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A note on flow shop scheduling problems with a learning effect on no-idle dominant machines
Applied Mathematics and Computation 184 (2007) 945–949 www.elsevier.com/locate/amc
A note on flow shop scheduling problems with a learning effect on no-idle dominant machines MingBao Cheng
a,*
, ShiJie Sun
b,*
, Ying Yu
b
a
b
Management School, Jinan University, Guangzhou 510632, PR China Department of Mathematics, Shanghai University, Shanghai 200444, PR China
Abstract This paper considers some permutation flow shop scheduling problems with a learning effect on no-idle dominant machines. The objective is to minimize maximum completion time. This objective is considered under following dominant machines constraint: idm, ddm, idm–ddm and ddm–idm. We present a polynomial-time solution algorithm for each of the above cases. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Scheduling; Permutation flow shop; Learning effect; Algorithm
1. Introduction In the traditional scheduling theory, a job processing time does not independent on job position in a sequence. However, in many realistic scheduling settings, the production facility (a machine, a worker) improves continuously with time. As a result, the processing time of a given job is shorter if it is scheduled later in the production sequence. In literature, this phenomenon is known as learning effect. Biskup [1] was the first one who investigated the effect of learning in the framework of scheduling. He assumed a learning process as a function dependent on a number of repetitions during a production of similar items, in other words, processing times depend on a job position in the sequence, i.e. pjr = pjra, where pj is the normal processing time of job Jj, r is the position of job Jj in the sequence and a 6 0 is the learning index of job Jj. He studied the single-machine problem of minimizing the total flow time, the weighted sum of completion time deviations from a common due date and the sum of job completion times. Similar works can be found in Mosheiov [2], Mosheiov and Sidney [3], Bachman and Janiak [4], etc. For the classical flow shop scheduling problem with n jobs and m machines, it is well known that except for the Johnson’s two-machine case, where the objective is to minimize the makespan, most of the flow shop scheduling problems are known to be NP-hard. Realizing the NP-hard nature of general flow shop problems, *
Corresponding authors. E-mail addresses: [email protected] (M. Cheng), sunsj@staff.shu.edu.cn (S. Sun).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.206
946
M. Cheng et al. / Applied Mathematics and Computation 184 (2007) 945–949
some researchers studied approximation algorithms. Readers can refer to Gonzalez and Sahni [5] and Smutnicki [6], etc. Other researchers studied sone special cases of flow shop problems, such as in an environment of a series of dominating machines. The previous works can be found in Nouweland et al. [7], Ho and Gupta [8], ˇ epek et al. [10], etc. Recently, there are many researchers interested in flow shop scheduling Xiang et al. [9], C with a learning effect, the reader is referred in Wang and Xia [11], Lee and Wu [12], Wu [13]. In this paper, we consider permutation flow shop scheduling problems with a learning effect on no-idle dominant machines. That is, the job processing time is a function of its position r in the sequence and no machine is allowed to have no-idle time between processing any two operations. The objective is to minimize the maximum completion time. The paper is organized as follows. In Section 2, we give a general introduction to flow shop problem with a learning effect and dominant machines. In Section 3, we consider the makespan minimization under following dominant machines constraint: idm, ddm, idm–ddm and ddm–idm, and present a polynomial-time solution algorithm for each of the above cases. The final section includes conclusions and remarks about future research. 2. Formulation and notation The permutation flow shop scheduling problem considered in this paper may be stated as follows: There are given n jobs to be processed on an m machine permutation flow shop (the same sequence or permutation of jobs is maintained throughout the flow shop). The normal processing time of job Jj on machine i (operation Oij) is pij, the actual processing time of job Jj on the machine i is pijr if operation Oij is the rth operation on machine i. We are asked to find the order in which these n jobs should be processed on the m machines such that a given objective function is minimized. In this paper, we consider the jobs processing times characterized by position-dependent function: pijr = pijra, i = 1, 2, . . . , m; r, j = 1, 2, . . . , n, where a 6 0 denotes a learning index. For a given schedule r, let Cij = Cij(r) represent the completion time of operation Oij, Cj = Cmj represents the completion time of job Jj, r = ([1], [2], . . . , [n]) denote a schedule, where [j] denotes the job that occupies the jth position in r. In the rest part of this paper, all the problems considered will be denoted by the three-field notation scheme ajbjc introduced by Graham et al. [14]. Definition 1. No-idle constraint: no machine is allowed to have an idle time between processing any two operations. Definition 2. Mi is dominated by Mk, iff max{pijjj = 1, 2, . . . , n} 6 min{pkjjj = 1, 2, . . . , n}. In abbreviated notation, it is denoted as Mi < Mk. Based on the above concept of dominant machines, the four cases considered in this paper are as follows: Case 1. The machines form an increasing series of dominating machines (idm), that is, M 1 < M 2 < < M m: Case 2. The machines form a decreasing series of dominating machines (ddm), that is, M 1 > M 2 > > M m: Case 3. The machines form an increasing–decreasing series of dominating machines (idm–ddm), that is, M 1 < M 2 < < M h > > M m;
where 1 < h < m:
Case 4. The machines form a decreasing–increasing series of dominating machines (ddm–idm). That is, M 1 > M 2 > > M h < < M m;
where 1 < h < m:
3. Minimize maximum completion time The following results of Lemmas can be easily obtained, and the results can be used in latter:
M. Cheng et al. / Applied Mathematics and Computation 184 (2007) 945–949
947
Lemma 1 (Mosheiov [2]). For the problem 1jpjr = pjrajCmax, an optimal schedule can be obtained by SPT (shortest processing time first) rule. Lemma 2. For the problem Fmjpijr = pijra,no idle,idmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
m X
pi½1 þ
i¼1
j X
pm½k k a :
k¼2
Lemma 3. For the problem Fmjpijr = pijra,no idle,ddmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
n1 X
p1½k k a þ
k¼1
m X
n X
pk½n na
k¼1
pm½k k a :
k¼jþ1
Lemma 4. For the problem Fmjpijr = pijra,no idle,idm ddmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
h1 X
pi½1 þ
i¼1
n1 X
ph½k k a þ
m X
pi½n na
i¼h
k¼1
n X
pm½k k a :
k¼jþ1
Lemma 5. For the problem Fmjpijr = pijra,no idle,ddm idmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
n1 X
p1½k k a þ
k¼1
h1 X
pi½n na
i¼1
n1 X k¼2
ph½k k a þ
m1 X i¼hþ1
pi½1 þ
j X
pm½k k a :
k¼1
Theorem 6. For the problem Fmjpijr = pijra,no idle,idmjCmax, if the first processed job Jt ascertained, then the schedule r = {Jt, r1} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 1) jobs in non-decreasing order of {pmj}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 2, we have m n X X C max ¼ pi½1 þ pm½k k a : i¼1
k¼2
Pn If the job processed first ascertained, then combining k¼2 pm½k k a can be minimized by sequencing the remaining (n 1) jobs in non-decreasing order of their normal processing times on the last machine by Lemma 1, an optimal schedule for the problem Fmjpijr = pijra,no idle,idmjCmax is obtained. h Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,idmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the first processed job in turn, then the remaining (n 1) jobs are sequenced in nondecreasing order of {pmj} on the last machine, respectively, thus n schedules are generated. The one with the minimum makespan among these n schedules is an optimal one. Obviously, it requires O(mn2log n) time. Theorem 7. For the problem Fmjpijr = pijra,no idle,ddmjCmax, if the last processed job Js ascertained, then the schedule r = {r1, Js} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 1) jobs in non-decreasing order of {p1j}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 3, we have C max ¼
n1 X k¼1
p1½k k a þ
n X i¼1
pi½n na :
948
M. Cheng et al. / Applied Mathematics and Computation 184 (2007) 945–949
Pn1 a If the last processed job ascertained, then combining k¼1 p 1½k k can be minimized by sequencing the remaining (n 1) jobs in non-decreasing order of their normal processing times on the first machine by Lemma 1, an optimal schedule for the problem Fmjpij = pijra,no idle,ddmjCmax is obtained. h Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,ddmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the last processed job in turn, then the remaining (n 1) jobs are sequenced in nondecreasing order of {p1j} on the last machine, respectively, thus n schedules are generated. The one with the minimum makespan among these n schedules is an optimal one. Obviously, it requires O(mn2log n) time. Theorem 8. For the problem Fmjpijr = pijra,no idle,idm ddmjCmax, if the first and the last processed job ascertained, then the schedule r = {Jt, r1, Js} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 2) jobs in non-decreasing order of {phj}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 4, we have C max ¼
h X
pi½1 þ
i¼1
n1 X k¼2
ph½k k a þ
m X
pi½n na :
i¼h
Pn1 If the first and the last processed job ascertained, then combining k¼2 ph½k k a can be minimized by sequencing the remaining (n 2) jobs in non-decreasing order of their normal processing times on machine Mh by Lemma 1, an optimal schedule for the problem Fmjpij = pijra,no idle,ddmjCmax is obtained. h Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,idm ddmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the first processed job in turn, and select one of the remaining (n 1) job processed in the last position in turn. Then the remaining (n 2) jobs are sequenced in non-decreasing order of {phj} on machine Mh, respectively, thus n(n 1) schedules are generated. The one with the minimum makespan among these n(n 1) schedules is an optimal one. Obviously, it requires O(mn3log n) time. Similar to Theorem 6, for the problem Fmjpijr = pijra,no idle,ddm idmjCmax, we have Theorem 9. For the problem Fmjpijr = pijra,no idle,ddm idmjCmax, if the first processed job Jt and the last processed job Js ascertained, then the schedule r = {Jt, r1, Js} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 2) jobs in non-decreasing order of {phj}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 5, we have n1 h1 X X p1½1 pm½n a Þþ Þn : ðp1½k þ pm½k ph½k Þk a þ ðpi½n þ mh h1 i¼hþ1 k¼2 i¼1 Pn1 If the first and the last processed job ascertained, then combining k¼2 ðp1½k þ pm½k ph½k Þk a can be minimized by sequencing the remaining (n 2) jobs in non-decreasing order of their normal processing times on machine Mh by Lemma 1, an optimal schedule for the problem Fmjpij = pijra,no idle,ddm idmjCmax is obtained. h
C max ¼
m X
ðpi½1 þ
Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,ddm idmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the first processed job in turn, and select one of the remaining (n 1) job processed in the last position in turn. Then the remaining (n 2) jobs are sequenced in non-decreasing order of {phj} on machine Mh, respectively, thus n(n 1) schedules are generated. The one with the minimum makespan among these n(n 1) schedules is an optimal one. Obviously, it requires O(mn3log n) time. 4. Conclusion This paper considers some permutation flow shop scheduling problems with a learning effect on no-idle dominant machines. The objective is to minimize maximum completion time. For the objective, the following
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949
dominant machines constraint: idm, ddm, idm–ddm and ddm–idm are considered. We present a polynomialtime solution algorithm for each of the four cases. Scheduling problems with such a learning effect in some other machine settings are also interesting and significant for future research. References [1] D. Biskup, Single-machine scheduling with learning consideration, European Journal of Operational Research 115 (1999) 173–178. [2] G. Mosheiov, Scheduling problems with a learning effect, European Journal of Operational Research 132 (2001) 687–693. [3] G. Mosheiov, J.B. Sidney, Scheduling with general job-dependent learning curve, European Journal of Operational Research 147 (2003) 665–670. [4] J. Bachman, A. Janiak, Scheduling jobs with position-dependent processing time, Journal of the Operational Research Society 55 (2004) 254–257. [5] T. Gonzalez, S. Sahni, Flow shop and job-shop schedules: complexity and approximation, Operations Research 26 (1978) 36–52. [6] C. Smutnicki, Some results of the worst-case analysis for flow shop scheduling, European Journal of Operational Research 109 (1998) 66–87. [7] A.V.D. Nouweland, M. Krabbenborg, J. Potters, Flow shop with a dominant machine, European Journal of Operational Research 62 (1992) 38–46. [8] J.C. Ho, J.N.D. Gupta, Flow shop scheduling with dominant machines, Computers and Operations Research 22 (1995) 237–246. [9] S. Xiang, G. Tang, T.C.E. Cheng, Solvable cases of permutation flow shop scheduling with dominating machines, International Journal Production Economics 66 (2000) 53–57. ˇ epek, M. Okada, M. Vlach, Non-preemptive flow shop scheduling with machine dominance, European Journal of Operational [10] O. C Research 139 (2002) 245–261. [11] J.-B. Wang, Z.-Q. Xia, Flow shop scheduling with a learning effect, Journal of the Operational Research Society 56 (2005) 1325–1330. [12] W.-C. Lee, C.-C. Wu, Minimizing total completion time in a two-machine flow-shop with a learning effect, International Journal Production Economics 88 (2004) 85–93. [13] C.-C. Wu, A makespan study of the two-machine flow-shop scheduling problem with a learning effect, Journal of Statistics and Management Systems 8 (2005) 13–25. [14] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics 5 (1979) 287–326.
A note on flow shop scheduling problems with a learning effect on no-idle dominant machines MingBao Cheng
a,*
, ShiJie Sun
b,*
, Ying Yu
b
a
b
Management School, Jinan University, Guangzhou 510632, PR China Department of Mathematics, Shanghai University, Shanghai 200444, PR China
Abstract This paper considers some permutation flow shop scheduling problems with a learning effect on no-idle dominant machines. The objective is to minimize maximum completion time. This objective is considered under following dominant machines constraint: idm, ddm, idm–ddm and ddm–idm. We present a polynomial-time solution algorithm for each of the above cases. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Scheduling; Permutation flow shop; Learning effect; Algorithm
1. Introduction In the traditional scheduling theory, a job processing time does not independent on job position in a sequence. However, in many realistic scheduling settings, the production facility (a machine, a worker) improves continuously with time. As a result, the processing time of a given job is shorter if it is scheduled later in the production sequence. In literature, this phenomenon is known as learning effect. Biskup [1] was the first one who investigated the effect of learning in the framework of scheduling. He assumed a learning process as a function dependent on a number of repetitions during a production of similar items, in other words, processing times depend on a job position in the sequence, i.e. pjr = pjra, where pj is the normal processing time of job Jj, r is the position of job Jj in the sequence and a 6 0 is the learning index of job Jj. He studied the single-machine problem of minimizing the total flow time, the weighted sum of completion time deviations from a common due date and the sum of job completion times. Similar works can be found in Mosheiov [2], Mosheiov and Sidney [3], Bachman and Janiak [4], etc. For the classical flow shop scheduling problem with n jobs and m machines, it is well known that except for the Johnson’s two-machine case, where the objective is to minimize the makespan, most of the flow shop scheduling problems are known to be NP-hard. Realizing the NP-hard nature of general flow shop problems, *
Corresponding authors. E-mail addresses: [email protected] (M. Cheng), sunsj@staff.shu.edu.cn (S. Sun).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.206
946
M. Cheng et al. / Applied Mathematics and Computation 184 (2007) 945–949
some researchers studied approximation algorithms. Readers can refer to Gonzalez and Sahni [5] and Smutnicki [6], etc. Other researchers studied sone special cases of flow shop problems, such as in an environment of a series of dominating machines. The previous works can be found in Nouweland et al. [7], Ho and Gupta [8], ˇ epek et al. [10], etc. Recently, there are many researchers interested in flow shop scheduling Xiang et al. [9], C with a learning effect, the reader is referred in Wang and Xia [11], Lee and Wu [12], Wu [13]. In this paper, we consider permutation flow shop scheduling problems with a learning effect on no-idle dominant machines. That is, the job processing time is a function of its position r in the sequence and no machine is allowed to have no-idle time between processing any two operations. The objective is to minimize the maximum completion time. The paper is organized as follows. In Section 2, we give a general introduction to flow shop problem with a learning effect and dominant machines. In Section 3, we consider the makespan minimization under following dominant machines constraint: idm, ddm, idm–ddm and ddm–idm, and present a polynomial-time solution algorithm for each of the above cases. The final section includes conclusions and remarks about future research. 2. Formulation and notation The permutation flow shop scheduling problem considered in this paper may be stated as follows: There are given n jobs to be processed on an m machine permutation flow shop (the same sequence or permutation of jobs is maintained throughout the flow shop). The normal processing time of job Jj on machine i (operation Oij) is pij, the actual processing time of job Jj on the machine i is pijr if operation Oij is the rth operation on machine i. We are asked to find the order in which these n jobs should be processed on the m machines such that a given objective function is minimized. In this paper, we consider the jobs processing times characterized by position-dependent function: pijr = pijra, i = 1, 2, . . . , m; r, j = 1, 2, . . . , n, where a 6 0 denotes a learning index. For a given schedule r, let Cij = Cij(r) represent the completion time of operation Oij, Cj = Cmj represents the completion time of job Jj, r = ([1], [2], . . . , [n]) denote a schedule, where [j] denotes the job that occupies the jth position in r. In the rest part of this paper, all the problems considered will be denoted by the three-field notation scheme ajbjc introduced by Graham et al. [14]. Definition 1. No-idle constraint: no machine is allowed to have an idle time between processing any two operations. Definition 2. Mi is dominated by Mk, iff max{pijjj = 1, 2, . . . , n} 6 min{pkjjj = 1, 2, . . . , n}. In abbreviated notation, it is denoted as Mi < Mk. Based on the above concept of dominant machines, the four cases considered in this paper are as follows: Case 1. The machines form an increasing series of dominating machines (idm), that is, M 1 < M 2 < < M m: Case 2. The machines form a decreasing series of dominating machines (ddm), that is, M 1 > M 2 > > M m: Case 3. The machines form an increasing–decreasing series of dominating machines (idm–ddm), that is, M 1 < M 2 < < M h > > M m;
where 1 < h < m:
Case 4. The machines form a decreasing–increasing series of dominating machines (ddm–idm). That is, M 1 > M 2 > > M h < < M m;
where 1 < h < m:
3. Minimize maximum completion time The following results of Lemmas can be easily obtained, and the results can be used in latter:
M. Cheng et al. / Applied Mathematics and Computation 184 (2007) 945–949
947
Lemma 1 (Mosheiov [2]). For the problem 1jpjr = pjrajCmax, an optimal schedule can be obtained by SPT (shortest processing time first) rule. Lemma 2. For the problem Fmjpijr = pijra,no idle,idmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
m X
pi½1 þ
i¼1
j X
pm½k k a :
k¼2
Lemma 3. For the problem Fmjpijr = pijra,no idle,ddmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
n1 X
p1½k k a þ
k¼1
m X
n X
pk½n na
k¼1
pm½k k a :
k¼jþ1
Lemma 4. For the problem Fmjpijr = pijra,no idle,idm ddmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
h1 X
pi½1 þ
i¼1
n1 X
ph½k k a þ
m X
pi½n na
i¼h
k¼1
n X
pm½k k a :
k¼jþ1
Lemma 5. For the problem Fmjpijr = pijra,no idle,ddm idmjCmax and a given schedule r = {[1], [2], . . . , [n]}, the completion time of job J[j] is C ½j ¼
n1 X
p1½k k a þ
k¼1
h1 X
pi½n na
i¼1
n1 X k¼2
ph½k k a þ
m1 X i¼hþ1
pi½1 þ
j X
pm½k k a :
k¼1
Theorem 6. For the problem Fmjpijr = pijra,no idle,idmjCmax, if the first processed job Jt ascertained, then the schedule r = {Jt, r1} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 1) jobs in non-decreasing order of {pmj}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 2, we have m n X X C max ¼ pi½1 þ pm½k k a : i¼1
k¼2
Pn If the job processed first ascertained, then combining k¼2 pm½k k a can be minimized by sequencing the remaining (n 1) jobs in non-decreasing order of their normal processing times on the last machine by Lemma 1, an optimal schedule for the problem Fmjpijr = pijra,no idle,idmjCmax is obtained. h Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,idmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the first processed job in turn, then the remaining (n 1) jobs are sequenced in nondecreasing order of {pmj} on the last machine, respectively, thus n schedules are generated. The one with the minimum makespan among these n schedules is an optimal one. Obviously, it requires O(mn2log n) time. Theorem 7. For the problem Fmjpijr = pijra,no idle,ddmjCmax, if the last processed job Js ascertained, then the schedule r = {r1, Js} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 1) jobs in non-decreasing order of {p1j}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 3, we have C max ¼
n1 X k¼1
p1½k k a þ
n X i¼1
pi½n na :
948
M. Cheng et al. / Applied Mathematics and Computation 184 (2007) 945–949
Pn1 a If the last processed job ascertained, then combining k¼1 p 1½k k can be minimized by sequencing the remaining (n 1) jobs in non-decreasing order of their normal processing times on the first machine by Lemma 1, an optimal schedule for the problem Fmjpij = pijra,no idle,ddmjCmax is obtained. h Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,ddmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the last processed job in turn, then the remaining (n 1) jobs are sequenced in nondecreasing order of {p1j} on the last machine, respectively, thus n schedules are generated. The one with the minimum makespan among these n schedules is an optimal one. Obviously, it requires O(mn2log n) time. Theorem 8. For the problem Fmjpijr = pijra,no idle,idm ddmjCmax, if the first and the last processed job ascertained, then the schedule r = {Jt, r1, Js} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 2) jobs in non-decreasing order of {phj}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 4, we have C max ¼
h X
pi½1 þ
i¼1
n1 X k¼2
ph½k k a þ
m X
pi½n na :
i¼h
Pn1 If the first and the last processed job ascertained, then combining k¼2 ph½k k a can be minimized by sequencing the remaining (n 2) jobs in non-decreasing order of their normal processing times on machine Mh by Lemma 1, an optimal schedule for the problem Fmjpij = pijra,no idle,ddmjCmax is obtained. h Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,idm ddmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the first processed job in turn, and select one of the remaining (n 1) job processed in the last position in turn. Then the remaining (n 2) jobs are sequenced in non-decreasing order of {phj} on machine Mh, respectively, thus n(n 1) schedules are generated. The one with the minimum makespan among these n(n 1) schedules is an optimal one. Obviously, it requires O(mn3log n) time. Similar to Theorem 6, for the problem Fmjpijr = pijra,no idle,ddm idmjCmax, we have Theorem 9. For the problem Fmjpijr = pijra,no idle,ddm idmjCmax, if the first processed job Jt and the last processed job Js ascertained, then the schedule r = {Jt, r1, Js} is an optimal one, where r1 is a partial sequence obtained by sequencing the remaining (n 2) jobs in non-decreasing order of {phj}. Proof. Consider the sequence r = (J[1], J[2], . . . , J[n]), by Lemma 5, we have n1 h1 X X p1½1 pm½n a Þþ Þn : ðp1½k þ pm½k ph½k Þk a þ ðpi½n þ mh h1 i¼hþ1 k¼2 i¼1 Pn1 If the first and the last processed job ascertained, then combining k¼2 ðp1½k þ pm½k ph½k Þk a can be minimized by sequencing the remaining (n 2) jobs in non-decreasing order of their normal processing times on machine Mh by Lemma 1, an optimal schedule for the problem Fmjpij = pijra,no idle,ddm idmjCmax is obtained. h
C max ¼
m X
ðpi½1 þ
Therefore, an optimal schedule of the problem Fmjpijr = pijra,no idle,ddm idmjCmax can be constructed as follows: Select J1, J2, . . . , Jn as the first processed job in turn, and select one of the remaining (n 1) job processed in the last position in turn. Then the remaining (n 2) jobs are sequenced in non-decreasing order of {phj} on machine Mh, respectively, thus n(n 1) schedules are generated. The one with the minimum makespan among these n(n 1) schedules is an optimal one. Obviously, it requires O(mn3log n) time. 4. Conclusion This paper considers some permutation flow shop scheduling problems with a learning effect on no-idle dominant machines. The objective is to minimize maximum completion time. For the objective, the following
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dominant machines constraint: idm, ddm, idm–ddm and ddm–idm are considered. We present a polynomialtime solution algorithm for each of the four cases. Scheduling problems with such a learning effect in some other machine settings are also interesting and significant for future research. References [1] D. Biskup, Single-machine scheduling with learning consideration, European Journal of Operational Research 115 (1999) 173–178. [2] G. Mosheiov, Scheduling problems with a learning effect, European Journal of Operational Research 132 (2001) 687–693. [3] G. Mosheiov, J.B. Sidney, Scheduling with general job-dependent learning curve, European Journal of Operational Research 147 (2003) 665–670. [4] J. Bachman, A. Janiak, Scheduling jobs with position-dependent processing time, Journal of the Operational Research Society 55 (2004) 254–257. [5] T. Gonzalez, S. Sahni, Flow shop and job-shop schedules: complexity and approximation, Operations Research 26 (1978) 36–52. [6] C. Smutnicki, Some results of the worst-case analysis for flow shop scheduling, European Journal of Operational Research 109 (1998) 66–87. [7] A.V.D. Nouweland, M. Krabbenborg, J. Potters, Flow shop with a dominant machine, European Journal of Operational Research 62 (1992) 38–46. [8] J.C. Ho, J.N.D. Gupta, Flow shop scheduling with dominant machines, Computers and Operations Research 22 (1995) 237–246. [9] S. Xiang, G. Tang, T.C.E. Cheng, Solvable cases of permutation flow shop scheduling with dominating machines, International Journal Production Economics 66 (2000) 53–57. ˇ epek, M. Okada, M. Vlach, Non-preemptive flow shop scheduling with machine dominance, European Journal of Operational [10] O. C Research 139 (2002) 245–261. [11] J.-B. Wang, Z.-Q. Xia, Flow shop scheduling with a learning effect, Journal of the Operational Research Society 56 (2005) 1325–1330. [12] W.-C. Lee, C.-C. Wu, Minimizing total completion time in a two-machine flow-shop with a learning effect, International Journal Production Economics 88 (2004) 85–93. [13] C.-C. Wu, A makespan study of the two-machine flow-shop scheduling problem with a learning effect, Journal of Statistics and Management Systems 8 (2005) 13–25. [14] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics 5 (1979) 287–326.