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Parallel-machine scheduling with machine-dependent maintenance periodic recycles
Author’s Accepted Manuscript Parallel-machine scheduling with machinedependent maintenance periodic recycles Guo Li, Mengqi Liu, Suresh P. Sethi, Dehua Xu
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S0925-5273(17)30014-2 http://dx.doi.org/10.1016/j.ijpe.2017.01.014 PROECO6642
To appear in: Intern. Journal of Production Economics Received date: 1 November 2016 Revised date: 27 December 2016 Accepted date: 20 January 2017 Cite this article as: Guo Li, Mengqi Liu, Suresh P. Sethi and Dehua Xu, Parallelmachine scheduling with machine-dependent maintenance periodic recycles, Intern. Journal of Production Economics, http://dx.doi.org/10.1016/j.ijpe.2017.01.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Parallel-machine scheduling with machine-dependent maintenance periodic recycles Guo Li a , Mengqi Liu b,∗ , Suresh P. Sethi c , Dehua Xu d a
School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China b
c
Business School, Hunan University, Changsha, Hunan 410082, China
Naveen Jindal School of Management, The University of Texas at Dallas, Dallas, TX 75080, United States d
School of Science, East China University of Technology, Nanchang, Jiangxi 330013, China (*Corresponding E-mail addresses: [email protected])
Abstract Parallel machine structure is very common in modern production systems. Its performance sometimes has a decisive impact on the whole productivity. In this paper, we consider a parallel-machine scheduling problem where each machine is subject to periodic maintenance. Instead of assuming all the machines have the same maintenance periodic cycle, we assume the maintenance periodic cycles are machine-dependent. The objective is to schedule all the jobs to the machines such that the makespan is minimized. We first provide computational complexity and non-approximability analyses of the problem and then present two mathematical programming models to tackle small-sized instances. Thereafter, we present a worst-case analysis of the classical LPT and LS algorithms for the problem. Then we propose two improved heuristic algorithms based on some observations of large-sized instances. In order to evaluate the performance of the heuristic algorithms, we resort to a lower bound of the optimal makespan, and find that it has a very interesting characteristic. Numerical experiments show that the improved heuristic algorithm MLPT improves the objective value of the LPT schedule by about 6.75% on average and that the MLPT heuristic algorithm has an average-case relative error less than 1.57% for all combinations of instances, which means that it is very suitable for real world applications. Keywords: Scheduling; Maintenance; Periodic recycle; Makespan; Heuristic algorithm
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1
Introduction
Parallel machine structure is very common in modern production systems. It is a natural extension of the single-machine structure and also the basic unit of the flexible job shop structure. Due to its importance, a lot of work has been done on the study of various parallel machine scenarios. To the best of our knowledge, Graham (1966) is among the first set of researchers who consider the optimal operations on parallel machine environment. He studies a problem where a number of jobs need to be scheduled onto some machines such that the makespan is minimized. And the proposed Longest Processing Time (LPT) first rule is now a well-known dispatching rule not only to the scheduling community but also to the production managers. Much of the past research on machine scheduling assumes that machines are available all the time. However, as machines are maintained according to their user manuals in order to function properly, there is a need to develop methodologies to handle this issue. To address this, we consider a parallel-machine scheduling problem with periodic maintenance where the maintenance periodic recycles may be different for each machine. We provide computational complexity and non-approximability analyses, and two mathematical programming models for the problem. We obtain worst-case ratios of the classical LPT and LS algorithms for the problem and propose two improved heuristic algorithms based on some observations of largesized instances. We also conduct numerical experiments to evaluate their performance. The results obtained may help production managers improve the productivity of their machine shops. Formally, our scheduling problem can be described as follows. We have to schedule n independent non-preemptive jobs J1 , J2 , . . . , Jn to m parallel machines M1 , M2 , . . ., Mn , where the processing time of job Ji is pi . The maintenance periodic recycle of machine Mi is ui + wi where ui is the length of the time between two consecutive maintenance activities on machine Mi and wi is the length of the time that is needed to perform one maintenance activity on the same machine. Let the interval between two consecutive maintenance activities on the same machine be called a working interval. Note that a machine cannot process any jobs during its maintenance. Therefore, jobs can only be schedule into these working intervals. Assume that all machines have just finished their maintenance at time zero. The objective is to schedule all the jobs to the machines such that the makespan, i.e, the completion time of the last finished job, is minimized.
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Extending the well-known three-field α|β|γ classification scheme suggested by Graham et al. (1979), we denote this problem by P m, mdmpr||Cmax , where mdmpr stands for machinedepended maintenance periodic recycle. Without loss of generality, in what follows, we assume that max{pi } ≤ max{ui }, otherwise there is clearly no feasible schedule for our problem. Moreover, we assume that jobs that scheduled in the same working interval are processed consecutively one after another from the start time of the interval. In what follows, we shall briefly review some works on machine scheduling with periodic maintenance. The single machine scenario has been investigated extensively by scholars from different perspectives. To the best of our knowledge, Liao and Chen (2003) are the first group of researchers that take periodic maintenance into consideration in the scheduling community. They study a single machine scheduling problem with the objective of minimizing maximum tardiness. Chen (2007a) and Sbihi and Varnier (2008) study the same scheduling problem as Liao and Chen (2003). They independently propose a heuristic algorithm and a branch-and-bound algorithm. Following similar approaches as those in Liao and Chen (2003), Chen (2006, 2007b, 2009) studies three single machine scheduling problems with the following objectives, i.e., minimizing total flow time, minimizing the sum of weighted total flow time and weighted maximum tardiness, and minimizing the number of tardy jobs, respectively. For parallel machine scheduling with periodic maintenance, there are relatively fewer papers. To the best of our knowledge, Xu et al. (2009) are the first group of researchers that consider parallel machine scheduling with periodic maintenance. They address a parallel machine scheduling problem with two machines where one of which is subject to periodic maintenance. The objective is minimizing makespan. They show that the worst-case ratio of the classical LPT algorithm is 3/2 for their problem. Liu et al. (2011) extend the scheduling model proposed in Xu et al. (2009) to the m machine case where only one machine is subject to periodic maintenance. For the online version of the problem, they give an optimal algorithm with competitive ratio 2. Sun and Li (2010) study a two-parallel-machine scenario where the machines are subject to the same periodic maintenance, i.e., they are maintained simultaneously. They propose an approximation algorithm with some worst-case analyses. He et al. (2016) study a two-parallel-machine scheduling problem where one machine is subjected to
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tool changes and the other follows periodic maintenance. They propose nine heuristic algorithms that employ two classical dispatching rules. For recent advances in machine scheduling, the readers are referred to Ma and Yuan (2014), Wang and Cheng (2015), Xu et al. (2015), Xu and Xu (2015), and Yin et al. (2014a, 2014b, 2016a, 2016b, 2016c), to name just a few. The rest of this paper is organized as follows. We provide computational complexity and non-approximability analyses in the next section. We present two mathematical programming models in Section 3. We study the worst-case ratios of the classical LPT and LS algorithms, and propose two improved heuristic algorithms in Section 4. In order to evaluate the performance of the algorithms, we focus on a lower bound of the optimal makespan in Section 5. We conduct numerical experiments and present some numerical results in Section 6. Finally, in section 7, we conclude the paper.
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Computational Complexity and Non-approximability
The classical version of the two-parallel-machine scheduling problem with the objective of minimizing makespan, known as P 2||Cmax even without the maintenance issue considered in this paper, is NP-hard (see Pinedo, 2012). Hence, our problem is obviously NP-hard. Ji et al. (2007) show that there is no polynomial time approximation algorithm with a worstcase ratio less than 2 for the single machine case unless P=NP. This motivates us to investigate the non-approximability of our problem. In fact, by simple reduction, we show that the nonapproximability also holds for our problem. Theorem 1. There is no polynomial time approximation algorithm with the worst-case ratio less than 2 for the parallel machine scheduling problem P m, mdmpr||Cmax unless P=NP. Proof. It suffices to reduce the problem P m, mdmpr||Cmax to the single machine case 1, pr||Cmax . This can be done by letting ui < pj for i = 2, 3, . . . , m and j = 1, 2, . . . , n and letting u1 ≥ pj for j = 1, 2, . . . , n. In other words, all the jobs must be scheduled on the first machine because the working intervals on the other machines are too small to fit in. This completes the proof.
Let σi = (ui + wi )/ui , σmin = min1≤i≤m {σi }, and σmax = max1≤i≤m {σi }. In what follow, we shall consider the special case where ui = u1 for i = 2, 3, . . . , m. In fact, we can show that
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there is also no polynomial time approximation algorithm with the worst-case ratio less than σmin for the parallel machine scheduling problem P m, mdmpr||Cmax unless P=NP. Theorem 2.
If ui = u1 for i = 2, 3, . . . , m, then there is no polynomial time approximation
algorithm with the worst-case ratio less than σmin for the parallel machine scheduling problem P m, mdmpr||Cmax unless P=NP. Proof. Consider the parallel machine scheduling problem P 2, mdmpr||Cmax with u1 = u2 , and an instance where ni=1 pi = 2u1 . It is easy to see that the optimal makespan of this instance is u1 if and only if we can partition the jobs into two sets such that the total processing time of the jobs in the first set equals to that of the second one. In other words, the optimal makespan of this instance is u1 if and only if the corresponding partition problem has a “YES” solution. However, as is well-known, the partition problem is NP-hard, which implies that there is no polynomial time algorithm for it unless P=NP. This means that no polynomial time algorithm can ensure a schedule with a makespan of u1 , even if the corresponding partition problem has a “YES” solution. This means that every polynomial time approximation algorithm will have a schedule with a makespan greater than min1≤i≤n {ui + wi }, even if the corresponding partition problem has a “YES” solution. This completes the proof.
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Mathematical Programming Models
In this section, we shall present two mathematical programming models, which can be used for solving small-sized instances. Both models are based on the idea of reconstructing the parallel machine structure into a single machine structure. However, they use different methodologies when solving the reconstructed single machine problem. Note that for a given machine, we may use at most n working intervals. This means that, for a given machine, we may only consider its first n working intervals or equivalently at most n maintenance activities of the machine rather than an infinite number of working intervals of the machine. This motivates us to construct a flexible single machine whose first n maintenance periodic recycles are the same as the first n maintenance periodic recycles of machine M1 and the next n maintenance periodic recycles are the same as the first n maintenance periodic recycles of machine M2 ,...,the ith set of n maintenance periodic recycles are the same as the first n 5
maintenance periodic recycles of machine Mi ,...,the mth set of n maintenance periodic recycles are the same as the first n maintenance periodic recycles of machine Mm . In other words, the constructed single machine imitates every machine according to increasing number of their index from the parallel machine structure consecutively for n maintenance periodic recycles. Clearly, this constructed single machine is non-traditional and therefore need investigation. Note that there are totally mn working intervals in the constructed single machine setting. Let xij be a binary decision variable such that xij = 1 if and only if job Ji is scheduled to be processed in the jth working interval of the constructed single machine setting where i = 1, 2, . . . , n and j = 1, 2, . . . , mn. Let yj be a binary decision variable such that yj = 1 if and only if there is at least one job in the jth working interval of the constructed single machine setting where j = 1, 2, . . . , mn. Let M be a sufficient large number. Now, we have the following mathematical programming model. Mathematical Programming Model 1 Objective: Min Cmax Constraints: mn xij = 1, j=1 n i=1 n i=1 n i=1 n
i = 1, 2, . . . , n
xij ≤ M yj , xij ≥ yj ,
(1)
j = 1, 2, . . . , mn j = 1, 2, . . . , mn
xij pi ≤ uk ,
(2) (3)
k = 1, 2, . . . , m, j = (k − 1)n + 1, (k − 1)n + 2, . . . , kn
(4)
xi,(k−1)n+j pi + (j − 1)uk + (j − 1)wk ≤ Cmax + M (1 − y(k−1)n+j )
i=1
(5) k = 1, 2, . . . , m, j = 1, 2, . . . , n
xij ∈ {0, 1}, yj ∈ {0, 1},
i = 1, 2, . . . , n, j = 1, 2, . . . , mn j = 1, 2 . . . , mn
(6) (7)
Constraint set (1) ensures that every job has to be scheduled into one of the mn working 6
intervals of the constructed single machine setting. Constraint sets (2) and (3) define variables y1 , y2 , . . . , ymn . Constraint set (4) ensures that the total processing times of the jobs that scheduled to each working interval does not violate its capacity. Constraint set (5) defines the makespan. Constraint sets (6) and (7) set the ranges of decision variables xij and yj where i = 1, 2, . . . , m and j = 1, 2, . . . , n. We now turn to the next mathematical programming model which is based on the methodology that there must be some order for any two jobs. For simplicity, let the the maintenance activities be viewed as jobs whose processing times are the corresponding maintenance durations. Specifically, let the jth maintenance activity of the constructed single machine setting be viewed as job Jn+j whose processing time is uj/m where j = 1, 2, . . . , mn. Let xij be a binary decision variable such that xij = 1 if and only if job Ji is processed before job Jj . Let si and Ci be the start time and completion time of job Ji in the constructed single machine setting where i = 1, 2, . . . , n + mn. We now have the following mathematical programming model.
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Mathematical Programming Model 2 Objective: Min
Cmax
Constraints: s i + pi = C i ,
i = 1, 2, . . . , n
skn+i + wk = Ckn+i , si ≥ 0,
(8)
k = 1, 2, . . . , m, i = 1, 2, . . . , n
i = 1, 2 . . . , n
(9) (10)
sn+1 = u1
(11)
si − Ci−1 = u1 ,
i = n + 2, n + 3, . . . , 2n
(12)
skn+i − Ckn+i−1 = uk ,
k = 2, 3, . . . , m, i = 1, 2, . . . , n
(13)
Ci ≤ sj + M (1 − xij ),
i = 1, 2, . . . , n, j = 1, 2, . . . , mn + n, i < j
(14)
Cj ≤ si + M xij , xi,mn+n = 1,
i = 1, 2, . . . , n, j = 1, 2, . . . , mn + n, i < j
i = 1, 2, . . . , n
Ci ≤ Cmax + M (1 − xi,2n ),
(16) i = 1, 2, . . . , n
Ci − Ckn ≤ Cmax + M (1 − xi,(k+1)n ), xij ∈ {0, 1},
(15)
k = 2, 3, . . . , m, i = 1, 2, . . . , n
i = 1, 2, . . . , n, j = 1, 2, . . . , mn + n, i < j
(17) (18) (19)
Constraint set (8) defines the relationship between the start time of each real job and its completion time. Constraint set (9) defines the relationship between the start time of each maintenance activity and its completion time. Constraint set (10) ensures that the start times of the jobs are non-negative, i.e., each job should be schedule no earlier than the planning horizon. Constraint (11) defines the start time of the first maintenance activity. Constraint sets (12) and (13) defines the working intervals. Constraint sets (14) and (15) defines the orders of any two jobs. Constraint set (16) ensures that all jobs must be schedule before the last maintenance of the constructed single machine setting. Constraint sets (17) and (18) define the makespan. Constraint set (19) sets the ranges of decision variable xij where i = 1, 2, . . . , m and j = 1, 2, . . . , n.
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4
Heuristic algorithms
Recall that the LPT algorithm is a well-known scheduling algorithm in machine scheduling. It first re-orders all the jobs in decreasing order of their processing times in a job sequence and then pick a job each time from the head of the sequence to the end of the sequence. When a job is picked, it is scheduled to an working interval of some machine where it can be finished as soon as possible. In this section, we shall first take a look at the performance of the classical LPT and LS algorithms from the theoretical point of view. And then we propose some heuristic algorithms based on the theoretical analysis and some observations. Recall that the LS algorithm differs from the LPT algorithm in that the former does not need to re-order jobs, i.e., the jobs can be ordered arbitrarily.
4.1
Performance of LPT and LS algorithms
Assume that ki batches are used on the ith machine of the LPT schedule. Recall that if the instance size is small, we may resort to the two mathematical programming models. Hence, in what follow, we assume that ki ≥ 2 for i = 1, 2, . . . , m. Moreover, without loss of generality, we assume that the makespan is obtained on the first machine. It is easy to see that the makespan of the LPT schedule CLPT ≤ k1 (u1 + w1 )
(20)
ki (ui + wi ) ≥ (k1 − 1)(u1 + w1 ), i = 2, 3, . . . , m.
(21)
and that
By simple algebra, we have ki ≥ (k1 − 1)
u1 + w 1 , i = 2, 3, . . . , m. ui + w i
(22)
Let Λi be the total processing time of the jobs that are scheduled on the ith machine of the LPT schedule. Recall that ki ≥ 2 for i = 1, 2, . . . , m. So, according to the LPT algorithm, we have Λi ≥
ki ui , i = 1, 2, . . . , m. 2
Particularly, we have
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(23)
k1 u1 . 2
(24)
(k1 − 1)(u1 + w1 ) ui , i = 2, 3, . . . , m. 2 ui + w i
(25)
Λ1 ≥ Substituting (22) into (23), we have Λi ≥ Let A=
m i=1
and uk = min uk + w k
ui ui + w i
(26)
u1 u2 um , ,..., u1 + w 1 u2 + w 2 um + w k
.
(27)
Clearly, we have uk uk + w k
(28)
uk (u1 + w1 ). uk + w k
(29)
A≥m and u1 ≥
Let Λ be the total processing time of all the jobs, i.e., Λ = Λ=
m
n
i=1 pi .
It is easy to see that
Λi .
(30)
i=1
Substituting (24) and (23) into (30), we have m
k1 (k1 − 1)(u1 + w1 ) ui Λ ≥ u1 + . 2 2 ui + w i
(31)
(k1 − 1)(u1 + w1 ) 1 Λ ≥ u1 + A. 2 2
(32)
i=2
That is
Substituting (28) into (32), we have m(k1 − 1)(u1 + w1 ) uk 1 . Λ ≥ u1 + 2 2 uk + w k
(33)
Note that Λ . m
(34)
1 (k1 − 1)(u1 + w1 ) uk . u1 + 2m 2 uk + w k
(35)
COPT ≥ Substituting (33) into (34), we have COPT ≥
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Substituting (29) and (26) into (35), we have COPT ≥
u1 + w 1 uk (k1 − 1)(u1 + w1 ) uk + . 2m uk + wk 2 uk + w k
That is
COPT ≥ (u1 + w1 )
1 k1 − 1 + 2m 2
uk . uk + w k
(36)
(37)
Combining (20) and (37), we have k1 (u1 + w1 ) CLPT ≤ . 1 COPT k (u1 + w1 ) 2m + k12−1 uku+w k
(38)
CLPT 2mk1 uk + w k ≤ . COPT 1 + m(k1 − 1) uk
(39)
CLPT 4m uk + wk . ≤ COPT 2m + 1 uk
(40)
That is
Note that k1 ≥ 2. So we have
Recall that
ui +wi ui
= σi and that σmax = max1≤i≤m {σi }. Hence, we have the following
theorem. Theorem 3. If at least two batches are used on each machine, then the worst-case ratio of the LPT algorithm for the parallel machine scheduling problem P m, mdmpr||Cmax is no more than 4mσk /(2m + 1). By similar analysis, we have the following theorem. Theorem 4. If at least two batches are used on each machine, then the worst-case ratio of the LS algorithm for the parallel machine scheduling problem P m, mdmpr||Cmax is no more than 4mσmax /(2m + 1). Note that in some real production environment the ratio wi /ui is usually less than or equal to 1/7 for that the machine is usually maintained once every week or once every month. This means that the worst-case ratios of the two algorithm are all less than 16/7 < 2.29 in some cases. In what follows, we shall look into the LPT and LS schedules and propose some improved heuristic algorithms.
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4.2
LPT and LS based Heuristic algorithms
The underlying idea of our improved heuristic algorithms is based on the following observation: if we can interchange a job from the batch which reaches the makespan of the schedule with a smaller job (a job with a smaller processing time) from another batch, then we get a new schedule with a smaller makespan or with the same makespan (this may happen when there are two or more machines with the same completion time which coincides with the makespan before the interchange). If we always choose to interchange jobs when we can get a smaller makespan by the above procedure, then we can repeatedly improve the schedule. This adjustment stops when there are no interchangeable jobs. For convenience, we call this heuristic algorithm MLPT if LPT algorithm and the above adjustment procedure is applied. Similarly, we have heuristic algorithm MLS if LS algorithm and the above adjustment procedure is applied.
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Lower bound
In order to evaluate the performance of these two heuristic algorithms, we shall employ a lower bound of the optimal makespan. Note that there may be cases where all machines are busy with processing jobs or under maintenance without idle times before the makespan. In order to get a lower bound, we may assume that all jobs are preemptive and that each job can be handled by more than one machine at the same time. With these assumptions, there must be a time before which all machines are busy with processing jobs or under maintenance without idle times. Clearly, such a time is a lower bound of the optimal makespan. In what follows, we call this lower bound ideal lower bound and we shall focus on the determination of this lower bound. Before going on, we shall look into some property of the ideal lower bound. Property 1. The ideal lower bound is some integer or some integer plus an irreducible fraction with a denominator not greater than m. Proof.
For the convenience of description, suppose that the ideal lower bound is l. If l is an
integer, then we are done. So, in what follows, we assume that l is not an integer.
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Without loss of generality, we assume that the completion times of the first m machines coincide with the ideal lower bound. By the definition of the ideal lower bound, we know that the other m − m machines are under maintenance at that time. Now, let take a look at the total processing time of the job that are scheduled to be processed by the last m − m machines. Clearly, each of these machines performs an integer number of maintenance periodic recycles. Note that all machines are busy with processing jobs or under maintenance without idle times before the time of the ideal lower bound. So the total processing time of the job that are scheduled to be processed by the last m − m machines is an integer. Therefore, the total processing time of the job that are scheduled to be processed by the first m machines are also an integer. Hence, in what follows, we only need to consider the completion time of the first m machines. Note that the total processing time of the jobs scheduled to be processed by the first m machines before time l is an integer. So, the total processing time of the jobs scheduled to be processed in time interval (l, l] is also an integer. Note that the first m machines are busy processing jobs in (l, l]. Hence, (l − l) ∗ m is an integer. This implies that l must be some integer plus an irreducible fraction with a denominator not greater than m.
This completes the proof.
This property motivates us to develop an enumeration based algorithm to get the ideal lower bound. Specifically, for a given integer k, we may split all the jobs into kΛ subjobs which have the same processing time of 1/k. Then, scan the machines in increasing order of their indexes every 1/k time unit, schedule a subjob onto the machine in that time interval if it is not under maintenance. This procedure ends if there are no unscheduled subjobs. If the procedure ends when it reaches the last available machine of that time interval, then we get the ideal lower bound. Otherwise, try the other integer value of k not greater than m.
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Numerical Experiments
We shall conduct numerical experiments to evaluate the performance of the four heuristic algorithms, i.e., LS algorithm, MLS algorithm, LPT algorithm, and MLPT algorithm. The experiment is divided into three parts according to the number of machines, i.e., m = 3, 13
m = 5, and m = 10. For each part, ten problem sizes, namely n = 20, 40, 60, 80, 100, 200, 500, 1000, 5000, and 10000 are considered. The length ui of the working interval on machine Mi was generated using discrete uniform distribution over [1,20]. For a fixed ui , four values of maintenance duration wi are considered, i.e., wi is generated randomly using discrete uniform distribution over [1,5], wi = ui /4, wi = ui /2, and wi = 3ui /4. The processing times of the jobs are assumed to be integers and were generated using discrete uniform distribution over [1, mini=1,2,...,m {ui }]. The proposed heuristic algorithms were coded in Matlab. For each problem size, one hundred instances were randomly generated and computed for each combination of m, n, ui , and wi . All the instances were solved by Matlab on an Intel(R) Core(TM)2 CPU SP9400 @ 2.4GHz with 2GB RAM on Windows XP. The average relative error ratios with respect to the lower bound presented in the previous section of the four algorithms for each combination are reported in Tables 1-3. The average relative error ratios on average for each of the four heuristic algorithms when m = 3, m = 5, and m = 10 are presented in Fig. 1.
Insert Tables 1-3 About Here.
Insert Figure 1 About Here. From Tables 1-3, one may find that (i) the LPT algorithm beats the LS algorithm for all combinations; (ii) the MLPT algorithm beats the MLS algorithm for almost all combinations; (iii) the average relative error ratios decrease in general as the number of jobs increases; (iv) the maximum average relative error ratio of the MLS is 12.93% for all combinations; (v) the average relative error ratios of the MLS are less than 6.23% for all combinations; (vi) the maximum average relative error ratio of the MLPT is less than 7.44% for all combinations; (vii) the average relative error ratios of the MLPT are less than 1.57% for all combinations. From Fig. 1, one may find that the MLPT algorithm performs the best while the LS algorithm performs the worst among the four heuristic algorithms from the average relative error ratio on average point of view. 14
One can also find that by simple computation that, compared with the LS algorithm, the MLS algorithm improves the objective value by about 21.85% on average. And that is about 6.75% for the MLPT algorithm. This indicates that our adjustment procedure is promising.
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Conclusion
In this paper, we have considered an important parallel machine scheduling problem with machinedependent maintenance periodic recycles. This is one of the fundamental problems in the field of scheduling. We provide the non-approximability analysis and two mathematical programming models for the problem. We also give the worst-case analysis of the classical LPT and LS algorithms for the problem. Based on an adjustment procedure, we propose two improved heuristic algorithms: MLS and MLPT. Numerical experiments show that the improved heuristic algorithm MLPT is promising and is very suitable for real world applications. Note that there is still a gap between the non-approximablity bound max{2, σmax } and the worst-case ratio upper bound 4mσmax /(2m + 1). So, further research may focus on reducing this gap by designing more sophisticated approximation algorithms. It is also worth considering other objectives such as minimizing maximum lateness or minimizing total number of late jobs if there is a due date for each job.
Acknowledgements This work was supported by the National Natural Science Foundation of China (grant nos.71372019, 71471057, 71102174), Beijing Higher Education Young Elite Teacher Project (grant no. YETP1173), and Beijing Philosophy and Social Science Foundation of China (grant no. 11JGC106).
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[11] Ma, R., & Yuan, J. (2014). Online tradeoff scheduling on a single machine to minimize makespan and total weighted completion time. International Journal of Production Economics, 158, 114-119. [12] Pinedo, M. L. (2012). Scheduling theory: algorithms, and systems(4th ed.). Springer-Verlag, New York. [13] Sbihi, M., & Varnier, C. (2008). Single-machine scheduling with periodic and flexible periodic maintenance to minimize maximum tardiness. Computers & Industrial Engineering, 55(4), 830-840. [14] Sun, K., & Li, H. (2010). Scheduling problems with multiple maintenance activities and non-preemptive jobs on two identical parallel machines. International Journal of Production Economics, 124(1), 151-158. [15] Wang, X., & Cheng, T. C. E. (2015). A heuristic for scheduling jobs on two identical parallel machines with a machine availability constraint. International Journal of Production Economics, 161, 74-82. [16] Xu, D., Cheng, Z., Yin, Y., & Li, H. (2009). Makespan minimization for two parallel machines scheduling with a periodic availability constraint. Computers & Operations Research, 36(6), 1809-1812. [17] Xu, D., Wan, L., Liu, A., & Yang, D. L. (2015). Single machine total completion time scheduling problem with workload-dependent maintenance duration. Omega, 52, 101-106. [18] Xu, Z., & Xu, D. (2015). Single-machine scheduling with preemptive jobs and workloaddependent maintenance durations. Operational Research - An International Journal, 15, 423436. [19] Yin, Y., Wu, W. H., Cheng, T. C. E., & Wu, C. C. (2014a). Due-date assignment and singlemachine scheduling with generalised position-dependent deteriorating jobs and deteriorating multi-maintenance activities. International Journal of Production Research, 52(8), 23112326.
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[20] Yin Y., Cheng, T. C. E., & Wang, D. J. (2016a). Rescheduling on identical parallel machines with machine disruptions to minimize total completion time. European Journal of Operational Research, 252, 737-749. [21] Yin, Y., Xu, J., Cheng, T. C. E., Wu, C. C., & Wang, D. J. (2016b). Approximation schemes for single-machine scheduling with a fixed maintenance activity to minimize the total amount of late work. Naval Research Logistics, 63(2), 172-183. [22] Yin Y., Wang, Y., Cheng, T. C. E., Liu, W., & Li, J. (2016c) Parallelmachine scheduling of deteriorating jobs with potential machine disruptions. Omega, http://dx.doi.org/10.1016/j.omega.2016.07.006. [23] Yin, Y., Ye, D., & Zhang, G. (2014b). Single machine batch scheduling to minimize the sum of total flow time and batch delivery cost with an unavailability interval. Information Sciences, 274, 310-322.
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Average relative error ratios on average
0.14 LS MLS LPT MLPT
0.12
0.1
0.08
0.06
0.04
0.02
0 3
5
10
Number of machines Fig. 1 Average relative error ratios on average.
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PII: DOI: Reference:
S0925-5273(17)30014-2 http://dx.doi.org/10.1016/j.ijpe.2017.01.014 PROECO6642
To appear in: Intern. Journal of Production Economics Received date: 1 November 2016 Revised date: 27 December 2016 Accepted date: 20 January 2017 Cite this article as: Guo Li, Mengqi Liu, Suresh P. Sethi and Dehua Xu, Parallelmachine scheduling with machine-dependent maintenance periodic recycles, Intern. Journal of Production Economics, http://dx.doi.org/10.1016/j.ijpe.2017.01.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Parallel-machine scheduling with machine-dependent maintenance periodic recycles Guo Li a , Mengqi Liu b,∗ , Suresh P. Sethi c , Dehua Xu d a
School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China b
c
Business School, Hunan University, Changsha, Hunan 410082, China
Naveen Jindal School of Management, The University of Texas at Dallas, Dallas, TX 75080, United States d
School of Science, East China University of Technology, Nanchang, Jiangxi 330013, China (*Corresponding E-mail addresses: [email protected])
Abstract Parallel machine structure is very common in modern production systems. Its performance sometimes has a decisive impact on the whole productivity. In this paper, we consider a parallel-machine scheduling problem where each machine is subject to periodic maintenance. Instead of assuming all the machines have the same maintenance periodic cycle, we assume the maintenance periodic cycles are machine-dependent. The objective is to schedule all the jobs to the machines such that the makespan is minimized. We first provide computational complexity and non-approximability analyses of the problem and then present two mathematical programming models to tackle small-sized instances. Thereafter, we present a worst-case analysis of the classical LPT and LS algorithms for the problem. Then we propose two improved heuristic algorithms based on some observations of large-sized instances. In order to evaluate the performance of the heuristic algorithms, we resort to a lower bound of the optimal makespan, and find that it has a very interesting characteristic. Numerical experiments show that the improved heuristic algorithm MLPT improves the objective value of the LPT schedule by about 6.75% on average and that the MLPT heuristic algorithm has an average-case relative error less than 1.57% for all combinations of instances, which means that it is very suitable for real world applications. Keywords: Scheduling; Maintenance; Periodic recycle; Makespan; Heuristic algorithm
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1
Introduction
Parallel machine structure is very common in modern production systems. It is a natural extension of the single-machine structure and also the basic unit of the flexible job shop structure. Due to its importance, a lot of work has been done on the study of various parallel machine scenarios. To the best of our knowledge, Graham (1966) is among the first set of researchers who consider the optimal operations on parallel machine environment. He studies a problem where a number of jobs need to be scheduled onto some machines such that the makespan is minimized. And the proposed Longest Processing Time (LPT) first rule is now a well-known dispatching rule not only to the scheduling community but also to the production managers. Much of the past research on machine scheduling assumes that machines are available all the time. However, as machines are maintained according to their user manuals in order to function properly, there is a need to develop methodologies to handle this issue. To address this, we consider a parallel-machine scheduling problem with periodic maintenance where the maintenance periodic recycles may be different for each machine. We provide computational complexity and non-approximability analyses, and two mathematical programming models for the problem. We obtain worst-case ratios of the classical LPT and LS algorithms for the problem and propose two improved heuristic algorithms based on some observations of largesized instances. We also conduct numerical experiments to evaluate their performance. The results obtained may help production managers improve the productivity of their machine shops. Formally, our scheduling problem can be described as follows. We have to schedule n independent non-preemptive jobs J1 , J2 , . . . , Jn to m parallel machines M1 , M2 , . . ., Mn , where the processing time of job Ji is pi . The maintenance periodic recycle of machine Mi is ui + wi where ui is the length of the time between two consecutive maintenance activities on machine Mi and wi is the length of the time that is needed to perform one maintenance activity on the same machine. Let the interval between two consecutive maintenance activities on the same machine be called a working interval. Note that a machine cannot process any jobs during its maintenance. Therefore, jobs can only be schedule into these working intervals. Assume that all machines have just finished their maintenance at time zero. The objective is to schedule all the jobs to the machines such that the makespan, i.e, the completion time of the last finished job, is minimized.
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Extending the well-known three-field α|β|γ classification scheme suggested by Graham et al. (1979), we denote this problem by P m, mdmpr||Cmax , where mdmpr stands for machinedepended maintenance periodic recycle. Without loss of generality, in what follows, we assume that max{pi } ≤ max{ui }, otherwise there is clearly no feasible schedule for our problem. Moreover, we assume that jobs that scheduled in the same working interval are processed consecutively one after another from the start time of the interval. In what follows, we shall briefly review some works on machine scheduling with periodic maintenance. The single machine scenario has been investigated extensively by scholars from different perspectives. To the best of our knowledge, Liao and Chen (2003) are the first group of researchers that take periodic maintenance into consideration in the scheduling community. They study a single machine scheduling problem with the objective of minimizing maximum tardiness. Chen (2007a) and Sbihi and Varnier (2008) study the same scheduling problem as Liao and Chen (2003). They independently propose a heuristic algorithm and a branch-and-bound algorithm. Following similar approaches as those in Liao and Chen (2003), Chen (2006, 2007b, 2009) studies three single machine scheduling problems with the following objectives, i.e., minimizing total flow time, minimizing the sum of weighted total flow time and weighted maximum tardiness, and minimizing the number of tardy jobs, respectively. For parallel machine scheduling with periodic maintenance, there are relatively fewer papers. To the best of our knowledge, Xu et al. (2009) are the first group of researchers that consider parallel machine scheduling with periodic maintenance. They address a parallel machine scheduling problem with two machines where one of which is subject to periodic maintenance. The objective is minimizing makespan. They show that the worst-case ratio of the classical LPT algorithm is 3/2 for their problem. Liu et al. (2011) extend the scheduling model proposed in Xu et al. (2009) to the m machine case where only one machine is subject to periodic maintenance. For the online version of the problem, they give an optimal algorithm with competitive ratio 2. Sun and Li (2010) study a two-parallel-machine scenario where the machines are subject to the same periodic maintenance, i.e., they are maintained simultaneously. They propose an approximation algorithm with some worst-case analyses. He et al. (2016) study a two-parallel-machine scheduling problem where one machine is subjected to
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tool changes and the other follows periodic maintenance. They propose nine heuristic algorithms that employ two classical dispatching rules. For recent advances in machine scheduling, the readers are referred to Ma and Yuan (2014), Wang and Cheng (2015), Xu et al. (2015), Xu and Xu (2015), and Yin et al. (2014a, 2014b, 2016a, 2016b, 2016c), to name just a few. The rest of this paper is organized as follows. We provide computational complexity and non-approximability analyses in the next section. We present two mathematical programming models in Section 3. We study the worst-case ratios of the classical LPT and LS algorithms, and propose two improved heuristic algorithms in Section 4. In order to evaluate the performance of the algorithms, we focus on a lower bound of the optimal makespan in Section 5. We conduct numerical experiments and present some numerical results in Section 6. Finally, in section 7, we conclude the paper.
2
Computational Complexity and Non-approximability
The classical version of the two-parallel-machine scheduling problem with the objective of minimizing makespan, known as P 2||Cmax even without the maintenance issue considered in this paper, is NP-hard (see Pinedo, 2012). Hence, our problem is obviously NP-hard. Ji et al. (2007) show that there is no polynomial time approximation algorithm with a worstcase ratio less than 2 for the single machine case unless P=NP. This motivates us to investigate the non-approximability of our problem. In fact, by simple reduction, we show that the nonapproximability also holds for our problem. Theorem 1. There is no polynomial time approximation algorithm with the worst-case ratio less than 2 for the parallel machine scheduling problem P m, mdmpr||Cmax unless P=NP. Proof. It suffices to reduce the problem P m, mdmpr||Cmax to the single machine case 1, pr||Cmax . This can be done by letting ui < pj for i = 2, 3, . . . , m and j = 1, 2, . . . , n and letting u1 ≥ pj for j = 1, 2, . . . , n. In other words, all the jobs must be scheduled on the first machine because the working intervals on the other machines are too small to fit in. This completes the proof.
Let σi = (ui + wi )/ui , σmin = min1≤i≤m {σi }, and σmax = max1≤i≤m {σi }. In what follow, we shall consider the special case where ui = u1 for i = 2, 3, . . . , m. In fact, we can show that
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there is also no polynomial time approximation algorithm with the worst-case ratio less than σmin for the parallel machine scheduling problem P m, mdmpr||Cmax unless P=NP. Theorem 2.
If ui = u1 for i = 2, 3, . . . , m, then there is no polynomial time approximation
algorithm with the worst-case ratio less than σmin for the parallel machine scheduling problem P m, mdmpr||Cmax unless P=NP. Proof. Consider the parallel machine scheduling problem P 2, mdmpr||Cmax with u1 = u2 , and an instance where ni=1 pi = 2u1 . It is easy to see that the optimal makespan of this instance is u1 if and only if we can partition the jobs into two sets such that the total processing time of the jobs in the first set equals to that of the second one. In other words, the optimal makespan of this instance is u1 if and only if the corresponding partition problem has a “YES” solution. However, as is well-known, the partition problem is NP-hard, which implies that there is no polynomial time algorithm for it unless P=NP. This means that no polynomial time algorithm can ensure a schedule with a makespan of u1 , even if the corresponding partition problem has a “YES” solution. This means that every polynomial time approximation algorithm will have a schedule with a makespan greater than min1≤i≤n {ui + wi }, even if the corresponding partition problem has a “YES” solution. This completes the proof.
3
Mathematical Programming Models
In this section, we shall present two mathematical programming models, which can be used for solving small-sized instances. Both models are based on the idea of reconstructing the parallel machine structure into a single machine structure. However, they use different methodologies when solving the reconstructed single machine problem. Note that for a given machine, we may use at most n working intervals. This means that, for a given machine, we may only consider its first n working intervals or equivalently at most n maintenance activities of the machine rather than an infinite number of working intervals of the machine. This motivates us to construct a flexible single machine whose first n maintenance periodic recycles are the same as the first n maintenance periodic recycles of machine M1 and the next n maintenance periodic recycles are the same as the first n maintenance periodic recycles of machine M2 ,...,the ith set of n maintenance periodic recycles are the same as the first n 5
maintenance periodic recycles of machine Mi ,...,the mth set of n maintenance periodic recycles are the same as the first n maintenance periodic recycles of machine Mm . In other words, the constructed single machine imitates every machine according to increasing number of their index from the parallel machine structure consecutively for n maintenance periodic recycles. Clearly, this constructed single machine is non-traditional and therefore need investigation. Note that there are totally mn working intervals in the constructed single machine setting. Let xij be a binary decision variable such that xij = 1 if and only if job Ji is scheduled to be processed in the jth working interval of the constructed single machine setting where i = 1, 2, . . . , n and j = 1, 2, . . . , mn. Let yj be a binary decision variable such that yj = 1 if and only if there is at least one job in the jth working interval of the constructed single machine setting where j = 1, 2, . . . , mn. Let M be a sufficient large number. Now, we have the following mathematical programming model. Mathematical Programming Model 1 Objective: Min Cmax Constraints: mn xij = 1, j=1 n i=1 n i=1 n i=1 n
i = 1, 2, . . . , n
xij ≤ M yj , xij ≥ yj ,
(1)
j = 1, 2, . . . , mn j = 1, 2, . . . , mn
xij pi ≤ uk ,
(2) (3)
k = 1, 2, . . . , m, j = (k − 1)n + 1, (k − 1)n + 2, . . . , kn
(4)
xi,(k−1)n+j pi + (j − 1)uk + (j − 1)wk ≤ Cmax + M (1 − y(k−1)n+j )
i=1
(5) k = 1, 2, . . . , m, j = 1, 2, . . . , n
xij ∈ {0, 1}, yj ∈ {0, 1},
i = 1, 2, . . . , n, j = 1, 2, . . . , mn j = 1, 2 . . . , mn
(6) (7)
Constraint set (1) ensures that every job has to be scheduled into one of the mn working 6
intervals of the constructed single machine setting. Constraint sets (2) and (3) define variables y1 , y2 , . . . , ymn . Constraint set (4) ensures that the total processing times of the jobs that scheduled to each working interval does not violate its capacity. Constraint set (5) defines the makespan. Constraint sets (6) and (7) set the ranges of decision variables xij and yj where i = 1, 2, . . . , m and j = 1, 2, . . . , n. We now turn to the next mathematical programming model which is based on the methodology that there must be some order for any two jobs. For simplicity, let the the maintenance activities be viewed as jobs whose processing times are the corresponding maintenance durations. Specifically, let the jth maintenance activity of the constructed single machine setting be viewed as job Jn+j whose processing time is uj/m where j = 1, 2, . . . , mn. Let xij be a binary decision variable such that xij = 1 if and only if job Ji is processed before job Jj . Let si and Ci be the start time and completion time of job Ji in the constructed single machine setting where i = 1, 2, . . . , n + mn. We now have the following mathematical programming model.
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Mathematical Programming Model 2 Objective: Min
Cmax
Constraints: s i + pi = C i ,
i = 1, 2, . . . , n
skn+i + wk = Ckn+i , si ≥ 0,
(8)
k = 1, 2, . . . , m, i = 1, 2, . . . , n
i = 1, 2 . . . , n
(9) (10)
sn+1 = u1
(11)
si − Ci−1 = u1 ,
i = n + 2, n + 3, . . . , 2n
(12)
skn+i − Ckn+i−1 = uk ,
k = 2, 3, . . . , m, i = 1, 2, . . . , n
(13)
Ci ≤ sj + M (1 − xij ),
i = 1, 2, . . . , n, j = 1, 2, . . . , mn + n, i < j
(14)
Cj ≤ si + M xij , xi,mn+n = 1,
i = 1, 2, . . . , n, j = 1, 2, . . . , mn + n, i < j
i = 1, 2, . . . , n
Ci ≤ Cmax + M (1 − xi,2n ),
(16) i = 1, 2, . . . , n
Ci − Ckn ≤ Cmax + M (1 − xi,(k+1)n ), xij ∈ {0, 1},
(15)
k = 2, 3, . . . , m, i = 1, 2, . . . , n
i = 1, 2, . . . , n, j = 1, 2, . . . , mn + n, i < j
(17) (18) (19)
Constraint set (8) defines the relationship between the start time of each real job and its completion time. Constraint set (9) defines the relationship between the start time of each maintenance activity and its completion time. Constraint set (10) ensures that the start times of the jobs are non-negative, i.e., each job should be schedule no earlier than the planning horizon. Constraint (11) defines the start time of the first maintenance activity. Constraint sets (12) and (13) defines the working intervals. Constraint sets (14) and (15) defines the orders of any two jobs. Constraint set (16) ensures that all jobs must be schedule before the last maintenance of the constructed single machine setting. Constraint sets (17) and (18) define the makespan. Constraint set (19) sets the ranges of decision variable xij where i = 1, 2, . . . , m and j = 1, 2, . . . , n.
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4
Heuristic algorithms
Recall that the LPT algorithm is a well-known scheduling algorithm in machine scheduling. It first re-orders all the jobs in decreasing order of their processing times in a job sequence and then pick a job each time from the head of the sequence to the end of the sequence. When a job is picked, it is scheduled to an working interval of some machine where it can be finished as soon as possible. In this section, we shall first take a look at the performance of the classical LPT and LS algorithms from the theoretical point of view. And then we propose some heuristic algorithms based on the theoretical analysis and some observations. Recall that the LS algorithm differs from the LPT algorithm in that the former does not need to re-order jobs, i.e., the jobs can be ordered arbitrarily.
4.1
Performance of LPT and LS algorithms
Assume that ki batches are used on the ith machine of the LPT schedule. Recall that if the instance size is small, we may resort to the two mathematical programming models. Hence, in what follow, we assume that ki ≥ 2 for i = 1, 2, . . . , m. Moreover, without loss of generality, we assume that the makespan is obtained on the first machine. It is easy to see that the makespan of the LPT schedule CLPT ≤ k1 (u1 + w1 )
(20)
ki (ui + wi ) ≥ (k1 − 1)(u1 + w1 ), i = 2, 3, . . . , m.
(21)
and that
By simple algebra, we have ki ≥ (k1 − 1)
u1 + w 1 , i = 2, 3, . . . , m. ui + w i
(22)
Let Λi be the total processing time of the jobs that are scheduled on the ith machine of the LPT schedule. Recall that ki ≥ 2 for i = 1, 2, . . . , m. So, according to the LPT algorithm, we have Λi ≥
ki ui , i = 1, 2, . . . , m. 2
Particularly, we have
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(23)
k1 u1 . 2
(24)
(k1 − 1)(u1 + w1 ) ui , i = 2, 3, . . . , m. 2 ui + w i
(25)
Λ1 ≥ Substituting (22) into (23), we have Λi ≥ Let A=
m i=1
and uk = min uk + w k
ui ui + w i
(26)
u1 u2 um , ,..., u1 + w 1 u2 + w 2 um + w k
.
(27)
Clearly, we have uk uk + w k
(28)
uk (u1 + w1 ). uk + w k
(29)
A≥m and u1 ≥
Let Λ be the total processing time of all the jobs, i.e., Λ = Λ=
m
n
i=1 pi .
It is easy to see that
Λi .
(30)
i=1
Substituting (24) and (23) into (30), we have m
k1 (k1 − 1)(u1 + w1 ) ui Λ ≥ u1 + . 2 2 ui + w i
(31)
(k1 − 1)(u1 + w1 ) 1 Λ ≥ u1 + A. 2 2
(32)
i=2
That is
Substituting (28) into (32), we have m(k1 − 1)(u1 + w1 ) uk 1 . Λ ≥ u1 + 2 2 uk + w k
(33)
Note that Λ . m
(34)
1 (k1 − 1)(u1 + w1 ) uk . u1 + 2m 2 uk + w k
(35)
COPT ≥ Substituting (33) into (34), we have COPT ≥
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Substituting (29) and (26) into (35), we have COPT ≥
u1 + w 1 uk (k1 − 1)(u1 + w1 ) uk + . 2m uk + wk 2 uk + w k
That is
COPT ≥ (u1 + w1 )
1 k1 − 1 + 2m 2
uk . uk + w k
(36)
(37)
Combining (20) and (37), we have k1 (u1 + w1 ) CLPT ≤ . 1 COPT k (u1 + w1 ) 2m + k12−1 uku+w k
(38)
CLPT 2mk1 uk + w k ≤ . COPT 1 + m(k1 − 1) uk
(39)
CLPT 4m uk + wk . ≤ COPT 2m + 1 uk
(40)
That is
Note that k1 ≥ 2. So we have
Recall that
ui +wi ui
= σi and that σmax = max1≤i≤m {σi }. Hence, we have the following
theorem. Theorem 3. If at least two batches are used on each machine, then the worst-case ratio of the LPT algorithm for the parallel machine scheduling problem P m, mdmpr||Cmax is no more than 4mσk /(2m + 1). By similar analysis, we have the following theorem. Theorem 4. If at least two batches are used on each machine, then the worst-case ratio of the LS algorithm for the parallel machine scheduling problem P m, mdmpr||Cmax is no more than 4mσmax /(2m + 1). Note that in some real production environment the ratio wi /ui is usually less than or equal to 1/7 for that the machine is usually maintained once every week or once every month. This means that the worst-case ratios of the two algorithm are all less than 16/7 < 2.29 in some cases. In what follows, we shall look into the LPT and LS schedules and propose some improved heuristic algorithms.
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4.2
LPT and LS based Heuristic algorithms
The underlying idea of our improved heuristic algorithms is based on the following observation: if we can interchange a job from the batch which reaches the makespan of the schedule with a smaller job (a job with a smaller processing time) from another batch, then we get a new schedule with a smaller makespan or with the same makespan (this may happen when there are two or more machines with the same completion time which coincides with the makespan before the interchange). If we always choose to interchange jobs when we can get a smaller makespan by the above procedure, then we can repeatedly improve the schedule. This adjustment stops when there are no interchangeable jobs. For convenience, we call this heuristic algorithm MLPT if LPT algorithm and the above adjustment procedure is applied. Similarly, we have heuristic algorithm MLS if LS algorithm and the above adjustment procedure is applied.
5
Lower bound
In order to evaluate the performance of these two heuristic algorithms, we shall employ a lower bound of the optimal makespan. Note that there may be cases where all machines are busy with processing jobs or under maintenance without idle times before the makespan. In order to get a lower bound, we may assume that all jobs are preemptive and that each job can be handled by more than one machine at the same time. With these assumptions, there must be a time before which all machines are busy with processing jobs or under maintenance without idle times. Clearly, such a time is a lower bound of the optimal makespan. In what follows, we call this lower bound ideal lower bound and we shall focus on the determination of this lower bound. Before going on, we shall look into some property of the ideal lower bound. Property 1. The ideal lower bound is some integer or some integer plus an irreducible fraction with a denominator not greater than m. Proof.
For the convenience of description, suppose that the ideal lower bound is l. If l is an
integer, then we are done. So, in what follows, we assume that l is not an integer.
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Without loss of generality, we assume that the completion times of the first m machines coincide with the ideal lower bound. By the definition of the ideal lower bound, we know that the other m − m machines are under maintenance at that time. Now, let take a look at the total processing time of the job that are scheduled to be processed by the last m − m machines. Clearly, each of these machines performs an integer number of maintenance periodic recycles. Note that all machines are busy with processing jobs or under maintenance without idle times before the time of the ideal lower bound. So the total processing time of the job that are scheduled to be processed by the last m − m machines is an integer. Therefore, the total processing time of the job that are scheduled to be processed by the first m machines are also an integer. Hence, in what follows, we only need to consider the completion time of the first m machines. Note that the total processing time of the jobs scheduled to be processed by the first m machines before time l is an integer. So, the total processing time of the jobs scheduled to be processed in time interval (l, l] is also an integer. Note that the first m machines are busy processing jobs in (l, l]. Hence, (l − l) ∗ m is an integer. This implies that l must be some integer plus an irreducible fraction with a denominator not greater than m.
This completes the proof.
This property motivates us to develop an enumeration based algorithm to get the ideal lower bound. Specifically, for a given integer k, we may split all the jobs into kΛ subjobs which have the same processing time of 1/k. Then, scan the machines in increasing order of their indexes every 1/k time unit, schedule a subjob onto the machine in that time interval if it is not under maintenance. This procedure ends if there are no unscheduled subjobs. If the procedure ends when it reaches the last available machine of that time interval, then we get the ideal lower bound. Otherwise, try the other integer value of k not greater than m.
6
Numerical Experiments
We shall conduct numerical experiments to evaluate the performance of the four heuristic algorithms, i.e., LS algorithm, MLS algorithm, LPT algorithm, and MLPT algorithm. The experiment is divided into three parts according to the number of machines, i.e., m = 3, 13
m = 5, and m = 10. For each part, ten problem sizes, namely n = 20, 40, 60, 80, 100, 200, 500, 1000, 5000, and 10000 are considered. The length ui of the working interval on machine Mi was generated using discrete uniform distribution over [1,20]. For a fixed ui , four values of maintenance duration wi are considered, i.e., wi is generated randomly using discrete uniform distribution over [1,5], wi = ui /4, wi = ui /2, and wi = 3ui /4. The processing times of the jobs are assumed to be integers and were generated using discrete uniform distribution over [1, mini=1,2,...,m {ui }]. The proposed heuristic algorithms were coded in Matlab. For each problem size, one hundred instances were randomly generated and computed for each combination of m, n, ui , and wi . All the instances were solved by Matlab on an Intel(R) Core(TM)2 CPU SP9400 @ 2.4GHz with 2GB RAM on Windows XP. The average relative error ratios with respect to the lower bound presented in the previous section of the four algorithms for each combination are reported in Tables 1-3. The average relative error ratios on average for each of the four heuristic algorithms when m = 3, m = 5, and m = 10 are presented in Fig. 1.
Insert Tables 1-3 About Here.
Insert Figure 1 About Here. From Tables 1-3, one may find that (i) the LPT algorithm beats the LS algorithm for all combinations; (ii) the MLPT algorithm beats the MLS algorithm for almost all combinations; (iii) the average relative error ratios decrease in general as the number of jobs increases; (iv) the maximum average relative error ratio of the MLS is 12.93% for all combinations; (v) the average relative error ratios of the MLS are less than 6.23% for all combinations; (vi) the maximum average relative error ratio of the MLPT is less than 7.44% for all combinations; (vii) the average relative error ratios of the MLPT are less than 1.57% for all combinations. From Fig. 1, one may find that the MLPT algorithm performs the best while the LS algorithm performs the worst among the four heuristic algorithms from the average relative error ratio on average point of view. 14
One can also find that by simple computation that, compared with the LS algorithm, the MLS algorithm improves the objective value by about 21.85% on average. And that is about 6.75% for the MLPT algorithm. This indicates that our adjustment procedure is promising.
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Conclusion
In this paper, we have considered an important parallel machine scheduling problem with machinedependent maintenance periodic recycles. This is one of the fundamental problems in the field of scheduling. We provide the non-approximability analysis and two mathematical programming models for the problem. We also give the worst-case analysis of the classical LPT and LS algorithms for the problem. Based on an adjustment procedure, we propose two improved heuristic algorithms: MLS and MLPT. Numerical experiments show that the improved heuristic algorithm MLPT is promising and is very suitable for real world applications. Note that there is still a gap between the non-approximablity bound max{2, σmax } and the worst-case ratio upper bound 4mσmax /(2m + 1). So, further research may focus on reducing this gap by designing more sophisticated approximation algorithms. It is also worth considering other objectives such as minimizing maximum lateness or minimizing total number of late jobs if there is a due date for each job.
Acknowledgements This work was supported by the National Natural Science Foundation of China (grant nos.71372019, 71471057, 71102174), Beijing Higher Education Young Elite Teacher Project (grant no. YETP1173), and Beijing Philosophy and Social Science Foundation of China (grant no. 11JGC106).
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Average relative error ratios on average
0.14 LS MLS LPT MLPT
0.12
0.1
0.08
0.06
0.04
0.02
0 3
5
10
Number of machines Fig. 1 Average relative error ratios on average.
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