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Worst-case analysis for flow shop scheduling with a learning effect
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Int. J. Production Economics 113 (2008) 748–753 www.elsevier.com/locate/ijpe
Worst-case analysis for flow shop scheduling with a learning effect Zhiyong Xu, Linyan Sun, Juntao Gong Management School, The State Key Laboratory for Manufacturing System Engineering, Xi’an Jiaotong University, 710049, PR China Received 26 March 2007; accepted 8 November 2007 Available online 21 November 2007
Abstract We consider flow shop scheduling problems with a learning effect. In this model the processing times of jobs are defined as functions of their positions in a permutation. The objective is to minimize one of the three regular performance criteria, namely, the total weighted completion time, the discounted total weighted completion time, and the sum of the quadratic job completion times. We present algorithms by using the optimal permutations for the corresponding single machine scheduling problems. We also analyze the worst-case bound of our algorithms. r 2007 Elsevier B.V. All rights reserved. Keywords: Scheduling; Flow shop; Learning effect; Worst-case analysis
1. Introduction In traditional scheduling problems, most research assumes that the production time of a given product is independent of its position in the production sequence. However, in many realistic settings, because firms and employees perform a task over and over, they learn how to perform more efficiently. The production facility (a machine, a worker) improves continuously over time. As a result, the production time of a given product is shorter if it is scheduled later, rather than earlier in the sequence. This phenomenon is known as a ‘‘learning effect’’ in the literature (Badiru, 1992; Biskup, 1999). Biskup (1999) and Cheng and Wang (2000) were among the pioneers that brought the concept of learning into the field of scheduling, although it has Corresponding author.
E-mail address: [email protected] (Z. Xu).
been widely employed in management science since its discovery by Wright (1936). Biskup (1999) proved that single-machine scheduling with a learning effect remains polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow time. Cheng and Wang (2000) considered a single machine scheduling problem in which the job processing times decrease as a result of learning. A volume-dependent piecewise linear processing time function was used to model the learning effects. The objective is to minimize the maximum lateness. They showed that the problem is NP-hard in the strong sense and then identified two special cases that are polynomially solvable. They also proposed two heuristics and analyzed their worst-case performance. Mosheiov (2001a, b) investigated several other single-machine problems, and the problem of minimizing the total flow time on identical parallel machines. Mosheiov and Sidney (2003) considered the case of a job-dependent
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learning curve, where the learning in the production process of some jobs is faster than that of others. Lee and Wu (2004) proposed a heuristic algorithm to solve the total completion time minimization problem in a two-machine flow shop with a learning effect. Mosheiov and Sidney (2005) introduced a polynomial time solution for the single machine scheduling problem to minimize the number of tardy jobs with general non-increasing job-dependent learning curves and a common due-date. Wang and Xia (2005) considered flow shop scheduling problems with a learning effect. The objective is to minimize one of two regular performance criteria, namely makespan and total flow time. They gave a heuristic algorithm with a worst-case error bound of m for each criterion, where m is the number of machines. They also found polynomial time solutions to two special cases of the problems, i.e., identical processing times on each machine and an increasing series of dominating machines. Wang (2005) considered flow shop scheduling problems with a learning effect. He suggested the use of Johnson’s rule as a heuristic algorithm for two-machine flow shop scheduling to minimize the makespan. He also developed polynomial time solution algorithms for some special cases of the following objective functions: the weighted sum of completion times and the maximum lateness. Wang et al. (in press) considered single-machine scheduling problem with a time-dependent learning effect. They showed by several examples that the weighted sum of completion times minimization problem, the maximum lateness minimization problem and the number of tardy jobs minimization problem cannot be optimally solved by the corresponding classical scheduling rules. But for some special cases, the problems can be solved in polynomial time. They also used the classical rules as heuristic algorithms for these three general problems, respectively, and analyzed their worst-case error bounds. A survey of
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this line of the scheduling research could be found in Bachman and Janiak (2004). In the case of 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 (Johnson, 1954), most of the flow shop scheduling problems are known to be NP-hard. It implies that it is highly unlikely that polynomial solution techniques can be found for the solution of general flow shop scheduling problem (Garey and Johnson, 1979). Realizing the NP-hard nature of flow shop problems some researches study approximation algorithms (Smutnicki, 1998). One of the earliest ideas in flow shop was the heuristics by using the optimal permutation for the corresponding single machine scheduling problems. Gonzalez and Sahni (1978) executed this idea by using the shortest processing time first (SPT) rule as a heuristic algorithm for the total completion time minimization flow shop problem. About other algorithms with tight worst-case bounds for the classical flow shop scheduling listed in Table 1. Wang and Xia (2005) implemented the same idea by utilizing the SPT rule as a heuristic algorithm for the total completion time minimization and makespan minimization flow shop problems with a learning effect, respectively. In this paper we generalize this idea to more complex flow shop scheduling problems with a learning effect. The objective is to minimize one of the three regular performance criteria, namely, the total weighted completion time, the discounted total weighted completion time, and the sum of the quadratic job completion times. We present a heuristic algorithm with worst-case bound for each criteria by utilizing the optimal permutations for the corresponding single machine problems. In Section 2 we give some general notations and assumptions. In Section 3, we propose a heuristic algorithm with a worst-case bound for the flow shop
Table 1 Worst-case analysis for the problem FmkF ðCÞ Flow shop problem FmkF ðCÞ
Tight ratio bound for FmkF ðCÞ
Permutation schedule
Time complexity
Reference
FmkC max P Fmk nj¼1 C j Pn Fmk j¼1 wj C j P Fmk nj¼1 wj ð1 erC j Þ P Fmk nj¼1 C 2j
m m m m
ARB SPT WSPT WDSPT
Oð1Þ Oðn log nÞ Oðn log nÞ Oðn log nÞ
Gonzalez and Sahni (1978) Gonzalez and Sahni (1978) Smutnicki (1998) Koulamas and Kyparisis (2005)
m2
SPT
Oðn log nÞ
Koulamas and Kyparisis (2005)
ARB denotes an arbitrary permutation for the problem FmkC max .
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total weighted completion time minimization, discounted total weighted completion time minimization, and the sum of the quadratic job completion times minimization, respectively. The last section contains some concluding remarks. 2. Notations and assumptions
3. Worst-case behavior for the general case In the following we give several lemmas for the single machine scheduling problems. P Lemma 1 (Smith, 1956). For the problem 1k nj¼1 wj C j ; an optimal schedule can be obtained by sequenp cing the jobs in non-decreasing order of wj (i.e., the j
The flow shop scheduling problem consists of scheduling n jobs J 1 ; J 2 ; . . . ; J n on m machines M 1 ; M 2 ; . . . ; M m . Each job J j consists of a chain operations ðO1;j ; O2;j ; . . . ; Om;j Þ. Operation Oi;j has to be processed on machine M i ; i ¼ 1; 2; . . . ; m. Processing of operation Oiþ1;j may start only after Oi;j has been completed and all machines process the jobs in the same order, i.e. a permutation schedule. The (normal) processing time of operation Oi;j is denoted by pi;j . The actual processing time pi;j;r of job J j is a function dependent on its position r in a schedule. As in Wang and Xia (2005), we consider two special models of job processing time characterized by position-dependent function, namely pi;j;r ¼ pi;j ða brÞ;
i ¼ 1; 2; . . . ; m,
r; j ¼ 1; 2; . . . ; n,
ð1Þ
where a40, bX0, ða brÞ is a decreasing function; pi;j;r ¼ pi;j rc ;
i ¼ 1; 2; . . . ; m; r; j ¼ 1; 2; . . . ; n,
(2)
where cp0 is a learning ratio. Since job processing time is some positive value, for model (1) it is assumed that a ðn þ 1Þb40. It is easy to notice that the processing time of any job characterized by model (2) is always positive. Therefore, no additional assumptions are required there. For a given permutation p, C i;j ¼ C i;j ðpÞ represents the completion time of operation Oi;j , C j ¼ C mj represents the completion time of job J j , p ¼ ð½1; ½2; . . . ; ½nÞ represents a permutation of ð1; 2; . . . ; nÞ, where ½j denotes the job that occupies the jth position in p, and f ðCÞ ¼ f ðC 1 ; C 2 ; . . . ; C n Þ represents a regular measure of performance. WeP consider three n regular objective functions, namely j¼1 wj C j (the total weighted completion time, where wj 40 is a P weight associated with job J j ), nj¼1 wj ð1 erC j Þ (the discounted total weighted completion time, where r 2 ð0; 1Þ is the P discount factor (see Pinedo, 1995, Section n 2 3.1)), and j¼1 C j (the sum of the quadratic job completion times, Townsend, 1978). In the remaining part of the paper, all the problems considered will be denoted using the three-field notation schema ajbjg introduced by Graham et al. (1979).
weighted shortest processing time first (WSPT) rule). Lemma 2 (Pinedo, 1995). For the problem P 1k nj¼1 wj ð1 erC j Þ; an optimal schedule can be obtained by sequencing the jobs in non-decreasing rpj order of 1e (i.e., the weighted discounted shortest w erpj j
processing time first (WDSPT) rule). Lemma 3 (Townsend, 1978). For the problem P 1k nj¼1 C 2j ; an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of pj (i.e., the shortest processing time first (SPT) rule). It is well P known that problems Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 wj C j ðmX2Þ, Fmjpi;j;r ¼ pi;j ða brÞj Pn wj ð1 erC j Þ ðmX2Þ, Fmjpi;j;r ¼ pi;j ða brÞj Pn Pj¼1 n 2 c j¼1 C j ðmX2Þ, Fmjpi;j;r ¼ pi;j r j j¼1 wj C j ðmX2Þ, Pn rC j c Fmjpi;j;r ¼ pi;j r j j¼1 wj ð1 e Þ ðmX2Þ and Fmj P pi;j;r ¼ pi;j rc j nj¼1 C 2j ðmX2Þ is NP-complete, respectively. So we turn our attention to obtain schedules whose performance approximates that of optimal schedules. In examining ‘‘worst’’ schedules, we restrict ourselves to busy schedules (Gonzalez and Sahni, 1978). A busy schedule is a schedule in which at all times from start to finish at least one machine is processing an operation. As stated Pn earlier, for the problems Fmjp ¼ p ða brÞj i;j;r i;j j¼1 C j , Fmjpi;j;r ¼ P pi;j rc j nj¼1 C j , Fmjpi;j;r ¼ pi;j ða brÞjC max , and Fmjpi;j;r ¼ pi;j rc jC max , Wang and Xia (2005) provided an SPT heuristic for these P problems, i.e., in order of increasing Lj (Lj ¼ m i¼1 pij ). From the above analysis and Lemma 1, we can use the WSPT (in order of increasing
Lj wj )
rule
as an approximate algorithm for problem Fmjpi;j;r ¼ P pi;j ða brÞj nj¼1 wj C j . Theorem 1. Let S be an optimal schedule and S be a WSPT schedule for the problem Fmjpi;j;r ¼ Pn w C ðSÞ Pn mðabÞ j¼1 j j P pi;j ða brÞj j¼1 wj C j . Then n p abn , w C ðS Þ j¼1 j j
and this bound is tight.
ARTICLE IN PRESS Z. Xu et al. / Int. J. Production Economics 113 (2008) 748–753
Pm
Proof. Let Lj ¼
i¼1 pij . Without loss L p w2 p p Lwnn . Let 2
L1 w1
of generality
C i ðSÞ be the we assume that completion time P of job J i using WSPT schedule S. Then C i ðSÞp ij¼1 Lj ða bjÞ and so n X
wi C i ðSÞp
i¼1
n X
wi
i X
i¼1
n X
wi
i¼1
i X
wi C i ðS ÞX
i¼1
n X
w½i
i X
i¼1
L½j ða bjÞ=m
n X
w½i
i¼1
Xða bnÞ
L½j =m
j¼1
n X
wi
i¼1
Pn
i X
i X
Lj =m,
j¼1
Pi
as the term i¼1 wL½i j¼1 L½j is minimized by the increasing order of wj (Lemma 1). Consequently, we j have Pn wj C j ðSÞ mða bÞ Pnj¼1 . p a bn w C ðS Þ j j j¼1 We show that the bound mðabÞ abn is tight. Consider the following instance. Learning takes place by the 100%-learning curve (a learning rate of 100% means that no learning is taking place), thus b ¼ 0, i.e., the bound mðabÞ ¼ m. The bound m of the WSPT rule abn P for Fmk nj¼1 wj C j is tight as shown in Smutnicki (1998) and therefore P the bound for the problem Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 wj C j is also tight. & Using similar strategy as in the proof of Theorem 1, the following theorem can be easily obtained. Theorem 2. Let S be an optimal schedule and S be a WSPT schedule for the problem Fmjpi;j;r ¼ Pn wj C j ðSÞ P m pi;j rc j nj¼1 wj C j . Then Pnj¼1 p nc , and this j¼1
and f 00 ðaÞ ¼ r2 era p0
Hence f ðaÞX0. This completes the proof.
j¼1
j¼1
Xða bnÞ
f 0 ðaÞ ¼ rera ð1 er Þ
f ð0Þ ¼ f ð1Þ ¼ 0.
Lj .
Let ðJ ½1 ; J ½2 ; . . . ; J ½n Þ be the order in which jobs complete in optimal schedule S . For S we have Pthe i C ½i ðS ÞX j¼1 L½j ða bjÞ=m and so n X
we obtain
for 0prp1. Hence f ðaÞ is a concave function on 0prp1 and 0pap1 for f 00 ðaÞp0. In addition,
Lj ða bjÞ
j¼1
pða bÞ
751
wj C j ðS Þ
bound is tight. Lemma 4. 1 era Xað1 er Þ if 0prp1 and 0pap1. Proof. Let f ðaÞ ¼ 1 era að1 er Þ. To take the first and second derivatives of f ðaÞ with respect to a,
&
From Lemma 2, we can use the WDSPT (in order rLj of increasing 1erLj ) rule as an approximate wj e algorithm for problem Fmjpi;j;r ¼ pi;j ða brÞj P n rC j Þ. j¼1 wj ð1 e Theorem 3. Let S be an optimal schedule and S be a WDSPT schedule for the problem Fmjpi;j;r ¼ pi;j Pn wj ð1erC j ðSÞ Þ Pn rC j ða brÞj j¼1 wj ð1 e Þ. Then Pnj¼1 rC j ðS Þ w ð1e j¼1 j
leqslant mðabÞ abn ,
Þ
and this bound is tight.
Proof.PSimilar to the proof of Theorem 1. Let Lj ¼ m i¼1 pij . Without loss of generality we assume rL2 rLn 1erL1 that w erL1 p 1e p p 1e . Let C i ðSÞ be w erL2 w erLn 1
n
2
the completion time ofPjob J i using WDSPT schedule S. Then C i ðSÞp ij¼1 Lj ða bjÞ and so Pi n n X X r L ðabjÞ j¼1 j wi ð1 erC i ðSÞ Þp wi ð1 e Þ i¼1
i¼1
p
n X
wi ð1 e
rðabÞ
Pi
L j¼1 j Þ.
i¼1
Let ðJ ½1 ; J ½2 ; . . . ; J ½n Þ be the order in which jobs complete in optimal schedule S . For S we have Pthe i C ½i ðS ÞX j¼1 L½j ða bjÞ=m and so n X
wi ð1 erC i ðS Þ Þ
i¼1
X
n X
w½i ð1 e
Pi
r
L ðabjÞ=m j¼1 ½j Þ
i¼1
X
n X
w½i ð1 e
rðabÞðabnÞ=mðabÞ
Pi
L j¼1 ½j Þ
i¼1
Pi n ða bnÞ X rðabÞ L j¼1 ½j Þ w½i ð1 e mða bÞ i¼1 Pi n ða bnÞ X rðabÞ L j¼1 j Þ. X wi ð1 e mða bÞ i¼1 X
The first two inequalities are derived from 1 erC j is an increasing function on C j , the third inequality
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is obtained from Lemma 4, while the last inequality is obtained by Lemma 2. Consequently, we have Pn w ð1 erC i ðSÞ Þ mða bÞ Pni¼1 i . Þ p rC ðS i a bn Þ i¼1 wi ð1 e
n X
C 2i ðS ÞX
i¼1
We show that the bound mðabÞ abn is tight. Consider the following instance. Learning takes place by the 100%-learning curve (a learning rate of 100% means that no learning is taking place), thus b ¼ 0, i.e., the bound mðabÞ abn ¼ m. The bound m of P the WDSPT rule for Fmk nj¼1 wj ð1 erC j Þ is tight as shown in Koulamas and Kyparisis (2005) and therefore P the bound for the problem Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 wj ð1 erC j Þ is also tight. & Using similar strategy as in the proof of Theorem 3, the following theorem can be easily obtained. Theorem 4. Let S be an optimal schedule and S be a WDSPT schedule for the problem Fmjpi;j;r ¼ Pn wj ð1erC j ðSÞ Þ Pn rC j c Þ. Then Pnj¼1 pi;j r j j¼1 wj ð1 e rC j ðS Þ w ð1e j¼1 j
p nmc ,
P C ½i ðS ÞX ij¼1 L½j ða bjÞ=m and so
Þ
and this bound is tight.
From Lemma 3, we can use the SPT (in order of algorithm for increasing Lj ) rule as an approximate P problem Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 C 2j . Theorem 5. Let S be an optimal schedule and S be an SPT schedule for the problem Fmjpi;j;r ¼ Pn 2 C ðSÞ P 2 pi;j ða brÞj nj¼1 C 2j . Then Pnj¼1 2j pðmðabÞ abn Þ ,
n X i X
L2½j ða bjÞ2 =m2
i¼1 j¼1
X
n X i ða bnÞ2 X L2½j m2 i¼1 j¼1
X
n X i ða bnÞ2 X L2j , 2 m i¼1 j¼1
Pn Pi 2 as the term i¼1 j¼1 L½j is minimized by the increasing order of Lj (Lemma 3). Consequently, we have Pn 2 mða bÞ 2 j¼1 C j ðSÞ p . Pn 2 a bn j¼1 C j ðS Þ 2 is tight. We show that the bound ðmðabÞ abn Þ Consider the following instance. Learning takes place by the 100%-learning curve (a learning rate of 100% means that no learning is taking place), thus 2 2 2 b ¼ 0, i.e., the bound ðmðabÞ abn Þ ¼ m . The bound m Pn of the SPT rule for Fmk j¼1 C 2j is tight as shown in Koulamas and Kyparisis (2005) and therefore the bound for the problem Fmjpi;j;r ¼ pi;j ða brÞj P n 2 j¼1 C j is also tight. &
Using similar strategy as in the proof of Theorem 5, the following theorem can be easily obtained. Theorem 6. Let S be an optimal schedule and S be an SPT schedule for the problem Fmjpi;j;r ¼ pi;j rc j Pn 2 C ðSÞ Pn j¼1 j 2 m 2 P n j¼1 C j . Then 2 pðnc Þ , and this bound is C ðS Þ
j¼1 j
tight.
C ðS Þ
j¼1 j
and this bound is tight.
4. Conclusions
Proof.PSimilar to the proof of Theorem 1. Let Lj ¼ m i¼1 pij . Without loss of generality we assume that L1 pL2 p pLn . Let C i ðSÞ be the completion time ofPjob J i using SPT schedule S. Then C i ðSÞp ij¼1 Lj ða bjÞ and so n X i¼1
C 2i ðSÞp
n X i X i¼1 j¼1
L2j ða bjÞ2 pða bÞ2
n X i X
L2j .
i¼1 j¼1
Let ðJ ½1 ; J ½2 ; . . . ; J ½n Þ be the order in which jobs complete in the optimal schedule S . For S we have
In this paper, we developed heuristic algorithms with tight worst-case bound for the flow shop scheduling with a learning effect and regular objective functions listed in Table 2. Our heuristic algorithms are based on the idea that the optimal permutation for the corresponding single machine scheduling problems can be used as a heuristic for the flow shop problem with a learning effect. Future research may focus on considering the other objective functions. Scheduling with learning effects in an other machine setting are also interesting and significant topics for the future research.
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Table 2 Worst-case analysis for the problem Fmjpi;j;r jF ðCÞ Flow shop problem FmkF ðCÞ
Fmjpi;j;r Fmjpi;j;r Fmjpi;j;r Fmjpi;j;r Fmjpi;j;r
¼ pi;j ða brÞjC max ¼ pi;j rc jC max P ¼ pi;j ða brÞj nj¼1 C j Pn c ¼ pi;j r j j¼1 C j P ¼ pi;j ða brÞj nj¼1 wj C j
Fmjpi;j;r ¼ pi;j rc j
Pn
j¼1 wj C j
Fmjpi;j;r ¼ pi;j ða brÞj Fmjpi;j;r ¼ pi;j rc j
j¼1 wj ð1
Pn
j¼1 wj ð1
Fmjpi;j;r ¼ pi;j ða brÞj Fmjpi;j;r ¼ pi;j rc j
Pn
Pn
Pn
2 j¼1 C j
e
erC j Þ
2 j¼1 C j
rC j
Þ
Tight ratio bound for FmkF ðCÞ
Permutation schedule
Time complexity
Reference
m m m m mða bÞ a bn m nc mða bÞ a bn m nc mða bÞ 2 a bn m 2 nc
SPT SPT SPT SPT WSPT
Oðn log nÞ Oðn log nÞ Oðn log nÞ Oðn log nÞ Oðn log nÞ
Wang and Xia Wang and Xia Wang and Xia Wang and Xia Theorem 1
WSPT
Oðn log nÞ
Theorem 2
WDSPT
Oðn log nÞ
Theorem 3
WDSPT
Oðn log nÞ
Theorem 4
SPT
Oðn log nÞ
Theorem 5
SPT
Oðn log nÞ
Theorem 6
Acknowledgment This research was supported by the National Natural Science Foundation of China (NSFC) under Grant 70433003. References Bachman, A., Janiak, A., 2004. Scheduling jobs with positiondependent processing times. Journal of the Operational Research Society 55, 257–264. Badiru, A.B., 1992. Computational survey of univariate and multivariate learning curve models. IEEE Transactions on Engineering Management 39, 176–188. Biskup, D., 1999. Single-machine scheduling with learning considerations. European Journal of Operational Research 115, 173–178. Cheng, T.C.E., Wang, G., 2000. Single machine scheduling with learning effect considerations. Annals of Operations Research 98, 273–290. Garey, M.R., Johnson, D.S., 1979. Computers and Intractability: A Guided Tour to the Theory of NP-completeness. San Francisco, Freeman. Gonzalez, T., Sahni, S., 1978. Flowshop and jobshop schedules: Complexity and approximation. Operations Research 26, 36–52. 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 of Discrete Mathematics 5, 287–326. Johnson, S.M., 1954. Optimal two-and-three-stage production schedules. Naval Research Logistics 1, 61–68. Koulamas, C., Kyparisis, G., 2005. Algorithms with performance guarantees for flow shops with regular objective functions. IIE Transactions 37, 1107–1111.
(2005) (2005) (2005) (2005)
Lee, W.-C., Wu, C.-C., 2004. Minimizing total completion time in a two-machine flowshop with a learning effect. International Journal of Production Economics 88, 85–93. Mosheiov, G., 2001a. Scheduling problems with a learning effect. European Journal of Operational Research 132, 687–693. Mosheiov, G., 2001b. Parallel machine scheduling with a learning effect. Journal of the Operational Research Society 52, 1165–1169. Mosheiov, G., Sidney, J.B., 2003. Scheduling with general jobdependent learning curves. European Journal of Operational Research 147, 665–670. Mosheiov, G., Sidney, J.B., 2005. Note on scheduling with general learning curves to minimize the number of tardy jobs. Journal of the Operational Research Society 56, 110–112. Pinedo, M., 1995. Scheduling: Theory, Algorithms and Systems. Prentice-Hall, Englewood Cliffs, NJ. Smith, W.E., 1956. Various optimizers for single-stage production. Naval Research Logistics 3, 59–66. Smutnicki, C., 1998. Some results of worst-case analysis for flow shop scheduling. European Journal of Operational Research 109, 66–87. Townsend, W., 1978. The single machine problem with quadratic penalty function of completion times: A branch-and-bound solution. Management Science 24, 530–534. Wang, J.-B., 2005. Flow shop scheduling jobs with positiondependent processing times. Journal of Applied Mathematics and Computing 18, 383–391. Wang, J.-B., Xia, Z.-Q., 2005. Flow shop scheduling with a learning effect. Journal of the Operational Research Society 56, 1325–1330. Wang, J.-B., Ng, C.T., Cheng, T.C.E., Liu, L.L. Single-machine scheduling with a time-dependent learning effect. International Journal of Production Economics, in press, doi:10.1016/j.ijpe.2007.03.013. Wright, T.P., 1936. Factors affecting the cost of airplanes. Journal of the Aeronautical Sciences 3, 122–128.
Int. J. Production Economics 113 (2008) 748–753 www.elsevier.com/locate/ijpe
Worst-case analysis for flow shop scheduling with a learning effect Zhiyong Xu, Linyan Sun, Juntao Gong Management School, The State Key Laboratory for Manufacturing System Engineering, Xi’an Jiaotong University, 710049, PR China Received 26 March 2007; accepted 8 November 2007 Available online 21 November 2007
Abstract We consider flow shop scheduling problems with a learning effect. In this model the processing times of jobs are defined as functions of their positions in a permutation. The objective is to minimize one of the three regular performance criteria, namely, the total weighted completion time, the discounted total weighted completion time, and the sum of the quadratic job completion times. We present algorithms by using the optimal permutations for the corresponding single machine scheduling problems. We also analyze the worst-case bound of our algorithms. r 2007 Elsevier B.V. All rights reserved. Keywords: Scheduling; Flow shop; Learning effect; Worst-case analysis
1. Introduction In traditional scheduling problems, most research assumes that the production time of a given product is independent of its position in the production sequence. However, in many realistic settings, because firms and employees perform a task over and over, they learn how to perform more efficiently. The production facility (a machine, a worker) improves continuously over time. As a result, the production time of a given product is shorter if it is scheduled later, rather than earlier in the sequence. This phenomenon is known as a ‘‘learning effect’’ in the literature (Badiru, 1992; Biskup, 1999). Biskup (1999) and Cheng and Wang (2000) were among the pioneers that brought the concept of learning into the field of scheduling, although it has Corresponding author.
E-mail address: [email protected] (Z. Xu).
been widely employed in management science since its discovery by Wright (1936). Biskup (1999) proved that single-machine scheduling with a learning effect remains polynomially solvable if the objective is to minimize the deviation from a common due date or to minimize the sum of flow time. Cheng and Wang (2000) considered a single machine scheduling problem in which the job processing times decrease as a result of learning. A volume-dependent piecewise linear processing time function was used to model the learning effects. The objective is to minimize the maximum lateness. They showed that the problem is NP-hard in the strong sense and then identified two special cases that are polynomially solvable. They also proposed two heuristics and analyzed their worst-case performance. Mosheiov (2001a, b) investigated several other single-machine problems, and the problem of minimizing the total flow time on identical parallel machines. Mosheiov and Sidney (2003) considered the case of a job-dependent
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learning curve, where the learning in the production process of some jobs is faster than that of others. Lee and Wu (2004) proposed a heuristic algorithm to solve the total completion time minimization problem in a two-machine flow shop with a learning effect. Mosheiov and Sidney (2005) introduced a polynomial time solution for the single machine scheduling problem to minimize the number of tardy jobs with general non-increasing job-dependent learning curves and a common due-date. Wang and Xia (2005) considered flow shop scheduling problems with a learning effect. The objective is to minimize one of two regular performance criteria, namely makespan and total flow time. They gave a heuristic algorithm with a worst-case error bound of m for each criterion, where m is the number of machines. They also found polynomial time solutions to two special cases of the problems, i.e., identical processing times on each machine and an increasing series of dominating machines. Wang (2005) considered flow shop scheduling problems with a learning effect. He suggested the use of Johnson’s rule as a heuristic algorithm for two-machine flow shop scheduling to minimize the makespan. He also developed polynomial time solution algorithms for some special cases of the following objective functions: the weighted sum of completion times and the maximum lateness. Wang et al. (in press) considered single-machine scheduling problem with a time-dependent learning effect. They showed by several examples that the weighted sum of completion times minimization problem, the maximum lateness minimization problem and the number of tardy jobs minimization problem cannot be optimally solved by the corresponding classical scheduling rules. But for some special cases, the problems can be solved in polynomial time. They also used the classical rules as heuristic algorithms for these three general problems, respectively, and analyzed their worst-case error bounds. A survey of
749
this line of the scheduling research could be found in Bachman and Janiak (2004). In the case of 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 (Johnson, 1954), most of the flow shop scheduling problems are known to be NP-hard. It implies that it is highly unlikely that polynomial solution techniques can be found for the solution of general flow shop scheduling problem (Garey and Johnson, 1979). Realizing the NP-hard nature of flow shop problems some researches study approximation algorithms (Smutnicki, 1998). One of the earliest ideas in flow shop was the heuristics by using the optimal permutation for the corresponding single machine scheduling problems. Gonzalez and Sahni (1978) executed this idea by using the shortest processing time first (SPT) rule as a heuristic algorithm for the total completion time minimization flow shop problem. About other algorithms with tight worst-case bounds for the classical flow shop scheduling listed in Table 1. Wang and Xia (2005) implemented the same idea by utilizing the SPT rule as a heuristic algorithm for the total completion time minimization and makespan minimization flow shop problems with a learning effect, respectively. In this paper we generalize this idea to more complex flow shop scheduling problems with a learning effect. The objective is to minimize one of the three regular performance criteria, namely, the total weighted completion time, the discounted total weighted completion time, and the sum of the quadratic job completion times. We present a heuristic algorithm with worst-case bound for each criteria by utilizing the optimal permutations for the corresponding single machine problems. In Section 2 we give some general notations and assumptions. In Section 3, we propose a heuristic algorithm with a worst-case bound for the flow shop
Table 1 Worst-case analysis for the problem FmkF ðCÞ Flow shop problem FmkF ðCÞ
Tight ratio bound for FmkF ðCÞ
Permutation schedule
Time complexity
Reference
FmkC max P Fmk nj¼1 C j Pn Fmk j¼1 wj C j P Fmk nj¼1 wj ð1 erC j Þ P Fmk nj¼1 C 2j
m m m m
ARB SPT WSPT WDSPT
Oð1Þ Oðn log nÞ Oðn log nÞ Oðn log nÞ
Gonzalez and Sahni (1978) Gonzalez and Sahni (1978) Smutnicki (1998) Koulamas and Kyparisis (2005)
m2
SPT
Oðn log nÞ
Koulamas and Kyparisis (2005)
ARB denotes an arbitrary permutation for the problem FmkC max .
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total weighted completion time minimization, discounted total weighted completion time minimization, and the sum of the quadratic job completion times minimization, respectively. The last section contains some concluding remarks. 2. Notations and assumptions
3. Worst-case behavior for the general case In the following we give several lemmas for the single machine scheduling problems. P Lemma 1 (Smith, 1956). For the problem 1k nj¼1 wj C j ; an optimal schedule can be obtained by sequenp cing the jobs in non-decreasing order of wj (i.e., the j
The flow shop scheduling problem consists of scheduling n jobs J 1 ; J 2 ; . . . ; J n on m machines M 1 ; M 2 ; . . . ; M m . Each job J j consists of a chain operations ðO1;j ; O2;j ; . . . ; Om;j Þ. Operation Oi;j has to be processed on machine M i ; i ¼ 1; 2; . . . ; m. Processing of operation Oiþ1;j may start only after Oi;j has been completed and all machines process the jobs in the same order, i.e. a permutation schedule. The (normal) processing time of operation Oi;j is denoted by pi;j . The actual processing time pi;j;r of job J j is a function dependent on its position r in a schedule. As in Wang and Xia (2005), we consider two special models of job processing time characterized by position-dependent function, namely pi;j;r ¼ pi;j ða brÞ;
i ¼ 1; 2; . . . ; m,
r; j ¼ 1; 2; . . . ; n,
ð1Þ
where a40, bX0, ða brÞ is a decreasing function; pi;j;r ¼ pi;j rc ;
i ¼ 1; 2; . . . ; m; r; j ¼ 1; 2; . . . ; n,
(2)
where cp0 is a learning ratio. Since job processing time is some positive value, for model (1) it is assumed that a ðn þ 1Þb40. It is easy to notice that the processing time of any job characterized by model (2) is always positive. Therefore, no additional assumptions are required there. For a given permutation p, C i;j ¼ C i;j ðpÞ represents the completion time of operation Oi;j , C j ¼ C mj represents the completion time of job J j , p ¼ ð½1; ½2; . . . ; ½nÞ represents a permutation of ð1; 2; . . . ; nÞ, where ½j denotes the job that occupies the jth position in p, and f ðCÞ ¼ f ðC 1 ; C 2 ; . . . ; C n Þ represents a regular measure of performance. WeP consider three n regular objective functions, namely j¼1 wj C j (the total weighted completion time, where wj 40 is a P weight associated with job J j ), nj¼1 wj ð1 erC j Þ (the discounted total weighted completion time, where r 2 ð0; 1Þ is the P discount factor (see Pinedo, 1995, Section n 2 3.1)), and j¼1 C j (the sum of the quadratic job completion times, Townsend, 1978). In the remaining part of the paper, all the problems considered will be denoted using the three-field notation schema ajbjg introduced by Graham et al. (1979).
weighted shortest processing time first (WSPT) rule). Lemma 2 (Pinedo, 1995). For the problem P 1k nj¼1 wj ð1 erC j Þ; an optimal schedule can be obtained by sequencing the jobs in non-decreasing rpj order of 1e (i.e., the weighted discounted shortest w erpj j
processing time first (WDSPT) rule). Lemma 3 (Townsend, 1978). For the problem P 1k nj¼1 C 2j ; an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of pj (i.e., the shortest processing time first (SPT) rule). It is well P known that problems Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 wj C j ðmX2Þ, Fmjpi;j;r ¼ pi;j ða brÞj Pn wj ð1 erC j Þ ðmX2Þ, Fmjpi;j;r ¼ pi;j ða brÞj Pn Pj¼1 n 2 c j¼1 C j ðmX2Þ, Fmjpi;j;r ¼ pi;j r j j¼1 wj C j ðmX2Þ, Pn rC j c Fmjpi;j;r ¼ pi;j r j j¼1 wj ð1 e Þ ðmX2Þ and Fmj P pi;j;r ¼ pi;j rc j nj¼1 C 2j ðmX2Þ is NP-complete, respectively. So we turn our attention to obtain schedules whose performance approximates that of optimal schedules. In examining ‘‘worst’’ schedules, we restrict ourselves to busy schedules (Gonzalez and Sahni, 1978). A busy schedule is a schedule in which at all times from start to finish at least one machine is processing an operation. As stated Pn earlier, for the problems Fmjp ¼ p ða brÞj i;j;r i;j j¼1 C j , Fmjpi;j;r ¼ P pi;j rc j nj¼1 C j , Fmjpi;j;r ¼ pi;j ða brÞjC max , and Fmjpi;j;r ¼ pi;j rc jC max , Wang and Xia (2005) provided an SPT heuristic for these P problems, i.e., in order of increasing Lj (Lj ¼ m i¼1 pij ). From the above analysis and Lemma 1, we can use the WSPT (in order of increasing
Lj wj )
rule
as an approximate algorithm for problem Fmjpi;j;r ¼ P pi;j ða brÞj nj¼1 wj C j . Theorem 1. Let S be an optimal schedule and S be a WSPT schedule for the problem Fmjpi;j;r ¼ Pn w C ðSÞ Pn mðabÞ j¼1 j j P pi;j ða brÞj j¼1 wj C j . Then n p abn , w C ðS Þ j¼1 j j
and this bound is tight.
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Pm
Proof. Let Lj ¼
i¼1 pij . Without loss L p w2 p p Lwnn . Let 2
L1 w1
of generality
C i ðSÞ be the we assume that completion time P of job J i using WSPT schedule S. Then C i ðSÞp ij¼1 Lj ða bjÞ and so n X
wi C i ðSÞp
i¼1
n X
wi
i X
i¼1
n X
wi
i¼1
i X
wi C i ðS ÞX
i¼1
n X
w½i
i X
i¼1
L½j ða bjÞ=m
n X
w½i
i¼1
Xða bnÞ
L½j =m
j¼1
n X
wi
i¼1
Pn
i X
i X
Lj =m,
j¼1
Pi
as the term i¼1 wL½i j¼1 L½j is minimized by the increasing order of wj (Lemma 1). Consequently, we j have Pn wj C j ðSÞ mða bÞ Pnj¼1 . p a bn w C ðS Þ j j j¼1 We show that the bound mðabÞ abn is tight. Consider the following instance. Learning takes place by the 100%-learning curve (a learning rate of 100% means that no learning is taking place), thus b ¼ 0, i.e., the bound mðabÞ ¼ m. The bound m of the WSPT rule abn P for Fmk nj¼1 wj C j is tight as shown in Smutnicki (1998) and therefore P the bound for the problem Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 wj C j is also tight. & Using similar strategy as in the proof of Theorem 1, the following theorem can be easily obtained. Theorem 2. Let S be an optimal schedule and S be a WSPT schedule for the problem Fmjpi;j;r ¼ Pn wj C j ðSÞ P m pi;j rc j nj¼1 wj C j . Then Pnj¼1 p nc , and this j¼1
and f 00 ðaÞ ¼ r2 era p0
Hence f ðaÞX0. This completes the proof.
j¼1
j¼1
Xða bnÞ
f 0 ðaÞ ¼ rera ð1 er Þ
f ð0Þ ¼ f ð1Þ ¼ 0.
Lj .
Let ðJ ½1 ; J ½2 ; . . . ; J ½n Þ be the order in which jobs complete in optimal schedule S . For S we have Pthe i C ½i ðS ÞX j¼1 L½j ða bjÞ=m and so n X
we obtain
for 0prp1. Hence f ðaÞ is a concave function on 0prp1 and 0pap1 for f 00 ðaÞp0. In addition,
Lj ða bjÞ
j¼1
pða bÞ
751
wj C j ðS Þ
bound is tight. Lemma 4. 1 era Xað1 er Þ if 0prp1 and 0pap1. Proof. Let f ðaÞ ¼ 1 era að1 er Þ. To take the first and second derivatives of f ðaÞ with respect to a,
&
From Lemma 2, we can use the WDSPT (in order rLj of increasing 1erLj ) rule as an approximate wj e algorithm for problem Fmjpi;j;r ¼ pi;j ða brÞj P n rC j Þ. j¼1 wj ð1 e Theorem 3. Let S be an optimal schedule and S be a WDSPT schedule for the problem Fmjpi;j;r ¼ pi;j Pn wj ð1erC j ðSÞ Þ Pn rC j ða brÞj j¼1 wj ð1 e Þ. Then Pnj¼1 rC j ðS Þ w ð1e j¼1 j
leqslant mðabÞ abn ,
Þ
and this bound is tight.
Proof.PSimilar to the proof of Theorem 1. Let Lj ¼ m i¼1 pij . Without loss of generality we assume rL2 rLn 1erL1 that w erL1 p 1e p p 1e . Let C i ðSÞ be w erL2 w erLn 1
n
2
the completion time ofPjob J i using WDSPT schedule S. Then C i ðSÞp ij¼1 Lj ða bjÞ and so Pi n n X X r L ðabjÞ j¼1 j wi ð1 erC i ðSÞ Þp wi ð1 e Þ i¼1
i¼1
p
n X
wi ð1 e
rðabÞ
Pi
L j¼1 j Þ.
i¼1
Let ðJ ½1 ; J ½2 ; . . . ; J ½n Þ be the order in which jobs complete in optimal schedule S . For S we have Pthe i C ½i ðS ÞX j¼1 L½j ða bjÞ=m and so n X
wi ð1 erC i ðS Þ Þ
i¼1
X
n X
w½i ð1 e
Pi
r
L ðabjÞ=m j¼1 ½j Þ
i¼1
X
n X
w½i ð1 e
rðabÞðabnÞ=mðabÞ
Pi
L j¼1 ½j Þ
i¼1
Pi n ða bnÞ X rðabÞ L j¼1 ½j Þ w½i ð1 e mða bÞ i¼1 Pi n ða bnÞ X rðabÞ L j¼1 j Þ. X wi ð1 e mða bÞ i¼1 X
The first two inequalities are derived from 1 erC j is an increasing function on C j , the third inequality
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is obtained from Lemma 4, while the last inequality is obtained by Lemma 2. Consequently, we have Pn w ð1 erC i ðSÞ Þ mða bÞ Pni¼1 i . Þ p rC ðS i a bn Þ i¼1 wi ð1 e
n X
C 2i ðS ÞX
i¼1
We show that the bound mðabÞ abn is tight. Consider the following instance. Learning takes place by the 100%-learning curve (a learning rate of 100% means that no learning is taking place), thus b ¼ 0, i.e., the bound mðabÞ abn ¼ m. The bound m of P the WDSPT rule for Fmk nj¼1 wj ð1 erC j Þ is tight as shown in Koulamas and Kyparisis (2005) and therefore P the bound for the problem Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 wj ð1 erC j Þ is also tight. & Using similar strategy as in the proof of Theorem 3, the following theorem can be easily obtained. Theorem 4. Let S be an optimal schedule and S be a WDSPT schedule for the problem Fmjpi;j;r ¼ Pn wj ð1erC j ðSÞ Þ Pn rC j c Þ. Then Pnj¼1 pi;j r j j¼1 wj ð1 e rC j ðS Þ w ð1e j¼1 j
p nmc ,
P C ½i ðS ÞX ij¼1 L½j ða bjÞ=m and so
Þ
and this bound is tight.
From Lemma 3, we can use the SPT (in order of algorithm for increasing Lj ) rule as an approximate P problem Fmjpi;j;r ¼ pi;j ða brÞj nj¼1 C 2j . Theorem 5. Let S be an optimal schedule and S be an SPT schedule for the problem Fmjpi;j;r ¼ Pn 2 C ðSÞ P 2 pi;j ða brÞj nj¼1 C 2j . Then Pnj¼1 2j pðmðabÞ abn Þ ,
n X i X
L2½j ða bjÞ2 =m2
i¼1 j¼1
X
n X i ða bnÞ2 X L2½j m2 i¼1 j¼1
X
n X i ða bnÞ2 X L2j , 2 m i¼1 j¼1
Pn Pi 2 as the term i¼1 j¼1 L½j is minimized by the increasing order of Lj (Lemma 3). Consequently, we have Pn 2 mða bÞ 2 j¼1 C j ðSÞ p . Pn 2 a bn j¼1 C j ðS Þ 2 is tight. We show that the bound ðmðabÞ abn Þ Consider the following instance. Learning takes place by the 100%-learning curve (a learning rate of 100% means that no learning is taking place), thus 2 2 2 b ¼ 0, i.e., the bound ðmðabÞ abn Þ ¼ m . The bound m Pn of the SPT rule for Fmk j¼1 C 2j is tight as shown in Koulamas and Kyparisis (2005) and therefore the bound for the problem Fmjpi;j;r ¼ pi;j ða brÞj P n 2 j¼1 C j is also tight. &
Using similar strategy as in the proof of Theorem 5, the following theorem can be easily obtained. Theorem 6. Let S be an optimal schedule and S be an SPT schedule for the problem Fmjpi;j;r ¼ pi;j rc j Pn 2 C ðSÞ Pn j¼1 j 2 m 2 P n j¼1 C j . Then 2 pðnc Þ , and this bound is C ðS Þ
j¼1 j
tight.
C ðS Þ
j¼1 j
and this bound is tight.
4. Conclusions
Proof.PSimilar to the proof of Theorem 1. Let Lj ¼ m i¼1 pij . Without loss of generality we assume that L1 pL2 p pLn . Let C i ðSÞ be the completion time ofPjob J i using SPT schedule S. Then C i ðSÞp ij¼1 Lj ða bjÞ and so n X i¼1
C 2i ðSÞp
n X i X i¼1 j¼1
L2j ða bjÞ2 pða bÞ2
n X i X
L2j .
i¼1 j¼1
Let ðJ ½1 ; J ½2 ; . . . ; J ½n Þ be the order in which jobs complete in the optimal schedule S . For S we have
In this paper, we developed heuristic algorithms with tight worst-case bound for the flow shop scheduling with a learning effect and regular objective functions listed in Table 2. Our heuristic algorithms are based on the idea that the optimal permutation for the corresponding single machine scheduling problems can be used as a heuristic for the flow shop problem with a learning effect. Future research may focus on considering the other objective functions. Scheduling with learning effects in an other machine setting are also interesting and significant topics for the future research.
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Table 2 Worst-case analysis for the problem Fmjpi;j;r jF ðCÞ Flow shop problem FmkF ðCÞ
Fmjpi;j;r Fmjpi;j;r Fmjpi;j;r Fmjpi;j;r Fmjpi;j;r
¼ pi;j ða brÞjC max ¼ pi;j rc jC max P ¼ pi;j ða brÞj nj¼1 C j Pn c ¼ pi;j r j j¼1 C j P ¼ pi;j ða brÞj nj¼1 wj C j
Fmjpi;j;r ¼ pi;j rc j
Pn
j¼1 wj C j
Fmjpi;j;r ¼ pi;j ða brÞj Fmjpi;j;r ¼ pi;j rc j
j¼1 wj ð1
Pn
j¼1 wj ð1
Fmjpi;j;r ¼ pi;j ða brÞj Fmjpi;j;r ¼ pi;j rc j
Pn
Pn
Pn
2 j¼1 C j
e
erC j Þ
2 j¼1 C j
rC j
Þ
Tight ratio bound for FmkF ðCÞ
Permutation schedule
Time complexity
Reference
m m m m mða bÞ a bn m nc mða bÞ a bn m nc mða bÞ 2 a bn m 2 nc
SPT SPT SPT SPT WSPT
Oðn log nÞ Oðn log nÞ Oðn log nÞ Oðn log nÞ Oðn log nÞ
Wang and Xia Wang and Xia Wang and Xia Wang and Xia Theorem 1
WSPT
Oðn log nÞ
Theorem 2
WDSPT
Oðn log nÞ
Theorem 3
WDSPT
Oðn log nÞ
Theorem 4
SPT
Oðn log nÞ
Theorem 5
SPT
Oðn log nÞ
Theorem 6
Acknowledgment This research was supported by the National Natural Science Foundation of China (NSFC) under Grant 70433003. References Bachman, A., Janiak, A., 2004. Scheduling jobs with positiondependent processing times. Journal of the Operational Research Society 55, 257–264. Badiru, A.B., 1992. Computational survey of univariate and multivariate learning curve models. IEEE Transactions on Engineering Management 39, 176–188. Biskup, D., 1999. Single-machine scheduling with learning considerations. European Journal of Operational Research 115, 173–178. Cheng, T.C.E., Wang, G., 2000. Single machine scheduling with learning effect considerations. Annals of Operations Research 98, 273–290. Garey, M.R., Johnson, D.S., 1979. Computers and Intractability: A Guided Tour to the Theory of NP-completeness. San Francisco, Freeman. Gonzalez, T., Sahni, S., 1978. Flowshop and jobshop schedules: Complexity and approximation. Operations Research 26, 36–52. 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 of Discrete Mathematics 5, 287–326. Johnson, S.M., 1954. Optimal two-and-three-stage production schedules. Naval Research Logistics 1, 61–68. Koulamas, C., Kyparisis, G., 2005. Algorithms with performance guarantees for flow shops with regular objective functions. IIE Transactions 37, 1107–1111.
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