The setup times are assumed to be sequence-independent, i.e. they only depend on the job to be processed, regardless of its preceding job. The objective is to ...
Bounding Schemes for Scheduling a Two-Machine Flowshop with Set-up Times Anis Gharbia,c , Talel Ladharia,b , Mohamed Kais Msaknib , Mehdi Serairid (a) Princess Fatimah Al-Nijriss Research Chair for Advanced Manufacturing Technologies, Department of Industrial Engineering, College of Engineering, King Saud University, PO Box 800 Riyadh 11421, Saudi Arabia. (b) Combinatorial Optimization Research Group - ROI, Ecole Polytechnique de Tunisie, BP 743, 2078, La Marsa, Tunisia. (c) Research Fellow, BEM Bordeaux Management School, France. (d) Heudiasyc UMR 6599, Centre de Recherches de Royallieu, Université de Technologie de Compiègne, France.
Abstract Numerous real-world industrial applications are modeled as a two-machine flowshop scheduling problem with sequence-independent setup times. These include scheduling multimedia data objects for World Wide Web applications, scheduling automated manufacturing cells with computer-controlled transportation robots, and scheduling client requests in a three-tiered Internet/database connectivity model. Despite the non-negligible theoretical research efforts that have been devoted to minimize total completion time under this scheduling environment, there is still a big hole in the related literature in terms of lower bounding procedures. Indeed, the two only existing lower bounds are much simplistic and prove to have a low performance. In this paper, we propose new lower bounding schemes that not only consistently outperform those of the literature, but require negligible CPU time for up to 300 jobs. This will considerably impact the development of efficient exact methods, and will offer a more accurate performance measure for potentially good heuristics.
1. Introduction Setup activities include several important tasks such as tools acquirement and adjustment, machine cleaning up, and material inspection. Recent studies show the significant impact of reducing setup times with respect to several performance criteria encompassing production speed increase, competitiveness improvement, lead times shortening, stock reduction, and customer satisfaction increase [3]. For the sake of illustration, Trovinger and Bohn [24] reported that combining the SMED method with IT-based techniques in a printed circuit board assembly yielded a setup time reduction of more than 80%, resulting to an annual benefit of $1.8 million. Not only it is crucial to optimize setup times, but more importantly, it is often essential to explicitly take them into
account while scheduling numerous production systems. Allahverdi and Souroush [3] listed various real-world scheduling applications where setup times are considered separately from processing times. In addition to industrial applications (e.g. metal processing, brake manufacturing, electronics, etc.), the list covers a wide range of areas including finance, information processing, health care, cosmetics, and satellite imaging, to quote just a few. In this paper, we investigate a two-machine permutation flowshop scheduling environment with setup time consideration. The setup times are assumed to be sequence-independent, i.e. they only depend on the job to be processed, regardless of its preceding job. The objective is to find a feasible job sequence which minimizes the total completion time. More formally, a set of n jobs has to be processed on machines M1 and M2 in that order. Processing job j (j = 1, ..., n) on machine Mi (i = 1, 2) requires si,j units of setup time and pi,j units of non-preemptive processing time. All data are assumed to be deterministic and integer. Each machine processes at most one job at one time and each job cannot be processed by more than one machine at oneP time. According to Allahverdi et al. [8], this strongly N P-hard problem is denoted by F 2 | STsi | Cj .
Interestingly, several real-world industrial applications can be modeled as a two-machine flowshop scheduling problem with sequence-independent setup times. These include scheduling multimedia data objects for World Wide Web applications [9], and scheduling automated manufacturing P cells with computer-controlled transportation robots [19]. In particular, the F 2 | STsi | Cj arises in the context of three-tiered Internet/database connectivity model which provides an interface between the enterprise service and clients [1]. In this model, client requests need to be processed by the application server and then by the database server (which represent M1 and M2 , respectively). For the sake of efficiency, the queued requests are to be scheduled so that the average completion time of a client request is minimized. Before a request can be processed, the resources needed for this request are to be made ready in the application server. Also, an access to the database must be preceded by a preprocessing operation which consists in a database query decomposition and a partial optimization. Since this latter preprocessing operation can start while the application server is processing the query, then it can be seen as its setup operation on the second server. From a theoretical point of view, the two-machine flowshop with sequence-independent setup times has been the scope of numerous investigations. The last decade witnessed a particular interest arousal in this scheduling environment with various optimization criteria and machine/job settings [2, 4, 5, 6, 10, 13, 16]. For an exhaustive overview of scheduling literature that deals with setup times, the reader is referred to the excellent survey papers of Allahverdi et al. [7, 8].
P Bagga and Khurana [11] were the first to address the F 2|STsi | Cj problem. They proposed two lower bounds, a dominance rule and a branch-and-bound algorithm which could P only solve small-sized instances with up to 9 jobs. The most important contribution in F 2|STsi | Cj related research is probably that of Allahverdi [6]. He implemented the lower bounds of Bagga and Khurana [11] together with two new dominance rules in a branch-and-bound algorithm which was able to solve instances with up to 35 jobs. He also proposed three constructive heuristics which have been improved later on by Al-Anzi and Allahverdi [1]. Recently, Msakni et al. [20] proposed a dynamic priority rule-based constructive heuristic which proved to consistently outperform those of [1].
2
Moreover, Msakni et al. [20] devised a genetic local search algorithm that provided near-optimal solutions in reasonable CPU time for large-sized instances with up to 500 jobs. P Despite the non-negligible research efforts that have been devoted to the F 2|STsi | Cj , there is still a big hole in the related literature in terms ofP lower bounding procedures. As far as we know, the only developed lower bounds for F 2 | STsi | Cj are those of Bagga and Khurana [11]. As it will be detailed in the next section and assessed in our experimental analysis, these bounds are much simplistic and prove to have a low performance. Undoubtedly, efficient and tight lower bounds constitute the bedrock of highly performant branch-and-bound algorithms. Consequently, the lack of strong lower bounding procedures considerably obstructs the development of efficient P exact methods for the F 2 | STsi | Cj . On the other hand, it is well-known that a critical performance measure of an approximate solution is its relative gap with respect to a lower bound. Therefore, relying on loose lower P bounds would yield inaccurate assessments of potentially good heuristics for the F 2 | STsi | Cj .
In aPpreliminary version of this work [17], we presented two bounding strategies for the F 2 | STsi | Cj . The first one considers the time a job has to wait between the two machines, while the second one is based on a newly derived equivalence result for the single machine problem. The present paper briefly recalls the proposed lower bounds and introduce newly developed ones. Our experimental results show that, not only our lower bounds consistently outperform those of the literature, but they require negligible CPU time. For instance, the average CPU time of one of our best derived lower bounds is about 0.1 seconds for 300 jobs.
2. Lower bounds of Bagga and Khurana [11] In this section, we describe the two lower bounds of Bagga and Khurana [11], denoted hereafter by 1 2 . and LBBK LBBK If the capacity of the secondPmachine is relaxed, then the obtained problem is a single machine one (denoted by 1 | STsi , qj | Cj ) with setup P times sj = s1,j , processing times pj = p1,j and delivery times qj = p2,jP(j = 1, ..., n). Since nj=1 qj is constant, then the obtained problem is Cj . P The optimal total completion time of this latter problem is a valid equivalent to 1 | STsi | Cj . Let Cj1 denote the completion time of job j in the Shortest lower bound for F 2 | STsi | P Processing Time (SPT) sequence of the 1 || Cj problem, which is obtained after merging the setup time and the processing time of each job j (j = 1, ..., n). Therefore, we have 1 LBBK
=
n X j=1
Cj1
+
n X
qj
j=1
1 The computation of LBBK requires sorting the jobs according to the nondecreasing order of their 1 s1,j + p1,j . Thus, LBBK can be computed in O(n log n) time. 2 In the computation P of LBBK , it is assumed that s1,j = p1,j = 0. Consequently, a single machine Cj with setup times sj = s2,j and processing times pj = p2,j (j = 1, ..., n) problem 1 | STsi | 2 is equal to the corresponding optimal total completion time which is is obtained. Then, LBBK obtained using the SPT rule after merging the setup time and the processing time of each job j 2 (j = 1, ..., n). LBBK can be computed in O(n log n) time.
3
3. Waiting time-based lower bounds The lower bounding scheme presented in this section is based on the minimum amount of time, denoted by δ j , that a job j (j = 1, ..., n) has to wait between its finishing time from M1 and its starting time on M2 . Note that trivially setting δ jP= 0 for all j = 1, ..., n yields the first lower 1 of Bagga and Khurana [11]. Let ∆ = nj=1 δ j denote the total waiting time. Clearly, bound LBBK P P P a valid lower bound for F 2 | STsi | Cj is LBW T = nj=1 Cj1 + nj=1 p2,j + ∆.
Firstly, if it is assumed that a job j is not preceded by any job, then the setup operation of job j on M2 starts at the same time as its setup operation on M1 . Consequently, a lower bound on δ j Pn 1 1 is δ j = max(0, s2,j − s1,j −P p1,j ), and a lower bound on ∆ is ∆1 = j=1 δ j . Therefore, a valid lower bound for the F 2 | STsi | Cj , which can be computed in O(n log n) time, is 1 LBW T
=
n X
Cj1
j=1
+
n X
p2,j + ∆1
j=1
Actually, except for the first scheduled job, the setup operation of a job j on M2 has to wait until its predecessor, say job h, finishes its processing. Note that there are at least δ 1h + p2,h units of time that must elapse between the finishing time of job h on M1 (which corresponds to the starting time of job j on M1 ) and its finishing time on M2 . Consequently, if a job j is not the first scheduled job, then a better lower bound on δ j is δ 2j = max(0, s2,j − s1,j − p1,j + minh6=j (δ 1h + p2,h )). Therefore, if a particularPjob j0 is the first one to be processed, then a lower bound on the total P waiting time is δ 1j0 + j6=j0 δ 2j . Hence, a valid lower bound onP∆ is ∆2 = minj0 =1,...,n (δ 1j0 + j6=j0 δ 2j ). Cj that can be computed in O(n log n) Consequently, a valid lower bound for the F 2 | STsi | time is n n X X 2 1 LBW T = Cj + p2,j + ∆2 j=1
j=1
Interestingly, the waiting time-based approach can be advantageously extended by taking into consideration all of the job predecessors. Indeed, a lower bound on δ j if job j is scheduled in [k] [k−1] [1] position k ≥ 2 is δ j = max(minh6=j (δ h + p2,h ) + s2,j − s1,j − p1,j , 0) (obviously, δ j = δ 1j ). Now, since each job has to be assigned exactly one position, then a lower bound on ∆ can be obtained [k] by solving the assignment problem where the cost of assigning job j to position k is δ j . Let ∆3 denote the P minimum total assignment cost. Hence, the following O(n3 ) time lower bound for the F 2 | STsi | Cj holds: n n X X 3 1 = C + p2,j + ∆3 LBW T j j=1
j=1
A second way to compute a multiple predecessor-based lower bound on δ j can be described as follows. Clearly, if job j is preceded by a job h, then a lower bound on the minimum waiting time δ j is δ h,j = max(δ 1h + p2,h + s2,j − s1,j − p1,j , 0). Note that, except the first (resp. last) scheduled job, each job has to be preceded (resp. succeeded) by another one. The first scheduled job is assumed to be preceded by a dummy job, denoted by job 0, with δ 0,j = δ 1j (j = 1, ..., n). Moreover, the last scheduled job is assumed to be succeeded by a dummy job, denoted by job n + 1, with δ h,n+1 = 0 (h = 1, ..., n). Obviously, since job 0 cannot precede job n + 1, then we set δ 0,n+1 = ∞. 4
Consequently, a lower bound on ∆ can be obtained by solving the assignment problem where the assignment costs are δ h,j (h = 0, ..., n; j = 1, ..., n + 1). Obviously, we set δ j,j = ∞ for all total assignment cost. Therefore, the following O(n3 ) j = 1, ..., n. Let ∆4 denote the minimum P time lower bound for the F 2 | STsi | Cj holds: 4 LBW T
=
n X
Cj1
+
j=1
n X
p2,j + ∆4
j=1
4 2 Clearly, since δ h,j ≥ δ 2j for all h, j = 1, ..., n, then we have LBW T ≥ LBW T . 4 Interestingly, LBW T can be further improved in the following way. Note that cycles may occur 4 . Obviously, these cycles should not occur in any feasible between jobs in the computation of LBW T job sequence. Therefore, the computation of ∆4 can be improved by determining a job sequence n n P P (rather than an assignment) that minimizes δ h,j . This turns out to minimize the makespan h=0 j=1 j6=h
on a single machine with sequence dependent setup times sh,j = δ h,j and zero processing times, with the additional constraint that job 0 must be the first scheduled job. It is well-known that the latter problem can be equivalently formulated as an Asymmetric Traveling Salesman Problem (ATSP). Indeed, it suffices to set the distance between two vertices h and j equal to dh,j = δ h,j (h = 0, ..., n; j = 1, ..., n), and dh,0 = 0 (h = 1, ..., n). Let ∆5 denote a lower bound on the optimal Ptour length of the obtained ATSP instance. Therefore, a valid lower bound for the F 2 | STsi | Cj is 5 LBW T =
n X
Cj1 +
j=1
n X
p2,j + ∆5
j=1
According to the recent comparative study of mathematical formulations for the ATSP [21], there are numerous mixed integer programs whose linear relaxations constitute a tight lower bound for the ATSP. In our implementation, the value of ∆5 is equal to the optimal objective value of the relaxed linear program that is proposed by Desrochers and Laporte [15]. Actually, we experimentally found that this relaxation provides the better effectiveness/efficiency trade-off for the derived ATSP instances. This is not to exclude that any development in ATSP-related optimization/lower bounding 5 . procedures could be advantageously used for more efficient/effective computation of LBW T Note that the mathematical formulation of the ATSP that is proposed by Desrochers and Laporte [15] consists in adding subtour elimination constraints to the classical 0-1 linear programming formulation of the assignment problem. Since the latter problem is equivalent to its linear relax5 4 ation, then the implemented computation of ∆5 yields LBW T ≥ LBW T . On the other hand, we 3 5 . found that there is no dominance relationship between LBW T and LBW T
4. Single machine-based lower bounds In this section, we assume that the capacity of the first machine is Prelaxed. The obtained problem Cj , with setup times sj = s2,j is a single machine scheduling problem, denoted by 1 | STsi , rj | 5
and release dates well as any lower bound for P rj = s1,j + p1,j . Therefore, the optimal solution asP 1 | STsi , rj | Cj constitutes a valid lower bound for the F 2 | ST | si P Cj . At this level, it should P Cj is N P-hard, since the 1 | rj | Cj is N P-hard [22]. Note be noticed that the 1 | STsi , rj | 2 proposed by Bagga and Khurana [11] can be seen as a trivial lower that the lower bound LBBK P bound for the 1 | STsi , rj | Cj where all the release dates are set to zero. P P In [17], we proved that the 1|STsi , rj | Cj can be equivalently reformulated as a 1|rj | Cj by setting the processing times equal to pj + sj and the release dates equal to max(rj − sj , 0).PAn interesting consequence of this equivalence result is that valid P lower bounds for the F 2 | STsi | Cj can be derived through relaxing/solving the derived 1|rj | Cj instance.
P 1 , can A first single machine-based lower bound for F 2 | STsi | Cj , denoted hereafter by LBSM be derived P by computing the 1optimal total completion time of the preemptive version of the obtained Cj problem. LBSM can be computed in O(n log n) by using the Shortest Remaining 1 | rj | Processing Time (SRPT) rule: at any time, schedule an available job having the shortest processing time [12].
P Interestingly, Della Croce and T’Kindt [14] exploited the properties of the preemptive 1 | rj | Cj solution and proposed an improved version of the preemptive lower bound which can be computed in O(n2 ) time. Clearly, their a second single machine-based lower P result can be used to derive 2 , that dominates LB 1 . For the sake Cj , denoted hereafter by LBSM bound for the F 2 | STsi | SM 2 is omitted. of clarity, the description of the computational details of LBSM 3 , can be derived by optimally A third single machine-based lower bound, denoted by LBSM P 3 Cj . Since the latter problem is NP-hard [22], then LBSM requires solving the obtained 1 | rj | an exponential computation time. Nonetheless, several efficient branch-and-bound algorithms for P Cj exist in the literature. In our implementation, we used the branch-and-bound the 1 | rj | 3 2 2 . algorithm presented in [23]. Obviously, LBSM dominates LBSM and LBSM
Actually, any job that is scheduled at position k cannot start processing on M2 before r[k] = k P
i=1
[i]
[i]
P1 , where P1 denotes the ith smallest s1,i + p1,i . An immediate consequence is that an
interesting relaxation scheme P can be derived by adding the position-based release dates P r[k] (k = 1, ..., n) to the 1|STsi , rj | Cj . The obtained problem is denoted by 1|STsi , rj , r[j] | Cj . Let Ci denote the completion time of the job that is scheduled at position i (i = 1,P ..., n), and xji = 1 if job j is scheduled at position i; 0 otherwise (i, j = 1, ..., n). The 1|STsi , rj , r[j] | Cj can be formulated as follows:
6
M in n P
j=1 n P
n P
Ci
(1)
i=1
xji = 1
∀i = 1, ..., n
(2)
xji = 1
∀j = 1, ..., n
(3)
i=1
C1 ≥
n P
(sj + pi )xj1
j=1
Ci ≥ Ci−1 + Ci ≥
n P
n P
(4)
(sj + pi )xji
j=1
(max(rj , r[i] ) + pj )xji
j=1
Ci ≥ 0 ∀i = 1, ..., n; xji ∈ {0, 1} ∀i, j = 1, ..., n
∀i = 2, ..., n
(5)
∀i = 1, ..., n
(6) (7)
Clearly, the linear relaxation of the above mathematical program (1)-(7) yields a valid lower P 4 . It is worth mentioning that bound for the F 2|STsi | Cj problem, denoted hereafter by LBSM 4 and the other single machine-based lower bounds. there is no dominance relationship between LBSM
5. Computational results All the discussed lower bounds were coded in C and compiled with the Microsoft Visual C++.net 8.0. All the experiments were conducted on a laptop with an Intel Core Duo T2350 1.86 GHz processor and 1 GB RAM. The assignment problems were solved using the code of Goldberg and Kennedy [18] available at http://www.avglab.com/andrew/soft.html. The ATSP-related linear programs were solved by CPLEX 11.1 with "concurrent" option. All the experiments were carried on a test bed of 2200 instances which were randomly generated in a fashion similar to that adopted in [6]. More precisely, the processing times were randomly generated using the discrete uniform distribution on [1, 100]. The setup times were randomly generated using the discrete uniform distribution on [1, 100K], where K ∈ {0.25, 0.5, 0.75, 1}. The number of jobs n was taken equal to 10, 15, 20, 25, 30, 35, 50, 70, 100, 200, and 300 jobs. For each combination of n and K, 50 instances were generated. Actually, one major aim of this section is to numerically assess the theoretical dominance of the newly proposed lower bounds with respect to those of the literature. Recall that all of the 1 , and that all of the new single machinenew waiting time-based lower bounds dominate LBBK 2 . Table 1 (resp. Table 2) illustrates the results of a pairwise based lower bounds dominate LBBK comparison between the waiting time-based (resp. single machine-based) lower bounds, including 1 2 ). In these tables, we provide the percentage of times the lower bound displayed (resp. LBBK LBBK in the corresponding row improves that displayed in the corresponding column.
7
> 1 LBBK 1 LBW T 2 LBW T 3 LBW T 4 LBW T 5 LBW T
1 LBBK − 88.45% 90.36% 90.50% 96.05% 96.09%
1 LBW T 0.00% − 80.86% 81.05% 92.73% 92.86%
2 LBW T 0.00% 0.00% − 4.00% 89.95% 90.09%
3 LBW T 0.00% 0.00% 0.00% − 89.77% 89.91%
4 LBW T 0.00% 0.00% 0.00% 0.05% − 1.45%
5 LBW T 0.00% 0.00% 0.00% 0.05% 0.00% −
Table 1: Performance of the waiting time-based lower bounds > 2 LBBK 1 LBSM 2 LBSM 3 LBSM 4 LBSM
2 LBBK − 99.91% 99.91% 99.91% 99.91%
1 LBSM 0.00% − 75.68% 87.18% 80.00%
2 LBSM 0.00% 0.00% − 81.14% 79.23%
3 LBSM 0.00% 0.00% 0.00% − 72.82%
4 LBSM 0.00% 19.45% 20.14% 25.77% −
Table 2: Performance of the single machine-based lower bounds Clearly, the results depicted in Tables 1 and 2 strongly support the theoretical dominance of our 1 2 . Indeed, even the simplest of the waiting and LBBK proposed lower bounds with respect to LBBK 1 1 time-based lower bounds (namely LBW T ) outperforms LBBK in 88.45% of the instances. More 2 in dramatically, all of the single machine-based lower bounds provide better results than LBBK all of the 2200 instances, except two (99.91%). Moreover, most of the dominance results provided 2 , LB 4 , in this paper are robustly validated by the results of Tables 1 and 2. Indeed, LBW T WT 2 3 1 3 1 2 LBSM and LBSM do better than LBW T , LBW T , LBSM and LBSM in 80.86%, 89.77%, 75.68% and 81.14%of the instances, respectively. The two only (slightly) disappointing results are related 3 5 3 to the worth of implementing LBW T and LBW T . Also, although LBW T has the "theoretical" 4 5 advantage of being competitive with LBW T and LBW T , it outperforms them in only one instance, 4 5 while being dominated by LBW T in 1975 instances, and by LBW T in1978 instances. It should 4 be noticed that LBSM outperforms all of the single machine-based lower bounds in 72.82% of the instances. The average required CPU time (in 10−3 seconds) of all the discussed lower bounds according to the variation of the number of jobs is displayed in Table 3. The obtained extremely short computation times exhibit very promising perspectives with respect to embedding the proposed lower 5 4 bounds within a branch-and-bound algorithm. Indeed, except for LBW T and LBSM , the computation of all of the proposed lower bounds requires less than 0.02 seconds, on average. It should be 3 (theoretically) requires an exponential computation time, it only highlighted that although LBSM takes 0.02 seconds, on average, for large-sized instances with 300 jobs. In accordance with Tables 1 and 2, the obtained results suggest that the best effectiveness/efficiency trade-off is obtained by 4 , LB 3 ). On the other hand, although LB 4 max(LBW T SM SM requires an excessive computation time that precludes it to be implemented at each node of a branch-and-bound algorithm, it could be P useful for assessing heuristic procedures for the F 2|STsi | Cj . 8
Nb. Jobs 10 15 20 25 30 35 50 70 100 200 300 Average
1 LBBK 0.01 0.02 0.03 0.01
2 LBBK 0.01 0.02 0.03 0.01
1 LBW T 0.01 0.01 0.02 0.03 0.01
2 LBW T 0.01 0.01 0.02 0.04 0.01
3 LBW T 0.02 0.03 0.06 0.14 0.16 0.23 0.59 1.47 3.19 10.11 20.03 3.28
4 LBW T 0.03 0.05 0.15 0.16 0.31 0.33 1.06 4.37 6.74 12.90 24.49 4.60
5 LBW T 7.33 5.78 9.68 14.15 19.54 32.03 64.53 141.66 304.17 2582.17 16630.73 1801.07
1 LBSM 0.01 0.01 0.01 0.01 0.02 0.03 0.04 0.11 0.21 0.05
2 LBSM 0.01 0.02 0.03 0.04 0.05 0.06 0.09 0.15 0.29 0.66 1.34 0.25
3 LBSM 0.03 0.07 0.13 0.18 0.27 0.35 1.11 2.04 4.55 30.54 104.47 11.98
(-) means that the average CPU time is less than 0.01 milliseconds
Table 3: Average CPU time of the lower bounds
Acknowledgement: Dr. Anis Gharbi and Dr. Talel Ladhari are grateful to Princess Fatimah Al-Nijriss Research Chair for Advanced Manufacturing Technologies for supporting this research.
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4 LBSM 4.60 9.30 17.25 36.97 59.78 93.15 293.26 985.55 3395.08 38000.56 151061.62 17632.46
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