- Email: [email protected]
On the hybrid flowshop scheduling problem
ARTICLE IN PRESS
Int. J. Production Economics 113 (2008) 495–497 www.elsevier.com/locate/ijpe
On the hybrid flowshop scheduling problem Mohamed Haouari, Lotfi Hidri Combinatorial Optimization Research Group - ROI, Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa, Tunisia Received 18 September 2006; accepted 8 October 2007 Available online 13 November 2007
Abstract The hybrid flowshop scheduling problem requires minimizing the makespan in a serial multiple-stage manufacturing system, where each stage consists of parallel identical machines. The purpose of this note is to pinpoint several inaccuracies contained in a recent paper and to propose a valid lower bound. r 2007 Elsevier B.V. All rights reserved. Keywords: Hybrid flowshop; Lower bounds
1. Introduction The hybrid flowshop scheduling problem (HFS) can be defined as follows. Each of n jobs from the job set J ¼ f1; 2; . . . ; ng has to be processed nonpreemptively on S production stages Z 1 , Z2 ,y, ZS in that order. The processing time of job j 2 J on center Z s ðs ¼ 1; . . . ; SÞ is pðj; sÞ. Each stage Z s consists of ms parallel identical machines. At any time, each machine can process at most one job and each job can be processed on at most one machine. The objective is to construct a schedule for which the maximum completion time, or makespan, is minimized. During the last decade, both the HFS and its numerous variations alike have been intensely investigated in the scheduling literature. The reader is referred to Linn and Zhang (1999) and Kis and Pesch (2005) for comprehensive surveys. Recent contributions are the papers by Vandevelde
Corresponding author. Fax: +216 71 748 843.
E-mail address: [email protected] (M. Haouari).
et al. (2005), Haouari et al. (2006), Jin et al. (2006), and Zandieh et al. (2006). In this note, we show that both a lower bound and a dominance result recently proposed by Jin et al. (2006) are incorrect and we propose a new valid lower bound. 2. Lower bounds for the hybrid flowshop problem 2.1. The lower bound of Jin et al. Prior to describing this lower bound, we present the following notation. 8 s1 > < P pðj; kÞ; j 2 J; s41; LSðj; sÞ : k¼1 > : 0; j 2 J; s ¼ 1: 8 S > < P pðj; kÞ; RSðj; sÞ : k¼sþ1 > : 0;
j 2 J; soS; j 2 J; s ¼ S:
JLðl; sÞ: the lth smallest value of LSðj; sÞ.
0925-5273/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2007.10.007
ARTICLE IN PRESS M. Haouari, L. Hidri / Int. J. Production Economics 113 (2008) 495–497
496
JRðl; sÞ: the lth smallest value of RSðj; sÞ. SPTs1 ðkÞ: the minimum-sum of completion times of the k smallest ðs 1Þ-stage jobs. Jin et al. (2006) claim that " ms 1 X LBs ¼ JLðk; s 1Þ þ SPTs1 ðms Þ ms k¼1 # ms n X X þ pðj; sÞ þ JRðk; sÞ ð1Þ j¼1
k¼1
is a lower bound on the problem. Actually, LBs is not a valid lower bound. This can be demonstrated with the following example. Example 1. n ¼ 6, S ¼ 3; m1 ¼ 3, m2 ¼ 1, m3 ¼ 3. The processing times are displayed in Table 1. P s For s ¼ P 3, we have m 1Þ ¼ 6, SPTs1 k¼1 JLðk; sP n ms ðms Þ ¼ 6, pðj; sÞ ¼ 15, and k¼1 JRðk; sÞ ¼ 0. j¼1 Thus, from (1) we get LBs ¼ 9. A feasible schedule having a makespan equal to 8 is depicted in Fig. 1. Table 1 Data of a six job-3 stage instance j
1
2
3
4
5
6
pðj; 1Þ pðj; 2Þ pðj; 3Þ
1 1 5
2 1 3
3 1 1
3 1 3
3 1 2
3 1 1
2.2. The lower bound of Santos et al. Santos et al. (1995) have proposed the following stage-based lower bound: " # ms ms n X X 1 X JLðk; sÞ þ pðj; sÞ þ JRðk; sÞ . lbs ¼ ms k¼1 j¼1 k¼1 (2) Thus, a valid global lower bound is ( ( ) ) S X glb ¼ max max pðj; sÞ ; max flbs g . j2J
rl ðs; IÞ: the lth value in the ascending order list of LSðj; sÞ taken over all of the jobs belonging to I, s ¼ 1; . . . ; S. ql ðs; IÞ: the lth value in the ascending order list of RSðj; sÞ taken over all of the jobs belonging to I, s ¼ 1; . . . ; S.
Following, Jin et al. (2006), we can derive a further lower bound based on the so-called Jackson pseudopreemptive bound (Carlier and Pinson, 1998). Thus, we get the following stage-based lower bound: ( ( )) S X plbs ¼ max max pðj; sÞ , (
Center 1 1
4 4 5
þ
5
2 3
M1.3
6 6 Center 2 2
1 1
2
3 3
4 4
5 5
1
3
6 3
0
1
2
3
.
ð4Þ
5
4
5
6
7
(5)
8 8
Proposition 1. Bound plb dominates glb.
8
Proof. It suffices to observe that (2) reads " # ms ms X X 1 X lbs ¼ rk ðs; JÞ þ pðj; sÞ þ qk ðs; JÞ . ms k¼1 j2J k¼1
4
4
The corresponding global lower bound is
Jin et al. claim that glb dominates plb. The following proposition shows that this claim is not correct.
6 7 5
2
M3.3
qk ðs; IÞ
1pspS
7
2
M3.2
" ms X 1 X rk ðs; IÞ þ pðj; sÞ ms k¼1 j2I #)
plb ¼ max fplbs g.
6 6
Center 3 M3.1
ms X
s¼1
k¼1
3
M2.1
max
IJ:jIjXms
1 2
M1.2
(3)
For the sake of clarity, we introduce the following additional notation. For a given job subset I J (with jIjXms Þ, we define
j2J
M1.1
1pspS
s¼1
8
9
Fig. 1. Gantt chart of a feasible schedule having a makespan equal to 8.
(6)
ARTICLE IN PRESS M. Haouari, L. Hidri / Int. J. Production Economics 113 (2008) 495–497
Thus, lbs p
( max
IJ:jIjXms
þ
ms X
"
then & !’ ms ms ms X X 1 X ai þ bi þ gi pC max . ms i¼1 i¼1 i¼1
ms X
X 1 rk ðs; IÞ þ pðj; sÞ ms k¼1 j2I #)
qk ðs; IÞ
Clearly, we have ms n X X gi ¼ pðj; sÞ
8s ¼ 1; . . . ; S.
k¼1
Therefore glbpplb.
Interestingly, for some instances plb strictly dominates glb. This can be demonstrated with the following example. Example 2. n ¼ 6, S ¼ 3; ms ¼ 2 for s ¼ 1; 2; 3. The processing n
times
are
the
following
for j2f1;2g; s¼1;2;3 1 2 for j2f3;4;5;6g; s¼1;2;3: In this case, P f Ss¼1 pðj; sÞg ¼ 6 and lbs ¼ 7 for
pðj; sÞ ¼
we have maxj2J
s ¼ 1; 2; 3. Thus glb ¼ 7. Now, let I ¼ f3; 4; 5; 6g. We have " #) ms ms X X 1 X rk ðs; IÞ þ pðj; sÞ þ qk ðs; IÞ ms k¼1 j2I k¼1 ¼8
for s ¼ 1; 2; 3.
Thus, plb4glb: 2.3. A new lower bound The following lemma provides a stage-based lower bound which takes into account the flowshop constraints. Lemma 2. A valid lower bound for the HFS is LB ¼ max LB0s , 2pspS
where &
1 SPTs1 ðms Þ ms !’ ms n X X þ pðj; sÞ þ JRðk; sÞ .
LB0s ¼ JLð1; s 1Þ þ
j¼1
k¼1
Proof. It suffices to prove that LB0s is a valid lower bound. Note that each machine M i ði ¼ 1; . . . ; ms Þ in stage s has to wait an amount of time ai before starting processing a first job and remains idle for an amount of time bi after finishing processing the jobs. Let gi denote the total workload of M i and C max denote the minimal completion time. Since we have ai þ bi þ gi pC max
for all i ¼ 1; . . . ; ms ,
(7)
and ms X i¼1
(8)
(9)
j¼1
i¼1
&
497
bi X
ms X
JRðk; sÞ.
(10)
k¼1
Moreover, since the earliest start time at center s 1 is JLð1; s 1Þ, then the minimum sum of total completion time of the ms smallest ðs 1Þ-stage jobs is bounded from below by ms JLð1; s 1Þþ SPTs1 ðms Þ. Thus, ms X ai Xms JLð1; s 1Þ þ SPTs1 ðms Þ. (11) i¼1
Combining (9)–(11) completes the proof. & Example 3. Consider the instance of example 1. For s ¼ 3, we have JLð1; 1Þ ¼ 1, SPTs1 ðms Þ ¼ 6, Psm P n s pðj; sÞ ¼ 15, and k¼1 JRðk; sÞ ¼ 0. Thus, from j¼1 0 (1) we get LB3 ¼ 8. Thus, the schedule which is depicted in Fig. 1 is optimal. References Carlier, J., Pinson, E., 1998. Jackson’s pseudo preemptive schedule for the Pmjri ; qi jC max scheduling problem. Annals of Operations Research 83 (1), 41–58. Haouari, M., Hidri, L., Gharbi, A., 2006. Optimal scheduling of a two-stage hybrid flow shop. Mathematical Methods of Operations Research 64, 107–124. Jin, Z., Yang, Z., Ito, T., 2006. Metaheuristic algorithms for the multistage hybrid flowshop scheduling problem. International Journal of Production Economics 100, 322–334. Kis, T., Pesch, E., 2005. A review of exact solution methods for the non-preemptive multiprocessor flowshop problem. European Journal of Operational Research 164, 592–608. Linn, R., Zhang, W., 1999. Hybrid flow shop scheduling: A survey. Computers and Industrial Engineering 37, 57–61. Santos, D.L., Hunsucker, J.L., Deal, D.E., 1995. Global lower bounds for flow shops with multiple processors. European Journal of Operational Research 80, 112–120. Vandevelde, A., Hoogeveen, H., Hurkens, C., Lenstra, J.K., 2005. Lower bounds for the head-body-tail problem on parallel machines: A computational study of the multiprocessor flow shop. INFORMS Journal on Computing 17, 305–320. Zandieh, M., Fatemi Ghomi, S.M.T., Moattar Husseini, S.M., 2006. An immune algorithm approach to hybrid flow shops scheduling with sequence-dependent setup times. Applied Mathematics and Computation 180, 111–127.
Int. J. Production Economics 113 (2008) 495–497 www.elsevier.com/locate/ijpe
On the hybrid flowshop scheduling problem Mohamed Haouari, Lotfi Hidri Combinatorial Optimization Research Group - ROI, Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa, Tunisia Received 18 September 2006; accepted 8 October 2007 Available online 13 November 2007
Abstract The hybrid flowshop scheduling problem requires minimizing the makespan in a serial multiple-stage manufacturing system, where each stage consists of parallel identical machines. The purpose of this note is to pinpoint several inaccuracies contained in a recent paper and to propose a valid lower bound. r 2007 Elsevier B.V. All rights reserved. Keywords: Hybrid flowshop; Lower bounds
1. Introduction The hybrid flowshop scheduling problem (HFS) can be defined as follows. Each of n jobs from the job set J ¼ f1; 2; . . . ; ng has to be processed nonpreemptively on S production stages Z 1 , Z2 ,y, ZS in that order. The processing time of job j 2 J on center Z s ðs ¼ 1; . . . ; SÞ is pðj; sÞ. Each stage Z s consists of ms parallel identical machines. At any time, each machine can process at most one job and each job can be processed on at most one machine. The objective is to construct a schedule for which the maximum completion time, or makespan, is minimized. During the last decade, both the HFS and its numerous variations alike have been intensely investigated in the scheduling literature. The reader is referred to Linn and Zhang (1999) and Kis and Pesch (2005) for comprehensive surveys. Recent contributions are the papers by Vandevelde
Corresponding author. Fax: +216 71 748 843.
E-mail address: [email protected] (M. Haouari).
et al. (2005), Haouari et al. (2006), Jin et al. (2006), and Zandieh et al. (2006). In this note, we show that both a lower bound and a dominance result recently proposed by Jin et al. (2006) are incorrect and we propose a new valid lower bound. 2. Lower bounds for the hybrid flowshop problem 2.1. The lower bound of Jin et al. Prior to describing this lower bound, we present the following notation. 8 s1 > < P pðj; kÞ; j 2 J; s41; LSðj; sÞ : k¼1 > : 0; j 2 J; s ¼ 1: 8 S > < P pðj; kÞ; RSðj; sÞ : k¼sþ1 > : 0;
j 2 J; soS; j 2 J; s ¼ S:
JLðl; sÞ: the lth smallest value of LSðj; sÞ.
0925-5273/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2007.10.007
ARTICLE IN PRESS M. Haouari, L. Hidri / Int. J. Production Economics 113 (2008) 495–497
496
JRðl; sÞ: the lth smallest value of RSðj; sÞ. SPTs1 ðkÞ: the minimum-sum of completion times of the k smallest ðs 1Þ-stage jobs. Jin et al. (2006) claim that " ms 1 X LBs ¼ JLðk; s 1Þ þ SPTs1 ðms Þ ms k¼1 # ms n X X þ pðj; sÞ þ JRðk; sÞ ð1Þ j¼1
k¼1
is a lower bound on the problem. Actually, LBs is not a valid lower bound. This can be demonstrated with the following example. Example 1. n ¼ 6, S ¼ 3; m1 ¼ 3, m2 ¼ 1, m3 ¼ 3. The processing times are displayed in Table 1. P s For s ¼ P 3, we have m 1Þ ¼ 6, SPTs1 k¼1 JLðk; sP n ms ðms Þ ¼ 6, pðj; sÞ ¼ 15, and k¼1 JRðk; sÞ ¼ 0. j¼1 Thus, from (1) we get LBs ¼ 9. A feasible schedule having a makespan equal to 8 is depicted in Fig. 1. Table 1 Data of a six job-3 stage instance j
1
2
3
4
5
6
pðj; 1Þ pðj; 2Þ pðj; 3Þ
1 1 5
2 1 3
3 1 1
3 1 3
3 1 2
3 1 1
2.2. The lower bound of Santos et al. Santos et al. (1995) have proposed the following stage-based lower bound: " # ms ms n X X 1 X JLðk; sÞ þ pðj; sÞ þ JRðk; sÞ . lbs ¼ ms k¼1 j¼1 k¼1 (2) Thus, a valid global lower bound is ( ( ) ) S X glb ¼ max max pðj; sÞ ; max flbs g . j2J
rl ðs; IÞ: the lth value in the ascending order list of LSðj; sÞ taken over all of the jobs belonging to I, s ¼ 1; . . . ; S. ql ðs; IÞ: the lth value in the ascending order list of RSðj; sÞ taken over all of the jobs belonging to I, s ¼ 1; . . . ; S.
Following, Jin et al. (2006), we can derive a further lower bound based on the so-called Jackson pseudopreemptive bound (Carlier and Pinson, 1998). Thus, we get the following stage-based lower bound: ( ( )) S X plbs ¼ max max pðj; sÞ , (
Center 1 1
4 4 5
þ
5
2 3
M1.3
6 6 Center 2 2
1 1
2
3 3
4 4
5 5
1
3
6 3
0
1
2
3
.
ð4Þ
5
4
5
6
7
(5)
8 8
Proposition 1. Bound plb dominates glb.
8
Proof. It suffices to observe that (2) reads " # ms ms X X 1 X lbs ¼ rk ðs; JÞ þ pðj; sÞ þ qk ðs; JÞ . ms k¼1 j2J k¼1
4
4
The corresponding global lower bound is
Jin et al. claim that glb dominates plb. The following proposition shows that this claim is not correct.
6 7 5
2
M3.3
qk ðs; IÞ
1pspS
7
2
M3.2
" ms X 1 X rk ðs; IÞ þ pðj; sÞ ms k¼1 j2I #)
plb ¼ max fplbs g.
6 6
Center 3 M3.1
ms X
s¼1
k¼1
3
M2.1
max
IJ:jIjXms
1 2
M1.2
(3)
For the sake of clarity, we introduce the following additional notation. For a given job subset I J (with jIjXms Þ, we define
j2J
M1.1
1pspS
s¼1
8
9
Fig. 1. Gantt chart of a feasible schedule having a makespan equal to 8.
(6)
ARTICLE IN PRESS M. Haouari, L. Hidri / Int. J. Production Economics 113 (2008) 495–497
Thus, lbs p
( max
IJ:jIjXms
þ
ms X
"
then & !’ ms ms ms X X 1 X ai þ bi þ gi pC max . ms i¼1 i¼1 i¼1
ms X
X 1 rk ðs; IÞ þ pðj; sÞ ms k¼1 j2I #)
qk ðs; IÞ
Clearly, we have ms n X X gi ¼ pðj; sÞ
8s ¼ 1; . . . ; S.
k¼1
Therefore glbpplb.
Interestingly, for some instances plb strictly dominates glb. This can be demonstrated with the following example. Example 2. n ¼ 6, S ¼ 3; ms ¼ 2 for s ¼ 1; 2; 3. The processing n
times
are
the
following
for j2f1;2g; s¼1;2;3 1 2 for j2f3;4;5;6g; s¼1;2;3: In this case, P f Ss¼1 pðj; sÞg ¼ 6 and lbs ¼ 7 for
pðj; sÞ ¼
we have maxj2J
s ¼ 1; 2; 3. Thus glb ¼ 7. Now, let I ¼ f3; 4; 5; 6g. We have " #) ms ms X X 1 X rk ðs; IÞ þ pðj; sÞ þ qk ðs; IÞ ms k¼1 j2I k¼1 ¼8
for s ¼ 1; 2; 3.
Thus, plb4glb: 2.3. A new lower bound The following lemma provides a stage-based lower bound which takes into account the flowshop constraints. Lemma 2. A valid lower bound for the HFS is LB ¼ max LB0s , 2pspS
where &
1 SPTs1 ðms Þ ms !’ ms n X X þ pðj; sÞ þ JRðk; sÞ .
LB0s ¼ JLð1; s 1Þ þ
j¼1
k¼1
Proof. It suffices to prove that LB0s is a valid lower bound. Note that each machine M i ði ¼ 1; . . . ; ms Þ in stage s has to wait an amount of time ai before starting processing a first job and remains idle for an amount of time bi after finishing processing the jobs. Let gi denote the total workload of M i and C max denote the minimal completion time. Since we have ai þ bi þ gi pC max
for all i ¼ 1; . . . ; ms ,
(7)
and ms X i¼1
(8)
(9)
j¼1
i¼1
&
497
bi X
ms X
JRðk; sÞ.
(10)
k¼1
Moreover, since the earliest start time at center s 1 is JLð1; s 1Þ, then the minimum sum of total completion time of the ms smallest ðs 1Þ-stage jobs is bounded from below by ms JLð1; s 1Þþ SPTs1 ðms Þ. Thus, ms X ai Xms JLð1; s 1Þ þ SPTs1 ðms Þ. (11) i¼1
Combining (9)–(11) completes the proof. & Example 3. Consider the instance of example 1. For s ¼ 3, we have JLð1; 1Þ ¼ 1, SPTs1 ðms Þ ¼ 6, Psm P n s pðj; sÞ ¼ 15, and k¼1 JRðk; sÞ ¼ 0. Thus, from j¼1 0 (1) we get LB3 ¼ 8. Thus, the schedule which is depicted in Fig. 1 is optimal. References Carlier, J., Pinson, E., 1998. Jackson’s pseudo preemptive schedule for the Pmjri ; qi jC max scheduling problem. Annals of Operations Research 83 (1), 41–58. Haouari, M., Hidri, L., Gharbi, A., 2006. Optimal scheduling of a two-stage hybrid flow shop. Mathematical Methods of Operations Research 64, 107–124. Jin, Z., Yang, Z., Ito, T., 2006. Metaheuristic algorithms for the multistage hybrid flowshop scheduling problem. International Journal of Production Economics 100, 322–334. Kis, T., Pesch, E., 2005. A review of exact solution methods for the non-preemptive multiprocessor flowshop problem. European Journal of Operational Research 164, 592–608. Linn, R., Zhang, W., 1999. Hybrid flow shop scheduling: A survey. Computers and Industrial Engineering 37, 57–61. Santos, D.L., Hunsucker, J.L., Deal, D.E., 1995. Global lower bounds for flow shops with multiple processors. European Journal of Operational Research 80, 112–120. Vandevelde, A., Hoogeveen, H., Hurkens, C., Lenstra, J.K., 2005. Lower bounds for the head-body-tail problem on parallel machines: A computational study of the multiprocessor flow shop. INFORMS Journal on Computing 17, 305–320. Zandieh, M., Fatemi Ghomi, S.M.T., Moattar Husseini, S.M., 2006. An immune algorithm approach to hybrid flow shops scheduling with sequence-dependent setup times. Applied Mathematics and Computation 180, 111–127.