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A discrete artificial bee colony

algorithm incorporating differential

evolution for the flow-shop scheduling

problem with blockingYu-Yan Hana, Dunwei Gonga & Xiaoyan Suna

a School of Information and Electrical Engineering, China

University of Mining and Technology, Xuzhou, PR China

Published online: 18 Jun 2014.

To cite this article: Yu-Yan Han, Dunwei Gong & Xiaoyan Sun (2014): A discrete artificial bee colonyalgorithm incorporating differential evolution for the flow-shop scheduling problem with blocking,

Engineering Optimization, DOI: 10.1080/0305215X.2014.928817

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Engineering Optimization, 2014

http://dx.doi.org/10.1080/0305215X.2014.928817

A discrete artificial bee colony algorithm incorporating

differential evolution for the flow-shop scheduling problem

with blocking

Yu-Yan Han, Dunwei Gong∗ and Xiaoyan Sun

School of Information and Electrical Engineering, China University of Mining and Technology,

Xuzhou, PR China

( Received 26 December 2013; accepted 19 May 2014)

A flow-shop scheduling problem with blocking has important applications in a variety of industrial systemsbut is underrepresented in the research literature. In this study, a novel discrete artificial bee colony (ABC)algorithm is presented to solve the above scheduling problem with a makespan criterion by incorporatingtheABC with differential evolution (DE). The proposed algorithm (DE-ABC) contains three key operators.One is related to the employed bee operator (i.e. adopting mutation and crossover operators of discrete DEto generate solutions with good quality); the second is concerned with the onlooker bee operator, whichmodifies the selected solutions using insert or swap operators based on the self-adaptive strategy; and thelast is for the local search, that is, the insert-neighbourhood-based local search with a small probability is

adopted to improve the algorithm’s capability in exploitation. The performance of the proposed DE-ABCalgorithm is empirically evaluated by applying it to well-known benchmark problems. The experimentalresults show thatthe proposedalgorithmis superiorto the comparedalgorithmsin minimizingthe makespancriterion.

Keywords: artificial bee colony; flow-shop scheduling; blocking; differential evolution; makespan

1. Introduction

Flow-shop scheduling problems can be generally classified into the following two categories

according to whether there are buffers or not: one with infinite buffers, and the other with finitebuffers. The former does not result in blocking any job since there are enough intermediate

buffers to store those completed jobs. The latter only maintains a limited capacity of in-process

inventories, which means that there are either no buffers or buffers with a limited capacity owing

to the finite storage facilities. The flow-shop scheduling problem with no intermediate buffers

considered here, named the blocking flow-shop (BFS) scheduling problem, is a special case of

the latter. For this case, a job must remain in the current machine until the next machine is available

for processing, which increases the waiting time or the productive cycle, and thus decreases the

production efficiency.

Owing to the above process characteristics, the BFS scheduling problem has been a typical

problem with a strong engineering background. In the chemical industry, partially processed jobs(i.e. physical or chemical materials) are held in machines because there is no intermediate storage

∗Corresponding author. Email: [email protected]

© 2014 Taylor & Francis

mailto:[email protected]

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2 Y.-Y. Han et al.

(Suhami and Mah 1981). For various processes in a manufacturing enterprise (Grabowski and

Pempera 2000), the intermediate product cannot be stored in some stages. In the case of the iron

and steel industry (Gong, Tang, and Duin 2010), the blocking of ingots in the soaking pit will result

in extra energy consumption since the blocked ingot requires a high temperature. With respect to

the computational complexity, it has been proved that the BFS scheduling problem with more than

two machines is NP hard (Storn and Price 1997, Allahverdi, Ng, and Cheng 2008). Therefore, it

is of great importance to seek appropriate methods to tackle the BFS problem.

With regard to the algorithms for solving the BFS scheduling problem with a makespan cri-

terion, the existing studies can be broadly classified into constructive heuristics and improved

metaheuristics. The former type uses specific rules to assign each job with a priority index to con-

struct a sequence, such as Nawaz–Enscore–Ham (NEH), profile fitting (PF) and MinMax (MM).

Nawaz, Enscore, and Ham (1983) first designed an NEH heuristic for the traditional flow-shop

problem with a makespan criterion, and the experimental results demonstrated the effectiveness

and efficiency of the proposed heuristic in comparison with existing algorithms. Since then, the

NEH heuristic has gained much attention and has been successfully applied to the BFS schedul-ing problem (Ribas, Companys, and Tort-Martorell 2011). McCormich et al. (1989) developed

a constructive heuristic, known as PF, for solving permutation problems in an assembly line.

In this heuristic, PF tries to generate a partial sequence by adding an unscheduled job so as to

minimize the sum of idle and blocking time on machines. Ronconi (2004) presented the MM

heuristic based on the makespan properties. In addition, they proposed two composite construc-

tive heuristics, called MME (the combination of MM and NEH) and PFE (the combination of PF

and NEH). Their empirical results showed that MME and PFE heuristics are superior to the NEH

heuristic. Recently, Pan and Wang (2012) designed two simple constructive heuristics for the BFS

scheduling problem, called the weighted profile fitting (wPF) and Pan–Wang (PW) heuristics,

based on the PF approach. They also developed three improved constructive heuristics, namely,PF-NEH, wPF-NEH and PW-NEH, by combining the procedure of the NEH heuristic with the

PF, wPF and PW, respectively. A series of BFS scheduling problem instances demonstrated that

the presented constructive heuristics perform significantly better than the existing ones.

The above constructive heuristics can rapidly yield feasible solutions. However, the quality

of these solutions is somewhat worse than that obtained by the metaheuristic algorithms (Li,

Wang, and Wu 2009). Caraffa et al. (2001) proposed a genetic algorithm (GA) to minimize the

makespan criterion for the BFS scheduling problem. In this work, each job sequence of the prob-

lem is viewed as a chromosome. Grabowski and Pempera (2007) presented tabu search algorithms

with and without multimoves (TS and TS + M, respectively) for the flow-shop scheduling prob-

lem with blocking, where a dynamic tabu list is used to overcome the local optima. Qian et al.

(2009) developed an effective hybrid differential evolution (HDE) to solve the flow-shop schedul-

ing problem with limited buffers between consecutive machines. Thereafter, Wang et al. (2010)

designed a novel hybrid discrete differential evolution (HDDE) to optimize the makespan criterion

for the BFS scheduling problem. To avoid the regions with local optima, Davendra et al. (2012)

adopted an enhanced differential evolution (EDEc) to solve the flow-shop scheduling problem

with blocking based on the segregation bias rules. The proposed EDEc algorithm obtained 49 new

upper bounds for the Taillard problems. Since then, DE has been successfully applied to different

problems owing to its powerful performance (Ramesh, Kannan, and Baskar 2012; Zhu, Yan, and

Zhao 2013). In addition, for the same problem, Wang and Tang (2012) presented a discrete particle

swarm optimization (DPSO) to minimize the makespan objective of the BFS scheduling problem

with m machines. In this work, they adopted a self-adaptive strategy to control the diversity of population. In addition, they provided a stochastic variable neighbourhood search approach to

improve the exploration.

The artificial bee colony (ABC) algorithm, one of the most recent swarm intelligence

approaches, was presented by Karaboga (2005). As the name implies, this algorithm simulates



Engineering Optimization 3

the foraging behaviour of honey bee colonies. ABC was commonly used for optimization prob-

lems with continuous variables, and showed promise in terms of accuracy and efficiency (Kang,

Li, and Ma 2013). However, owing to the continuous nature of the basic ABC algorithm, it

cannot be directly used to generate a feasible job permutation for the flow-shop scheduling

problem. To overcome such a drawback, Pan et al. (2011) first proposed a discrete artificial

bee colony (DABC) algorithm for the lot-streaming flow-shop scheduling problem. There-

after, Han, Duan, and Zhang (2012) applied an improved DABC algorithm to tackle the BFS

scheduling problem, and the experimental results demonstrated that the proposed algorithm

outperforms the HDDE, DABC, GA, TS and TS + M algorithms in minimizing the makespan

criterion.

Among the aforementioned metaheuristic algorithms, the DDE and DABC algorithms are the

most powerful and have been successfully applied to the BFS scheduling problem. However, both

the employed and onlooker bees in the existing DABC algorithms adopted the insert or swap

operator to produce new neighbouring solutions, which may be local optima. Besides, the DABC

algorithm spends a lot of computation time on repeated search, which greatly reduces the con-vergence speed and the efficiency of the algorithm. To tackle these problems, a hybrid algorithm,

integrating DDE and DABC algorithms, was proposed. In the literature, hybrid algorithms often

obtain results of high quality. Fan, Liang, and Zahara (2004) employed a hybrid simplex search and

particle swarm optimization (PSO) to solve multimodal optimization problems. Xu et al. (2014)

embedded a local search into the artificial immune algorithm for solving a distributed permutation

flow-shop scheduling problem. Huang and Shiau (2008) presented a hybrid algorithm based on the

column generation and the constructive heuristic for a proportionate flexible flow-shop scheduling

problem. Following that, for the same problem, Shiau and Huang (2012) proposed a hybrid two-

phase encoding particle swarm optimization (TPEPSO) algorithm and the experimental results

demonstrated its robustness. Han et al. (2014) embedded the estimation of distribution algorithm(EDA) into NSGA-II to replace traditional crossover and mutation operators and the experimental

results demonstrated the superiority of the hybrid algorithm in terms of quality. In this study, the

proposed hybrid algorithm, DE-ABC, was also empirically demonstrated to perform better in

exploration and exploitation. According to the notation introduced by Graham et al. (1979), the

BFS scheduling problem with makespan criterion under study is denoted as Fm|blocking|C max

throughout this article.

To sum up, in this study, the proposed algorithm makes three main contributions: (1) the

mutation and crossover operators are adopted to generate good solutions, instead of the insert

or swap operator in the employed bee stage; (2) the insert or swap operator based on the self-

adaptive strategy is employed to modify the given solutions in the onlooker bee stage; and (3) the

insert-neighbourhood-based local search with a small probability is performed so as to improve

the algorithm’s capability in exploitation.

The remainder of this article is organized as follows. In Section 2, the description of

Fm|blocking|C max is given. Section 3 addresses the basic ABC algorithm. The proposed algorithm

is presented in detail in Section 4. Section 5 provides the experimental results. Finally, the article

ends with some conclusions in Section 6.

2. Formulation of the blocking flow-shop scheduling problem

Table 1 lists the symbols and notations that will be used throughout this article.

The BFS scheduling problem with no buffers is formulated as follows. There are n jobs and m

machines. Each job from sequence π has to be processed on m machines without intermediate

buffers in the same series. Jobπ( j) has a sequence of m operations O j,k . Operation O j,k corresponds



4 Y.-Y. Han et al.

Table 1. Symbols and notation.

n Total number of jobs

m Total number of machines

PS Population size

π = {π(1),π(2), . . . ,π(n)} Job permutationπ( j) jth job of sequence π

Set of all sequences, = {π1,π2, ...,πPS }

πi ith sequence i ∈ {1,2, ..., PS }

O j,k Operation of job ( j = 1, 2,..., n) on machine k , k = 1,2,...m

P j,k Processing time of job ( j = 1, 2,..., n) on machine k , k = 1,2,...m

D j,k Departure time of job ( j = 1, 2,..., n) on machine k , k = 1,2, ...m

S j,0 Start time of job ( j = 1, 2,..., n) on the first machine

to the processing of jobπ( j) on machine k during an uninterrupted processing time, P j,k . Moreover,

the following constraints are considered for the Fm|blocking|C max problem in this article:

• At any time, each machine can process at most one job and each job can be processed on at

most one machine.

• Since the flow shop has no intermediate buffers, a job cannot leave a machine until the next

machine downstream is available for processing.

• Both the set-up time and transportation time of each job are included in the processing time.

The purpose is to seek a schedule for the processing sequence of jobs on all machines under the

above constraints so that the maximal completion time (i.e. makespan) is minimized. According

to Ronconi (2004), the departure time of each job on each machine can be calculated using the

following equations:

S 1,0 = 0 (1)

D1,k = D1,k −1 + P1,k k = 1, 2, . . . , m − 1 (2)

S j,0 = D j−1,1 j = 2, 3, . . . . n (3)

D j,k = max{ D j,k −1 + P j,k , D j−1,k +1} k = 1,2, . . . , m − 1 j = 2,3, . . . , n (4)

D j,m = D j,m−1 + P j,m j = 1, 2, . . . , n (5)

Since the start time of the first job on the first machine, S 1,0, is 0, the makespan of the job

permutation, π = {π(1),π(2), . . . ,π(n)}, is equal to the time when the last job in the processing

sequence is finished at machine m, and its value can be represented by Equation (6):

C max(π) = Dn,m (6)

Therefore, the objective of the formulated optimization problem can be formulated as follows:

minπ∈C max(π) (7)

3. Basic artificial bee colony algorithm

As the name implies, the basic ABC algorithm simulates the foraging behaviour of three kinds

of honey bee, i.e. employed bees, onlooker bees and scout bees. The ABC algorithm is an iter-

ative process like other swarm intelligence-based algorithms (Karaboga and Akay 2009). First,

a population representing a number of food sources is randomly initialized; each food source is




Figure 1. Pseudo-code of the basic artificial bee colony (ABC) algorithm.

a candidate solution of an optimization problem. Then, these food sources are updated by three

kinds of honey bee. For the first kind, a new food source is generated based on the current one

using an employed bee, and its fitness value is calculated. With respect to the second kind, the

food source sharing with the employed bee is further modified using an onlooker bee. Then, the

fitness value of the modified food source is calculated. For the last kind, a new food source is

randomly produced to update the unchanged one in the entire population using the scout bee.

After a number of iterations, the best food source obtained so far is saved and taken as the optimal

solution of the optimization problem. Figure 1 depicts the pseudo-code of the above process.

4. The proposed differential evolution–artificial bee colony algorithm

In this study, a DE-ABC algorithm incorporating DABC and DDE algorithms is designed to

enhance the ability of DABC in exploration and exploitation. The proposed algorithm adopts

both the MME heuristic and the random method to initialize the population. Then, the employed

bees explore new and unknown areas in the search space using the DDE approach. Following

that, the onlooker bees share these solutions and adopt the insert or swap operator to modify them

according to a self-adaptive strategy. Finally, the scout bees perform several insert operators on

the unchanged solutions in the population and replace them.

4.1. Initializing population

The MME heuristic, first proposed by Ronconi and Armentano (2001), has aroused much interest

and been successfully applied to the BFS scheduling problem with the makespan criterion. The

MME heuristic contains two main components, i.e. the MM and NEH heuristics, and employs

the shortest critical path to reduce the blocking time of a job on machines. Because of the good

performance of the MME heuristic, an initial solution with high quality is generated by adopting

the MME heuristic in this study. In addition, to maintain the diversity of the initial population, the

other solutions are randomly generated in the entire search space. Figure 2 states the pseudo-code

of the MME heuristic.

With the above heuristic, the procedure of initializing a population is summarized as follows.

Step 1: Perform the MME heuristic given in Figure 2 to yield a solution.



6 Y.-Y. Han et al.

Figure 2. Pseudo-code of the MME heuristic.

Step 2: Randomly generate a solution in the search space. If it is distinct from all the existing

solutions in the population, then put it into the population; otherwise, discard it.

Step 3: Repeat Step 2 until the population has PS individuals.

4.2. Exploring new and unknown areas using the discrete differential evolution approach

The insert and swap operators, through which the neighbouring solutions are generated, show the

superiority of the DABC algorithm. However, in the existing DABC algorithm, both the employed

and onlooker bees adopt the insert or swap operator to yield new solutions. Because of such a

repeated search, the neighbouring solutions obtained by them may be local optima. Thus, the

previous DABC algorithm finds it difficult to explore the entire solution space, which indicates

that it is weak in exploration. The merit of DDE is that it takes advantage of the differences among

individuals in the population to seek the global optimal solution. That is, DDE has the capability

to escape from the local optima. Therefore, the DDE algorithm is incorporated into DABC to

yield new solutions instead of the insert or swap operator in the employed bee stage. Since DDE

contains two critical operators, i.e. the mutation and crossover operators, an individual or jobpermutation π is first generated by performing the mutation operator. The job permutation, π ,

itself may not include all jobs since some jobs may exist repeatedly, whereas others may be lost.

Therefore, the crossover operator is applied to generate a complete sequence of jobs. Figure 3

gives the pseudo-code of the mutation and crossover operators.

To simply illustrate the aforementioned steps, suppose that there are five jobs, and the values

of the parameters pmu and pc are equal to 0.9 and 0.1, respectively. Let πref = {1,2,3,4,5}. An

example of constructing a new solution using the mutation and crossover operators is shown in

the following.

(1) Randomly select three solutions, πa = {2,1,4,3,5}, πb = {4,2,5,1,3} and πc =

{2,5,4,3,1}, from the population.

(2) Perform the mutation operator.

Set i = 1 and rand () = 0.8 < 0.9

π(1) = (πb(1) − πc(1)) ∗ 1 ◦ πa(1) = ((4 − 2) + 2 + 5)%5 + 1 = 5




Figure 3. Pseudo-code of the mutation and crossover operators.

i = 2 and rand () = 0.6 < 0.9

π(2) = (πb(2) − πc(2)) ∗ 1 ◦ πa(2) = ((2 − 5) + 1 + 5)%5 + 1 = 4

i = 3 and rand () = 0.95 > 0.9

π(3) = (πb(3) − πc(3)) ∗ 0 ◦ πa(3) = (0 + 4 + 5)%5 + 1 = 5

i = 4 and rand () = 0.1 < 0.9

π(4) = (πb(4) − πc(4)) ∗ 1 ◦ πa(4) = ((1 − 3) + 3 + 5)%5 + 1 = 2

i = 5 and rand () = 0.9

π(5) = (πb(5) − πc(5)) ∗ 0 ◦ πa(5) = (0 + 5 + 5)%5 + 1 = 1

So the job permutation, π = {5,4,5,2,1}, is generated using the mutation operator.

(3) Execute the crossover operator.

Set j = 1 and rand () = 0.3 > 0.1, so put the job, π(1), into πtemp, that is, πtemp = {5}

Set j = 2 and rand () = 0.05 < 0.1, so no job is selected.

Set j = 3 and rand () = 0.6 > 0.1; as π (3) already exists in πtemp, this job is discarded.

Set j = 4 and rand () = 0.08 < 0.1, so no job is selected.

Set j = 5 and rand () = 0.75 > 0.1, so put job π (5) into πtemp, that is, πtemp = {5, 1}.

In the following, generate a sub-sequence, πnew, by removing the jobs included in πtemp from

πref , soπnew = {2,3,4}. Take job 5 fromπtemp and insert it into the best position of πnew to obtain a

sub-sequence, πnew = {2,3,4,5}. Similarly, insert job 1 into the current sub-sequence, and obtain

the complete sequence, πnew = {2,3,1,5,4}, with the smallest value of makespan.



8 Y.-Y. Han et al.

Figure 4. Pseudo-code of the self-adaptive strategy.

4.3. Modifying the selected solutions using a self-adaptive strategy

The onlooker bees use the tournament selection to share some solutions generated by the employed

bees, and adopt the insert or swap operator to modify them based on a self-adaptive strategy so as to

search for outstanding neighbouring solutions. The insert operator randomly selects two positions, p1 and p2 ( p1 < p2), from a sequence π and moves all jobs between the positions p1 + 1 and p2

forward a position in turn, whereas the swap operator just interchanges the corresponding jobs

between positions p1 and p2 ( p1 < p2). Figure 4 describes the pseudo-code of the self-adaptive

strategy based on the insert and swap operators.

4.4. Updating unchanged solutions

As addressed in the basic ABC algorithm, after the employed bees and onlooker bees have finished

their search, the algorithm checks whether or not there are any exhausted or unchanged solutions.

Next, these unchanged solutions are replaced with the new ones discovered by the scout bees.The scout bees play a role in realizing the negative feedback mechanism and the fluctuation

property in the self-organization of the ABC algorithm (Akay and Karaboga 2012). In the basic

ABC algorithm, the scout bees randomly generate solutions to replace those unchanged ones.

This process enhances the diversity of the population, but it also reduces the search efficiency.

Therefore, each scout bee performs several insert operators on the unchanged solutions in this

study.

4.5. Local search

As a simple neighbourhood search approach, the local search seeks a better solution in the neigh-bourhood of a given solution. It has been shown that the insert-neighbourhood-based local search

(Ruben and Stutzle 2008) is superior to TS and the simulated annealing (SA) algorithm (Ruben and

Concepcion 2005). Therefore, this local search is adopted to enhance the ability of the proposed

algorithm in exploitation. In this study, the local search is performed on the solutions selected by




Figure 5. Pseudo-code of the local search.

the onlooker bees stage with a small probability of pls, which controls whether or not a solutionperforms the local search. Specifically, a random value, r , is generated in the range of [0,1]. If r

is less than pls, then the solution performs the local search. Figure 5 states the pseudo-code of the

local search.

4.6. Summary of the proposed algorithm

In summary, the steps of the proposed algorithm are as follows:

Step 1: Set the values of the parameters used in the proposed algorithm and initialize the

population by using MME and the random method stated in Subsection 4.1.Step 2: Perform the mutation and crossover operators presented in Subsection 4.2 to produce

outstanding solutions.

Step 3: Modify the selected solutions using the self-adaptive strategy given in Subsection 4.3.

Step 4: Carry out the local search to seek a better solution from the neighbourhood of a given

solution with the probability of pls.

Step 5: Obtain new solutions by applying several insert operators to the unchanged solutions

in the whole population and replace the exhausted ones according to the approach described

in Subsection 4.4.

Step 6: Judge whether the termination criterion of the algorithm is met or not. If it is, stop the

evolution of the population and output the optimal solutions; otherwise, go to Step 2.

5. Experiments

In this section, the proposed algorithm is applied to some typical instances of the BFS scheduling

problem and compared with six existing available algorithms to evaluate the performance of the

proposed algorithm. The proposed algorithm is written with C++, and implemented on a personal

computer with Pentium® Dual 2.79 GHz and 1.96 GB of memory.

5.1. Experimental setting

There are 120 instances from the well-known benchmark set provided by Taillard (1993). These

instances have also been used by the existing algorithms by treating them as Fm|blocking|C max.

They are divided into 12 groups of different sizes, each consisting of 10 instances of the same



10 Y.-Y. Han et al.

size. The values of parameters, the size of instances and the maximal computation time adopted

by the proposed algorithm are set as follows:

• The population size, PS , the mutation probability, pmu, the crossover probability, pc, and the

local search probability, pls, are set to 20, 0.9, 0.1 and 0.2, respectively.• The number of jobs for each instance is chosen from the set {20,50,100,200,500} and that of

machines is chosen from {5,10,20}.

• For each instance, the maximal computation time is set to 5 × m × n milliseconds.

The proposed DE-ABC algorithm is compared with the existing metaheuristics, i.e. TS + M

(Grabowski and Pempera 2007),HDDE (Wang et al. 2010), DABC (Han et al. 2011), IABC (Han,

Duan, and Zhang 2012), DPSOsvns (Wang and Tang 2012) and EDEc (Davendra et al. 2012). For

each instance, each method is independently run five times, the minimal makespan is recorded and

compared with the referenced makespan taken from the branch-and-bound method of Ronconi

(2005), and the average relative percentage difference of the five runs is obtained. For all instances

in a group, such average relative percentage differences are computed and denoted as the ARPD.The makespan of the jth instance provided by the ith algorithm in the t th run is denoted as

C i j,t , the referenced value of which, provided by Ronconi (2005), is C R j . In addition, the average

relative percentage difference obtained by the ith algorithm, denoted as ARPDi, can be expressed

as follows:

ARPDi =1

50

10

j=1

5

t =1

C R j − C i j,t

C R j× 100% (8)

It is clear that the larger the value of ARPDi, the better the result produced by the algorithm.

During the solving of the above instances, if a better solution of an instance is obtained using

the proposed algorithm than that of the comparative algorithms, its upper bound is updated. Thecomputational results related to the following aspects are reported in the experiments:

• comparison results between the proposed algorithm and the six comparative ones

• update of upper bounds of some benchmark problems

• convergence of different algorithms for six instances

• non-parametric test on ARPD indicator

• sensitivity analysis on parameters pmu, pc and pls used in the proposed algorithm.

5.2. Results and analysis

5.2.1. Comparison of DABC, IABC and DE-ABC

In this section, the proposed DE-ABC algorithm is compared with two existing ABC algorithms,

DABC and IABC. The essential difference between DABC and DE-ABC is that the former uses

the insert or swap operator in the employed bee stage, whereas the latter adopts the mutation

and crossover operators of DDE. In addition to the above difference, for the IABC and DE-

ABC algorithms, the other difference is that the latter employs only one of the four initialization

strategies provided by the former. Although the four initialization strategies of IABC have been

recognized as superior to the single one, DDE is beneficial in generating good solutions if the

proposed algorithm is better than IABC. Thus, this section provides the following experimental

results to demonstrate the performance of DDE in exploration.Table 2 reports the corresponding comparison results on the premise of the same computational

time and experimental environment. From Table 2: (1) for instances with a small scale, such as

20 × 5, 20 × 10 and 20 × 20, the proposed algorithm is slightly inferior to DABC and IABC

in the value of ARPD; (2) except for the above cases, the proposed algorithm is superior to




Table 2. Average relative percentage difference (ARPD) values of DABCa , IABCa and DE-ABCa .

DABC IABC DE-ABC

n × m ARPD MinAD MaxAD ARPD MinAD MaxAD ARPD MinAD MaxAD Time (s)

20 × 5 0.44 0.40 0.46 0.41 0.32 0.46 0.34 0.21 0.46 0.50

20 × 10 2.40 2.38 2.40 2.38 2.35 2.39 2.34 2.25 2.40 1.00

20 × 20 3.30 3.30 3.30 3.30 3.29 3.30 3.29 3.26 3.30 2.00

50 × 5 4.60 4.29 4.90 4.68 4.32 5.03 4.89 4.52 5.21 1.2550 × 10 6.09 5.71 6.44 6.15 5.80 6.46 6 .25 5.93 6.55 2.50

50 × 20 6.27 5.99 6.54 6.29 6.03 6.57 6 .45 6.15 6.72 5.00

100 × 5 2.08 1.74 2.43 2.31 2.07 2.54 2.67 2.29 3.09 2.50

100 × 10 5.63 5.40 5.88 5.97 5.68 6.34 6 .27 5.93 6.62 5.00100 × 20 5.11 4.73 5.41 5.39 5.13 5.72 5.53 5.15 5.89 10.00

200 × 10 3.49 3.22 3.81 4.14 3.86 4.44 4.37 3.91 4.74 10.00

200 × 20 3.91 3.62 4.23 4.46 4.24 4.67 4.70 4.45 5.02 20.00

500 × 20 3.25 3.10 3.46 3.70 3.56 3.89 3.72 3.51 3.94 50.00

Average 3.88 3.65 4.11 4.10 3.89 4.32 4.24 3.96 4.50 9.15

Note: Data in italic indicate the best value of ARPD for each group of instances among these algorithms.a Pentium Dual 2.79 GHz and 1.96 GB memory.

the comparative ones; and (3) on average, the ARPD value of DE-ABC is equal to 4.22, much

better than the values of 3.88 and 4.10 gained by DABC and IABC, respectively. The minimal

and maximal values of ARPD (MinAD and MaxAD, respectively) produced by DE-ABC are also

much larger than those generated by the DABC andIABC algorithms. Furthermore, the superiority

of the DE-ABC algorithm over the comparative ones increases along with the increment of the

problem size.

The following conclusion can be drawn from the above experimental results: the proposed

algorithm can take full advantage of the differentiation information among the population in theemployed bee stage to enhance the capability in exploration.

5.2.2. Comparison of EDEc, HDDE and DE-ABC

The HDDE and EDEc algorithms proposed by Wang et al. (2010) and Davendra et al. (2012),

respectively, can obtain better solutions than some existing algorithms, i.e. TS + M, GA and HDE,

suggesting that the algorithms based on DDE are more powerful. Although DDE has the capability

in exploration, it lacks the capability in exploitation. However, DABC can successfully search

for local optima in the neighbourhood of a solution, so the DE-ABC algorithm, by incorporating

DDE with DABC, can balance the capability in exploration and exploitation, which can be verifiedfrom Table 3.

As illustrated in Table 3: (1) for instances of 20 × 5, 20 × 10 and 20 × 20, the proposed

algorithm is slightly inferior to EDEc and HDDE; (2) except for the above cases and the 100 × 5

instance, the proposed algorithm outperforms the comparative ones; (3) the ARPD value, 4.22,

generated by DE-ABC is much larger than the values of 3.79 and 3.75 obtained by EDEc and

HDDE, respectively, indicating that the hybrid algorithm based on DDE and DABC does better

in exploration and exploitation than the above comparative ones.

From the above experimental results, it can be concluded that DE-ABC overcomes the shortage

of DDE in exploitation by incorporating DABC and obtains solutions with high quality.

5.2.3. Comparison of TS + M , DPSOsvns and DE-ABC

For blocking scheduling problems, there are two typical algorithms, i.e. TS + M and DPSOsvns.

The former is a benchmark and the latter has been recently developed. In this study, the proposed



12 Y.-Y. Han et al.

Table 3. Average relative percentage difference (ARPD) values of HDDEa , DE-ABCa and EDEcb.

EDEc HDDE DE-ABC

n × m ARPD MinAD MaxAD ARPD MinAD MaxAD ARPD MinAD MaxAD

20 × 5 0.46 0.00 1.83 0.45 0.42 0.46 0.34 0.21 0.46

20 × 10 2.39 0.57 5.36 2.38 2.37 2.39 2.34 2.25 2.40

20 × 20 3.30 2.13 4.48 3.30 3.29 3.30 3.29 3.26 3.30

50 × 5 4.80 3.03 6.06 4.58 4.37 4.85 4.89 4.52 5.21

50 × 10 5.78 4.50 6.95 5.94 5.66 6.19 6 .25 5.93 6.5550 × 20 5.82 3.68 8.13 6.19 6.02 6.44 6 .45 6.15 6.72

100 × 5 3.23 1.46 3.92 1.78 1.47 2.09 2.67 2.29 3.09

100 × 10 5.92 3.62 6.24 5.44 5.22 5.72 6 .27 5.93 6.62

100 × 20 4.74 3.28 5.25 4.86 4.66 5.14 5.53 5.15 5.89200 × 10 3.23 2.58 4.84 3.41 3.21 3.68 4.37 3.91 4.74

200 × 20 3.32 3.09 3.73 3.83 3.65 4.08 4.70 4.45 5.02

500 × 20 2.56 2.12 3.05 2.91 2.72 3.13 3.72 3.51 3.94

Average 3.79 2.51 4.98 3.75 3.59 3.97 4.24 3.96 4.50

Note: Data in italic indicate the best value of ARPD for each group of instances among these algorithms.a Pentium Dual 2.79 GHz and 1.96 GB memory.bMacBook Pro, 2.4 GHz Intel Core 2 Duo, 2 GB RAM.

Table 4. Average relative percentage difference (ARPD) values of TS + Ma , DPSObsvns and DE-ABC.

TS + M DPSOsvns DE-ABC

n × m ARPD Time (s) ARPD Time (s) ARPD MinAD MaxAD Time (s)

20 × 5 −0.24 2.70 −0.07 1.00 0.34 0.21 0.46 0.5020 × 10 1.77 4.60 1.94 2.00 2.34 2.25 2.40 1.00

20 × 20 2.94 7.60 3.12 3.00 3.29 3.26 3.30 2.00

50 × 5 0.55 6.20 3.36 2.50 4.89 4.52 5.21 1.25

50 × 10 3.52 10.80 4.25 5.00 6 .25 5.93 6.55 2.5050 × 20 4.26 19.30 4.60 10.00 6 .45 6.15 6.72 5.00

100 × 5 2.62 12.40 1.48 5.00 2.67 2.29 3.09 2.50

100 × 10 2.66 22.10 4.42 10.00 6 .27 5.93 6.62 5.00

100 × 20 3.02 39.40 3.38 20.00 5.53 5.15 5.89 10.00200 × 10 0.58 44.30 2.53 20.00 4.37 3.91 4.74 10.00

200 × 20 2.31 79.40 2.82 40.00 4.70 4.45 5.02 20.00

500 × 20 1.47 209.00 1.63 100.00 3.72 3.51 3.94 50.00

Average 1.68 38.15 2.79 18.20 4.24 3.96 4.50 9.15

Note: Data in italic indicate the best value of ARPD for each group of instances among these algorithms.a Pentium P-IV, 1000 MHz and 30,000 iterations.bIntel 2.33GHz CPU and 2 GB memory.

algorithm is compared with these, and associated comparison results are reported in Table 4. It

should be noted that the TS + M algorithm employed 30,000 iterations on an Intel 1 GHz CPU,

whereas both DPSOsvns and DE-ABC performed fewer than 2000 iterations on a Pentium 2.33

and 2.79 GHz CPU, respectively. Although the implementation environment in this study is better

than that of TS + M, fewer iterations are adopted here. Therefore, it can be seen that the value of

ARPD produced by DE-ABC is comparable with that of TS + M.From Table 4: (1) for all instances, DE-ABC clearly prevails over TS + M and DPSOsvns in the

value of ARPD; and (2) even the MinAD value of DE-ABC is much larger than that generated

by the TS + Mand DPSOsvns algorithms. These values demonstrate that the proposed algorithm

performs better in the value of ARPD than the comparative algorithms.




5.2.4. New upper bounds

The excellent performance of the proposed algorithm encouraged the authors to seek new upper

bounds of Taillard’s benchmarks for the Fm|blocking|C max scheduling problem. Ronconi (2005)

first reported the upper bounds of 120 instances; following that, Grabowski and Pempera (2007)obtained 94 out of 120 upper bounds using the TS+M algorithm. Later, more algorithms, i.e.

HDDE, DABC, DPSOsvns, EDEc and IABC, provided newer upper bounds. Thus, in Tables 5 and

6, for each instance, the best solution yielded by the proposed algorithm is reported to update

some upper bounds.

As stated in Tables 5 and 6: (1) for the instances with a small scale, i.e. 20 × 5, 20 × 10 and

20 × 20, 30 best values of makespan obtained by DE-ABC are equal to the corresponding best

ones yielded by HDDE, DABC, DPSOsvns, EDEc and IABC, respectively; (2) for the remaining

90 instances, zero, six (6/90 = 7%), one (1/90 = 1%), zero, 11 (11/90 = 12%) and 10 (10/90 =

11%) new solutions produced by TS + M, DABC, HDDE, DPSOsvns, EDEc and IABC are found,

respectively, while 62 out of 90 (62/90 = 69%) new upper bounds provided by the proposedalgorithm are further improved. These experimental results clearly demonstrate that the proposed

algorithm can update more upper bounds of Taillard’s benchmarks than the comparative ones.

5.2.5. Convergence curves of different algorithms

To further evaluate the makespan values of different algorithms, in this section, the convergence

curves of the makespan values are investigated. For the Fm |blocking|C max scheduling problem,

Ta53, Ta72, Ta86, Ta96, Ta109 and Ta115 of Taillard’s benchmarks are selected, and their best val-

ues of makespan obtained by HDDE, DABC, DPSOsvns, EDEc, IABC andDE-ABC are calculated,

respectively, with the increment of computation time, shown as Figures 6–8.From Figures 6–8: (1) for each instance, each algorithm can obtain better and better makespan

values as the computation time increases; and (2) for the same computation time, the makespan

values of the proposed algorithm are the smallest among these algorithms.

In summary, compared with the other algorithms for the Fm|blocking|C max schedulingproblem,

the algorithm proposed in this study does better in both exploration and exploitation. Therefore,

it can achieve more outperforming solutions.

5.2.6. Non-parametric testing in the value of ARPD

To evaluate whether the differences in the value of ARPD between the algorithm proposed in this

study and the comparative ones are significant, the Mann–Whitney U distribution test, a non-

parametric testing method, was conducted. Table 7 lists the experimental results; the symbols ‘+’

and ‘−’ in the table denote that the proposed algorithm is significantly superior and inferior to

the comparative method, respectively, whereas the symbol ‘0’ indicates that there is no significant

difference between them.

Table 7 reports that: (1) the proposed algorithm is significantly inferior to HDDE, DABC, EDEc

and IABC for the instance of 20 × 5; (2) for the 20 × 10 and 20 × 20 instances, the proposed

algorithm is not significantly different from the comparative algorithms except for TS + M; (3)

for the 100 × 5 instance, the proposed algorithm is inferior to EDEc; (4) there is no significant

difference between DE-ABC and IABC for the 500 × 20 instance; and (5) except for the abovecases, the proposed algorithm is significantly superior to TS + M, HDDE, DPSOsvns, DABC,

EDEc and IABC.

From the above experimental results, it can be concluded that the proposed algorithm is

significantly superior to the comparative ones for most instances.



14 Y.-Y. Han et al.

Table 5. Values of makespan for the instances with n = 20 and 50.

Data set Instance TS + M DABC HDDE DPSOsvns EDEc IABC DE-ABC

20 × 5 Ta01 1387 1374 1374 1374 1374 1374 1374

Ta02 1408 1408 1408 1408 1408 1408 1408Ta03 1280 1280 1280 1280 1280 1280 1280Ta04 1448 1448 1448 1448 1448 1448 1448Ta05 1341 1341 1341 1341 1341 1341 1341Ta06 1363 1363 1363 1363 1363 1363 1363Ta07 1381 1381 1381 1381 1381 1381 1381Ta08 1379 1379 1379 1379 1379 1379 1379Ta09 1373 1373 1373 1373 1373 1373 1373Ta10 1283 1283 1283 1283 1283 1283 1283

20 × 10 Ta11 1698 1698 1698 1698 1698 1698 1698Ta12 1836 1833 1833 1833 1833 1833 1833Ta13 1674 1659 1659 1659 1659 1659 1659Ta14 1555 1535 1535 1535 1535 1535 1535

Ta15 1631 1617 1617 1617 1617 1617 1617Ta16 1603 1590 1590 1590 1590 1590 1590Ta17 1629 1622 1622 1622 1622 1622 1622Ta18 1754 1731 1731 1731 1731 1731 1731Ta19 1759 1749 1749 1749 1749 1749 1749

Ta20 1782 1782 1782 1782 1782 1782 178220 × 20 Ta21 2449 2436 2436 2436 2436 2436 2436

Ta22 2242 2234 2234 2234 2234 2234 2234Ta23 2483 2479 2479 2479 2479 2479 2479Ta24 2348 2348 2348 2348 2348 2348 2348Ta25 2450 2435 2435 2435 2435 2435 2435Ta26 2398 2383 2383 2383 2383 2383 2383Ta27 2397 2390 2390 2390 2390 2390 2390Ta28 2345 2328 2328 2328 2328 2328 2328Ta29 2363 2363 2363 2363 2363 2363 2363Ta30 2334 2323 2323 2323 2323 2323 2323

50 × 5 Ta31 3163 3009 3027 3038 3028 3021 3016Ta32 3348 3215 3225 3233 3227 3203 3219

Ta33 3173 3040 3032 3050 3031 3022 3025

Ta34 3277 3142 3145 3181 3140 3128 3136

Ta35 3338 3180 3188 3196 3173 3186 3162Ta36 3330 3192 3190 3213 3194 3193 3177Ta37 3168 3046 3049 3055 3053 3031 3024Ta38 3228 3084 3081 3084 3089 3081 3059Ta39 3068 2923 2917 2935 2936 2921 2921

Ta40 3285 3134 3127 3137 3127 3135 3125

50 × 10 Ta41 3776 3654 3665 3686 3681 3661 3647Ta42 3641 3508 3522 3548 3534 3512 3496Ta43 3588 3487 3497 3517 3524 3497 3495

Ta44 3786 3687 3682 3732 3710 3680 3672Ta45 3745 3643 3650 3669 3663 3636 3637Ta46 3747 3607 3621 3674 3634 3600 3619

Ta47 3778 3707 3716 3729 3731 3708 3688Ta48 3708 3587 3591 3615 3603 3572 3589

Ta49 3668 3527 3553 3573 3563 3541 3534

Ta50 3729 3637 3642 3672 3656 3631 3623

50 × 20 Ta51 4627 4515 4516 4547 4541 4521 4517Ta52 4411 4307 4298 4349 4329 4289 4278Ta53 4388 4278 4282 4343 4319 4289 4266Ta54 4479 4382 4379 4456 4416 4378 4344

Ta55 4359 4282 4283 4283 4304 4278 4276Ta56 4372 4295 4301 4334 4328 4290 4299

Ta57 4402 4324 4325 4371 4369 4312 4311Ta58 4444 4322 4341 4384 4361 4328 4336

Ta59 4423 4318 4328 4343 4365 4320 4323

Ta60 4609 4429 4443 4520 4452 4433 4415

Note: Data in bold indicate the best upper bound for each instance among the comparative algorithms.




Table 6. Values of makespan for the instances with n = 100, 200 and 500.

Data set Instance TS + M DABC HDDE DPSOsvns EDEc IABC DE-ABC

100 × 5 Ta61 6639 6226 6274 6270 6155 6232 6179

Ta62 6481 6117 6121 6098 6042 6105 6072Ta63 6299 6017 6035 6018 5941 6007 5976

Ta64 6120 5835 5839 5842 5772 5810 5783

Ta65 6340 6056 6060 6026 5997 6056 6001Ta66 6244 5910 5956 5951 5876 5903 5891

Ta67 6346 6074 6101 6094 6008 6063 6049

Ta68 6289 5997 6022 5964 5908 5980 5948

Ta69 6559 6201 6234 6221 6153 6218 6187Ta70 6509 6247 6247 6209 6160 6237 6195

100 × 10 Ta71 7320 7119 7140 7146 7053 7097 7069

Ta72 7108 6824 6853 6882 6806 6806 6781Ta73 7233 6959 6957 6997 6925 6951 6911Ta74 7413 7213 7246 7283 7201 7173 7188

Ta75 7168 6899 6895 6940 6853 6852 6840Ta76 6993 6719 6754 6768 6702 6697 6661Ta77 7092 6874 6869 6924 6841 6835 6826Ta78 7143 6927 6931 6961 6897 6886 6859Ta79 7327 7144 7133 7192 7059 7090 7039Ta80 7299 7037 7056 7086 6990 6982 6993

100 × 20 Ta81 8101 7866 7878 8068 7894 7829 7841Ta82 8105 7919 7931 8015 7902 7881 7883Ta83 8071 7918 7944 7985 7897 7873 7856Ta84 8081 7875 7923 7999 7919 7883 7857Ta85 8074 7899 7900 7977 7905 7873 7859Ta86 8151 7940 7971 8054 7979 7923 7870Ta87 8273 8009 8031 8122 8041 8002 7993Ta88 8248 8059 8086 8185 8061 8042 8026Ta89 8116 7980 8018 8084 7984 7952 7928Ta90 8261 8005 8017 8102 8025 7945 7973

200 × 10 Ta91 14,220 13,711 13,722 13,806 13,725 13,622 13,617Ta92 14,089 13,647 13,596 13,721 13,624 13,567 13,514Ta93 14,149 13,732 13,762 13,867 13,721 13,632 13,584Ta94 14,156 13,678 13,686 13,706 13,675 13,605 13,532Ta95 14,130 13,666 13,702 13,832 13,723 13,528 13,504Ta96 13,963 13,410 13,492 13,517 13,420 13,352 13,311Ta97 14,386 13,901 13,925 13,943 13,898 13,859 13,814Ta98 14,256 13,798 13,839 13,885 13,809 13,722 13,644Ta99 13,954 13,626 13,656 13,726 13,585 13,507 13,505

Ta100 14,224 13,764 13,733 13,818 13,703 13,638 13,623

200 × 20 Ta101 15,334 15,006 15,057 15,117 15,111 14,931 14,901Ta102 15,522 15,229 15,263 15,309 15,326 15,180 15,087Ta103 15,713 15,329 15,360 15,495 15,440 15,218 15,205Ta104 15,687 15,233 15,276 15,452 15,400 15,188 15,109Ta105 15,443 15,149 15,123 15,257 15,215 15,076 15,001Ta106 15,472 15,224 15,223 15,370 15,372 15,220 15,108Ta107 15,522 15,199 15,288 15,289 15,327 15,185 15,113Ta108 15,540 15,322 15,283 15,402 15,410 15,203 15,139Ta109 15,394 15,162 15,207 15,337 15,280 15,093 15,066Ta110 15,523 15,239 15,254 15,379 15,357 15,132 15,141

500 × 20 Ta111 37,860 36,960 37,206 37,614 37,139 36,908 36,865Ta112 38,044 37,359 37,485 37,965 37,474 37,192 37,132Ta113 37,732 36,827 37,142 37,477 37,154 36,872 36,778Ta114 38,062 37,328 37,291 37,986 37,471 37,245 37,132

Ta115 37,991 37,032 37,267 37,592 37,162 36,936 36,722Ta116 38,132 37,221 37,477 37,969 37,372 37,244 37,131Ta117 37,561 36,970 37,133 37,528 37,324 36,845 36,825Ta118 37,750 37,219 37,161 37,790 37,562 37,045 37,019Ta119 37,730 37,060 36,939 37,455 37,112 36,816 36,885

Ta120 38,014 37,243 37,389 37,818 37,294 37,137 37,056

Note: Data in bold indicate the best upper bound for each instance among the comparative algorithms.



16 Y.-Y. Han et al.

0 5 10 15 20 254200

4250

4300

4350

4400

4450

4500

4550

4600

4650

4700

(s)

m a k e s p a n

Ta 53 (50×20 problem)

DE-ABCDABCHDDEIABCDPSOsvnsEDE

0 5 10 15 20 25 30 35 40 45 506700

6800

6900

7000

7100

7200

7300

7400

(s)

m a k e s p a n


DE-ABCDABCHDDEIABCDPSOsvnsEDE

Figure 6. Convergence curves of instances Ta 53 and Ta72.

0 5 10 15 20 25 30 35 40 45 501.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

1.4

1.41x 104

(s)

M a k e s p a n


DE-ABCDABCHDDEIABCDPSOsvnsEDEc


0 5 10 15 20 25 307800

7900

8000

8100

8200

8300

8400

(s)

M a k e s p a n


Figure 7. Convergence curves of instances Ta86 and Ta96.

0 10 20 30 40 50 60 70 80 90 1001.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56

1.57

1.58

1.59

x 104

(s)

M a k e s p a n

Ta 109 (200×20 problem)

0 20 40 60 80 100 120 1403.66

3.68

3.7

3.72

3.74

3.76

3.78

3.8

3.82

3.84x 104

(s)

M a k e s p a n

Ta 115 (500×20 prolbem)



Figure 8. Convergence curves of instances Ta109 and Ta115.




Table 7. Non-parametric testing results in the average relative percentage difference (ARPD) value.

DE-ABC vs TS + M HDDE DPSOsvns DABC EDEc IABC

20 × 5 + − + − − −

20 × 10 + 0 0 0 0 020 × 20 + 0 0 0 0 0

50 × 5 + + + + 0 +

50 × 10 + + + + + +

50 × 20 + + + + + +

100 × 5 + + + + − +

100 × 10 + + + + + +

100 × 20 + + + + + +

200 × 10 + + + + + +

200 × 20 + + + + + +

500 × 20 + + + + + 0

–0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.1

4.15

4.2

4.25

4.3

Parameter pmu

A R P D

Figure 9. Influence of parameter pmu on the average relative percentage difference (ARPD).

5.2.7. Sensitivity analysis on parameters pmu, pls and pc

There are three important parameters, pmu, pc and pls, in the proposed algorithm. To investigate

the influences of these parameters on the performances of DE-ABC, their values were changed

from 0 to 1.0 with a step size of 0.1, and the corresponding values of ARPD were obtained and

are shown in Figures 9–11. In these figures, the horizontal coordinate denotes the value of a

parameter, and the vertical coordinate represents the average value of ARPD for 120 instances.

Figure 9 illustrates that: (1) the trajectory tendency is relatively stable when the value of pmu

varies in the range of [0, 0.1]; (2) the value of ARPD gradually reduces to a low level when thevalue of pmu changes from 0.1 to 0.2 and from 0.3 to 0.8; and (3) when pmu is equal to 0.9, the

ARPD value reaches the largest value, 4.22. It can be concluded that the parameter pmu has no

evident influence on the value of ARPD of the proposed algorithm. Therefore, in this study, the

value of pmu is set as 0.9.



18 Y.-Y. Han et al.

–0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14

4.05

4.1

4.15

4.2

4.25

4.3

4.35

4.4

Parameter pc

A R P D

Figure 10. Influence of parameter pc on the average relative percentage difference (ARPD).

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.2

3.4

3.6

3.8

4

4.2

4.4

Parameter pls

A R P D

Figure 11. Influence of parameter pls on the average relative percentage difference (ARPD).

Figure 10 reports that: (1) when the value of pc is in the range of [0.3, 1], the value of ARPD

becomes smaller and smaller along with the increment in pc. The reason is that as the solution

obtained by the mutation operator can be considered with a smaller and smaller probability,




equivalently, the mutation operator works with the same probability; and (2) if pc = 1, then the

value of ARPD is the smallest, indicating that only the crossover operator is used to generate a

new solution based on the current reference one, resulting in trapping into local optima of the

optimization problem. Thus, the value of pc is set as 0.1 in this study.

Figure 11 demonstrates that: (1) when the value of pls is equal to 0, suggesting that the selected

solutions do not undergo the local search at all, the proposed algorithm will have a bad perfor-

mance; and (2) the larger the value of pls, the smaller the value of ARPD obtained by the proposed

algorithm. This is the reason why the selected solutions will perform the local search with a large

probability, which needs to spend a long time and loses opportunities to generate promising solu-

tions by a number of iterations. Therefore, an appropriate value of pls leading to a good ARPD is

set as 0.2.

6. Conclusions

Since the BFS scheduling problem plays a key role in real-world applications, it is necessary to

develop effective methods for this problem. In recent years, both ABC and DE have proven to be

effective algorithms for solving the BFS scheduling problem. Thus, a novel algorithm, DE-ABC,

is presented in this study by incorporating the merits of DDE with those of DABC to solve the

BFS problem with makespan criterion. First, the mutation and crossover operators with powerful

exploration are adopted to generate solutions in the employed bee stage, and then the insert or swap

operator based on the self-adaptive strategy is proposed to modify the selected solutions in the

onlooker bee stage. Finally, an efficient local search operator based on the insert neighbourhood

is designed to improve the algorithm’s capability in exploitation.

The performances of the proposed algorithm in this article are measured on a set of 120benchmark instances proposed by Taillard. The experimental results demonstrate the superiority

of the proposed algorithm in terms of the ARPD indicator by comparison with the results yielded

by some algorithms presented in the literature, such as TS + M, DPSOsvns, DABC, HDDE, EDEc

and IABC.

It is worth mentioning that the excellent performance of the proposed algorithm in exploration

and exploitation may be attributed to the combination of DDE, the insert or swap operator based

on the self-adaptive strategy and the local search. Future work could apply the proposed algorithm

to scheduling problems with other criteria and extend the ideas proposed in this study to multi-

objective scheduling problems.

Funding

This researchis jointly supported by the NationalNaturalScience Foundation of China [grant no. 61375067]; FundamentalResearch Funds for the Central Universities [grant no. 2013XK09]

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