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Single machine scheduling with past-sequence-dependent setup times and learning effects
Information Processing Letters 102 (2007) 22–26 www.elsevier.com/locate/ipl
Single machine scheduling with past-sequence-dependent setup times and learning effects Wen-Hung Kuo, Dar-Li Yang ∗ Department of Information Management, National Formosa University, Yun-Lin 632, Taiwan Received 7 May 2006; received in revised form 2 August 2006 Available online 28 November 2006 Communicated by W.-L. Hsu
Abstract This paper studies a single machine scheduling problem with setup times and learning considerations. The setup times are proportional to the length of the already scheduled jobs. That is, the setup times are past-sequence-dependent. It is assumed that the learning process reflects a decrease in the process time as a function of the number of repetitions, i.e., as a function of the job position in the sequence. The following objectives are considered: the makespan, the total completion time, the total absolute differences in completion times and the sum of earliness, tardiness and common due-date penalty. Polynomial time algorithms are proposed to optimally solve the above objective functions. © 2006 Elsevier B.V. All rights reserved. Keywords: Scheduling; Single-machine; Setup times; Learning effect
1. Introduction In classical scheduling problems, the setup times are considered either sequence independent or sequence dependent [1]. In the first case, the setup times are usually added to the job processing times while in the second case the setup times depend not only on the job currently being scheduled but also on the last scheduled job. Koulamas and Kyparisis [9] first introduced a scheduling problem with past-sequence-dependent (p-s-d, for short) setup times. In this problem, the setup time is dependent on all already scheduled jobs. They showed that the standard single machine scheduling with p-s-d setup * Corresponding author. Tel.: +886 5 631-5100; fax: +886 5 632 7291. E-mail address: [email protected] (D.-L. Yang).
0020-0190/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2006.11.002
times can be solvable in polynomial time when the objectives are the makespan, the total completion time and the total absolute differences in completion times, respectively. Some extensions with nonlinear p-s-d setup times are also considered. Recently, there is a growing interest in the literature to study scheduling problems with a learning effect. Biskup [3] was the first to analyze the learning effect in single machine scheduling problems. 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. Later, Mosheiov [14] applied similar solution techniques to several other single machine problems. Lee et al. [12] considered the learning effect in a bi-criterion single machine scheduling problem. The objective is
W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26
23
to minimize the linear combination of total completion time and maximum tardiness. They created some dominance properties and applied them to enhance the performance of a proposed algorithm. Mosheiov and Sidney [15] considered a more general learning effect model in which the learning effects of some jobs are better than those of others in a sequence, i.e., the learning effects are job-dependent. They showed that some scheduling problems with the job-dependent learning effect remain polynomially solvable. Later, Mosheiov and Sidney [16] provided a polynomial time solution for the single-machine scheduling problem to minimize the number of tardy jobs with general nonincreasing job-dependent learning curves and common due-date. Lin [13] further showed that the problem with jobdependent due-dates is strongly NP-hard and that the problem remains NP-hard even when there are only two due-dates. On the other hand, Cheng and Wang [4] introduced a volume-dependent processing time function to model the learning effects on job processing times in a single machine scheduling problem. The objective is to minimize the maximum lateness. They showed that the problem is NP-hard and identified two special cases which are polynomially solvable. A survey on this kind of the scheduling research could be found in Bachman and Janiak [2] and Cheng et al. [5]. The concepts of separated setup time and learning effect have been extensively studied independently in the literature. However, the scheduling problem simultaneously with the effects of setup and learning has not been studied yet. Therefore, in this paper, we study a single machine scheduling problem simultaneously with pastsequence-dependent setup times and a learning effect. The remaining part of the paper is organized as follows. In Section 2, we formulate the model. In Section 3, several single machine scheduling problems are considered and shown to be solvable in polynomial time. In Section 4, all of the scheduling problems are extended to the problems with the job-dependent learning effect and are proved to remain polynomially solvable. The conclusions are given in the last section.
constitute our model. The problem is formally described as follows: There are n jobs to be processed on a single machine. All jobs are non-preemptive and available for processing at time zero. Let pr denote the normal processing time of job r (Jr , r = 1, 2, . . . , n); also, let J[r] and p[r] denote the job occupying position r in a particular sequence and its normal processing time, respectively. As in Biskup, it is assumed that the actual processing time of J[r] when scheduled in position r is given as follows.
2. Problem formulation
3. Preliminary results
To model the effect of p-s-d setup times, we follow Koulamas and Kyparisis [9] by assuming that the setup time of a job is proportional to the length of the already scheduled jobs. The learning effect is modeled in its popular form of the log-linear curve (see Biskup [3]). In order to investigate the effects of p-s-d setup and learning simultaneously, we combine the above models to
A = p[r] r a , p[r]
r = 1, 2, . . . , n,
(1)
where a 0 is a constant learning index. Also, as in Koulamas and Kyparisis [9], it is assumed that the p-s-d setup time of J[r] when scheduled in position r is given as follows. s[1] = 0 and s[r] = b
r−1 j =1
A p[j ],
r = 2, . . . , n,
(2)
where b 0 is a normalizing constant. For convenience, we denote the learning effect given in Eq. (1) by LE and denote the p-s-d setup given in Eq. (2) by spsd . In addition, let Cr denote the completion time of job r in a sequence. Let d denote a common due date and Tr and Er denote the tardiness of job r and the earliness of job r, respectively, where Tr = max{Cr − d, 0}, Er = max{d − Cr , 0}. In this paper we will consider the minimization of the following objective functions: the makespan Cmax = maxr {Cr }, the total completion differences in time TC = nr=1 Cr , the total absolute completion times TADC = ni=1 nj=i |Ci − Cj | and the sum of earliness, tardiness and common due-date penalty ETCP = nr=1 (αEr + βTr + γ d) where α, β and γ are the unit earliness, tardiness and due date penalty, respectively. Thus, using the three-field notation introduced by Graham et al. [6], the corresponding scheduling problems are denoted by 1|LE, spsd |Cmax , 1|LE, spsd |TADC
1|LE, spsd |TC, and 1|LE, spsd |ETCP,
respectively.
First, a useful lemma is given as follows. Lemma 1. Let there be two sequences of numbers xi and yi . In addition, the two sequences are of the same length. The sum i xi yi of products of the corresponding elements is the least if the sequences are monotonic in the opposite sense.
24
W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26
Proof. See p. 261 in Hardy et al. [7].
2
TADC =
|Ci − Cj |
i=1 j =i
3.1. The 1|LE, spsd |Cmax scheduling problem We first derive the makespan of the problem. Clearly, n n A s[r] + p[r] = (n − r)b + 1 r a p[r] . Cmax = r=1
n n
=
r=1
=
r=1
n A (r − 1)(n − r + 1) s[r] + p[r]
n
(r − 1)(n − r + 1)
r=1
(3) Eq. (3) can be viewed as the scalar product of two vectors, the wr = [(n − r)b + 1]r a and p[r] vectors, respectively (r = 1, . . . , n). Once the elements of the wr vector have been sorted in strictly decreasing order, from Lemma 1, the elements of the p[r] vector should be accordingly sorted in strictly increasing order. Based on the above analysis, the optimal sequence for the 1|LE, spsd |Cmax problem is the well-known shortest processing time (SPT) first sequence.
+b
n
(l − 1)(n − l + 1) r a p[r] .
(5)
l=r+1
Eq. (5) can be viewed as the scalar product of two vectors, the wr = (r − 1)(n − r + 1) +b
n
(l − 1)(n − l + 1) r a
(6)
l=r+1
3.2. The 1|LE, spsd |TC scheduling problem We derive the total completion time of the problem. Clearly, TC =
n A (n − r + 1) s[r] + p[r] r=1
n b(n − r)(n − r + 1) a (n − r + 1) + r p[r] = 2 r=1
n b(n − r) a r p[r] . = (n − r + 1) 1 + (4) 2 r=1
Eq. (4) can be viewed as the scalar product of two vectors, the wr = (n − r + 1)(1 + b(n − r)/2)r a and p[r] vectors respectively (r = 1, . . . , n). Since the elements of the wr vector are already sorted in strictly decreasing order, from Lemma 1, the elements of the p[r] vector should be sorted in strictly increasing order. Based on the above analysis, the optimal sequence for the 1|LE, spsd |TC problem is still the well-known shortest processing time (SPT) sequence. 3.3. The 1|LE, spsd |TADC scheduling problem We consider a scheduling problem with the objective of minimizing the total absolute variation in the job completion times (TADC). This scheduling measure was first considered by Kanet [8]. The TADC of the 1|LE, spsd |TADC scheduling problem can be calculated as follows:
and p[r] vectors respectively (r = 1, . . . , n). Based on the above analysis and Lemma 1, the optimal sequence for the 1|LE, spsd |TADC problem can be obtained in O(n log n) time by arranging the elements of the wr and p[r] vectors in opposite orders. 3.4. The 1|LE, spsd |ETCP scheduling problem This problem is divided into two parts. Firstly the unrestricted common due date problem with earliness, tardiness and due date penalties is introduced. Then a polynomial-time algorithm is given. The goal of the unrestricted common due date problem is to jointly minimize the weighted earliness, tardiness and due date penalty. An unrestricted common due date d is a decision variable whose value is to be determined. If there are no p-s-d setup and learning effects (i.e., a = b = 0), the 1|LE, spsd |ETCP problem reduces to the 1// (αEi + βTi + γ d) problem. Panwalkar et al. [17] provided some useful results of the 1// (αEi + βTi + γ d) problem as follows: Theorem 1. For the 1//
(αEi + βTi + γ d) problem,
(1) it is optimal to assign the due date at the completion time of the kth job, where k is the smallest integer greater than or equal to (nβ − nγ )/(α + β); (2) the optimal schedule is V-shaped, i.e., early jobs are arranged in nonincreasing order of their processing times and tardy jobs are arranged in nondecreasing order of their processing times;
W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26
(3) the positional weight of a job when scheduled in position r in the sequence is given by vr = min nγ + (r − 1)α, (n + 1 − r)β . According to Theorem 1, ETCP = βTi + γ d) can be rewritten as ETCP =
n
vr + b
+
n
n−1
r=1
vl+1 r a p[r]
l=r
r=1
=
i=1 (αEi
n n A (αE[i] + βT[i] + γ d) = vr s[r] + p[r] i=1
=
n
wr p[r] ,
(7)
r=1
a where wr = (vr + b n−1 l=r vl+1 )r . Based on the above analysis and Lemma 1, the following O(n log n) algorithm is provided to solve the 1|LE, spsd |ETCP problem. Algorithm 1. Step 1. Assign the optimal due-date at the completion time of the kth job, where k is the smallest integer greater than or equal to (nβ − nγ )/(α + β), that is, nβ − nγ . k= α+β Step 2. Calculate each value of wr , r = 1, . . . , n. Step 3. Assign the job with the longest normal processing time to the position with the smallest value of wr , the job with the second longer normal processing time to the position with the second smaller value of wr , etc. The following corollary follows directly from Algorithm 1 and Theorem 1. Corollary 1. For the 1|LE, spsd |ETCP problem, (1) if b 1, then there exists an optimal schedule in which the jobs are arranged in nondecreasing order of their processing times; (2) if b = 0, then the common due date problem reduces to the problem tackled by Mosheiov [14]. In addition, it could be solved by Algorithm 1 of which the complexity is O(n log n); (3) if a = b = 0, then there exists an optimal schedule in which early jobs are arranged in nonincreasing order of their processing times and tardy jobs are arranged in nondecreasing order of their processing times.
25
Proof. We provide a brief proof as follows. (1) If b 1, thenw1 > w2 > · · · > wn . Hence, the result follows from Lemma 1. (2) If b = 0, the problem reduces to that studied by Mosheiov [14]. Mosheiov solved the problem by formulating it as an assignment problem. The complexity of solving an assignment problem is O(n3 ). In fact, the formulation of an assignment problem is not necessary, the particular problem (b = 0) could be solved by defining the weights wr = vr r a first, and then applying a simple sorting procedure as Algorithm 1. (3) If a = b = 0, then the 1|LE, spsd |ETCP problem reduces to the 1|| (αEr + βTr + γ d) problem. Therefore, the result follows directly from Theorem 1. 2 4. Extensions In this section, all of the scheduling problems are extended by the introduction of job-dependent learning effects and they are shown that they are proved to remain polynomially solvable. Under the job-dependent learning environment, all of the jobs have different learning rates ai 0. As in Biskup, let xir be a 0/1 variable such that xir = 1 if Ji is the rth job to be processed and xir = 0 otherwise. Then all of the scheduling problems in Section 3 can be formulated as the following assignment problems: min
s.t.
n n
Wir pi xir
i=1 r=1 n
xir = 1,
r = 1, 2, . . . , n,
xir = 1, r=1 xir = 0 or 1,
i = 1, 2, . . . , n,
i=1 n
i, r = 1, 2, . . . , n,
where
Wir = (n − r)b + 1 r ai for the makespan problem,
b(n − r) ai r Wir = (n − r + 1) 1 + 2 for total completion time problem, Wir = (r − 1)(n − r + 1) +b
n
(l − 1)(n − l + 1) r ai
l=r+1
for the total absolute differences in completion times problem, and
26
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Wir = vr + b
n−1
vl+1 r ai
l=r
for the sum of earliness, tardiness and common duedate penalty problem. 5. Conclusions This paper considers a single machine scheduling problem with setup times and learning considerations. The setup times are proportional to the length of the already scheduled jobs. The learning effect is also investigated in the scheduling environments, it is assumed a learning process reflects a decrease in the process time as a function of the number of repetitions. The following objective functions are considered: the makespan, the total completion time, the total absolute differences in completion times and the sum of earliness, tardiness and common due-date penalty. The polynomial time algorithms are proposed to optimally solve the above objective functions. In our study, the job-independent/job-dependent learning effect of a job is assumed. However, in some other situations, a learning effect may be time-dependent [10,11]. Therefore, it is worthwhile for future research to investigate p-s-d-setup scheduling problems with different learning effects. In these cases, it may be necessary to resort to heuristic algorithms for obtaining optimal sequences. Acknowledgements The authors would like to thank two anonymous referees for their helpful comments and suggestions on an early version of this paper. This research is supported in part by the National Science Council of Taiwan, Republic of China, under grant number NSC-94-2213-E-150016.
References [1] A. Allahverdi, J.N.D. Gupta, T. Aldowaisan, A review of scheduling research involving setup considerations, Omega 27 (1999) 219–239. [2] A. Bachman, A. Janiak, Scheduling jobs with position-dependent processing times, J. Oper. Res. Soc. 55 (2004) 257–264. [3] D. Biskup, Single-machine scheduling with learning considerations, Eur. J. Oper. Res. 115 (1999) 173–178. [4] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Ann. Oper. Res. 98 (2000) 273–290. [5] T.C.E. Cheng, Q. Ding, B.M.T. Lin, A concise survey of scheduling with time-dependent processing times, Eur. J. Oper. Res. 152 (2004) 1–13. [6] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Ann. Discrete Math. 5 (1979) 287–326. [7] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, London, 1967. [8] J.J. Kanet, Minimizing variation of flow time in single machine systems, Management Sci. 27 (1981) 1453–1459. [9] C. Koulamas, G.J. Kyparisis, Single-machine scheduling with past-sequence-dependent setup times, Eur. J. Oper. Res. (2006), in press. [10] W.H. Kuo, D.L. Yang, Single-machine group scheduling with a time-dependent learning effect, Comput. Oper. Res. 33 (2006) 2099–2112. [11] W.H. Kuo, D.L. Yang, Minimizing the total completion time in a single machine scheduling problem with a time-dependent learning effect, Eur. J. Oper. Res. 174 (2006) 1184–1190. [12] W.C. Lee, C.C. Wu, H.J. Sung, A bi-criterion single-machine scheduling problem with learning considerations, Acta Inform. 40 (2004) 303–315. [13] B.M.T. Lin, Complexity results for single-machine scheduling with positional learning effects, J. Oper. Res. Soc. (2006), in press. [14] G. Mosheiov, Scheduling problems with learning effect, Eur. J. Oper. Res. 132 (2001) 687–693. [15] G. Mosheiov, J.B. Sidney, Scheduling with general jobdependent learning curves, Eur. J. Oper. Res. 147 (2003) 665– 670. [16] G. Mosheiov, J.B. Sidney, Note on scheduling with general learning curves to minimize the number of tardy jobs, J. Oper. Res. Soc. 52 (2005) 110–112. [17] S.S. Panwalkar, M.L. Smith, A. Seidmann, Common due date assignment to minimize total penalty for the one machine scheduling problem, Oper. Res. 30 (1982) 391–399.
Single machine scheduling with past-sequence-dependent setup times and learning effects Wen-Hung Kuo, Dar-Li Yang ∗ Department of Information Management, National Formosa University, Yun-Lin 632, Taiwan Received 7 May 2006; received in revised form 2 August 2006 Available online 28 November 2006 Communicated by W.-L. Hsu
Abstract This paper studies a single machine scheduling problem with setup times and learning considerations. The setup times are proportional to the length of the already scheduled jobs. That is, the setup times are past-sequence-dependent. It is assumed that the learning process reflects a decrease in the process time as a function of the number of repetitions, i.e., as a function of the job position in the sequence. The following objectives are considered: the makespan, the total completion time, the total absolute differences in completion times and the sum of earliness, tardiness and common due-date penalty. Polynomial time algorithms are proposed to optimally solve the above objective functions. © 2006 Elsevier B.V. All rights reserved. Keywords: Scheduling; Single-machine; Setup times; Learning effect
1. Introduction In classical scheduling problems, the setup times are considered either sequence independent or sequence dependent [1]. In the first case, the setup times are usually added to the job processing times while in the second case the setup times depend not only on the job currently being scheduled but also on the last scheduled job. Koulamas and Kyparisis [9] first introduced a scheduling problem with past-sequence-dependent (p-s-d, for short) setup times. In this problem, the setup time is dependent on all already scheduled jobs. They showed that the standard single machine scheduling with p-s-d setup * Corresponding author. Tel.: +886 5 631-5100; fax: +886 5 632 7291. E-mail address: [email protected] (D.-L. Yang).
0020-0190/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2006.11.002
times can be solvable in polynomial time when the objectives are the makespan, the total completion time and the total absolute differences in completion times, respectively. Some extensions with nonlinear p-s-d setup times are also considered. Recently, there is a growing interest in the literature to study scheduling problems with a learning effect. Biskup [3] was the first to analyze the learning effect in single machine scheduling problems. 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. Later, Mosheiov [14] applied similar solution techniques to several other single machine problems. Lee et al. [12] considered the learning effect in a bi-criterion single machine scheduling problem. The objective is
W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26
23
to minimize the linear combination of total completion time and maximum tardiness. They created some dominance properties and applied them to enhance the performance of a proposed algorithm. Mosheiov and Sidney [15] considered a more general learning effect model in which the learning effects of some jobs are better than those of others in a sequence, i.e., the learning effects are job-dependent. They showed that some scheduling problems with the job-dependent learning effect remain polynomially solvable. Later, Mosheiov and Sidney [16] provided a polynomial time solution for the single-machine scheduling problem to minimize the number of tardy jobs with general nonincreasing job-dependent learning curves and common due-date. Lin [13] further showed that the problem with jobdependent due-dates is strongly NP-hard and that the problem remains NP-hard even when there are only two due-dates. On the other hand, Cheng and Wang [4] introduced a volume-dependent processing time function to model the learning effects on job processing times in a single machine scheduling problem. The objective is to minimize the maximum lateness. They showed that the problem is NP-hard and identified two special cases which are polynomially solvable. A survey on this kind of the scheduling research could be found in Bachman and Janiak [2] and Cheng et al. [5]. The concepts of separated setup time and learning effect have been extensively studied independently in the literature. However, the scheduling problem simultaneously with the effects of setup and learning has not been studied yet. Therefore, in this paper, we study a single machine scheduling problem simultaneously with pastsequence-dependent setup times and a learning effect. The remaining part of the paper is organized as follows. In Section 2, we formulate the model. In Section 3, several single machine scheduling problems are considered and shown to be solvable in polynomial time. In Section 4, all of the scheduling problems are extended to the problems with the job-dependent learning effect and are proved to remain polynomially solvable. The conclusions are given in the last section.
constitute our model. The problem is formally described as follows: There are n jobs to be processed on a single machine. All jobs are non-preemptive and available for processing at time zero. Let pr denote the normal processing time of job r (Jr , r = 1, 2, . . . , n); also, let J[r] and p[r] denote the job occupying position r in a particular sequence and its normal processing time, respectively. As in Biskup, it is assumed that the actual processing time of J[r] when scheduled in position r is given as follows.
2. Problem formulation
3. Preliminary results
To model the effect of p-s-d setup times, we follow Koulamas and Kyparisis [9] by assuming that the setup time of a job is proportional to the length of the already scheduled jobs. The learning effect is modeled in its popular form of the log-linear curve (see Biskup [3]). In order to investigate the effects of p-s-d setup and learning simultaneously, we combine the above models to
A = p[r] r a , p[r]
r = 1, 2, . . . , n,
(1)
where a 0 is a constant learning index. Also, as in Koulamas and Kyparisis [9], it is assumed that the p-s-d setup time of J[r] when scheduled in position r is given as follows. s[1] = 0 and s[r] = b
r−1 j =1
A p[j ],
r = 2, . . . , n,
(2)
where b 0 is a normalizing constant. For convenience, we denote the learning effect given in Eq. (1) by LE and denote the p-s-d setup given in Eq. (2) by spsd . In addition, let Cr denote the completion time of job r in a sequence. Let d denote a common due date and Tr and Er denote the tardiness of job r and the earliness of job r, respectively, where Tr = max{Cr − d, 0}, Er = max{d − Cr , 0}. In this paper we will consider the minimization of the following objective functions: the makespan Cmax = maxr {Cr }, the total completion differences in time TC = nr=1 Cr , the total absolute completion times TADC = ni=1 nj=i |Ci − Cj | and the sum of earliness, tardiness and common due-date penalty ETCP = nr=1 (αEr + βTr + γ d) where α, β and γ are the unit earliness, tardiness and due date penalty, respectively. Thus, using the three-field notation introduced by Graham et al. [6], the corresponding scheduling problems are denoted by 1|LE, spsd |Cmax , 1|LE, spsd |TADC
1|LE, spsd |TC, and 1|LE, spsd |ETCP,
respectively.
First, a useful lemma is given as follows. Lemma 1. Let there be two sequences of numbers xi and yi . In addition, the two sequences are of the same length. The sum i xi yi of products of the corresponding elements is the least if the sequences are monotonic in the opposite sense.
24
W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26
Proof. See p. 261 in Hardy et al. [7].
2
TADC =
|Ci − Cj |
i=1 j =i
3.1. The 1|LE, spsd |Cmax scheduling problem We first derive the makespan of the problem. Clearly, n n A s[r] + p[r] = (n − r)b + 1 r a p[r] . Cmax = r=1
n n
=
r=1
=
r=1
n A (r − 1)(n − r + 1) s[r] + p[r]
n
(r − 1)(n − r + 1)
r=1
(3) Eq. (3) can be viewed as the scalar product of two vectors, the wr = [(n − r)b + 1]r a and p[r] vectors, respectively (r = 1, . . . , n). Once the elements of the wr vector have been sorted in strictly decreasing order, from Lemma 1, the elements of the p[r] vector should be accordingly sorted in strictly increasing order. Based on the above analysis, the optimal sequence for the 1|LE, spsd |Cmax problem is the well-known shortest processing time (SPT) first sequence.
+b
n
(l − 1)(n − l + 1) r a p[r] .
(5)
l=r+1
Eq. (5) can be viewed as the scalar product of two vectors, the wr = (r − 1)(n − r + 1) +b
n
(l − 1)(n − l + 1) r a
(6)
l=r+1
3.2. The 1|LE, spsd |TC scheduling problem We derive the total completion time of the problem. Clearly, TC =
n A (n − r + 1) s[r] + p[r] r=1
n b(n − r)(n − r + 1) a (n − r + 1) + r p[r] = 2 r=1
n b(n − r) a r p[r] . = (n − r + 1) 1 + (4) 2 r=1
Eq. (4) can be viewed as the scalar product of two vectors, the wr = (n − r + 1)(1 + b(n − r)/2)r a and p[r] vectors respectively (r = 1, . . . , n). Since the elements of the wr vector are already sorted in strictly decreasing order, from Lemma 1, the elements of the p[r] vector should be sorted in strictly increasing order. Based on the above analysis, the optimal sequence for the 1|LE, spsd |TC problem is still the well-known shortest processing time (SPT) sequence. 3.3. The 1|LE, spsd |TADC scheduling problem We consider a scheduling problem with the objective of minimizing the total absolute variation in the job completion times (TADC). This scheduling measure was first considered by Kanet [8]. The TADC of the 1|LE, spsd |TADC scheduling problem can be calculated as follows:
and p[r] vectors respectively (r = 1, . . . , n). Based on the above analysis and Lemma 1, the optimal sequence for the 1|LE, spsd |TADC problem can be obtained in O(n log n) time by arranging the elements of the wr and p[r] vectors in opposite orders. 3.4. The 1|LE, spsd |ETCP scheduling problem This problem is divided into two parts. Firstly the unrestricted common due date problem with earliness, tardiness and due date penalties is introduced. Then a polynomial-time algorithm is given. The goal of the unrestricted common due date problem is to jointly minimize the weighted earliness, tardiness and due date penalty. An unrestricted common due date d is a decision variable whose value is to be determined. If there are no p-s-d setup and learning effects (i.e., a = b = 0), the 1|LE, spsd |ETCP problem reduces to the 1// (αEi + βTi + γ d) problem. Panwalkar et al. [17] provided some useful results of the 1// (αEi + βTi + γ d) problem as follows: Theorem 1. For the 1//
(αEi + βTi + γ d) problem,
(1) it is optimal to assign the due date at the completion time of the kth job, where k is the smallest integer greater than or equal to (nβ − nγ )/(α + β); (2) the optimal schedule is V-shaped, i.e., early jobs are arranged in nonincreasing order of their processing times and tardy jobs are arranged in nondecreasing order of their processing times;
W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26
(3) the positional weight of a job when scheduled in position r in the sequence is given by vr = min nγ + (r − 1)α, (n + 1 − r)β . According to Theorem 1, ETCP = βTi + γ d) can be rewritten as ETCP =
n
vr + b
+
n
n−1
r=1
vl+1 r a p[r]
l=r
r=1
=
i=1 (αEi
n n A (αE[i] + βT[i] + γ d) = vr s[r] + p[r] i=1
=
n
wr p[r] ,
(7)
r=1
a where wr = (vr + b n−1 l=r vl+1 )r . Based on the above analysis and Lemma 1, the following O(n log n) algorithm is provided to solve the 1|LE, spsd |ETCP problem. Algorithm 1. Step 1. Assign the optimal due-date at the completion time of the kth job, where k is the smallest integer greater than or equal to (nβ − nγ )/(α + β), that is, nβ − nγ . k= α+β Step 2. Calculate each value of wr , r = 1, . . . , n. Step 3. Assign the job with the longest normal processing time to the position with the smallest value of wr , the job with the second longer normal processing time to the position with the second smaller value of wr , etc. The following corollary follows directly from Algorithm 1 and Theorem 1. Corollary 1. For the 1|LE, spsd |ETCP problem, (1) if b 1, then there exists an optimal schedule in which the jobs are arranged in nondecreasing order of their processing times; (2) if b = 0, then the common due date problem reduces to the problem tackled by Mosheiov [14]. In addition, it could be solved by Algorithm 1 of which the complexity is O(n log n); (3) if a = b = 0, then there exists an optimal schedule in which early jobs are arranged in nonincreasing order of their processing times and tardy jobs are arranged in nondecreasing order of their processing times.
25
Proof. We provide a brief proof as follows. (1) If b 1, thenw1 > w2 > · · · > wn . Hence, the result follows from Lemma 1. (2) If b = 0, the problem reduces to that studied by Mosheiov [14]. Mosheiov solved the problem by formulating it as an assignment problem. The complexity of solving an assignment problem is O(n3 ). In fact, the formulation of an assignment problem is not necessary, the particular problem (b = 0) could be solved by defining the weights wr = vr r a first, and then applying a simple sorting procedure as Algorithm 1. (3) If a = b = 0, then the 1|LE, spsd |ETCP problem reduces to the 1|| (αEr + βTr + γ d) problem. Therefore, the result follows directly from Theorem 1. 2 4. Extensions In this section, all of the scheduling problems are extended by the introduction of job-dependent learning effects and they are shown that they are proved to remain polynomially solvable. Under the job-dependent learning environment, all of the jobs have different learning rates ai 0. As in Biskup, let xir be a 0/1 variable such that xir = 1 if Ji is the rth job to be processed and xir = 0 otherwise. Then all of the scheduling problems in Section 3 can be formulated as the following assignment problems: min
s.t.
n n
Wir pi xir
i=1 r=1 n
xir = 1,
r = 1, 2, . . . , n,
xir = 1, r=1 xir = 0 or 1,
i = 1, 2, . . . , n,
i=1 n
i, r = 1, 2, . . . , n,
where
Wir = (n − r)b + 1 r ai for the makespan problem,
b(n − r) ai r Wir = (n − r + 1) 1 + 2 for total completion time problem, Wir = (r − 1)(n − r + 1) +b
n
(l − 1)(n − l + 1) r ai
l=r+1
for the total absolute differences in completion times problem, and
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W.H. Kuo, D.-L. Yang / Information Processing Letters 102 (2007) 22–26
Wir = vr + b
n−1
vl+1 r ai
l=r
for the sum of earliness, tardiness and common duedate penalty problem. 5. Conclusions This paper considers a single machine scheduling problem with setup times and learning considerations. The setup times are proportional to the length of the already scheduled jobs. The learning effect is also investigated in the scheduling environments, it is assumed a learning process reflects a decrease in the process time as a function of the number of repetitions. The following objective functions are considered: the makespan, the total completion time, the total absolute differences in completion times and the sum of earliness, tardiness and common due-date penalty. The polynomial time algorithms are proposed to optimally solve the above objective functions. In our study, the job-independent/job-dependent learning effect of a job is assumed. However, in some other situations, a learning effect may be time-dependent [10,11]. Therefore, it is worthwhile for future research to investigate p-s-d-setup scheduling problems with different learning effects. In these cases, it may be necessary to resort to heuristic algorithms for obtaining optimal sequences. Acknowledgements The authors would like to thank two anonymous referees for their helpful comments and suggestions on an early version of this paper. This research is supported in part by the National Science Council of Taiwan, Republic of China, under grant number NSC-94-2213-E-150016.
References [1] A. Allahverdi, J.N.D. Gupta, T. Aldowaisan, A review of scheduling research involving setup considerations, Omega 27 (1999) 219–239. [2] A. Bachman, A. Janiak, Scheduling jobs with position-dependent processing times, J. Oper. Res. Soc. 55 (2004) 257–264. [3] D. Biskup, Single-machine scheduling with learning considerations, Eur. J. Oper. Res. 115 (1999) 173–178. [4] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Ann. Oper. Res. 98 (2000) 273–290. [5] T.C.E. Cheng, Q. Ding, B.M.T. Lin, A concise survey of scheduling with time-dependent processing times, Eur. J. Oper. Res. 152 (2004) 1–13. [6] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Ann. Discrete Math. 5 (1979) 287–326. [7] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, London, 1967. [8] J.J. Kanet, Minimizing variation of flow time in single machine systems, Management Sci. 27 (1981) 1453–1459. [9] C. Koulamas, G.J. Kyparisis, Single-machine scheduling with past-sequence-dependent setup times, Eur. J. Oper. Res. (2006), in press. [10] W.H. Kuo, D.L. Yang, Single-machine group scheduling with a time-dependent learning effect, Comput. Oper. Res. 33 (2006) 2099–2112. [11] W.H. Kuo, D.L. Yang, Minimizing the total completion time in a single machine scheduling problem with a time-dependent learning effect, Eur. J. Oper. Res. 174 (2006) 1184–1190. [12] W.C. Lee, C.C. Wu, H.J. Sung, A bi-criterion single-machine scheduling problem with learning considerations, Acta Inform. 40 (2004) 303–315. [13] B.M.T. Lin, Complexity results for single-machine scheduling with positional learning effects, J. Oper. Res. Soc. (2006), in press. [14] G. Mosheiov, Scheduling problems with learning effect, Eur. J. Oper. Res. 132 (2001) 687–693. [15] G. Mosheiov, J.B. Sidney, Scheduling with general jobdependent learning curves, Eur. J. Oper. Res. 147 (2003) 665– 670. [16] G. Mosheiov, J.B. Sidney, Note on scheduling with general learning curves to minimize the number of tardy jobs, J. Oper. Res. Soc. 52 (2005) 110–112. [17] S.S. Panwalkar, M.L. Smith, A. Seidmann, Common due date assignment to minimize total penalty for the one machine scheduling problem, Oper. Res. 30 (1982) 391–399.