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A single-machine scheduling problem with learning effects in intermittent batch production
Computers & Industrial Engineering 57 (2009) 762–765
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
A single-machine scheduling problem with learning effects in intermittent batch production q Dar-Li Yang, Wen-Hung Kuo * Department of Information Management, National Formosa University, Yun-Lin 632, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 17 March 2008 Received in revised form 22 November 2008 Accepted 6 February 2009 Available online 14 February 2009 Keywords: Single-machine Scheduling Intermittent Learning Makespan
a b s t r a c t This paper studies a single-machine scheduling problem with three models of learning and forgetting effects in intermittent batch production. They are the models of no transmission, partial transmission and total transmission of learning from batch to batch. The phenomena exist in many realistic production systems. The objective is to minimize the makespan. We show that the problems with the models of no transmission and partial transmission of learning from batch to batch are polynomially solvable. We also provide two polynomial time algorithms for two special cases in the problem with the total transmission model. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In classical scheduling problems, processing times of jobs are assumed to be constant. However, in many realistic situations, because the firms and employees perform the same task repeatedly, they learn how to perform them more efficiently. Therefore, the actual processing time of a job is shorter when it is scheduled later, than earlier in the sequence. This phenomenon is known as the ‘‘learning effect” in the literature. The impact of learning on productivity in manufacturing was first found by Wright (1936). However, Biskup (1999) was the first to introduce learning effect into scheduling problems. He proposed a learning effect model in which the processing time of a job is a function of the job position in a sequence. He showed that single-machine scheduling problems with a learning effect still remain polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow times. Mosheiov (2001a) provided a polynomial time solution for the single-machine makespan minimization problem and solved two multi-criteria problems which can be formulated as assignment problems. He also showed that the SPT (the shortest processing time first) rule does not remain optimal for the minimum flow-time problem on parallel identical machines. Mosheiov (2001b) further showed that the flow-time minimization problem q
This manuscript was processed by Area Edtior Maged M. Dessouky. * Corresponding author. Tel.: +886 5 631 5733; fax: +886 4 2293 9659. E-mail address: [email protected] (W.-H. Kuo).
0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.02.003
with the learning effect on parallel identical machines has a polynomial time solution. Mosheiov and Sidney (2003) extended learning effect to be job-dependent, that is, learning rates are different from job to job. They showed that the problems of makespan and total flow-time minimization on a single machine, a due-date assignment problem and total flow-time minimization on unrelated parallel machines remain polynomially solvable. For more research results on scheduling problems with other learning effect models under different machine environments, the readers are referred to the review papers of Alidaee and Womer (1990), Biskup (2008) and Cheng, Ding, and Lin (2004). The above learning effect models tell us that, as cumulative jobs increase, the processing time of the subsequent job decreases in a continuous production system. It gives the implication of continuous production rather than intermittent batch production. However, many realistic production systems are intermittent, for example, production systems with the applications of ‘‘group technology”. That is, different products with similar designs and/or production processes are grouped together to produce in a production run. In such intermittent production, it is reasonable to assume that if plenty of time has elapsed between production runs, the learning effect would not continue to follow what it was left when production resumes, but that the processing time of the subsequent job would revert to a higher level. This suggests a phenomenon of forgetting between production runs. Keachie and Fontana (1966) also indicated that transmission of learning from period to period can depend on many variables, such as the type of work performed, the time between manufacturing periods, the
763
D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765
length of time required to manufacture the lot and the labor turnover. Therefore, there may be total, partial or even no transmission of learning from one production run to another one. In this paper, we study a scheduling problem with three models of learning and forgetting effects in intermittent batch production.
Theorem 1. For the problem of 1/B, LE, Tno/Cmax, there exists an optimal schedule that satisfies the following conditions: (a) the jobs within a batch are sequenced in non-decreasing order of their normal processing times and (b) the batches can be sequenced in any order.
2. Notations and assumptions
Proof. The problem of arranging the job sequence optimally within a batch is the same as the problem of 1/LE/Cmax. Mosheiov (2001a) proved that the optimal schedule is to sequence the jobs in non-decreasing order of their normal processing times. Therefore, the theorem follows because there is no transmission of learning from batch to batch. h
As mentioned above, we consider a single-machine scheduling problem with learning and forgetting effects in intermittent batch production. The problem is developed by using the following notations. m Bi ni n Jij ai bi pij prij pi[k] pki½k Cij Ci[k] Cmax
the number of batches (m P 2). the ith batch, i = 1, 2, . . ., m the number of jobs in batch Bi, i = 1, 2, . . ., m the total number of jobs (i.e. n1 + n2 + + nm = n). the jth job in batch Bi, j = 1, 2, . . ., ni. the learning factor of jobs within batch Bi (ai 6 0). the learning factor of batch Bi (bi 6 0). the normal processing time of Jij in the original sequence. the actual processing time of Jij which is scheduled in the rth position in a sequence in batch Bi the normal processing time of Ji[k]which is scheduled in the kth position in a sequence in batch Bi the actual processing time of Ji[k]which is scheduled in the kth position in a sequence in batch Bi the completion time of Jij. the completion time of Ji[k]which is scheduled in the kth position in a sequence in batch Bi. the makespan of all jobs.
There are n jobs grouped into m batches and processed on a single machine. All jobs are available at time zero. The normal processing time of a job is incurred when the job is scheduled first in the first production batch. The actual processing times of the subsequent jobs are smaller than their normal processing times because of the learning effect. Assume that the actual processing time of a job is a decreasing function of its position in a sequence. Usually, the learning effect can be accumulated through completing jobs. However, if plenty of time has elapsed between production runs, it may incur a forgetting effect. That is, there may be total, partial or even no transmission of learning from batch to batch. A single-machine scheduling problem with three models of learning and forgetting effects is studied in the following sections. 3. Model I: no transmission of learning from batch to batch In the first model, we consider that there is no transmission of learning from batch to batch. The objective of the single-machine scheduling problem is to minimize the makespan of all jobs. As mentioned in Biskup (1999), we assume that the actual processing time of job Jij when scheduled in position r of batch Bi, is given by
Hence, the makespan of all jobs is as follows: ni m X X i¼1
4. Model II: partial transmission of learning from batch to batch In the second model, we consider partial transmission of learning from batch to batch in the single-machine scheduling problem. We assume that the learning effect of jobs within a batch is the same as that in the first model. In addition, the actual processing time of batch Bi when scheduled in the rth batch is defined as follows:
Pir ¼ Pi r bi where Pi is the total processing time of jobs within batch Bi if there is no transmission of learning from batch to batch. That is, Pni a pi½k k i . Pi ¼ k¼1 Then, the makespan of all jobs is calculated as follows:
C max ¼
m X
a
a
pi½j j i :
j¼1
For convenience, let LE denote the learning effect, B denote that the problem is an intermittent batch production problem and Tno denote that there is no transmission of learning from batch to batch. Therefore, following the three-field notation of Graham, Lawler, Lenstra, and Rinnooy Kan (1979), the proposed problem is denoted by 1/B, LE, Tno/Cmax.
b
ðpi½1 1ai þ pi½2 2ai þ þ pi½ni ni i Þi i ¼
i¼1
m X
bi
Pi i
ð1Þ
i¼1
Let Tpart denote that there is partial transmission of learning from batch to batch. The proposed problem is denoted by 1/B, LE, Tpart/Cmax. As in Biskup (1999), let xir be a 0/1 variable such that xir = 1 if batch Bi is the rth batch to be processed and xir = 0 otherwise. Then the problem of 1/B, LE, Tpart/Cmax can be formulated as the following assignment problem:
min s:t:
m X m X
Pi rbi xir
i¼1 r¼1 m X i¼1 m X
ð2Þ
xir ¼ 1;
r ¼ 1; 2; . . . ; m;
ð3Þ
xir ¼ 1;
i ¼ 1; 2; . . . ; m;
ð4Þ
r¼1
xir ¼ 0 or 1;
prij ¼ pij r ai :
C max ¼
The optimal job sequence within a certain batch Bi can be obtained by a sorting algorithm and thus taking O(ni log ni) time. Hence, the total running time to sequence jobs of all batches is Pm i¼1 Oðni log ni Þ. On the other hand, the running time to sequence the batches in any order is O(1). Therefore, the overall complexity of the problem of 1/B, LE, Tno/Cmax is O(n log n) (see Kuo & Yang, 2006a).
i; r ¼ 1; 2; . . . ; m:
ð5Þ
Since Pi of batch Bi is not affected by the batch sequence, from Theorem 1, Pi can be minimized by sequencing the corresponding jobs in non-decreasing order of their normal processing times. Based on the above analysis, a simple algorithm to determine the optimal schedule for the problem of 1/B, LE, Tpart/Cmax is developed as follow. Algorithm 1 Step 1: Arrange jobs within each batch in non-decreasing order of their normal processing times.
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D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765
Step 2: Formulate the corresponding assignment problem as Eq. (2) and determine the batch sequence according to the solution of the corresponding problem. From the analysis in Section 3, the complexity of Step 1 is O(n log n). On the other hand, Step 2 is to solve an assignment problem and thus the complexity of Step 2 is O(m3). Thus, the overall complexity of Algorithm 1 is O(n log n + m3). Corollary 1. If the learning factors of all batches are equal, i.e. bi = b, then for the problem of 1/B, LE, Tpart/Cmax, there exists an optimal schedule that satisfies the following conditions: (a) the jobs within a batch are sequenced in non-decreasing order of their normal processing times and (b) the batches are sequenced in non-decreasing order of Pi. Proof. First, the proof of part (a) is the same as that in Theorem 1. Pm b is equivalent to Next, because bi = b, to minimize i¼1 P i i minimizing the makespan of the problem of 1/LE/Cmax if batches are taken as jobs. Then, the result in part (b) of Corollary 1 follows. h From Corollary 1, if bi = b, the complexity of the problem of 1/B, LE, Tpart /Cmax is reduced to O(n log n).
Theorem 3. If B1 B2 Bm and a1 = a2 = = am, then for the problem of 1/B, LE, Ttotal/Cmax, there exists an optimal schedule satisfies the following conditions: (a) the jobs within a batch are sequenced in non-decreasing order of their normal processing times. (b) the batches are arranged as an increasing sequence of dominating batches. Proof. The theorem can be easily proved by using simple job interchanging technique. h Next, if the job numbers of all batches are equal (i.e. ), then the proposed problem can be n1 ¼ n2 ¼ ¼ nm ¼ n=m ¼ n formulated as an assignment problem. Again, let xir be a 0/1 variable such that xir = 1 if batch Bi is the rth batch to be processed and xir = 0 otherwise. Then, the problem to minimize the makespan of all jobs is formulated as follows.
min
m X m X n X
þ jÞai xir pi½j ððr 1Þn
ð7Þ
xir ¼ 1;
r ¼ 1; 2; . . . ; m;
ð8Þ
xir ¼ 1;
i ¼ 1; 2; . . . ; m;
ð9Þ
i¼1 r¼1 j¼1
s:t:
m X i¼1
5. Model III: total transmission of learning from batch to batch
m X
In the third model, we consider total transmission of learning from batch to batch in the single-machine scheduling problem. Without loss of generality, assume that batch Bi is sequenced in the ith batch. Then, the actual processing time of job Jij when scheduled in position r in batch Bi is as follows:
r¼1
prij ¼ pij r þ
i1 X
!ai nk
xir ¼ 0 or 1;
i; r ¼ 1; 2; . . . ; m:
ð10Þ
Based on Theorem 2 and the above analysis, a simple algorithm to determine the optimal schedule for the problem of 1/B, LE, Ttotal/ Cmax is developed as follow. Algorithm 2
:
k¼1
Hence, the makespan of all jobs is calculated as follows:
C max ¼
ni m X X i¼1 r¼1
pi½r r þ
i1 X
!a i nk
ð6Þ
:
Step 1: Arrange jobs within each batch in non-decreasing order of their normal processing times. Step 2: Formulate the corresponding assignment problem as Eq. (3) and determine the batch sequence according to the solution of the corresponding problem.
k¼1
Let Ttotal denote total transmission of learning from batch to batch. Then the proposed problem is denoted by 1/B, LE, Ttotal/Cmax. Theorem 2. For any batch sequence of the 1/B, LE, Ttotal/Cmax problem, the total processing time of jobs within the batch is minimized by sequencing jobs in non-decreasing order of their normal processing times. Proof. The theorem can be easily proved by using simple job interchanging technique. h Corollary 2. For the problem of 1/B, LE, Ttotal/Cmax, there exists an optimal schedule by sequencing jobs within each batch in nondecreasing order of their normal processing times. Proof. The result follows directly from Theorem 2.
h
In the following, two special cases of the problem of 1/B, LE, Ttotal/Cmax are discussed in Theorem 3 and Algorithm 2, respectively. Definition 1. Bi is dominated by Bj, or Bj dominates Bi iff The symbol maxfpik jk ¼ 1; 2; :::; ni g 6 minfpjk jk ¼ 1; 2; :::; nj g. Bi Bj denotes that Bi is dominated by Bj. Definition 2. The batches form an increasing sequence of dominating batches iff B1 B2 Bm .
Note that Step 1 can be obtained by a sorting algorithm and log n Þ time. Step 2 is to solve an assignment probthus it takes Oðn lem and thus it takes O(m3) time. Thus, the overall time complexity log n þ m3 Þ. of Algorithm 2 is Oðn 6. Conclusions This paper studies a single-machine scheduling problem with three models of learning and forgetting effects in intermittent batch production. They are the models of no transmission, partial transmission and total transmission of learning from batch to batch, respectively. The objective is to minimize the makespan. We show that the problem with the models of no transmission and partial transmission of learning from batch to batch is polynomially solvable. We also provide two polynomial time algorithms to find the optimal solutions of two special cases in the problem with the model of total transmission of learning from batch to batch. However, the complexity of the problem with the model of total transmission of learning from batch to batch is an open question. Therefore, it is an interesting topic for the future research. Besides, in this study, the learning effect of a job depends on its position in a schedule. There are other learning effect models studied in the literature. Thus, it is also worthwhile to consider another learning effect model in intermittent batch production, for example, a time-dependent learning effect model proposed by Kuo and Yang (2006a, 2006b).
D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765
Acknowledgements The authors are grateful to the referees for their constructive comments on an earlier version of this paper. This research is supported in part by the National Science Council of Taiwan, Republic of China, under Grant Number NSC-96-2221-E-150-060. References Alidaee, B., & Womer, N. K. (1990). Scheduling with time dependent processing times: Review and extensions. Journal of the Operational Research Society, 50, 711–729. Biskup, D. (1999). Single-machine scheduling with learning considerations. European Journal of Operational Research, 115, 173–178. Biskup, D. (2008). A state-of-the-art review on scheduling with learning effects. European Journal of Operational Research, 188, 315–329. Cheng, T. C. E., Ding, Q., & Lin, B. M. T. (2004). A concise survey of scheduling with time-dependent processing times. European Journal of Operational Research, 152, 1–13.
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Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals Discrete Mathematics, 5, 287–326. Keachie, E. C., & Fontana, R. J. (1966). Effects of learning on optimal lot size. Management Science, 13(2), B102–B108. Kuo, W. H., & Yang, D. L. (2006a). Single-machine group scheduling with a timedependent learning effect. Computers & Operations Research, 33(8), 2099–2112. Kuo, W. H., & Yang, D. L. (2006b). Minimizing the total completion time in a single machine scheduling problem with a time-dependent learning effect. European Journal of Operational Research, 174(2), 1184–1190. Mosheiov, G. (2001a). Scheduling problems with learning effect. European Journal of Operational Research, 132, 687–693. Mosheiov, G. (2001b). Parallel machine scheduling with learning effect. Journal of the Operational Research Society, 52, 1–5. Mosheiov, G., & Sidney, J. B. (2003). Scheduling with general job-dependent learning curves. European Journal of Operational Research, 147, 665–670. Wright, T. P. (1936). Factors affecting the cost of airplanes. Journal of Aeronautical Science, 3, 122–128.
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
A single-machine scheduling problem with learning effects in intermittent batch production q Dar-Li Yang, Wen-Hung Kuo * Department of Information Management, National Formosa University, Yun-Lin 632, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 17 March 2008 Received in revised form 22 November 2008 Accepted 6 February 2009 Available online 14 February 2009 Keywords: Single-machine Scheduling Intermittent Learning Makespan
a b s t r a c t This paper studies a single-machine scheduling problem with three models of learning and forgetting effects in intermittent batch production. They are the models of no transmission, partial transmission and total transmission of learning from batch to batch. The phenomena exist in many realistic production systems. The objective is to minimize the makespan. We show that the problems with the models of no transmission and partial transmission of learning from batch to batch are polynomially solvable. We also provide two polynomial time algorithms for two special cases in the problem with the total transmission model. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In classical scheduling problems, processing times of jobs are assumed to be constant. However, in many realistic situations, because the firms and employees perform the same task repeatedly, they learn how to perform them more efficiently. Therefore, the actual processing time of a job is shorter when it is scheduled later, than earlier in the sequence. This phenomenon is known as the ‘‘learning effect” in the literature. The impact of learning on productivity in manufacturing was first found by Wright (1936). However, Biskup (1999) was the first to introduce learning effect into scheduling problems. He proposed a learning effect model in which the processing time of a job is a function of the job position in a sequence. He showed that single-machine scheduling problems with a learning effect still remain polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow times. Mosheiov (2001a) provided a polynomial time solution for the single-machine makespan minimization problem and solved two multi-criteria problems which can be formulated as assignment problems. He also showed that the SPT (the shortest processing time first) rule does not remain optimal for the minimum flow-time problem on parallel identical machines. Mosheiov (2001b) further showed that the flow-time minimization problem q
This manuscript was processed by Area Edtior Maged M. Dessouky. * Corresponding author. Tel.: +886 5 631 5733; fax: +886 4 2293 9659. E-mail address: [email protected] (W.-H. Kuo).
0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.02.003
with the learning effect on parallel identical machines has a polynomial time solution. Mosheiov and Sidney (2003) extended learning effect to be job-dependent, that is, learning rates are different from job to job. They showed that the problems of makespan and total flow-time minimization on a single machine, a due-date assignment problem and total flow-time minimization on unrelated parallel machines remain polynomially solvable. For more research results on scheduling problems with other learning effect models under different machine environments, the readers are referred to the review papers of Alidaee and Womer (1990), Biskup (2008) and Cheng, Ding, and Lin (2004). The above learning effect models tell us that, as cumulative jobs increase, the processing time of the subsequent job decreases in a continuous production system. It gives the implication of continuous production rather than intermittent batch production. However, many realistic production systems are intermittent, for example, production systems with the applications of ‘‘group technology”. That is, different products with similar designs and/or production processes are grouped together to produce in a production run. In such intermittent production, it is reasonable to assume that if plenty of time has elapsed between production runs, the learning effect would not continue to follow what it was left when production resumes, but that the processing time of the subsequent job would revert to a higher level. This suggests a phenomenon of forgetting between production runs. Keachie and Fontana (1966) also indicated that transmission of learning from period to period can depend on many variables, such as the type of work performed, the time between manufacturing periods, the
763
D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765
length of time required to manufacture the lot and the labor turnover. Therefore, there may be total, partial or even no transmission of learning from one production run to another one. In this paper, we study a scheduling problem with three models of learning and forgetting effects in intermittent batch production.
Theorem 1. For the problem of 1/B, LE, Tno/Cmax, there exists an optimal schedule that satisfies the following conditions: (a) the jobs within a batch are sequenced in non-decreasing order of their normal processing times and (b) the batches can be sequenced in any order.
2. Notations and assumptions
Proof. The problem of arranging the job sequence optimally within a batch is the same as the problem of 1/LE/Cmax. Mosheiov (2001a) proved that the optimal schedule is to sequence the jobs in non-decreasing order of their normal processing times. Therefore, the theorem follows because there is no transmission of learning from batch to batch. h
As mentioned above, we consider a single-machine scheduling problem with learning and forgetting effects in intermittent batch production. The problem is developed by using the following notations. m Bi ni n Jij ai bi pij prij pi[k] pki½k Cij Ci[k] Cmax
the number of batches (m P 2). the ith batch, i = 1, 2, . . ., m the number of jobs in batch Bi, i = 1, 2, . . ., m the total number of jobs (i.e. n1 + n2 + + nm = n). the jth job in batch Bi, j = 1, 2, . . ., ni. the learning factor of jobs within batch Bi (ai 6 0). the learning factor of batch Bi (bi 6 0). the normal processing time of Jij in the original sequence. the actual processing time of Jij which is scheduled in the rth position in a sequence in batch Bi the normal processing time of Ji[k]which is scheduled in the kth position in a sequence in batch Bi the actual processing time of Ji[k]which is scheduled in the kth position in a sequence in batch Bi the completion time of Jij. the completion time of Ji[k]which is scheduled in the kth position in a sequence in batch Bi. the makespan of all jobs.
There are n jobs grouped into m batches and processed on a single machine. All jobs are available at time zero. The normal processing time of a job is incurred when the job is scheduled first in the first production batch. The actual processing times of the subsequent jobs are smaller than their normal processing times because of the learning effect. Assume that the actual processing time of a job is a decreasing function of its position in a sequence. Usually, the learning effect can be accumulated through completing jobs. However, if plenty of time has elapsed between production runs, it may incur a forgetting effect. That is, there may be total, partial or even no transmission of learning from batch to batch. A single-machine scheduling problem with three models of learning and forgetting effects is studied in the following sections. 3. Model I: no transmission of learning from batch to batch In the first model, we consider that there is no transmission of learning from batch to batch. The objective of the single-machine scheduling problem is to minimize the makespan of all jobs. As mentioned in Biskup (1999), we assume that the actual processing time of job Jij when scheduled in position r of batch Bi, is given by
Hence, the makespan of all jobs is as follows: ni m X X i¼1
4. Model II: partial transmission of learning from batch to batch In the second model, we consider partial transmission of learning from batch to batch in the single-machine scheduling problem. We assume that the learning effect of jobs within a batch is the same as that in the first model. In addition, the actual processing time of batch Bi when scheduled in the rth batch is defined as follows:
Pir ¼ Pi r bi where Pi is the total processing time of jobs within batch Bi if there is no transmission of learning from batch to batch. That is, Pni a pi½k k i . Pi ¼ k¼1 Then, the makespan of all jobs is calculated as follows:
C max ¼
m X
a
a
pi½j j i :
j¼1
For convenience, let LE denote the learning effect, B denote that the problem is an intermittent batch production problem and Tno denote that there is no transmission of learning from batch to batch. Therefore, following the three-field notation of Graham, Lawler, Lenstra, and Rinnooy Kan (1979), the proposed problem is denoted by 1/B, LE, Tno/Cmax.
b
ðpi½1 1ai þ pi½2 2ai þ þ pi½ni ni i Þi i ¼
i¼1
m X
bi
Pi i
ð1Þ
i¼1
Let Tpart denote that there is partial transmission of learning from batch to batch. The proposed problem is denoted by 1/B, LE, Tpart/Cmax. As in Biskup (1999), let xir be a 0/1 variable such that xir = 1 if batch Bi is the rth batch to be processed and xir = 0 otherwise. Then the problem of 1/B, LE, Tpart/Cmax can be formulated as the following assignment problem:
min s:t:
m X m X
Pi rbi xir
i¼1 r¼1 m X i¼1 m X
ð2Þ
xir ¼ 1;
r ¼ 1; 2; . . . ; m;
ð3Þ
xir ¼ 1;
i ¼ 1; 2; . . . ; m;
ð4Þ
r¼1
xir ¼ 0 or 1;
prij ¼ pij r ai :
C max ¼
The optimal job sequence within a certain batch Bi can be obtained by a sorting algorithm and thus taking O(ni log ni) time. Hence, the total running time to sequence jobs of all batches is Pm i¼1 Oðni log ni Þ. On the other hand, the running time to sequence the batches in any order is O(1). Therefore, the overall complexity of the problem of 1/B, LE, Tno/Cmax is O(n log n) (see Kuo & Yang, 2006a).
i; r ¼ 1; 2; . . . ; m:
ð5Þ
Since Pi of batch Bi is not affected by the batch sequence, from Theorem 1, Pi can be minimized by sequencing the corresponding jobs in non-decreasing order of their normal processing times. Based on the above analysis, a simple algorithm to determine the optimal schedule for the problem of 1/B, LE, Tpart/Cmax is developed as follow. Algorithm 1 Step 1: Arrange jobs within each batch in non-decreasing order of their normal processing times.
764
D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765
Step 2: Formulate the corresponding assignment problem as Eq. (2) and determine the batch sequence according to the solution of the corresponding problem. From the analysis in Section 3, the complexity of Step 1 is O(n log n). On the other hand, Step 2 is to solve an assignment problem and thus the complexity of Step 2 is O(m3). Thus, the overall complexity of Algorithm 1 is O(n log n + m3). Corollary 1. If the learning factors of all batches are equal, i.e. bi = b, then for the problem of 1/B, LE, Tpart/Cmax, there exists an optimal schedule that satisfies the following conditions: (a) the jobs within a batch are sequenced in non-decreasing order of their normal processing times and (b) the batches are sequenced in non-decreasing order of Pi. Proof. First, the proof of part (a) is the same as that in Theorem 1. Pm b is equivalent to Next, because bi = b, to minimize i¼1 P i i minimizing the makespan of the problem of 1/LE/Cmax if batches are taken as jobs. Then, the result in part (b) of Corollary 1 follows. h From Corollary 1, if bi = b, the complexity of the problem of 1/B, LE, Tpart /Cmax is reduced to O(n log n).
Theorem 3. If B1 B2 Bm and a1 = a2 = = am, then for the problem of 1/B, LE, Ttotal/Cmax, there exists an optimal schedule satisfies the following conditions: (a) the jobs within a batch are sequenced in non-decreasing order of their normal processing times. (b) the batches are arranged as an increasing sequence of dominating batches. Proof. The theorem can be easily proved by using simple job interchanging technique. h Next, if the job numbers of all batches are equal (i.e. ), then the proposed problem can be n1 ¼ n2 ¼ ¼ nm ¼ n=m ¼ n formulated as an assignment problem. Again, let xir be a 0/1 variable such that xir = 1 if batch Bi is the rth batch to be processed and xir = 0 otherwise. Then, the problem to minimize the makespan of all jobs is formulated as follows.
min
m X m X n X
þ jÞai xir pi½j ððr 1Þn
ð7Þ
xir ¼ 1;
r ¼ 1; 2; . . . ; m;
ð8Þ
xir ¼ 1;
i ¼ 1; 2; . . . ; m;
ð9Þ
i¼1 r¼1 j¼1
s:t:
m X i¼1
5. Model III: total transmission of learning from batch to batch
m X
In the third model, we consider total transmission of learning from batch to batch in the single-machine scheduling problem. Without loss of generality, assume that batch Bi is sequenced in the ith batch. Then, the actual processing time of job Jij when scheduled in position r in batch Bi is as follows:
r¼1
prij ¼ pij r þ
i1 X
!ai nk
xir ¼ 0 or 1;
i; r ¼ 1; 2; . . . ; m:
ð10Þ
Based on Theorem 2 and the above analysis, a simple algorithm to determine the optimal schedule for the problem of 1/B, LE, Ttotal/ Cmax is developed as follow. Algorithm 2
:
k¼1
Hence, the makespan of all jobs is calculated as follows:
C max ¼
ni m X X i¼1 r¼1
pi½r r þ
i1 X
!a i nk
ð6Þ
:
Step 1: Arrange jobs within each batch in non-decreasing order of their normal processing times. Step 2: Formulate the corresponding assignment problem as Eq. (3) and determine the batch sequence according to the solution of the corresponding problem.
k¼1
Let Ttotal denote total transmission of learning from batch to batch. Then the proposed problem is denoted by 1/B, LE, Ttotal/Cmax. Theorem 2. For any batch sequence of the 1/B, LE, Ttotal/Cmax problem, the total processing time of jobs within the batch is minimized by sequencing jobs in non-decreasing order of their normal processing times. Proof. The theorem can be easily proved by using simple job interchanging technique. h Corollary 2. For the problem of 1/B, LE, Ttotal/Cmax, there exists an optimal schedule by sequencing jobs within each batch in nondecreasing order of their normal processing times. Proof. The result follows directly from Theorem 2.
h
In the following, two special cases of the problem of 1/B, LE, Ttotal/Cmax are discussed in Theorem 3 and Algorithm 2, respectively. Definition 1. Bi is dominated by Bj, or Bj dominates Bi iff The symbol maxfpik jk ¼ 1; 2; :::; ni g 6 minfpjk jk ¼ 1; 2; :::; nj g. Bi Bj denotes that Bi is dominated by Bj. Definition 2. The batches form an increasing sequence of dominating batches iff B1 B2 Bm .
Note that Step 1 can be obtained by a sorting algorithm and log n Þ time. Step 2 is to solve an assignment probthus it takes Oðn lem and thus it takes O(m3) time. Thus, the overall time complexity log n þ m3 Þ. of Algorithm 2 is Oðn 6. Conclusions This paper studies a single-machine scheduling problem with three models of learning and forgetting effects in intermittent batch production. They are the models of no transmission, partial transmission and total transmission of learning from batch to batch, respectively. The objective is to minimize the makespan. We show that the problem with the models of no transmission and partial transmission of learning from batch to batch is polynomially solvable. We also provide two polynomial time algorithms to find the optimal solutions of two special cases in the problem with the model of total transmission of learning from batch to batch. However, the complexity of the problem with the model of total transmission of learning from batch to batch is an open question. Therefore, it is an interesting topic for the future research. Besides, in this study, the learning effect of a job depends on its position in a schedule. There are other learning effect models studied in the literature. Thus, it is also worthwhile to consider another learning effect model in intermittent batch production, for example, a time-dependent learning effect model proposed by Kuo and Yang (2006a, 2006b).
D.-L. Yang, W.-H. Kuo / Computers & Industrial Engineering 57 (2009) 762–765
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