for the JSP with 6 jobs and 6 machines.

Figs. 6–9 show the GATT charts for the schedule of the 6�6 JSPwith the PSO, the basic BBO and the HBBO respectively, andFig. 10 gives statistics of the CPLEX for the schedule of the firstinstance, while Fig. 11 presents the comparative evolution curvesof the three bio-inspired intelligence methods. From the simula-tion results, it is obvious that the performance of the proposedalgorithm HBBO is better. As the minimum of the makespan (55)is found by the CPLEX and the HBBO, and the basic BBO finds aworse solution with a higher makespan (57), while the PSO obtainsthe worst solution with the highest makespan (59).

To get the average performance of the bio-inspired computationalgorithms, more runs on the problem instance should be per-formed. Further experiments with the PSO, the basic BBO and the

HBBO are given from Figs. 12–14 to verify the stability and advan-tage of the proposed method. For the first instance, each of thealgorithms runs for 10 times independently and the evolutioncurves are obtained to test if the algorithm is stable.

Figs. 12–14 present the evolution curves of the PSO, the basicBBO and the HBBO for 10 runs. Among the three methods, the aver-age converging rate of the PSO is the fastest apparently. However,the PSO algorithm gets into local optimal easily and the averagemakespan value of it is the largest among that of the three meth-ods. Only in one of the 10 runs the PSO obtains the best knownsolution for the problems so far. The basic BBO algorithm has bet-ter performance than the PSO, but the schedule result is not thebest as it is not very stable. In 6 of 10 runs the basic BBO obtainsthe optimal solution and the HBBO gets the best schedule in allthe 10runs. From the comparative results, it is obvious that the

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0 100 200 300 400 500 600 700 800 900 1000620

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fitne

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PSOBBOHBBO



HBBO method is more reliable than the other two algorithms.Therefore, the HBBO can solve different JSP problems effectivelyand stably as it can find the best schedule in all runs.

The processing time and operation sequence for the 10�5 JSPinstance are given in Table 2. Figs. 15–20 present the experimentalresults of different methods for the JSP with 10 jobs and 5machines.

Figs. 15–18 show the GATT charts for the schedule of the 10�5JSP without optimization and with the PSO, the basic BBO andthe HBBO respectively. Fig. 19 gives statistics of the CPLEX forthe schedule of the second instance, and Fig. 20 compares the evo-lution curves of the three methods. From the experimental results,it is obvious that the proposed algorithm performs better than thePSO and the basic BBO as the makespan of it is least.

Test on a more complicated JSP instance is conducted.Figs. 21–26 present the experimental results of different methodsfor the JSP with 20 jobs and 10 machines. The processing timeand operation sequence for the 20�10 JSP instance is given inTable 3.

Figs. 21–24 give the GATT charts for the schedule of the 20�10JSP with the PSO, the basic BBO and the HBBO respectively, andFig. 25 gives statistics of the CPLEX for the schedule of the thirdinstance, while the evolution curves of the three bio-inspiredmethods are given in Fig. 26. The same to the results of the 6�6and 10�5 JSPs, the schedule of the HBBO algorithm is still the best.Moreover, the best makespan values of different algorithms for thethree problem instances are provided in Table 4 in comparing tomake the result more clear and convincing. From the experimental

results mention above, it is obvious that the CPLEX performs betterthan the PSO and the basic BBO as it obtains the best solution for6�6 and 10�5 instances. However for the 20�10 instance, the make-span value of the CPLEX is higher than the HBBO. The results showthat the HBBO is more powerful than the PSO and the basic BBO,and is at least not worse than the CPLEX in solving the JSP.

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Fig. 21. GATT chart for a possible schedule of the 20�10 JSP without optimization.

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6.2. Experiment 2

Previous work has shown that there is rarely an ultimatemethod that can outperform other methods in all the problemcases. The second series of experiments aims at illustrating theeffectiveness and performance of the HBBO for the JSP to minimizemakespan. Several well-known benchmark problems with differ-ent scales are selected, including FT06, FT10, and FT20 (Fisher &Thompson, 1963), LA01–LA26 (Lawrence, 1984). These problemsare widely used in the literature and available on the OR-Library

web site (Beasley, 1990) (URL: http://people.brunel.ac.uk/~mast-jjb/jeb/info.html). Numerical analysis based on those well-knownbenchmark problems can give some convincing explanations thatour proposed method shows advantages.

In this experiment the results of the proposed HBBO algorithmare compared with that of numerical comparative algorithmsfrom literatures, such as, GA(Zhao, 2011), SA (Lu, 2005), HEA(Ge, Du, & Qian, 2007), MPSO (Lin, Horng, Kao, Chen, & Run,2010), HIA (Ge, Sun, Liang, & Qian, 2008), HGA-Param(Goncalves, Magalhaes Mendes, & Mauricio, 2005), SBI (Joseph

http://people.brunel.ac.uk/~mastjjb/jeb/info.html

http://people.brunel.ac.uk/~mastjjb/jeb/info.html

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8

724747

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985 1061

15

1061 1114

16

1140 1178

1

1196 1254

4

1327 1395

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8931 5931

18

0 87

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87101

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101 128

8

128 166

11

166 213

13

213 276

10

276 332

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332 373

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541 576

20

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781792

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871 939

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1

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7811 8711

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1187 1255

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67215521

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1281 1367

1

0 27

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331 371

13

371 410

6

614 674

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674 728

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728 784

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784 852

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852 920

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43010201

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75014301

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1057 1121

2

8211 1211

10

1128 1204

5

1204 1276

4

1276 1327

14

0431 7231

20

1340 1397

15

13971423

Fig. 23. GATT chart for the schedule of the 20�10 JSP optimized by the basic BBO method.

0 200 400 600 800 1000 1200

1

2

3

4

5

6

7

8

9

10

Pro

cess

ing

Mac

hine

Processing Time

7

0 53

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76 149

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149 238

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238 313

4

336 421

12

421 520

16

520 607

17

607 665

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665 748

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825 883

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059739

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950973

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3

1054 1113

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1113 1188

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1251 1289

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12891302

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772862

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485505

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505 558

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625 688

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467 167

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865 914

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914934

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960 999

1

2301 6101

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1039 1088

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10881120

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1120 1189

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0 91

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91 147

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147 207

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207 267

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267 316

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316 407

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407 493

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493 561

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561 610

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610 687

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687 774

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774 837

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837 884

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884 940

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940 998

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998 1055

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1055 1113

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1123 1208

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1221 8021

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1221 1276

11

0 76

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89 136

4

237252

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271297

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331 408

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535025

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572 621

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436126

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634 707

7

707728

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728 789

1

789 858

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858 914

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914919

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919937

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1016 1059

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1059 1123

6

1123 1182

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1182 1223

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1251 1290

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130 187

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302 891

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203 266

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266 313

7

623313

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362 448

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448 486

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486 517

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517 582

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582 620

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661 746

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789 830

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830 919

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249 739

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942 981

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1098 1183

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0 14

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14 91

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91 123

3

123138

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138 237

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237 331

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331 403

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403428

19

428 479

7

479 532

10

532 617

9

617 686

2

688 777

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777 833

11

833 897

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897 960

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960 1016

8

1016 1089

18

1211 1265

6

1265 1306

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0 77

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7789

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89 125

9

125130

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130 198

14

198 227

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227 268

4

268 336

17

336 390

10

390 454

3

454 485

1

485 542

2

542 625

5

625 716

7

728754

8

765 824

20

824 863

11

897 982

18

982 1039

13

1055 1104

13

14 87

12

91 152

3

152163

10

163 250

17

250271

6

271 316

19

316 393

5

393 471

11

471 558

9

558 607

7

607 639

20

639667

18

667 742

8

742765

2

940 1016

15

1016 1069

4

1069 1137

16

1137 1175

1

1175 1233

14

12331236

9

0 27

8

27 65

18

65 152

16

152166

11

166 213

13

213 276

10

276 332

6

407 442

4

442 510

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275165

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572 613

3

620 680

20

680 743

5

938038

12

839 907

1

937 1016

7

1016 1074

17

10741109

15

11201141

2

1183 1269

1

0 27

16

313 353

13

353 392

6

486 546

12

546 614

11

614 670

19

670 701

4

701 752

3

752775

7

775 829

5

839 911

8

529119

9

925 989

17

989 1057

2

10591066

10

1066 1142

18

1142 1211

14

12111224

15

12241250

20

1250 1307




0 500 1000 1500 2000 2500 30001300

1400

1500

1600

1700

1800

1900

2000

2100

fitne

ss v

alue


PSOBBOHBBO




Machineassigned

Processingtime


Machineassigned

Processingtime

Job1 O11 10 27 Job11 O1101 3 76O12 6 25 O1102 8 73O13 7 57 O1103 7 47O14 3 49 O1104 2 85O15 4 69 O1105 6 86O16 1 54 O1106 1 87O17 9 79 O1107 4 56O18 2 16 O1108 10 64O19 5 66 O1109 5 85O110 8 58 O1110 9 13




Job5 O51 6 77 Job15 O1501 7 12O52 3 56 O1502 9 47O53 1 89 O1503 5 63O54 8 78 O1504 10 56O55 2 53 O1505 1 47O56 7 91 O1506 4 13O57 4 61 O1507 6 53

Table 3 (continued)


Machineassigned

Processingtime


Machineassigned

Processingtime

O58 5 41 O1508 2 32O59 9 9 O1509 8 21O510 10 72 O1510 3 26






Table 4The best makespan values of different algorithms for the three instances.

Problem instance Without optimization CPLEX PSO BBO HBBO

6*6 66 55 55 55 5510*5 776 630 685 644 63020*10 1797 1325 1694 1423 1307


et al.,1988), SBII (Joseph et al.,1988), TSAB (Nowicki & Smutnicki,1996), TSSB (Pezzella & Merelli, 2000), PSO-priority based (Sha &Hsu, 2006), PSO-permutation based (Sha & Hsu, 2006), HPSO(Sha & Hsu, 2006), and TS (Mauro & Marco, 1993). The CPLEX,the PSO, and the basic BBO are also tested on the benchmarks.The parameters of all the three bio-inspired intelligent algorithms(the PSO, the basic BBO and the HBBO) and the exact solver CPLEXare given in the beginning of Section 6.

Table 5Makespans of HBBO, PSO, BBO, CPLEX and some literature results.

Instance n�m BK GA SA HEA MPSO HIA HGA-Param SBI SBII TSAB TSSB

FT06 6�6 55 – – 55 55 55 55 55 55 55 55FT10 10�10 930 – – 930 930 930 930 1015 930 930 930FT20 20�5 1165 – – 1169 1165 1165 1165 1290 1178 1165 1165LA01 10�5 666 666 674 666 666 666 666 666 666 666 666LA02 10�5 655 666 671 655 655 655 655 720 669 655 655LA03 10�5 597 666 619 597 597 597 597 623 605 597 597LA04 10�5 590 – 602 590 590 590 590 597 593 590 590LA05 10�5 593 – 593 593 593 593 593 593 593 593 593LA06 15�5 926 926 926 926 926 926 926 926 926 926 926LA07 15�5 890 – – 890 890 890 890 890 890 890 890LA08 15�5 863 – – 863 863 863 863 868 863 863 863LA09 15�5 951 – – 951 951 951 951 951 951 951 951LA10 15�5 958 – – 958 958 958 958 959 959 958 958LA11 20�5 1222 1222 1222 1222 1222 1222 1222 1222 1222 1222 1222LA12 20�5 1039 1070 – 1039 1039 1039 1039 1039 1039 1039 1039LA13 20�5 1150 1180 – 1150 1150 1150 1150 1150 1150 1150 1150LA14 20�5 1292 1292 – 1292 1292 1292 1292 1292 1292 1292 1292LA15 20�5 1207 1215 – 1207 1207 1207 1207 1207 1207 1207 1207LA16 10�10 945 979 1005 945 945 945 945 1021 978 945 945LA17 10�10 784 784 793 784 784 784 784 796 787 784 784LA18 10�10 848 848 – 848 848 848 848 891 859 848 848LA19 10�10 842 855 – – 842 842 842 875 860 842 842LA20 10�10 902 950 1007 – 902 902 907 924 914 902 902LA21 15�10 1046 1097 1134 1046 1046 1046 1046 1172 1084 1047 1046LA22 15�10 927 940 1122 935 932 932 935 1040 944 927 927LA23 15�10 1032 1040 – 1032 1032 1032 1032 1061 1032 1032 1032LA24 15�10 935 953 – – 941 950 953 1000 976 939 938LA25 15�10 977 990 – – 977 979 986 1048 1017 977 979LA26 20�10 1218 1231 1355 1218 1218 1218 1218 1304 1224 1218 1218LA31 30�10 1784 – – 1784 1784 1784 1784 1784 1784 1784 1784LA36 15�15 1268 – – 1287 1278 1281 1279 1351 1305 1268 1268

Instance PSO-priority based PSO-permutation based HPSO TS CPLEX PSO BBO HBBO

MS⁄ Iter MS Iter MS Iter

FT06 55 55 55 55 55 55 1000 55 354 55 253FT10 1007 937 930 935 945 1040 3000 1001 3000 930 439FT20 1242 1165 1165 1165 1270 1238 3000 1182 3000 1165 497LA01 681 666 666 666 666 666 496 666 310 666 203LA02 694 655 655 655 666 665 1000 655 724 655 309LA03 633 597 597 597 597 619 1000 608 1000 597 373LA04 611 590 590 590 590 607 1000 603 1000 590 389LA05 593 593 593 593 593 593 785 593 873 593 243LA06 926 926 926 926 926 926 3000 926 1989 926 522LA07 890 890 890 890 890 890 1877 890 1520 890 580LA08 863 863 863 863 863 863 1722 863 1358 863 493LA09 953 951 951 951 951 956 3000 951 2105 951 649LA10 958 958 958 958 958 958 2055 958 1799 958 652LA11 1222 1222 1222 1222 1222 1222 3000 1222 2005 1222 973LA12 1039 1039 1039 1039 1039 1103 3000 1053 3000 1039 1169LA13 1150 1150 1150 1150 1150 1173 3000 1168 3000 1150 1204LA14 1292 1292 1292 1292 1292 1292 2300 1292 2298 1292 822LA15 1232 1207 1207 1207 1207 1231 3000 1223 3000 1207 1254LA16 1006 945 945 945 985 1033 3000 998 3000 945 1203LA17 833 784 784 784 785 803 3000 797 3000 784 1223LA18 901 848 848 848 861 870 3000 867 3000 848 925LA19 895 842 842 842 860 867 3000 866 3000 842 873LA20 963 907 902 902 907 1002 3000 918 3000 902 643LA21 1201 1055 1046 1046 1095 1173 3000 1146 3000 1046 799LA22 1046 935 927 933 937 977 3000 951 3000 933 3000LA23 1146 1032 1032 1032 1032 1082 3000 1045 3000 1032 1062LA24 1082 937 935 941 963 992 3000 979 3000 935 1179LA25 1107 983 977 979 995 1035 3000 1013 3000 977 1300LA26 1409 1218 1218 1218 1237 1358 3000 1275 3000 1218 1266LA31 1784 1784 1784 1784 1784 1868 3000 1813 3000 1784 2398LA36 1437 1291 1268 1278 1312 1373 3000 1331 3000 1268 2466

Note: MS represents makespan and Iter represents iteration.


The comparative results are shown in Tables 5–7 and Fig. 27.The first column of Table 5 gives the problem names. Thesecond column n�m means that the JSP instance involves njobs and m machines. BK represents the best known solutionfor each instance. In Table 5 the makespan time of 18 algorithmsare given. The parameters are set according to the rule that is

mentioned in the beginning of Section 6. As the iteration numberscan reflect the efficiency directly and also reveal the time con-suming proportion of the method, the iteration numbers of thePSO, the basic BBO and the HBBO are also presented. Eachinstance is executed for 5 runs. Since that the convergence char-acter of each method is different, not all the three algorithms can

Table 6Bias% of HBBO, PSO, BBO, CPLEX and some literature results.

Instance GA SA HEA MPSO HIA HGA-Param SBI SBII TSAB

FT06 – – 0 0 0 0 0 0 0FT10 – – 0 0 0 0 9.1398 0 0FT20 – – 0.3433 0 0 0 10.7296 1.1159 0LA01 0 1.2012 0 0 0 0 0 0 0LA02 1.6794 2.4427 0 0 0 0 9.9237 2.1374 0LA03 11.5578 3.6851 0 0 0 0 4.3551 1.3400 0LA04 – 2.0339 0 0 0 0 1.1864 0.5085 0LA05 – 0 0 0 0 0 0 0 0LA06 0 0 0 0 0 0 0 0 0LA07 – – 0 0 0 0 0 0 0LA08 – – 0 0 0 0 0.5794 0 0LA09 – – 0 0 0 0 0 0 0LA10 – – 0 0 0 0 0.1044 0.1044 0LA11 0 0 0 0 0 0 0 0 0LA12 2.9836 – 0 0 0 0 0 0 0LA13 2.6087 – 0 0 0 0 0 0 0LA14 0 – 0 0 0 0 0 0 0LA15 0.6628 – 0 0 0 0 0 0 0LA16 3.5979 6.3492 0 0 0 0 8.0423 3.4921 0LA17 0 1.1480 0 0 0 0 1.5306 0.3827 0LA18 5.3215 – 0 0 0 0 5.0708 1.2972 0LA19 4.8757 – – 0 0 0 3.9192 2.1378 0LA20 1.4024 11.6408 – 0 0 0.5543 2.4390 1.3304 0LA21 1.0673 8.4130 0 0 0 0 12.0459 3.6329 0.0956LA22 2.6087 21.0356 0.8630 0.5394 0.5394 0.8630 12.1899 1.8339 0LA23 0 – 0 0 0 0 2.8101 0 0LA24 0.6628 – – 0.6417 1.6043 1.9251 6.9519 4.3850 0.4278LA25 3.5979 – – 0 0.2047 0.9212 7.2671 4.0942 0LA26 0 11.2479 0 0 0 0 7.0608 0.4926 0LA31 – – 0 0 0 0 0 0 0LA36 – – 1.4984 0.7886 1.0252 0.8675 6.5457 2.9180 0Average 2.3838 5.3229 0.1002 0.0635 0.1088 0.1655 3.6094 1.0065 0.0169

Instance TSSB PSO-priority based PSO-permutation based HPSO TS CPLEX PSO BBO HBBO

FT06 0 0 0 0 0 0 0 0 0FT10 0 8.2796 0.7527 0 0.5376 1.6129 11.8280 7.6344 0FT20 0 6.6094 0 0 0 9.0129 6.2661 1.4592 0LA01 0 2.2523 0 0 0 0 0 0 0LA02 0 5.9542 0 0 0 1.6794 1.5267 0 0LA03 0 6.0302 0 0 0 0 3.6851 1.8425 0LA04 0 3.5593 0 0 0 0 2.8814 2.2034 0LA05 0 0 0 0 0 0 0 0 0LA06 0 0 0 0 0 0 0 0 0LA07 0 0 0 0 0 0 0 0 0LA08 0 0 0 0 0 0 0 0 0LA09 0 0.2103 0 0 0 0 0.5258 0 0LA10 0 0 0 0 0 0 0 0 0LA11 0 0 0 0 0 0 0 0 0LA12 0 0 0 0 0 0 6.1598 1.3474 0LA13 0 0 0 0 0 0 2.0000 1.5652 0LA14 0 0 0 0 0 0 0 0 0LA15 0 2.0713 0 0 0 0 1.9884 1.3256 0LA16 0 6.4550 0 0 0 4.2328 9.3122 5.6085 0LA17 0 6.2500 0 0 0 0.1276 2.4235 1.6582 0LA18 0 6.2500 0 0 0 1.5330 2.5943 2.2406 0LA19 0 6.2945 0 0 0 2.1378 2.9691 2.8504 0LA20 0 6.7627 0.5543 0 0 0.5543 11.0865 1.7738 0LA21 0 14.8184 0.8604 0 0 4.6845 12.1415 9.5602 0LA22 0 12.8371 0.8630 0 0.6472 1.0787 5.3937 2.5890 0.6472LA23 0 11.0465 0 0 0 0 4.8450 1.2597 0LA24 0.3209 15.7219 0.2139 0 0.6417 2.9947 6.0963 4.7059 0LA25 0.2047 13.3060 0.6141 0 0.2047 1.8424 5.9365 3.6847 0LA26 0 15.6814 0 0 0 1.5599 11.4943 4.6798 0LA31 0 0 0 0 0 0 4.7085 1.6256 0LA36 0 13.3281 1.8139 0 0.7886 3.4700 8.2808 4.9685 0Average 0.0170 5.2812 0.1830 0 0.0910 1.1781 4.0046 2.0833 0.0209


obtain the best solution of each instance. Therefore, if the finalbest solutions are obtained within the given iterations, the col-umns of ‘Iter’ in Table 5 give the least iterations that the algo-rithms cost to obtain the optimal solution. However if thealgorithm can never achieve the optima, the numbers of itera-tions set at the beginning of the algorithm are given in that

column. The bias% between every algorithm and the best knownsolution is given in Table 6 and it is computed according to thefollowing formula.

bias% ¼ MSxi � BKi

BKi� 100 ð8Þ

Table 8Makespans of CPLEX, PSO, BBO, and HBBO.

Instance n�m BK CPLEX PSO BBO HBBO

ORB1 10�10 1059 1089 1137 1126 1059ORB2 10�10 888 900 992 945 919ABZ5 10�10 1234 1270 1286 1272 1250ABZ6 10�10 943 948 984 979 948ABZ9 20�15 679 758 844 821 734

Fig. 28. Gaps between the obtained and the best known makepan of each method.

Fig. 27. The bias% of CPLEX, PSO, BBO and HBBO for each instance.

Table 7Statistics of HBBO, PSO, BBO, CPLEX and some literature results.

Algorithm GA SA HEA MPSO HIA HGA-Param SBI SBII TSAB TSSB

N_OBK 5 3 23 27 26 25 12 15 28 28N_IT 15 13 27 31 31 31 31 31 31 31N_OBK/N_IT 0.3333 0.2308 0.8519 0.8710 0.8387 0.8065 0.3871 0.4839 0.9032 0.9032sgap 308 652 31 21 35 51 1073 295 5 5sgap% 2.1131 5.8893 0.1211 0.0718 0.1196 0.1743 3.6675 1.0083 0.0171 0.0171

Algorithm PSO-priority based PSO-permutation based HPSO TS CPLEX PSO BBO HBBO

N_BK 11 23 31 25 17 9 11 30N_IT 31 31 31 31 31 31 31 31N_OBK/N_IT 0.3548 0.7419 1 0.8065 0.5484 0.2903 0.3548 0.9678sgap 1594 60 0 29 376 1273 651 6.0000sgap% 5.4483 0.2051 0 0.0991 1.2852 4.3511 2.2251 0.0205


where MSix is the makespan of instance i that is obtained by method

x, BKi is the best known makespan of instance i. The last row ofTable 6 gives the average bias% of each method of all the 31instances. The bias%s of the CPLEX, the PSO, the basic BBO and theHBBO for each instance are also given in Fig. 27 to make it moreconvenient for comparing.

Since that the number of instances tested by each method is dif-ferent, it is not very convincing to compare the result by averagebias% directly. Therefore, the statistics of the number that bestknown solution is obtained (N_OBK) and the total number ofinstance tested by each method (N_IT) is given in Table 7. The

sum of gaps (sgap) between the obtained and the best knownmakepan of all the instances that are tested by each method isgiven. Similar to the bias%, the sgap% is obtained by

Sgap% ¼P

igapiPiBKi

� 100 ð9Þ

where i represents that the instance i is tested by the correspondingmethod, sgap denotes the sum of all the gaps between the make-span obtained by each method and the best known.

From Tables 5–7, it is obvious that our proposed HBBO methodoutperforms all other algorithms except the HPSO. The HBBOobtained the best-known solution for 30 of the 31 instances, whilethe basic BBO obtained 11 of the 31 instances. The iterations of theHBBO are apparently less than that of the PSO and the basic BBO asgiven in Table 5 for each instance. In addition the bias% and sgap%of the HBBO is lower than most of the competitors.

6.3. Experiment 3

Our proposed HBBO is tested on the ORB (Joseph et al., 1988)and ABZ (Applegate & Cook, 1991) benchmarks to further illumi-nate the advantage of the HBBO. The computational results ofthe HBBO are particularly compared with that of the CPLEX, thePSO, and the basic BBO. The CPLEX, the PSO, the basic BBO andthe HBBO are tested on 5 tougher instances (ORB1, ORB2, ABZ5,ABZ6 and ABZ9) and the experimental results are presented inTable 8. The gaps between the makepan obtained by each methodand the best known makepan of all the instances are given inFig. 28. From the comparative results, we can see that the HBBOapproach dramatically outperforms the PSO and the basic BBOfor all the instances, and can also obtain better solutions than theCPLEX for four of the five instances.


From the three series of experiments, we can see that the HBBOcan make better performance than the CPLEX, the PSO, the basicBBO and most of the literature results. From the results of the basicBBO and other basic methods we can see that the basic BBOperforms better than the PSO, the GA and the SA as it can obtainsmaller makespan. Moreover, the convergence character of thebasic BBO is also better than that of the PSO. Therefore, the basicBBO is a competitive bio-inspired algorithm. Comparing the resultsof the HBBO algorithm with that of the basic BBO, we can see thatthe benefits from the chaos theory and the ‘‘searching around theoptimum’’ strategy for solving the job shop scheduling problemsis significant. The CPLEX can report feasible solutions for mostinstances, but the bias and gaps of it is larger than that of theHBBO. In addition, the number of the best-known solutionsobtained by the CPLEX is less than that of the HBBO, as indicatedin Tables 6 and 7. Results of other 14 methods from literatureare also given but most of their results are inferior to ours exceptthe HPSO which achieves the best-known solutions for all the31instances as shown in experiment 2. To sum up in conclusionthe proposed HBBO is demonstrated to be competitive as it canmake better schedule within less iterations.

7. Conclusions

This paper develops a novel HBBO algorithm to solve the JSPproblem. BBO is a new simulated bio-inspired intelligentalgorithm which has some features that are unique to other biol-ogy-based optimization methods. To reduce the computationtime and to guarantee avoidance of the local minimum, theproposed HBBO method integrates the chaos theory and the‘‘searching around the optimum’’ strategy into the basic BBOalgorithm and takes advantages of the efficiency, accuracy andstability of them.

Different types of experiments are conducted to highlight theadvantages of our approach over the existing methods. Our pro-posed HBBO is tested and approved in numerical simulation basedon both self-designed numerical examples and 36 standard bench-mark instances that are taken from the OR-Library. Series of thesimulation results highlight the advantages of our approach overnumerical existing methods. It should be noticed that the HBBOgenerates satisfactory job-shop scheduling successfully within lessiterations, while the basic BBO and the PSO algorithms converge tolocal best solutions easily. Therefore, the chaos theory and the‘‘searching around the optimum’’ strategy can consistentlyimprove the performance of the original algorithm. It may finallybe concluded that our aggregation method HBBO is a more effec-tive and robust method for the JSP as it outperforms the PSO, thebasic BBO, the CPLEX, and the other state-of-the-art numericalmethods.

Our future work will focus on applying this novel technique tomore complicated JSPs, and implement this hybrid approach inenterprise systems.

Acknowledgements

This work was partially supported by National Key BasicResearch Program of China (973 Project) under grant#2013CB035503 and #2014CB046401, Natural Science Founda-tion of China (NSFC) under grant #61333004, #61273054 and#61175109, National Magnetic Confinement Fusion ResearchProgram of China under grant #2012GB102006, Top-NotchYoung Talents Program of China, Aeronautical Foundation ofChina under grant #20135851042, and the FundamentalResearch Funds for the Central Universities of China under grant#YWF-11-03-Q-012.

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