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Parallel machine earliness and tardiness scheduling with proportional weights
Computers & Operations Research 30 (2003) 801 – 808
www.elsevier.com/locate/dsw
Parallel machine earliness and tardiness scheduling with proportional weights Hongyi Suna , Guoqing Wangb; ∗ a
Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong b Department of Business Administration, Jinan University, Guangzhou, People’s Republic of China Received 1 July 2001; received in revised form 1 February 2002; accepted 1 March 2002
Abstract In this paper we study the problem of scheduling n jobs with a common due date and proportional early and tardy penalties on m identical parallel machines. We show that the problem is NP-hard and propose a dynamic programming algorithm to solve it. We also propose two heuristics to tackle the problem and analyze their worst-case error bounds.
Scope and purpose Scheduling problems to minimize the total weighted earliness and tardiness (WET) arise in Just-in-time manufacturing systems, where one of the objectives is to complete each job as close to its due date as possible. The earliness and tardiness weights of a job in WET tend to increase with the value of the job. Because processing time is often a good surrogate for the value of a job, it is reasonable to consider weights that are proportional to job processing times. In this paper we study the parallel identical machine WET problem with proportional weights. We propose both exact and approximation algorithms to tackle the problem. ? 2002 Published by Elsevier Science Ltd. Keywords: Parallel machine scheduling; Earliness and tardiness; Heuristic
1. Introduction In this article, we study the following parallel machine total weighted earliness and tardiness (WET) scheduling problem. We are given n independent jobs J = {J1 ; : : : ; Jn } to be scheduled on m ∗
Corresponding author. E-mail addresses: [email protected] (H. Sun), [email protected] (G. Wang).
0305-0548/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd. PII: S 0 3 0 5 - 0 5 4 8 ( 0 2 ) 0 0 0 5 5 - 2
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identical parallel machines M = {M1 ; : : : ; Mm }. Associated with each job Jj are a processing time pj and a weight wj . All jobs have a common due date d. Given a schedule , let Cj be the completion time of job Jj . Jj is early, if Cj ¡ d; it is tardy, if Cj ¿ d; and it is on time, if Cj = d. The objective of the problem is to And a schedule that minimizes the total weighted earliness and tardiness, that is min z( ) = wj |Cj − d|: Jj ∈ J
In practice the weights wj , j = 1; : : : ; n, are chosen according to some internal and external considerations and limitations that can aBect the future of the company. Choosing appropriate weights is usually diCcult and time consuming. In many real applications the weights are set equal to one for all jobs. Another alternative approach is to use weights that are proportional to processing times of the jobs, since processing time is often a good surrogate for the value of the job [1,2]. The presentation in this paper is conAned to the WET scheduling problem with the proportional weights, that is wj = pj ; where is a given constant number. Without loss of generality, we can assume = 1. We refer to this problem as Q. The WET scheduling problems arise in the Just-in-time (JIT) systems, where the objective is to complete each job as close to its due date as possible. WET scheduling problems have been extensively investigated in the literature; however, the majority of prior research eBorts are conAned to the single machine settings. We refer to Baker and Scudder [3] for a survey of the single machine literature up to 1990. It is well known that the problem is NP-hard in the ordinary sense for m = 1 and the due date is unrestricted, i.e. it is suCciently large to have no impact on the job sequence on the machine [4]. However, Ahmed and Sundararaghavan [5], Hoogeveen and van de Velde [6], Hall and Posner [4] showed independently that the special case of proportional weights and a unrestricted due date is solvable by the longest processing time (LPT) rule. Rachamadugu [7] and Alidaee and Dragan [8] further extended the result to the case of a restricted due date. Note that a simple procedure to determined optimal jobs starting time is also proposed by Rachamadugu [7]. Alidaee [9] studied the WET scheduling problem with proportional weights and individual job due dates. There are only a handful of results on WET scheduling for the parallel machine settings, and almost all of these results focus on the unrestricted due date case. Sundararaghavan and Ahmed [10] and Hall [11] provided an O(n log n) algorithm to minimize the sum of absolute deviations. Emmons [12] extended this result to the case where the deviations of the completion times of the tardy jobs and those of the early jobs are weighted with two diBerent factors and to the settings with uniform machines. Kubiak et al. [13] showed that for unrelated machines the sum of deviations can be optimized in polynomial time by solving a transportation problem. Federgruen and Mosheiov [14] provided a heuristic to tackle the WET scheduling problem on identical parallel machines in which the cost of a job is given by a general nondecreasing, convex function F of the absolute deviation of its completion time from common unrestricted due date, and showed that the heuristic is asymptotically optimal. Chen and Powell [15] proposed a column generation based exact decomposition algorithm for the WET scheduling problem on identical parallel machines, and showed that
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the algorithm is capable of solving problems of moderate sizes. Heady and Zhu [16] studied the WET scheduling problem on identical parallel machines with individual job due dates. The rest of the paper is organized as follows. In the next section, we show that the problem is NP-hard, and provide a dynamic programming algorithm to solve it. We develop some heuristics to tackle the problem and analyze their worst-case error bounds in Section 3. Finally, we present the conclusions in Section 4. 2. Complexity In this section, we establish some NP-hardness results for the problem under consideration. For any given schedule , let Ni be the set of jobs assigned to Mi , and Ti be the subset of tardy jobs on Mi , and Ei = Ni \ Ti . From the Lemma 2 of [5], we have the following result. Lemma 1. Let be any schedule such that one of its jobs is completed at time d on Mi . Let be any schedule obtained from by permuting jobs in Ti and Ei among themselves. z( ) = z( ): Theorem 1. Q with unrestricted due date is NP-hard even when m = 2. Proof. Consider the following problem which is known to be binary NP-complete [17]. Given positive integers r, X and a set of positive integers : : ; xr } such that Partition: S = {x1 ; : r x = 2X , does there exist a partition of S into subset S such that x = l 1 l xl ∈ S1 x l ∈ S \ S1 xl = X ? l=1 For any given instance of Partition, construct an instance of Q with unrestricted due date as follows: N = {J1 ; : : : ; Jr+2 }; p j = xj ;
m = 2;
j = 1; : : : ; r;
pr+1 = pr+2 = 2X 2 ; d=
r+2
pj = 2X (2X + 1);
j=1
y = X2 +
r
xj2 =2:
l=1
Job set S = {J1 ; : : : ; Jr } are referred to as partition job set. We now show that there exists a solution for this instance with z( ) 6 y if and only if there exists a desired solution for Partition. If Part: Let S1 be a desired solution for Partition and S1 be the corresponding partition job set. Consider the permutation schedule as shown in Fig. 1, and sequence all jobs in S1 and S \ S1 in an arbitrary order. It is easy to check that z( ) = y. Only If Part: Let be a schedule such that z( ) 6 y. Since pr+1 = pr+2 ¿ y, it is clear that both Jr+1 and Jr+2 must be on time, and so all jobs in S must be tardy in . Let S1 be the set of
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Fig. 1. Schedule .
partition jobs assigned on M1 in , and A = xi ∈S1 xi . From Lemma 1, for any permutation of the partition jobs on both machines, it is easy to show that r z( ) = xj2 =2 + A2 =2 + (2X − A)2 =2: i=1
To have z( ) 6 y, we must have A = X . With a similar argument, we can easily show that Q is unary NP-hard for arbitrary m. From [7,9], we know the following result. Lemma 2. There exists an optimal schedule for Q such that all jobs assigned to any machine are in LPT order. Let P = Jj ∈J pj . Based on Lemma 1, we can develop a dynamic programming algorithm to solve the problem as follows. Algorithm DP (a) Renumber the jobs in LPT order, i.e. p1 ¿ p2 ¿ · · · ¿ pn . (b) DeAne Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) as the minimum total weighted earliness and tardiness if we have scheduled jobs J1 through Jj , such that on Ml the Arst job starts at el and the last job Anishes at tl , l = 1; : : : ; m. (c) Recursive relations: For j = 1; : : : ; n, el = 0; : : : ; d, tl = el ; : : : ; P, l = 1; : : : ; m, Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) = min16i6m {Fj−1 (e1 ; : : : ; em ; t1 ; : : : ; ti − pj ; : : : ; tm ) + wj |ti − d|}. (d) Initial conditions: For j = 0; : : : ; n, el = 0; : : : ; d, tl = el ; : : : ; P, l = 1; : : : ; m, 0; if j = 0 and el = tl ; Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) = ∞; otherwise: (e) Optimal solution: z( ∗ ) = Fn (e1 ; : : : ; em ; t1 ; : : : ; tm ), over all e1 = 0; : : : ; d, tl = el ; : : : ; P, and l = 1; : : : ; m. The optimality of DP can be easily justiAed as follows. From Lemma 1, we know that there exists an optimal solution in which Jj is assigned to the last position on one of the machines. If Jj is
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assigned to Mi , then Cj = ti , i = 1; : : : ; m. Since Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) is determined by the optimal assignment by deAnition, this justify the validity of the recursive relations. The time complexity of the algorithm can be established as follows. Since only m − 1 values of t1 ; : : : ; tm are independent, the number of diBerent states of the recursive relations is at most ndm P m−1 . For each state, the right-hand side of the recursive relations can be calculated in O(m) time. Thus the overall computational complexity of DP is O(nmdm P m−1 ). Since DP is pseudopolynomial for a given m, we know that problem Q is NP-hard in the ordinary sense in this case. 3. Heuristics and worst-case analysis In this section, we present two heuristics to tackle the problem. Our heuristics are all based on the list scheduling (LS) procedure to construct a schedule for parallel machines as follows: Given a certain list or permutation of a set of jobs, the next job of the list is scheduled on the machine available Arst. To use this rule, we deAne ri for Mi as the dummy initial machine available time, i.e. the available time for the Arst job on Mi for list scheduling. While the job assignment is given, we need to determine the optimal job starting times on each machine. As there is no inserted idle times between jobs, it is suCcient to determine the starting time of the Arst job on each machine. For any given schedule , if the Arst job on Mi does not start at time zero, then there must be an on time job, namely J i , on Mi . The optimal starting time of the Arst job on Mi can be determined according to the following result [5]. pk 6 pj 6 pk : (1) Jk ∈ Ti
Jj ∈ N i
Ei \{J i }
With these results, we can propose our heuristics as follows. Heuristic LPT (i) Renumber the jobs in the LPT order. (ii) Let rl = 0, and assign jobs according to the LS rule. (iii) Determine the optimal starting times for each machine according to (1). For Heuristic LPT, we have the following result. Lemma 3. There exists an instance for Q such that z( LPT )=z( ∗ ) 6 m − $; for any $ ¿ 0. Proof. Consider the following instance of the problem. The job set J = {J1 ; : : : ; Jn } such that n = m(K + 1); pj = 2mK;
d = 2m2 K;
j = 1; : : : ; m − 1;
pm = mK; pj = 1;
j = m + 1; : : : ; mK + m;
where K is a positive integer. It is easy to see that z( ∗ ) = mk(K + 1)=2 and z( LPT ) = m2 K 2 (mK + 1)=2. Hence z( LPT )=z( ∗ ) = m − (m − 1)=(K + 1). We see that z( LPT )=z( ∗ ) → m; as K → ∞.
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Now we propose a modiAed heuristic MLPT. Heuristic MLPT (i) Renumber the jobs in the LPT order. (ii) Assign job Jl , l = 1; : : : ; m to machine Ml . Let rl = max{0; pl − d}, and assign other jobs according to the LS rule. (iii) Determine the optimal starting time for each machine according to Lemma 1. Lemma 4. When p1 ¿ d for Q; then z( MLPT )=z( ∗ ) 6 25 . 12 Proof. Let Q be an instance for Q with job set N = {Jm+1 ; : : : ; Jn } and due date d = 0. For any feasible schedule for Q ; it is not diCcult to verify that 2 m 1 z( ) = pj + pj2 ; 2 i=1 J ∈N Jj ∈ N i
j
which means that Q is equivalent to the parallel machine scheduling problem P== Ci2 ; where Ci is the completion time of Mi . Now let LPT and o be the LPT and optimal schedules for Q , respectively. It is clear that ) z( MLPT ) 6 z( LPT
and from Chandra and Wong [18], we know that )− ) z( LPT 25 z( LPT Jj ∈ N pj 6 6 : o o z( ) z( ) − Jj ∈N pj 24 Let ∗ be an optimal schedule for Q in which all jobs on each machine are sequenced in the LPT order on each machine. Let pi∗ be the maximum job processing time on Mi , and Ji∗ be the last job which starts before d and Anishes at d + xi on Mi , where 0 6 xi 6 pi∗ . We can show that 2 2 m m 1 z( ∗ ) ¿ pj + pi∗ − xi − pi∗ + pj + x i + pj2 − pi2∗ 2 i=1 J ∈E J ∈T J ∈N i=1 j
i
j
i
2 m m 1 pj − pi∗ + pj2 − pi2∗ ¿ 4 i=1 J ∈N ∗ J ∈N i=1 j
i
j
2 m 1 pj − p i + pj2 ¿ 4 i=1 J ∈N ∗ J ∈N j
i
¿ z( )=2: Hence, we have the desired result.
j
j
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4. Conclusions In this paper, we have studied the identical parallel machine scheduling problem to minimize the total weighted earliness and tardiness with proportional weights. We have shown that the problem is NP-hard in the ordinary sense when m is given. We have also developed some heuristics to tackle the problem, and analyzed their worst-case performance. As the worst-case error bound of the improved longest processing time heuristic is only derived for a restrictive case, it would be of interest to investigate the worst-case error bound of the heuristic for general case of the problem for further research. Acknowledgements We are grateful to two anonymous referees whose comments and suggestions have greatly improved the presentation of the paper. References [1] Arkin EM, Roundy RO. Weighted-tardiness scheduling on parallel machines with proportional weights. Operations Research 1991;39:64–81. [2] Szwarc W, Liu JJ. Weighted tardiness single machine scheduling with proportional weights. Management Science 1993;57:167–78. [3] Baker KR, Scudder GD. Sequencing with earliness and tardiness penalties: a review. Operations Research 1990;38:22–36. [4] Hall NG, Posner ME. Earliness-tardiness scheduling problems. I: weighted deviation of completion times about a common due date. Operations Research 1991;39:836–46. [5] Ahmed MU, Sundararaghavan PS. Minimizing the weighted sum of late and early completion times in a single machine. IIE Transactions 1990;22:288–90. [6] Hoogeveen JA, van de Velde SL. Scheduling around a small common due date. European Journal of Operational Research 1991;55:237–42. [7] Rachamadugu R. Scheduling jobs with proportionate early=tardy penalties. IIE Transactions 1995;27:679–82. [8] Alidaee B, Dragan I. A note on minimizing the weighted sum of tardy and early completion penalties in a single machine: a case of small common due date. European Journal of Operational Research 1997;96:559–63. [9] Alidaee B. Minimizing absolute and squared deviation of completion times from due dates. Production and Operations Management 1994;3:133–47. [10] Sundararaghavan PS, Ahmed MU. Minimizing the sum of absolute lateness in single-machine and multimachine scheduling. Naval Research Logistics Quarterly 1984;31:325–33. [11] Hall NG. Single- and multiple-processor models for minimizing completion time variance. Naval Research Logistics 1986;33:49–54. [12] Emmons H. Scheduling to a common due date on parallel common processors. Naval Research Logistics Quarterly 1987;34:851–970. [13] Kubiak W, Low S, Sethi S. Equivalence of mean Qow time problems and mean absolute deviation problems. Operations Research Letters 1990;9:371–4. [14] Federgruen A, Mosheiov G. Heuristics for multi-machine scheduling problems with earliness and tardiness costs. Management Science 1996;42:1544–56. [15] Chen Z-L, Powell WB. A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem. European Journal of Operational Research 1999;116:220–32. [16] Heady RB, Zhu Z. Minimizing the sum of job earliness and tardiness in a multimachine system. International Journal of Production Research 1998;36:1619–32.
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[17] Garey MR, Johnson DS. Computers and intractability: a guide to the theory of NP-completeness. San Francisco: Freeman, 1979. [18] Chandra AK, Wong CK. Worst-case analysis of a placement algorithm related to storage allocation. SIAM Journal on Computing 1975;4:249–63. Hongyi Sun is an Associate Professor at the Department of Manufacturing Engineering & Engineering Management, City University of Hong Kong. He holds a Bachelor degree in Computer Science from Harbin University of Science and Technology, a Master degree in Engineering Management from Harbin Institute of Technology (HIT), both in China, and a Ph.D. in Industrial Management from Aalborg University in Denmark. His teaching and research areas include manufacturing=operations strategy, quality management, management of technological innovation and entrepreneurship, and international comparative study. Guoqing Wang is an Associate Professor at the Department of Business Administration, Jinan University, Guangzhou, China. He obtained a Bachelor degree in Mechanical Engineering and a Master degree in Engineering Management from Harbin Institute of Technology, and a Ph.D. in Operations Research from The Hong Kong Polytechnic University. His research interests are in production and operations management and machine scheduling. He has published more than a dozen papers in this and other international journals in the past four years.
www.elsevier.com/locate/dsw
Parallel machine earliness and tardiness scheduling with proportional weights Hongyi Suna , Guoqing Wangb; ∗ a
Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong b Department of Business Administration, Jinan University, Guangzhou, People’s Republic of China Received 1 July 2001; received in revised form 1 February 2002; accepted 1 March 2002
Abstract In this paper we study the problem of scheduling n jobs with a common due date and proportional early and tardy penalties on m identical parallel machines. We show that the problem is NP-hard and propose a dynamic programming algorithm to solve it. We also propose two heuristics to tackle the problem and analyze their worst-case error bounds.
Scope and purpose Scheduling problems to minimize the total weighted earliness and tardiness (WET) arise in Just-in-time manufacturing systems, where one of the objectives is to complete each job as close to its due date as possible. The earliness and tardiness weights of a job in WET tend to increase with the value of the job. Because processing time is often a good surrogate for the value of a job, it is reasonable to consider weights that are proportional to job processing times. In this paper we study the parallel identical machine WET problem with proportional weights. We propose both exact and approximation algorithms to tackle the problem. ? 2002 Published by Elsevier Science Ltd. Keywords: Parallel machine scheduling; Earliness and tardiness; Heuristic
1. Introduction In this article, we study the following parallel machine total weighted earliness and tardiness (WET) scheduling problem. We are given n independent jobs J = {J1 ; : : : ; Jn } to be scheduled on m ∗
Corresponding author. E-mail addresses: [email protected] (H. Sun), [email protected] (G. Wang).
0305-0548/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd. PII: S 0 3 0 5 - 0 5 4 8 ( 0 2 ) 0 0 0 5 5 - 2
802
H. Sun, G. Wang / Computers & Operations Research 30 (2003) 801 – 808
identical parallel machines M = {M1 ; : : : ; Mm }. Associated with each job Jj are a processing time pj and a weight wj . All jobs have a common due date d. Given a schedule , let Cj be the completion time of job Jj . Jj is early, if Cj ¡ d; it is tardy, if Cj ¿ d; and it is on time, if Cj = d. The objective of the problem is to And a schedule that minimizes the total weighted earliness and tardiness, that is min z( ) = wj |Cj − d|: Jj ∈ J
In practice the weights wj , j = 1; : : : ; n, are chosen according to some internal and external considerations and limitations that can aBect the future of the company. Choosing appropriate weights is usually diCcult and time consuming. In many real applications the weights are set equal to one for all jobs. Another alternative approach is to use weights that are proportional to processing times of the jobs, since processing time is often a good surrogate for the value of the job [1,2]. The presentation in this paper is conAned to the WET scheduling problem with the proportional weights, that is wj = pj ; where is a given constant number. Without loss of generality, we can assume = 1. We refer to this problem as Q. The WET scheduling problems arise in the Just-in-time (JIT) systems, where the objective is to complete each job as close to its due date as possible. WET scheduling problems have been extensively investigated in the literature; however, the majority of prior research eBorts are conAned to the single machine settings. We refer to Baker and Scudder [3] for a survey of the single machine literature up to 1990. It is well known that the problem is NP-hard in the ordinary sense for m = 1 and the due date is unrestricted, i.e. it is suCciently large to have no impact on the job sequence on the machine [4]. However, Ahmed and Sundararaghavan [5], Hoogeveen and van de Velde [6], Hall and Posner [4] showed independently that the special case of proportional weights and a unrestricted due date is solvable by the longest processing time (LPT) rule. Rachamadugu [7] and Alidaee and Dragan [8] further extended the result to the case of a restricted due date. Note that a simple procedure to determined optimal jobs starting time is also proposed by Rachamadugu [7]. Alidaee [9] studied the WET scheduling problem with proportional weights and individual job due dates. There are only a handful of results on WET scheduling for the parallel machine settings, and almost all of these results focus on the unrestricted due date case. Sundararaghavan and Ahmed [10] and Hall [11] provided an O(n log n) algorithm to minimize the sum of absolute deviations. Emmons [12] extended this result to the case where the deviations of the completion times of the tardy jobs and those of the early jobs are weighted with two diBerent factors and to the settings with uniform machines. Kubiak et al. [13] showed that for unrelated machines the sum of deviations can be optimized in polynomial time by solving a transportation problem. Federgruen and Mosheiov [14] provided a heuristic to tackle the WET scheduling problem on identical parallel machines in which the cost of a job is given by a general nondecreasing, convex function F of the absolute deviation of its completion time from common unrestricted due date, and showed that the heuristic is asymptotically optimal. Chen and Powell [15] proposed a column generation based exact decomposition algorithm for the WET scheduling problem on identical parallel machines, and showed that
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the algorithm is capable of solving problems of moderate sizes. Heady and Zhu [16] studied the WET scheduling problem on identical parallel machines with individual job due dates. The rest of the paper is organized as follows. In the next section, we show that the problem is NP-hard, and provide a dynamic programming algorithm to solve it. We develop some heuristics to tackle the problem and analyze their worst-case error bounds in Section 3. Finally, we present the conclusions in Section 4. 2. Complexity In this section, we establish some NP-hardness results for the problem under consideration. For any given schedule , let Ni be the set of jobs assigned to Mi , and Ti be the subset of tardy jobs on Mi , and Ei = Ni \ Ti . From the Lemma 2 of [5], we have the following result. Lemma 1. Let be any schedule such that one of its jobs is completed at time d on Mi . Let be any schedule obtained from by permuting jobs in Ti and Ei among themselves. z( ) = z( ): Theorem 1. Q with unrestricted due date is NP-hard even when m = 2. Proof. Consider the following problem which is known to be binary NP-complete [17]. Given positive integers r, X and a set of positive integers : : ; xr } such that Partition: S = {x1 ; : r x = 2X , does there exist a partition of S into subset S such that x = l 1 l xl ∈ S1 x l ∈ S \ S1 xl = X ? l=1 For any given instance of Partition, construct an instance of Q with unrestricted due date as follows: N = {J1 ; : : : ; Jr+2 }; p j = xj ;
m = 2;
j = 1; : : : ; r;
pr+1 = pr+2 = 2X 2 ; d=
r+2
pj = 2X (2X + 1);
j=1
y = X2 +
r
xj2 =2:
l=1
Job set S = {J1 ; : : : ; Jr } are referred to as partition job set. We now show that there exists a solution for this instance with z( ) 6 y if and only if there exists a desired solution for Partition. If Part: Let S1 be a desired solution for Partition and S1 be the corresponding partition job set. Consider the permutation schedule as shown in Fig. 1, and sequence all jobs in S1 and S \ S1 in an arbitrary order. It is easy to check that z( ) = y. Only If Part: Let be a schedule such that z( ) 6 y. Since pr+1 = pr+2 ¿ y, it is clear that both Jr+1 and Jr+2 must be on time, and so all jobs in S must be tardy in . Let S1 be the set of
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H. Sun, G. Wang / Computers & Operations Research 30 (2003) 801 – 808
Fig. 1. Schedule .
partition jobs assigned on M1 in , and A = xi ∈S1 xi . From Lemma 1, for any permutation of the partition jobs on both machines, it is easy to show that r z( ) = xj2 =2 + A2 =2 + (2X − A)2 =2: i=1
To have z( ) 6 y, we must have A = X . With a similar argument, we can easily show that Q is unary NP-hard for arbitrary m. From [7,9], we know the following result. Lemma 2. There exists an optimal schedule for Q such that all jobs assigned to any machine are in LPT order. Let P = Jj ∈J pj . Based on Lemma 1, we can develop a dynamic programming algorithm to solve the problem as follows. Algorithm DP (a) Renumber the jobs in LPT order, i.e. p1 ¿ p2 ¿ · · · ¿ pn . (b) DeAne Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) as the minimum total weighted earliness and tardiness if we have scheduled jobs J1 through Jj , such that on Ml the Arst job starts at el and the last job Anishes at tl , l = 1; : : : ; m. (c) Recursive relations: For j = 1; : : : ; n, el = 0; : : : ; d, tl = el ; : : : ; P, l = 1; : : : ; m, Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) = min16i6m {Fj−1 (e1 ; : : : ; em ; t1 ; : : : ; ti − pj ; : : : ; tm ) + wj |ti − d|}. (d) Initial conditions: For j = 0; : : : ; n, el = 0; : : : ; d, tl = el ; : : : ; P, l = 1; : : : ; m, 0; if j = 0 and el = tl ; Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) = ∞; otherwise: (e) Optimal solution: z( ∗ ) = Fn (e1 ; : : : ; em ; t1 ; : : : ; tm ), over all e1 = 0; : : : ; d, tl = el ; : : : ; P, and l = 1; : : : ; m. The optimality of DP can be easily justiAed as follows. From Lemma 1, we know that there exists an optimal solution in which Jj is assigned to the last position on one of the machines. If Jj is
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assigned to Mi , then Cj = ti , i = 1; : : : ; m. Since Fj (e1 ; : : : ; em ; t1 ; : : : ; tm ) is determined by the optimal assignment by deAnition, this justify the validity of the recursive relations. The time complexity of the algorithm can be established as follows. Since only m − 1 values of t1 ; : : : ; tm are independent, the number of diBerent states of the recursive relations is at most ndm P m−1 . For each state, the right-hand side of the recursive relations can be calculated in O(m) time. Thus the overall computational complexity of DP is O(nmdm P m−1 ). Since DP is pseudopolynomial for a given m, we know that problem Q is NP-hard in the ordinary sense in this case. 3. Heuristics and worst-case analysis In this section, we present two heuristics to tackle the problem. Our heuristics are all based on the list scheduling (LS) procedure to construct a schedule for parallel machines as follows: Given a certain list or permutation of a set of jobs, the next job of the list is scheduled on the machine available Arst. To use this rule, we deAne ri for Mi as the dummy initial machine available time, i.e. the available time for the Arst job on Mi for list scheduling. While the job assignment is given, we need to determine the optimal job starting times on each machine. As there is no inserted idle times between jobs, it is suCcient to determine the starting time of the Arst job on each machine. For any given schedule , if the Arst job on Mi does not start at time zero, then there must be an on time job, namely J i , on Mi . The optimal starting time of the Arst job on Mi can be determined according to the following result [5]. pk 6 pj 6 pk : (1) Jk ∈ Ti
Jj ∈ N i
Ei \{J i }
With these results, we can propose our heuristics as follows. Heuristic LPT (i) Renumber the jobs in the LPT order. (ii) Let rl = 0, and assign jobs according to the LS rule. (iii) Determine the optimal starting times for each machine according to (1). For Heuristic LPT, we have the following result. Lemma 3. There exists an instance for Q such that z( LPT )=z( ∗ ) 6 m − $; for any $ ¿ 0. Proof. Consider the following instance of the problem. The job set J = {J1 ; : : : ; Jn } such that n = m(K + 1); pj = 2mK;
d = 2m2 K;
j = 1; : : : ; m − 1;
pm = mK; pj = 1;
j = m + 1; : : : ; mK + m;
where K is a positive integer. It is easy to see that z( ∗ ) = mk(K + 1)=2 and z( LPT ) = m2 K 2 (mK + 1)=2. Hence z( LPT )=z( ∗ ) = m − (m − 1)=(K + 1). We see that z( LPT )=z( ∗ ) → m; as K → ∞.
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Now we propose a modiAed heuristic MLPT. Heuristic MLPT (i) Renumber the jobs in the LPT order. (ii) Assign job Jl , l = 1; : : : ; m to machine Ml . Let rl = max{0; pl − d}, and assign other jobs according to the LS rule. (iii) Determine the optimal starting time for each machine according to Lemma 1. Lemma 4. When p1 ¿ d for Q; then z( MLPT )=z( ∗ ) 6 25 . 12 Proof. Let Q be an instance for Q with job set N = {Jm+1 ; : : : ; Jn } and due date d = 0. For any feasible schedule for Q ; it is not diCcult to verify that 2 m 1 z( ) = pj + pj2 ; 2 i=1 J ∈N Jj ∈ N i
j
which means that Q is equivalent to the parallel machine scheduling problem P== Ci2 ; where Ci is the completion time of Mi . Now let LPT and o be the LPT and optimal schedules for Q , respectively. It is clear that ) z( MLPT ) 6 z( LPT
and from Chandra and Wong [18], we know that )− ) z( LPT 25 z( LPT Jj ∈ N pj 6 6 : o o z( ) z( ) − Jj ∈N pj 24 Let ∗ be an optimal schedule for Q in which all jobs on each machine are sequenced in the LPT order on each machine. Let pi∗ be the maximum job processing time on Mi , and Ji∗ be the last job which starts before d and Anishes at d + xi on Mi , where 0 6 xi 6 pi∗ . We can show that 2 2 m m 1 z( ∗ ) ¿ pj + pi∗ − xi − pi∗ + pj + x i + pj2 − pi2∗ 2 i=1 J ∈E J ∈T J ∈N i=1 j
i
j
i
2 m m 1 pj − pi∗ + pj2 − pi2∗ ¿ 4 i=1 J ∈N ∗ J ∈N i=1 j
i
j
2 m 1 pj − p i + pj2 ¿ 4 i=1 J ∈N ∗ J ∈N j
i
¿ z( )=2: Hence, we have the desired result.
j
j
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807
4. Conclusions In this paper, we have studied the identical parallel machine scheduling problem to minimize the total weighted earliness and tardiness with proportional weights. We have shown that the problem is NP-hard in the ordinary sense when m is given. We have also developed some heuristics to tackle the problem, and analyzed their worst-case performance. As the worst-case error bound of the improved longest processing time heuristic is only derived for a restrictive case, it would be of interest to investigate the worst-case error bound of the heuristic for general case of the problem for further research. Acknowledgements We are grateful to two anonymous referees whose comments and suggestions have greatly improved the presentation of the paper. References [1] Arkin EM, Roundy RO. Weighted-tardiness scheduling on parallel machines with proportional weights. Operations Research 1991;39:64–81. [2] Szwarc W, Liu JJ. Weighted tardiness single machine scheduling with proportional weights. Management Science 1993;57:167–78. [3] Baker KR, Scudder GD. Sequencing with earliness and tardiness penalties: a review. Operations Research 1990;38:22–36. [4] Hall NG, Posner ME. Earliness-tardiness scheduling problems. I: weighted deviation of completion times about a common due date. Operations Research 1991;39:836–46. [5] Ahmed MU, Sundararaghavan PS. Minimizing the weighted sum of late and early completion times in a single machine. IIE Transactions 1990;22:288–90. [6] Hoogeveen JA, van de Velde SL. Scheduling around a small common due date. European Journal of Operational Research 1991;55:237–42. [7] Rachamadugu R. Scheduling jobs with proportionate early=tardy penalties. IIE Transactions 1995;27:679–82. [8] Alidaee B, Dragan I. A note on minimizing the weighted sum of tardy and early completion penalties in a single machine: a case of small common due date. European Journal of Operational Research 1997;96:559–63. [9] Alidaee B. Minimizing absolute and squared deviation of completion times from due dates. Production and Operations Management 1994;3:133–47. [10] Sundararaghavan PS, Ahmed MU. Minimizing the sum of absolute lateness in single-machine and multimachine scheduling. Naval Research Logistics Quarterly 1984;31:325–33. [11] Hall NG. Single- and multiple-processor models for minimizing completion time variance. Naval Research Logistics 1986;33:49–54. [12] Emmons H. Scheduling to a common due date on parallel common processors. Naval Research Logistics Quarterly 1987;34:851–970. [13] Kubiak W, Low S, Sethi S. Equivalence of mean Qow time problems and mean absolute deviation problems. Operations Research Letters 1990;9:371–4. [14] Federgruen A, Mosheiov G. Heuristics for multi-machine scheduling problems with earliness and tardiness costs. Management Science 1996;42:1544–56. [15] Chen Z-L, Powell WB. A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem. European Journal of Operational Research 1999;116:220–32. [16] Heady RB, Zhu Z. Minimizing the sum of job earliness and tardiness in a multimachine system. International Journal of Production Research 1998;36:1619–32.
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[17] Garey MR, Johnson DS. Computers and intractability: a guide to the theory of NP-completeness. San Francisco: Freeman, 1979. [18] Chandra AK, Wong CK. Worst-case analysis of a placement algorithm related to storage allocation. SIAM Journal on Computing 1975;4:249–63. Hongyi Sun is an Associate Professor at the Department of Manufacturing Engineering & Engineering Management, City University of Hong Kong. He holds a Bachelor degree in Computer Science from Harbin University of Science and Technology, a Master degree in Engineering Management from Harbin Institute of Technology (HIT), both in China, and a Ph.D. in Industrial Management from Aalborg University in Denmark. His teaching and research areas include manufacturing=operations strategy, quality management, management of technological innovation and entrepreneurship, and international comparative study. Guoqing Wang is an Associate Professor at the Department of Business Administration, Jinan University, Guangzhou, China. He obtained a Bachelor degree in Mechanical Engineering and a Master degree in Engineering Management from Harbin Institute of Technology, and a Ph.D. in Operations Research from The Hong Kong Polytechnic University. His research interests are in production and operations management and machine scheduling. He has published more than a dozen papers in this and other international journals in the past four years.