The scheduling problems have been discussed in the literature extensively under the assumption that the machines are permanently available without any breakdown. However, in the real manufacturing environments, the machines could be unavailable inevitably for many reasons. In this paper, the authors introduce the hybrid flowshop scheduling problem with random breakdown (RBHFS) together with a discrete group search optimizer algorithm (DGSO). In particular, two different working cases, preemptresume case, and preemptrepeat case are considered under random breakdown. The proposed DGSO algorithm adopts the vector representation and several discrete operators, such as insert, swap, differential evolution, destruction, and construction in the producers, scroungers, and rangers phases. In addition, an orthogonal test is applied to configure the adjustable parameters in the DGSO algorithm. The computational results in both cases indicate that the proposed algorithm significantly improves the performances compared with other high performing algorithms in the literature.
The hybrid flowshop scheduling (HFS) problem, being also referred to as multiprocessor or flexible flowshop, is one kind of production scheduling problems, which has been widely used in the process industry such as the paper, oil, petrochemical, and pharmaceutical industries [
Compared with a lot of works on the HFS problem, the studies on the RBHFS problem are still in their infancy. Alcaide et al. [
Recently, a populationbased optimization algorithm group search optimizer (GSO) has been proposed by He et al. [
Considering the successes of the GSO algorithm, the authors proposed to use a discrete group search optimizer (DGSO) algorithm for the production scheduling problem. The proposed DGSO algorithm maintains the optimization mechanism of the basic GSO algorithm and abandons the angle evolution strategy to improve the effectiveness and efficiency of the algorithm. In the DGSO algorithm, an encoding scheme based on the vector representation is introduced in order to adapt the GSO algorithm to the discrete problems. Meanwhile, an improved variable neighborhood search, a novel differential evolution operation, and the destruction and construction procedures are proposed in the producer, scrounger, and ranger phases, respectively. In addition, in order to achieve a good performance of the DGSO algorithm, an orthogonal experiment design is carried out for getting a guideline on tuning of the parameters in the algorithm. In both the preemptresume case and the preemptrepeat case, our proposed DGSO algorithm shows the stateoftheart results on benchmarks.
The rest of the paper is organized as follows. In Section
The RBHFS problem with makespan criterion can be described as follows. There are
It is assumed that a machine keeps processing the jobs sequentially until it break down or it has finished all the jobs, and machine breakdowns may arise at any time in working periods. The authors find that the possible positions where the breakdown may happen in are shown in Figure
The Gantt chart of the RBHFS problem.
This paper considers two different cases while dealing with an interrupted job. Case 1 is the preemptresume case and Case 2 is the preemptrepeat case. In Case 1, after repairing, only the unprocessed part of the interrupted job needs to be processed. And in Case 2, after repairing, the whole interrupted job needs to be reprocessed. It is assumed that
if
Case 1:
Case 2:
if
Case 1 and Case 2:
if
Case 1 and Case 2:
if
Case 1 and Case 2:
The basic GSO algorithm adopts the framework of the biological model, ProducerScrounger (PS) model [
Procedure the GSO algorithm
Choose the producers and perform the producing
Choose the scroungers and perform the scrounging
Disperse the rest individuals to perform ranging
Evaluate the individuals
End procedure
Following the procedure for the continuous function optimization, in this paper the authors propose a discrete version of the GSO algorithm for the RBHFS problem. It is discussed in detail below.
Owing to its continuous nature, the GSO algorithm dose not directly fit for the discrete flowshop scheduling problem. So it is important to find a suitable mapping which can conveniently convert individuals to solutions. In the HFS problem, there are two formats to represent a solution: the matrix representation and the vector representation [
Recently, Couzin et al. [
Aiming at improving the efficiency of the proposed DGSO algorithm, the authors introduce an improved variable neighborhood search (IVNS) [
(a) the insert local search
while (
do {
for (
job
if (
producer =
endif
endfor
}
(b) the swap local search
while (
do {
for (
job
if (
producer =
endif
endfor
}
In the population, the left individuals except the producer are divided into the scroungers and the rangers. Each individual is set to be scrounger or ranger with the probability of
As for each scrounger, a novel discrete differential evolution scheme is employed for improving the scrounging performance, which consists of three steps: mutation, crossover, and selection.
In the mutation part, the
Next a crossover operation called CRO is employed in the crossover part, and it will be able to work effectively even though the individuals in the population are very close to each other in the later stage of evolution. The crossed individual
Following the crossover operation, the selection is conducted. The best one which has the lowest objective value among the two crossed individuals and the incumbent scrounger is accepted. In other words, if either of these two crossed individuals yields a better makespan than the scrounger, then the better individual becomes the scrounger; otherwise the old scrounger is retained.
In the basic GSO algorithm, the rangers search randomly in the predefined space to increase the population diversity and avoid getting trapped in local optima. Here the rangers employ the destruction and construction procedures of the iterated greedy (IG) algorithm with one parameter: destruction size (
Based on the above operations, the procedure of the DGSO algorithm for the RBHFS problem is summarized as follows.
Set the algorithm parameters PS,
The producer conducts an improved variable neighborhood search to search for a better solution if the producer is changed.
The scroungers employ the discrete differential evolution operation to keep searching for the highquality solutions, which includes mutation, crossover, and selection.
The rangers produce new solutions by using the destruction and construction procedures to avoid local optimum.
Evaluate each member in the population and utilize the best individual to update the producer.
If the given termination criterion is satisfied, end the procedure and return the producer; otherwise go back to Step
Considering the character of the RBHFS problem, some discrete operators are introduced for the producers, scroungers, and rangers. The DGSO algorithm has the producer and the scroungers to play the part of exploitation and employs the rangers to play the part of exploration. Since both the exploitation and exploration are improved and well balanced, it is expected to generate good results for the RBHFS problem under the criterion of makespan minimization. In the next section, the performance of the DGSO algorithm is investigated based on simulation results and comparisons.
To fully examine the performance of the DGSO algorithm, a parameter discussion, a preliminary experiment, and an extensive experimental comparison with other powerful methods are provided. As no test instances are available for the RBHFS problem, some benchmark problems for the HFS problem are modified. Ten benchmark problems proposed by Liao et al. [
Tuning parameters properly is critical for an evolutionary algorithm to achieve a good performance [
Factors and levels for orthogonal experiment.
Factor  Level 1  Level 2  Level 3  Level 4  Level 5 

PS(1)  10  20  30  40  50 

0.6  0.7  0.8  0.9  1 
MR(3)  0.2  0.4  0.6  0.8  1 

2  3  4  5  6 

1  2  3  4  5 

1  2  3  4  5 
Parameter experiments are conducted on ten Liao’s benchmark problems. The breakdowns in the parameter discussion are subject to preemptresume case (Case 1) for simplicity. Taking the test on instance j30c5e1, for example, Table
Orthogonal parameter table L_{25}(5^{6}) and results of j30c5e1.
Test  Factor  Mean value  

PS 

MR 


 
1  10(1)  0.6(1)  0.2(1)  2(1)  1(1)  1(1)  505.7 
2  10(1)  0.7(2)  0.4(2)  3(2)  2(2)  2(2)  503.5 
3  10(1)  0.8(3)  0.6(3)  4(3)  3(3)  3(3)  504.5 
4  10(1)  0.9(4)  0.8(4)  5(4)  4(4)  4(4)  503.8 
5  10(1)  1(5)  1(5)  6(5)  5(5)  5(5)  509.8 
6  20(2)  0.6(1)  0.4(2)  4(3)  4(4)  5(5)  503.2 
7  20(2)  0.7(2)  0.6(3)  5(4)  5(5)  1(1)  506.5 
8  20(2)  0.8(3)  0.8(4)  6(5)  1(1)  2(2)  504.7 
9  20(2)  0.9(4)  1(5)  2(1)  2(2)  3(3)  505.4 
10  20(2)  1(5)  0.2(1)  3(2)  3(3)  4(4)  512.7 
11  30(3)  0.6(1)  0.6(3)  6(5)  2(2)  4(4)  503.5 
12  30(3)  0.7(2)  0.8(4)  2(1)  3(3)  5(5)  503.4 
13  30(3)  0.8(3)  1(5)  3(2)  4(4)  1(1)  506.6 
14  30(3)  0.9(4)  0.2(1)  4(3)  5(5)  2(2)  506.3 
15  30(3)  1(5)  0.4(2)  5(4)  1(1)  3(3)  512.3 
16  40(4)  0.6(1)  0.8(4)  3(2)  5(5)  3(3)  503.4 
17  40(4)  0.7(2)  1(5)  4(3)  1(1)  4(4)  506.4 
18  40(4)  0.8(3)  0.2(1)  5(4)  2(2)  5(5)  506.0 
19  40(4)  0.9(4)  0.4(2)  6(5)  3(3)  1(1)  507.2 
20  40(4)  1(5)  0.6(3)  2(1)  4(4)  2(2)  512.0 
21  50(5)  0.6(1)  1(5)  5(4)  3(3)  2(2)  504.0 
22  50(5)  0.7(2)  0.2(1)  6(5)  4(4)  3(3)  505.3 
23  50(5)  0.8(3)  0.4(2)  2(1)  5(5)  4(4)  504.5 
24  50(5)  0.9(4)  0.6(3)  3(2)  1(1)  5(5)  507.1 
25  50(5)  1(5)  0.8(4)  4(3)  2(2)  1(1)  510.4 



505.46  503.96  507.20  506.20  507.24  507.28  

506.50  505.02  506.14  506.66  505.76  506.10  

506.42  505.26  506.72  506.16  506.36  506.18  

507.00  505.96  505.14  506.52  506.18  506.18  

506.26  511.44  506.44  506.10  506.10  505.90  


std  0.558  2.946  0.770  0.247  0.554  0.544 
Comparing with the group search optimizer (GSO) algorithm, there are three new elements in the proposed DGSO algorithm: the improved variable neighborhood search in the producer phase, the differential evolution in the scrounger phase, and the destruction and construction procedures in the ranger phase. The following abbreviations represent the variants considered: the GSOIVNS (GSO with improved variable neighborhood search) and the GSOIVNSDE (DHS with improved variable neighborhood search and differential evolution). To verify the effect of each element in the algorithm for solving the RBHFS problem, the authors test four algorithms: GSO, GSOIVNS, GSOIVNSDE, and DGSO in the preemptrepeat case (Case 2). Each method was run twenty times for each Liao’s benchmark problem and its performance, including the average and minimum value, was recorded in Table
Performance of HDHS with different versions.
Problem  GSO  GSOIVNS  GSOIVNSDE  DGSO  

AVE  MIN  AVE  MIN  AVE  MIN  AVE  MIN  
j30c5e1  564.2  555  561.9  558  561.3  555  552.7  549 
j30c5e2  721.4  716  720.0  716  722.5  716  700.7  697 
j30c5e3  752.8  743  741.9  736  740.5  732  723.3  712 
j30c5e4  704.5  695  702.5  695  700.5  688  685.3  674 
j30c5e5  748.1  726  733.6  726  727.6  726  725.1  709 
j30c5e6  743.2  737  737.3  730  731.1  721  718.1  708 
j30c5e7  751.9  745  748.0  743  737.2  733  732.1  723 
j30c5e8  859.0  824  842.1  801  822.2  788  786.8  771 
j30c5e9  799.9  786  796.0  779  791.8  770  766.9  760 
j30c5e10  750.2  738  738.4  718  729.6  715  718.5  710 


Average  739.5  726  732.2  720  726.4  714  710.9  701 
In Table
Several metaheuristics have been applied to the RBHFS problem. To evaluate the performance of the proposed DGSO algorithm in solving the RBHFS problem under the criterion makespan, the DGSO algorithm is compared with a PSO algorithm proposed by Liao et al. [
Comparison results on Liao’s benchmark problems of Case 1.
Problem  PSO  RKGA  IA  DGSO  

AVE  MIN  STD 

AVE  MIN  STD 

AVE  MIN  STD 

AVE  MIN  STD 


j30c5e1  511.7  507  2.4 

513.0  505  3.9  60.4  509.0  506 

53.6 



64.2 
j30c5e2  670.3 

0.4  29.8  670.5 

0.8  64.9 



44.9 




j30c5e3  663.6  657  3.7 

667.5  655  7.5  83.4  659.6  655  1.7  75.9 



72.3 
j30c5e4  631.3  628  2.1  85.3  630.7  626  3.1 

629.2  627  1.2  80.4 



82.3 
j30c5e5  668.5  666  1.7  64.1  671.7  666  4.5  68.5  662.8 


70.0 




j30c5e6  675.4  670  3.3  67.6  679.0  670  5.6  66.3  669.1  665  2.5  62.7 




j30c5e7  686.1  684  1.8  49.7  686.0  683  2.5  74.2  683.9  682  0.9  54.8 




j30c5e8  737.9  732  3.2  95.4  738.3  731  5.1  101.1  734.2  731  1.8  86.3 




j30c5e9  708.8  705  2.0 

709.8  703  5.1  103.7  705.4  702  1.6  90.2 



99.7 
j30c5e10  650.5  641  5.3  82.2  648.8  636  7.2  84.9  642.5  637  2.2 




83.0 


Average  660.4  656  2.6  68.1  661.5  655  4.5  78.0  656.6  654  1.4  69.0 




Comparison results on Liao’s benchmark problems of Case 2.
Problem  PSO  RKGA  IA  DGSO  

AVE  MIN  STD 

AVE  MIN  STD 

AVE  MIN  STD 

AVE  MIN  STD 


j30c5e1  561.6  557  5.1  80.2  562.1  555  8.7  107.0  559.6  554 

87.0 


2.7 

j30c5e2  720.0  701  4.4 

723.8  701  7.8  102.3  716.3  701  3.2  97.7 



111.6 
j30c5e3  747.5  739  6.0  69.6  748.6  735  6.2  89.0  743.0  736 

77.4 


4.6 

j30c5e4  702.2  696  7.5  101.9  701.3  694  8.1  121.6  697.5  688 





72.0 
j30c5e5  733.7  726  8.7  54.3  734.1  726  13.5  105.3  731.5  726  5.9  54.2 




j30c5e6  737.1  730  3.6  114.4  737.0  720  6.0 

730.7  718 

101.8 


8.8  124.8 
j30c5e7  752.2  745  4.3  91.7  751.8  743  6.9  110.1  747.9  741 




4.7  103.9 
j30c5e8  836.7  802  10.4  96.8  839.2  792  14.9  84.6  826.4  790  9.5 




90.0 
j30c5e9  799.4  791  6.8 

800.2  789  15.4  107.3  793.6  788  5.9  106.5 



101.6 
j30c5e10  748.9  740 

88.7  746.2  715  11.6  108.8  738.1  712  9.3  93.5 


8.7 



Average  733.9  723  6.0  88.3  734.4  717  9.9  100.8  728.5  715  5.4  85.6 




In Tables
To confirm whether the observed differences are indeed statistically significant, the authors carry out an analysis of variance (ANOVA). ARE is analyzed by multicompare method using least significant difference (LSD) procedure, where ARE denotes the average relative error to the best solution found by any of the compared algorithms. Obviously, the smaller ARE value is the better result the algorithm yields. The means plots of ARE for all the compared algorithms with LSD intervals at a 95% confidence level in Case 1 and Case 2 are shown in Figures
Means plot of ARE with 95% LSD intervals for different algorithms in Case 1.
Means plot of ARE with 95% LSD intervals for different algorithms in Case 2.
Figure
The computational results under two breakdown cases and no breakdown case.
This paper models the hybrid flowshop scheduling problem with random breakdown by analyzing the random breakdown time point and offering an approach to dealing with the breakdown. Then a discrete group search optimizer algorithm is proposed to minimize the makespan of the RBHFS problem. Several efficient operators are introduced for the producers, scroungers, and rangers in the DGSO algorithm. In addition, an orthogonal test is applied to configure the algorithm parameters after a small number of experiments. Compared with some stateoftheart algorithms on the same benchmark problems, the simulation results demonstrate the effectiveness and efficiency of the proposed DGSO algorithm. Future studies can focus on the replication of the DGSO algorithm for other kinds of scheduling problems, such as stochastic scheduling problem and multiobjective scheduling problem.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous reviewers for giving us helpful suggestions. This work is supported by National Natural Science Foundation of China (Grant nos. 61174040, 61104178) and Fundamental Research Funds for the Central Universities, Shanghai Commission of Science and Technology (Grant no. 12JC1403400).