A polynomial-time approximation scheme for the two-machine flow shop scheduling problem with an availability constraint
|
|
- Quentin Cain
- 6 years ago
- Views:
Transcription
1 A polynomial-time approximation scheme for the two-machine flow shop scheduling problem with an availability constraint Joachim Breit Department of Information and Technology Management, Saarland University, Saarbrücken, Germany Abstract We study the problem of scheduling n preemptable jobs in a two-machine flow shop where the first machine is not available for processing during a given time interval. The objective is to minimize the makespan. We propose a polynomialtime approximation scheme for this problem. The approach is extended to solve the problem in which the second machine is not continuously available. Keywords: Approximation algorithms; Flow shop scheduling; Availability constraints 1 Introduction In classical scheduling problems machines are assumed to be available throughout the planning period. In many practical situations, however, this assumption is not justified. Machine availability restrictions may be due to maintenance requirements or rest periods. They can also arise from the overlapping of two consecutive planning periods in a system with rolling horizons. In such a system a new planning period is triggered before all jobs of the previous period have been completed. If the assignment of these uncompleted jobs cannot be altered anymore, e.g. due to process preparation, machines are available in jb@itm.uni-sb.de; postal address: Saarland University, Department of Information and Technology Management, P.O. Box , D Saarbrücken, Germany; fax: +49(681)
2 several time intervals only. The scheduling procedure for the new planning period must take into account these availability restrictions. Real-life situations of this kind motivate the investigation of generalized scheduling models in which machine availability may be restricted. In the past years such models have received considerable attention from researchers. Schmidt [1] and Lee [2] give surveys of this research area. In this paper we will consider a flow shop problem with limited machine availability. In our problem which will be denoted by Π we are given a set N = {J 1,..., J n } of jobs to be processed on two machines A and B. A machine can process at most one job at a time, and a job can be processed by at most one of the machines at a time. While machine B is continuously available for processing, machine A is not available during the given time interval [s, t]. Let = t s > 0. For convenience we shall call the interval of non-availability on machine A the hole. Each job J j N, j = 1,..., n, consists of two operations O Aj and O Bj which have to be performed on machines A and B, respectively. We denote by a j and b j, respectively, the durations of operations O Aj and O Bj. Without loss of generality we assume that all input data (s, t, a j, b j ) are nonnegative integer values. We define a(q) = j Q{a j }, b(q) = j Q{b j } for a non-empty set Q of jobs, and a( ) = b( ) = 0. R ij (S) and F ij (S) denote the start and finish time of operation O ij, i {A, B}, j {1,..., n} in a schedule S. If schedule S is feasible we have R Bj (S) F Aj (S), j = 1,..., n. We may drop the reference to schedule S when no confusion can arise. Denote the makespan of a schedule S by C(S) = max j {F Bj }. An optimal schedule S is a schedule with a minimal makespan of C. We assume that jobs are preemptable. This means that an operation can be suspended at any time and resumed later on without any penalty. Our scenario differs from the resumable scenario as introduced by Lee [3] where a job may be preempted at time s on machine A only and then has to be resumed at time t. It can be easily shown that both models have the same optimal makespan. A polynomial-time algorithm guaranteeing a schedule S such that C(S)/C ρ is called a ρ-approximation algorithm. The value ρ is referred to as the relative worst-case error bound. A family of algorithms is called a polynomial-time approximation scheme, or PTAS for short, if for any given ε > 0 one of the algorithms guarantees a relative worstcase error bound of ρ = 1 + ε and its running time is polynomial in the length of the problem input. If the running time is also polynomial in 1/ε the family of algorithms is called a fully polynomial-time approximation scheme, or FPTAS for short. Our objective 2
3 is to find a PTAS for our problem. Lee [3] proves that the resumable variant of our problem is NP-hard. The same is true for the problem with a hole on machine B. Similar NP-hardness proofs can be easily found for the preemptable version of the problem. Lee proposes dynamic programming algorithms to solve the problems exactly and provides approximation algorithms with relative worst-case error bound of 3/2 (hole on machine A) and 4/3 (hole on machine B). Cheng and Wang [4] and Breit [5] present improved fast approximation algorithms for these two problems with relative worst-case error bounds of 4/3 (hole on machine A) and 5/4 (hole on machine B). Ng and Kovalyov [6] provide fully polynomial-time approximation schemes for both problems with time complexity O(n 5 /ε 4 ). Kubiak et al. [7] investigate the case where each of the machines may have an arbitrary number of holes. They show that this problem is not approximable even if only one hole occurs on each machine. To solve the problem exactly they propose a branch and bound algorithm. B lažewicz et al. [8] provide constructive and local search based heuristics for the same problem. Cheng and Wang [9] study the problem with two consecutive holes. In their problem the hole on machine A (B) starts at the same time at which the hole on machine B (A) ends. They provide a 5/3-approximation algorithm for the non-preemptable case of this problem. Notice that a feasible schedule for the resumable problem version is also feasible for the model with preemptable jobs. Approximation algorithms for the resumable problem version solve the preemptable case with the same performance guarantee. Furthermore the non-approximability result in Kubiak et al. [7] also holds for the preemptable case. The rest of this paper is organized as follows. In Section 2 we deal with some special cases of our problem. This enables us to make simplifying assumptions for the following sections. The proposed approximation scheme is based on an exact algorithm for a relaxation of our problem. In Section 3 we describe this relaxation and the optimization algorithm. In Section 4 we turn to the original problem and show how to transform the optimal solution for the relaxed problem into an approximate solution for the original problem. Section 5 describes an extension of the presented approach for the problem with a non-continuously available machine B. Section 6 finally contains some conclusions. 3
4 2 Preliminaries In this section we will separate from a few special cases of our problem. The classical variant of our problem in which both machines are continuously available is solved by Johnson s algorithm [10]. It is not difficult to see that Johnson s algorithm remains optimal, if the hole occurs at the beginning of the planning period or after all jobs on machine A have been completed, i.e. if s = 0 or a(n) s. We will therefore assume in the following that s > 0 and a(n) > s. Furthermore, following Lee [3], we assume that there exists an optimal permutation solution for our problem in which at most one operation is preempted, namely the one interrupted by the hole. From the integrality assumption we made in Section 1 it follows that in this optimal solution all operations on machine A are started at integer points in time. Suppose that some of the jobs in set N consist only of one operation or, equivalently, have an operation with zero processing time. We define N a = {J j N a j = 0} and N b = {J j N b j = 0} and consider an arbitrary problem instance I(Π) of our problem Π. Let I(Π 0 ) be a problem instance derived from I(Π) by removing all jobs of set {N a N b }. We denote by σ0 the optimal job permutation for problem instance I(Π 0 ). Then we have the following lemma. Lemma 1. There exists an optimal solution σ for I(Π) which is identical with the optimal solution σ0 for I(Π 0 ) with the exception that permuation σ0 is preceded (succeeded) by an arbitrary sequence of the jobs of set N a (N b ). Proof. Consider an optimal solution σ0 for instance I(Π 0) and the corresponding schedule. We insert an arbitrary sequence of the jobs of set N a at the beginning of the schedule, shifting the B-operations of permutation σ0 to the right if necessary. We also insert an arbitrary sequence of the jobs of set N b at the end of the schedule. All operations are started as early as possible. Suppose that adding the jobs of sets N a and N b does not lead to an increase of the makespan. Because the optimal makespan for I(Π) cannot be smaller than the optimal makespan for I(Π 0 ) the new schedule must be optimal. If the makespan increases then under our assumption a(n) > s it equals max{a(n)+, b(n)} which is obviously as well a lower bound for the optimal makespan for I(Π). Lemma 1 allows us to assume in the following that min {a j, b j } > 0 (1) j N 4
5 3 Relaxed problem Without loss of generality we assume that the jobs in set N are indexed such that a i b i a j b j, i < j (2) Let τ(q) be the permutation generated by Johnson s algorithm for a job set Q N. If we schedule the jobs in set Q using permuation τ(q) on two continuously available machines we obtain a schedule with a makespan of C(Q). A lower bound LB on the minimal makespan C is given by C C(N) LB = a(n) + b(n) 2 (3) For ε > 0 we define the job sets U = {J j N max{a j, b j } εlb} V = N \ U V = {J j V a j b j } V = {J j V a j > b j } For convenience we will refer to jobs of set U as big jobs while jobs of set V will be called small jobs. Note that we have U 2 for otherwise a(u) + b(u) > 2 εlb = a(n) + b(n), ε ε a contradiction. We will now formulate problem Π (ε), a relaxation of problem Π for a given ε. For convenience we drop the reference to parameter ε in the following. In problem Π the processing of a small job on machine B can start as soon as its processing on machine A starts, i.e. we allow R Bj R Aj, j V. In addition to this, a small job J j can be split into two partial jobs J j,1 and J j,2 where a j,1 + a j,2 = a j, b j,1 + b j,2 = b j and a j,1 /b j,1 = a j,2 /b j,2. For a feasible schedule we require R Bj,1 R Aj,1, R Bj,2 R Aj,2 (4) A job split in the described manner will be referred to simply as a split job in the following. Additionally, we require that all operations on machine A must start at integer points in time. Recall that there exists an optimal solution for problem Π with the same property. In all other details problem Π is identical with problem Π. We are looking for 5
6 an optimal solution S for problem Π with a makespan of C. It is obvious that C C (5) In order to state the optimization procedure for problem Π we define a block as a set of jobs scheduled consecutively, i.e. without interruption by the processing of other jobs. A big (small) job J j in a schedule S is called critical if F Aj (S) = R Bj (S) (R Aj (S) = R Bj (S)) and a delay of operation O Aj as well as a delay of operation O Bj results in an increase of the makespan. Procedure H. Determine sets U, V and V. Completely enumerate all possible partitions of set U into disjoint subsets U b and U a, U b U a = U. If one of the sets U b, U a, V, V is empty then introduce a respective dummy job J d,l, a d,l = b d,l = 0, l {U b, U a, V, V }, representing the set. For each partition (U b, U a ) generate a schedule by performing the following three steps. All operations are started as early as possible. 1. Find the permutations τ(u b ) and τ(u a ) and denote by J z the first job in permutation τ(u a ). The jobs of sets U b and U a, respectively, are scheduled as a block. The small jobs are sequenced in ascending order of their indices, possibly in several blocks. Under the restrictions F Aj s, j U b (6) F Aj > t, j U a (7) proceed as follows 2. Schedule as many jobs of set V as possible before the block U b. This may involve that the last job before block U b is a split job. Then schedule block U b. 3. Schedule the remaining jobs as follows (a) If a(v ) + a(u b ) s and scheduling the blocks in the order (V, U b, U a, V ) under restrictions (6) and (7) would not result in idle time on machine A between blocks U b and U a then continue by scheduling the blocks U a and V in this order. 6
7 (b) If a(v ) + a(u b ) s and scheduling the blocks in the order (V, U b, U a, V ) under restrictions (6) and (7) would result in idle time on machine A between blocks U b and U a then define s if a z 1 α = s a z + 1 otherwise If a(v ) + a(u b ) + a(v ) < α then stop and continue with the next partition (U b, U a ). Else continue by scheduling as many jobs of block V as possible after the last job of block U b until there is no idle time left on machine A before time α. This may involve that the last job of set V scheduled before the hole is a split job. Then schedule the block U a. Then schedule the remaining jobs of set V (the first job of this last block may be the second part of the split job). (c) If a(v ) + a(u b ) > s then the last job scheduled before the block U b may be a split job. After the hole continue by scheduling the second part of the split job (if it exists) and the remaining jobs of set V. Then schedule the blocks U a and V in this order. Select the best schedule among all generated schedules and call it S H. end. We want to prove the following theorem. Theorem 1. C(S H ) = C. Proof. It can be easily shown that there exists an optimal solution for problem Π in which the big jobs finished on machine A before the hole and also those finished on machine A after the hole appear in the order given by Johnson s rule. Because Procedure H enumerates all partitions of set U into subsets U b and U a we may restrict attention in the following to a step of the loop where the partition is optimal. We will first separate from two simple cases in sections (i) and (ii). Then in sections (iii) till (v) we will investigate the different structures of schedule S H obtained in steps (3a), (3b) and (3c) of Procedure H under the assumption that cases (i) and (ii) are irrelevant. (i) No idle time on machine B. Schedule S H may start with a non-empty subset X of job set V. These jobs are scheduled such that machine A (B) is busy during the interval [0, a(x)] ([0, b(x)]). If there is no idle time at all on machine B in schedule S H, and the schedule finishes on machine B, 7
8 then it is clearly optimal as C b(n) = C(S H ). If the schedule finishes on machine A we have C a(n) + = C(S H ). In the following we will therefore suppose that there exists idle time in schedule S H on machine B after the completion of job set X if it exists. (ii) Idle time on machine B after completion of block U a. Schedule S H may finish with a non-empty subset Y of job set V. In the case where there exists idle time on machine B after the processing of block U a the jobs of set Y can be scheduled such that machine A (B) is busy during the interval [a(n)+ a(y ), a(n)+ ] ([a(n)+ b(y ), a(n)+ ]) and we have C a(n)+ = C(S H ). Thus, we assume in the following that there is no idle time in schedule S H on machine B after the completion of block U a. (iii) Schedule S H is created in step (3a). Recall that we assumed in (1) that no (non-dummy) job contains an operation with zero processing time. Under restrictions (6) and (7) we then clearly have for an optimal partition (U b, U a ) max{f Bj } min{r Aj } max{a(u b ) + + C(U a ), C(U b ) + b(u a )} j U a j U b Furthermore, augmenting a schedule for set {U b U a } by partial schedules for the set of small jobs V cannot be done in a time span smaller than a(v ) + b(v ). Thus, a lower bound on the minimal makespan is C max{a(u b ) + + C(U a ), C(U b ) + b(u a )} + a(v ) + b(v ) (8) An example of schedule S H generated in step (3a) is depicted in Figure 1. Taking into account our assumptions after sections (i) and (ii) it is not difficult to verify that schedule S H meets lower bound (8). It follows that C = C(S H ). (iv) Schedule S H is created in step (3b). Let V b (V a ) denote the subset of jobs of set V which are processed before (after) the hole on machine A and denote by J s the last job (possibly a split job) in block V b. Neglecting all other jobs we can build a schedule for V b in which the processing on machine B finishes not later than the processing on machine A. As > 0 and R Bz t it is therefore not possible that a job of set V b is a critical job in the schedule generated in step (3b). Taking 8
9 furthermore into account our assumptions after sections (i) and (ii) it is then clear that a(v ) + C(U b ) + b(v b ) + b(u a ) + b(v a ) C(S H ) = max α + + C(U a ) + b(v a ) (9) Let us first suppose that the value C(S H ) is determined by the upper part of equation (9). Then a job among set U b is critical in schedule S H, see Figure (2a). In this case we obviously meet lower bound (8) so that schedule S H is optimal. Suppose now that the lower part of equation (9) determines the value C(S H ) and that a job among set U a is critical in schedule S H. This case is depicted in Figure (2b). It is obvious that the processing of block U a on machine B in an optimal schedule cannot end earlier than time α + + C(U a ). Because the jobs of set V are ordered according to non-decreasing ratio a j /b j we have C α + + C(U a ) + b(v a ) = C(S H ) (10) if there exists an optimal schedule S with min j Ua {R Aj (S )} = α. Suppose that there exists an optimal schedule S with min j Ua {R Aj (S )} = β > α. It is not difficult to derive a lower bound for the minimal makespan in this situation departing from schedule S H. We have to shift the block U a to the right by β α time units which leaves a gap of this size on machine A just before time β. Because the jobs of set V are ordered according to non-decreasing ratio a j /b j the best way to fill the gap is to use jobs of set V a in the order of their indices. Then we obviously have C β + + C(U a ) + b(v a ) (β α) bs a s = C(S H ) + (β α) (β α) bs a s C(S H ) Schedule S H is thus again optimal. Note that it is impossible that in each optimal solution we have a(v )+a(u b )+a(v ) < α while set U a consists of at least one (non-dummy) job because this implies that there exists idle time on machine A in each optimal schedule. Procedure H therefore does not generate a schedule for this situation. The case where there exists no big (non-dummy) job finished after the hole and a(v ) + a(u b ) + a(v ) s is solved by Johnson s algorithm and was precluded from consideration. 9
10 (v) Schedule S H is created in step (3c). Let V b (V a ) denote the subset of jobs of set V which are processed before (after) the hole on machine A. Taking into account our assumptions after sections (i) and (ii) we have C(S H ) = max a(v b ) + a(u b ) + + a(v a ) + C(U a ) + b(v ) a(v b ) + a(u b ) + + b(v a ) + b(u a ) + b(v ) a(v b ) + C(U b ) + b(v a ) + b(u a ) + b(v ) (11) The case in which the length of schedule S H is determined by the upper part of equation (11) is shown in Figure (3a). Here, a job in set U a is critical and we obviously meet lower bound (8). Suppose that the makespan is determined by the middle part of equation (11), see Figure (3b). The first job in block V a is critical. Denote by L(δ) (L (δ)) the idle time on machine B up to time δ in schedule S H (S ). We have C(S H ) = b(n) + L(t) and C b(n)+l (t). Because the jobs of set V are ordered according to non-decreasing ratio a j /b j and the choice of jobs for set U b is optimal we clearly have L (t) t b(u b ) b(v b ) = L(t) and thus C(S H ) = C. Consider finally the case where the lower part of equation (11) determines the value C(S H ). A job in set U b is critical, see Figure (3c). Let γ = s a(u b ) + C(U b ). We have C(S H ) = b(n) + L(γ) and C b(n) + L (γ). Again it is easy to see that L (γ) γ b(u b ) b(v b ) = L(γ) and thus C(S H ) = C. For a given ε the loop of Procedure H has to be executed 2 U 2 2 ε times. The steps within the loop can be implemented to run in O(n log n) time. Thus, for a given ε Procedure H runs in O(n log n) time. 4 Original Problem In this section we turn to the original problem and show how to transform schedule S H found by Procedure H into a new schedule S T which is feasible for the original problem. This task will be accomplished by Procedure T. Because all the operations of the big jobs and the A-operations of the small jobs are scheduled in a feasible manner in schedule S H Procedure T does not alter their assignment. It only adjusts the schedule for the B-operations of the small jobs. 10
11 Procedure T. Departing from schedule S H delay the processing of each small operation O Bj, j V, by εlb time units within the time windows assigned to small B-operations in schedule S H and the time window [C(S H ), C(S H ) + εlb]. end. Figure 4 illustrates Procedure T. The upper two parts of the figure show an example of schedule S H with three blocks of small jobs on machine B (marked V 1, V 2 and V 3 ). The lower part of the figure (machine schedule B-S T ) shows the assignment of these blocks after the application of Procedure T. Only the time windows allocated to small jobs in schedule S H and the time window [C(S H ), C(S H ) + εlb] are to be seen. The gap at the beginning of the schedule has a width of εlb time units. Note that schedule S H is lengthened by the same time span. We have the following lemma. Lemma 2. If Procedure T is applied to schedule S H generated by Procedure H we obtain a schedule S T such that C(S T ) C(S H ) + εlb (12) and schedule S T is feasible for problem Π. Proof. The first part of the lemma is trivial. Because we shift each processing time unit of the small B-operations by exactly εlb time units departing from schedule S H it is clear that the makespan does not increase by more than εlb time units. It remains to show that schedule S T is feasible for problem Π. First, observe that Procedure T does not alter the assignment of the big jobs in schedule S H and that no small operation is moved to a time window already assigned to a big job. Thus, all big jobs remain assigned in a feasible manner in schedule S T and there are no clashes of big and small operations. We must also assure that R Bj (S T ) F Aj (S T ) for all j V. Recall that we required R Bj (S H ) R Aj (S H ), j V. If a small operation O Aj is not split in schedule S H then we have R Aj (S H ) + εlb > F Aj (S H ) and Procedure T generates a schedule S T such that R Bj (S T ) R Bj (S H ) + εlb R Aj (S H ) + εlb > F Aj (S H ) = F Aj (S T ). We come to the case where operation O Aj is split. Recall that the splitting results in two partial jobs J j,1 and J j,2 and that condition (4) holds. We denote by J x the job immediately following job J j,2. Because the time windows assigned to operations O B,j1 and O B,j2 have a total size of b j and Procedure T delays these operations by εlb > b j time units within the time windows assigned to small operations on machine B we have 11
12 R B,j1 (S T ) > R Bx (S H ) R Ax (S H ) F Aj,2 (S H ) = F Aj,2 (S T ). Finally, observe that shifting a small operation O Bj by εlb time units within the time windows assigned to small jobs in schedule S H allows to shift the preceding small operation by the same amount of time units without causing clashes among B-operations of small jobs. It follows that schedule S T is feasible for problem Π. It is clear that Procedure T can be implemented to run in O(n) time and that schedule S T may be a non-permutation schedule. We are now ready to state the approximation scheme. Algorithm AS. Input: An instance of Problem Π and ε > 0. Call Procedure H with parameter ε to find schedule S H. Apply Procedure T with parameter ε to schedule S H and output schedule S T. end. Theorem 2. The family of algorithms PTAS for problem Π. given by Algorithm AS and parameter ε is a Proof. It follows immediately from Theorem 1, (3), (5) and (12) that C(S T ) (1+ε)C. We also found in the proof of Lemma 2 that schedule S T is feasible for problem Π. For a given ε Procedures H and T have a polynomial time complexity of O(n log n). 5 Problem with a hole on machine B We now show how to derive a PTAS for the case with a hole on machine B from the PTAS presented in the previous section. Let us denote the problem with the hole on machine B by Π B. We consider in the following an arbitrary instance of this problem which we denote by I B. The hole in this instance occurs in the time interval [s B, t B ]. Instance I B has an optimal makespan of CB > t B and we define q = (1 + ε)cb. Furthermore, let I A1 be an instance of problem Π derived from instance I B by interchanging the values a j and b j, j = 1,..., n, and defining the hole on machine A as [CB t B, CB s B]. Let CA1 be the optimal makespan for instance I A1. Then we obviously have CA1 = C B. We define another instance I A2 of problem Π which is identical with instance I A1 with the exception that the hole now occurs in the interval [q t B, q s B ], i.e. εca1 time units later than in instance I A1. Let CA2 be the optimal makespan for this instance. The following lemma is provided without proof. 12
13 Lemma 3. Let I 1 and I 2 be two instances of problem Π which are identical with the exception the hole of length occurs later in instance I 2. Then the optimal makespan of I 2 does not exceed that of I 1. It follows from Lemma 3 that CA1 CA2. Suppose we knew value q. Then we could generate instance I A2 from a given instance I B, apply Algorithm AS and would obtain a schedule, say S A2, of length C(S A2 ). Using Theorem 2 and Lemma 3 we obtain C(S A2 ) (1 + ε)ca2 (1 + ε)c A1 = (1 + ε)c B. Reversing schedule S A2 provides the desired approximate solution for instance I B. Unfortunately we do not know value q. However, using binary search we are able to find a smallest integer value p q and a corresponding instance I A3 with a hole [p t B, p s B ] on machine A such that the application of Algorithm AS on I A3 yields a schedule S A3 of length C(S A3 ) p. Appropriate lower and upper start values for the binary search procedure could be t B and (1 + ε)(t B + C(N)). The reversed schedule S A3 provides the desired approximation for instance I B. The above approach obviously constitutes a PTAS for the problem with a hole on machine B. 6 Conclusions We presented a polynomial-time approximation scheme for the problem of scheduling n preemptable jobs in a two-machine flow shop with an availability constraint imposed on the first machine. The approach was extended to the case where the second machine is non-continuously available. It constitutes an improvement compared with the known approximation algorithms by Cheng et al. [4] and Breit [5]. It is also an alternative to the FPTAS presented by Ng and Kovalyov [6], particularly for large n. Acknowledgements The author would like to acknowledge the constructive comments by an anonymous referee which helped to improve the paper. 13
14 References [1] Schmidt G. Scheduling with limited machine availability. European Journal of Operational Research 2000;121:1 15. [2] Lee C-Y. Machine scheduling with availability constraints. In: Leung JY-T (Ed.). Handbook of Scheduling. CRC Press, p [3] Lee C-Y. Minimizing the makespan in the two-machine flowshop scheduling problem with an availability constraint. Operations Research Letters 1997;20: [4] Cheng TCE, Wang G. An improved heuristic for two-machine flowshop scheduling with an availability constraint. Operations Research Letters 2000;26: [5] Breit J. An improved approximation algorithm for two-machine flow shop scheduling with an availability constraint. Information Processing Letters 2004;90: [6] Ng CT, Kovalyov MY. An FPTAS for scheduling a two-machine flowshop with one unavailability interval. Naval Research Logistics Quarterly 2004;51: [7] Kubiak W, B lažewicz J, Formanowicz P, Breit J, Schmidt G. Two-machine flow shops with limited machine availability. European Journal of Operational Research 2002;136: [8] B lažewicz J, Breit J, Formanowicz P, Kubiak W, Schmidt G. Heuristic algorithms for the two-machine flowshop with limited machine availability. Omega 2001;29: [9] Cheng TCE, Wang G. Two-machine flowshop scheduling with consecutive availability constraints. Information Processing Letters 1999;71: [10] Johnson SM. Optimal two- and three-stage production schedules with setup times included. Naval Research Logistics Quarterly 1954;1:
15 Vita Joachim Breit holds a Ph.D. in Business Administration from Saarland University, Saarbrücken, Germany. He is member of the scheduling group at the Department of Information and Technology Management at the same university. His primary research interests are deterministic scheduling and computer simulation. 15
16 Figure 1: Schedule S H generated in step (3a) 16
17 Figure 2: Schedule S H generated in step (3b) 17
18 Figure 3: Schedule S H generated in step (3c) 18
19 Figure 4: Generating schedule S T 19
An improved approximation algorithm for two-machine flow shop scheduling with an availability constraint
An improved approximation algorithm for two-machine flow shop scheduling with an availability constraint J. Breit Department of Information and Technology Management, Saarland University, Saarbrcken, Germany
More informationSTABILITY OF JOHNSON S SCHEDULE WITH LIMITED MACHINE AVAILABILITY
MOSIM 01 du 25 au 27 avril 2001 Troyes (France) STABILITY OF JOHNSON S SCHEDULE WITH LIMITED MACHINE AVAILABILITY Oliver BRAUN, Günter SCHMIDT Department of Information and Technology Management Saarland
More informationHEURISTICS FOR TWO-MACHINE FLOWSHOP SCHEDULING WITH SETUP TIMES AND AN AVAILABILITY CONSTRAINT
HEURISTICS FOR TWO-MACHINE FLOWSHOP SCHEDULING WITH SETUP TIMES AND AN AVAILABILITY CONSTRAINT Wei Cheng Health Monitor Network, Parmus, NJ John Karlof Department of Mathematics and Statistics University
More informationComplexity analysis of job-shop scheduling with deteriorating jobs
Discrete Applied Mathematics 117 (2002) 195 209 Complexity analysis of job-shop scheduling with deteriorating jobs Gur Mosheiov School of Business Administration and Department of Statistics, The Hebrew
More informationMinimizing Mean Flowtime and Makespan on Master-Slave Systems
Minimizing Mean Flowtime and Makespan on Master-Slave Systems Joseph Y-T. Leung,1 and Hairong Zhao 2 Department of Computer Science New Jersey Institute of Technology Newark, NJ 07102, USA Abstract The
More informationComplexity and Algorithms for Two-Stage Flexible Flowshop Scheduling with Availability Constraints
Complexity and Algorithms or Two-Stage Flexible Flowshop Scheduling with Availability Constraints Jinxing Xie, Xijun Wang Department o Mathematical Sciences, Tsinghua University, Beijing 100084, China
More informationHeuristics for two-machine flowshop scheduling with setup times and an availability constraint
Heuristics for two-machine flowshop scheduling with setup times and an availability constraint Wei Cheng A Thesis Submitted to the University of North Carolina Wilmington in Partial Fulfillment Of the
More informationResearch Article Batch Scheduling on Two-Machine Flowshop with Machine-Dependent Setup Times
Advances in Operations Research Volume 2009, Article ID 153910, 10 pages doi:10.1155/2009/153910 Research Article Batch Scheduling on Two-Machine Flowshop with Machine-Dependent Setup Times Lika Ben-Dati,
More informationScheduling Linear Deteriorating Jobs with an Availability Constraint on a Single Machine 1
Scheduling Linear Deteriorating Jobs with an Availability Constraint on a Single Machine 1 Min Ji a, b, 2 Yong He b, 3 T.C.E. Cheng c, 4 a College of Computer Science & Information Engineering, Zhejiang
More informationImproved Bounds for Flow Shop Scheduling
Improved Bounds for Flow Shop Scheduling Monaldo Mastrolilli and Ola Svensson IDSIA - Switzerland. {monaldo,ola}@idsia.ch Abstract. We resolve an open question raised by Feige & Scheideler by showing that
More informationOptimal on-line algorithms for single-machine scheduling
Optimal on-line algorithms for single-machine scheduling J.A. Hoogeveen A.P.A. Vestjens Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O.Box 513, 5600 MB, Eindhoven,
More informationScheduling jobs with agreeable processing times and due dates on a single batch processing machine
Theoretical Computer Science 374 007 159 169 www.elsevier.com/locate/tcs Scheduling jobs with agreeable processing times and due dates on a single batch processing machine L.L. Liu, C.T. Ng, T.C.E. Cheng
More informationScheduling Parallel Jobs with Linear Speedup
Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev, m.uetz}@ke.unimaas.nl
More informationMinimizing the total flow-time on a single machine with an unavailability period
Minimizing the total flow-time on a single machine with an unavailability period Julien Moncel (LAAS-CNRS, Toulouse France) Jérémie Thiery (DIAGMA Supply Chain, Paris France) Ariel Waserhole (G-SCOP, Grenoble
More informationOn Machine Dependency in Shop Scheduling
On Machine Dependency in Shop Scheduling Evgeny Shchepin Nodari Vakhania Abstract One of the main restrictions in scheduling problems are the machine (resource) restrictions: each machine can perform at
More informationSingle Machine Scheduling with Job-Dependent Machine Deterioration
Single Machine Scheduling with Job-Dependent Machine Deterioration Wenchang Luo 1, Yao Xu 2, Weitian Tong 3, and Guohui Lin 4 1 Faculty of Science, Ningbo University. Ningbo, Zhejiang 315211, China; and
More informationAlgorithm Design. Scheduling Algorithms. Part 2. Parallel machines. Open-shop Scheduling. Job-shop Scheduling.
Algorithm Design Scheduling Algorithms Part 2 Parallel machines. Open-shop Scheduling. Job-shop Scheduling. 1 Parallel Machines n jobs need to be scheduled on m machines, M 1,M 2,,M m. Each machine can
More informationAn FPTAS for parallel-machine scheduling under a grade of service provision to minimize makespan
An FPTAS for parallel-machine scheduling under a grade of service provision to minimize makespan Min Ji 1 College of Computer Science & Information Engineering, Zhejiang Gongshang University, Hangzhou
More informationThis means that we can assume each list ) is
This means that we can assume each list ) is of the form ),, ( )with < and Since the sizes of the items are integers, there are at most +1pairs in each list Furthermore, if we let = be the maximum possible
More informationarxiv: v1 [math.oc] 3 Jan 2019
The Product Knapsack Problem: Approximation and Complexity arxiv:1901.00695v1 [math.oc] 3 Jan 2019 Ulrich Pferschy a, Joachim Schauer a, Clemens Thielen b a Department of Statistics and Operations Research,
More informationThe coordinated scheduling of steelmaking with multi-refining and tandem transportation
roceedings of the 17th World Congress The International Federation of Automatic Control The coordinated scheduling of steelmaking with multi-refining and tandem transportation Jing Guan*, Lixin Tang*,
More informationBatch delivery scheduling with simple linear deterioration on a single machine 1
Acta Technica 61, No. 4A/2016, 281 290 c 2017 Institute of Thermomechanics CAS, v.v.i. Batch delivery scheduling with simple linear deterioration on a single machine 1 Juan Zou 2,3 Abstract. Several single
More informationSingle machine scheduling with forbidden start times
4OR manuscript No. (will be inserted by the editor) Single machine scheduling with forbidden start times Jean-Charles Billaut 1 and Francis Sourd 2 1 Laboratoire d Informatique Université François-Rabelais
More information4 Sequencing problem with heads and tails
4 Sequencing problem with heads and tails In what follows, we take a step towards multiple stage problems Therefore, we consider a single stage where a scheduling sequence has to be determined but each
More informationA half-product based approximation scheme for agreeably weighted completion time variance
A half-product based approximation scheme for agreeably weighted completion time variance Jinliang Cheng a and Wieslaw Kubiak b Faculty of Business Administration Memorial University of Newfoundland St.
More informationUniversity of Twente. Faculty of Mathematical Sciences. Scheduling split-jobs on parallel machines. University for Technical and Social Sciences
Faculty of Mathematical Sciences University of Twente University for Technical and Social Sciences P.O. Box 217 7500 AE Enschede The Netherlands Phone: +31-53-4893400 Fax: +31-53-4893114 Email: memo@math.utwente.nl
More informationHYBRID FLOW-SHOP WITH ADJUSTMENT
K Y BERNETIKA VOLUM E 47 ( 2011), NUMBER 1, P AGES 50 59 HYBRID FLOW-SHOP WITH ADJUSTMENT Jan Pelikán The subject of this paper is a flow-shop based on a case study aimed at the optimisation of ordering
More informationAccepted Manuscript. An Approximation Algorithm for the Three-Machine Scheduling Problem with the Routes Given by the Same Partial Order
Accepted Manuscript An Approximation Algorithm for the Three-Machine Scheduling Problem with the Routes Given by the Same Partial Order Richard Quibell, Vitaly A. Strusevich PII: S0360-8352(14)00258-7
More informationApproximation Schemes for Job Shop Scheduling Problems with Controllable Processing Times
Approximation Schemes for Job Shop Scheduling Problems with Controllable Processing Times Klaus Jansen 1, Monaldo Mastrolilli 2, and Roberto Solis-Oba 3 1 Universität zu Kiel, Germany, kj@informatik.uni-kiel.de
More informationA note on the complexity of the concurrent open shop problem
J Sched (2006) 9: 389 396 DOI 10.1007/s10951-006-7042-y A note on the complexity of the concurrent open shop problem Thomas A. Roemer C Science + Business Media, LLC 2006 Abstract The concurrent open shop
More informationTheoretical Computer Science
Theoretical Computer Science 411 (010) 417 44 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: wwwelseviercom/locate/tcs Resource allocation with time intervals
More informationarxiv: v2 [cs.dm] 2 Mar 2017
Shared multi-processor scheduling arxiv:607.060v [cs.dm] Mar 07 Dariusz Dereniowski Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, Gdańsk, Poland Abstract
More informationLecture 2: Scheduling on Parallel Machines
Lecture 2: Scheduling on Parallel Machines Loris Marchal October 17, 2012 Parallel environment alpha in Graham s notation): P parallel identical Q uniform machines: each machine has a given speed speed
More informationAlgorithms. Outline! Approximation Algorithms. The class APX. The intelligence behind the hardware. ! Based on
6117CIT - Adv Topics in Computing Sci at Nathan 1 Algorithms The intelligence behind the hardware Outline! Approximation Algorithms The class APX! Some complexity classes, like PTAS and FPTAS! Illustration
More informationAPTAS for Bin Packing
APTAS for Bin Packing Bin Packing has an asymptotic PTAS (APTAS) [de la Vega and Leuker, 1980] For every fixed ε > 0 algorithm outputs a solution of size (1+ε)OPT + 1 in time polynomial in n APTAS for
More informationApproximation Schemes for Parallel Machine Scheduling Problems with Controllable Processing Times
Approximation Schemes for Parallel Machine Scheduling Problems with Controllable Processing Times Klaus Jansen 1 and Monaldo Mastrolilli 2 1 Institut für Informatik und Praktische Mathematik, Universität
More informationPartial job order for solving the two-machine flow-shop minimum-length problem with uncertain processing times
Preprints of the 13th IFAC Symposium on Information Control Problems in Manufacturing, Moscow, Russia, June 3-5, 2009 Fr-A2.3 Partial job order for solving the two-machine flow-shop minimum-length problem
More informationSemi-Online Multiprocessor Scheduling with Given Total Processing Time
Semi-Online Multiprocessor Scheduling with Given Total Processing Time T.C. Edwin Cheng Hans Kellerer Vladimir Kotov Abstract We are given a set of identical machines and a sequence of jobs, the sum of
More informationEmbedded Systems 14. Overview of embedded systems design
Embedded Systems 14-1 - Overview of embedded systems design - 2-1 Point of departure: Scheduling general IT systems In general IT systems, not much is known about the computational processes a priori The
More informationCompletion Time Scheduling and the WSRPT Algorithm
Connecticut College Digital Commons @ Connecticut College Computer Science Faculty Publications Computer Science Department Spring 4-2012 Completion Time Scheduling and the WSRPT Algorithm Christine Chung
More informationHigh Multiplicity Scheduling on One Machine with Forbidden Start and Completion Times
High Multiplicity Scheduling on One Machine with Forbidden Start and Completion Times Michaël Gabay, Christophe Rapine, Nadia Brauner Abstract We are interested in a single machine scheduling problem where
More informationApproximation schemes for parallel machine scheduling with non-renewable resources
Approximation schemes for parallel machine scheduling with non-renewable resources Péter Györgyi a,b, Tamás Kis b, a Department of Operations Research, Loránd Eötvös University, H1117 Budapest, Pázmány
More informationComplexity analysis of the discrete sequential search problem with group activities
Complexity analysis of the discrete sequential search problem with group activities Coolen K, Talla Nobibon F, Leus R. KBI_1313 Complexity analysis of the discrete sequential search problem with group
More information5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
More informationScheduling linear deteriorating jobs with an availability constraint on a single machine
Theoretical Computer Science 362 (2006 115 126 www.elsevier.com/locate/tcs Scheduling linear deteriorating jobs with an availability constraint on a single machine Min Ji a,b, Yong He b, T.C.E. Cheng c,
More informationOnline Interval Coloring and Variants
Online Interval Coloring and Variants Leah Epstein 1, and Meital Levy 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. Email: lea@math.haifa.ac.il School of Computer Science, Tel-Aviv
More informationPartition is reducible to P2 C max. c. P2 Pj = 1, prec Cmax is solvable in polynomial time. P Pj = 1, prec Cmax is NP-hard
I. Minimizing Cmax (Nonpreemptive) a. P2 C max is NP-hard. Partition is reducible to P2 C max b. P Pj = 1, intree Cmax P Pj = 1, outtree Cmax are both solvable in polynomial time. c. P2 Pj = 1, prec Cmax
More informationBin packing and scheduling
Sanders/van Stee: Approximations- und Online-Algorithmen 1 Bin packing and scheduling Overview Bin packing: problem definition Simple 2-approximation (Next Fit) Better than 3/2 is not possible Asymptotic
More informationResearch Article Minimizing the Number of Tardy Jobs on a Single Machine with an Availability Constraint
Journal of Industrial Engineering, Article ID 568317, 13 pages http://dx.doi.org/10.1155/2014/568317 Research Article Minimizing the Number of Tardy Jobs on a Single Machine with an Availability Constraint
More informationScheduling for Parallel Dedicated Machines with a Single Server
Scheduling for Parallel Dedicated Machines with a Single Server Celia A. Glass, 1 Yakov M. Shafransky, 2 Vitaly A. Strusevich 3 1 City University, London, United Kingdom 2 Institute of Engineering Cybernetics,
More informationCMSC 451: Lecture 7 Greedy Algorithms for Scheduling Tuesday, Sep 19, 2017
CMSC CMSC : Lecture Greedy Algorithms for Scheduling Tuesday, Sep 9, 0 Reading: Sects.. and. of KT. (Not covered in DPV.) Interval Scheduling: We continue our discussion of greedy algorithms with a number
More informationMinimizing total weighted tardiness on a single machine with release dates and equal-length jobs
Minimizing total weighted tardiness on a single machine with release dates and equal-length jobs G. Diepen J.M. van den Akker J.A. Hoogeveen institute of information and computing sciences, utrecht university
More informationPolynomial time solutions for scheduling problems on a proportionate flowshop with two competing agents
Journal of the Operational Research Society (2014) 65, 151 157 2014 Operational Research Society Ltd All rights reserved 0160-5682/14 wwwpalgrave-journalscom/jors/ Polynomial time solutions for scheduling
More informationM 2 M 3. Robot M (O)
R O M A TRE DIA Universita degli Studi di Roma Tre Dipartimento di Informatica e Automazione Via della Vasca Navale, 79 { 00146 Roma, Italy Part Sequencing in Three Machine No-Wait Robotic Cells Alessandro
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should
More informationCS264: Beyond Worst-Case Analysis Lecture #18: Smoothed Complexity and Pseudopolynomial-Time Algorithms
CS264: Beyond Worst-Case Analysis Lecture #18: Smoothed Complexity and Pseudopolynomial-Time Algorithms Tim Roughgarden March 9, 2017 1 Preamble Our first lecture on smoothed analysis sought a better theoretical
More informationThe unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan
The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan J.J. YUAN 1,, Z.H. Liu 2,, C.T. NG and T.C.E. CHENG 1 Department of Mathematics, Zhengzhou
More informationLPT rule: Whenever a machine becomes free for assignment, assign that job whose. processing time is the largest among those jobs not yet assigned.
LPT rule Whenever a machine becomes free for assignment, assign that job whose processing time is the largest among those jobs not yet assigned. Example m1 m2 m3 J3 Ji J1 J2 J3 J4 J5 J6 6 5 3 3 2 1 3 5
More informationMINIMIZING TOTAL TARDINESS FOR SINGLE MACHINE SEQUENCING
Journal of the Operations Research Society of Japan Vol. 39, No. 3, September 1996 1996 The Operations Research Society of Japan MINIMIZING TOTAL TARDINESS FOR SINGLE MACHINE SEQUENCING Tsung-Chyan Lai
More informationCOSC 341: Lecture 25 Coping with NP-hardness (2)
1 Introduction Figure 1: Famous cartoon by Garey and Johnson, 1979 We have seen the definition of a constant factor approximation algorithm. The following is something even better. 2 Approximation Schemes
More informationLecture 13. Real-Time Scheduling. Daniel Kästner AbsInt GmbH 2013
Lecture 3 Real-Time Scheduling Daniel Kästner AbsInt GmbH 203 Model-based Software Development 2 SCADE Suite Application Model in SCADE (data flow + SSM) System Model (tasks, interrupts, buses, ) SymTA/S
More informationMachine Minimization for Scheduling Jobs with Interval Constraints
Machine Minimization for Scheduling Jobs with Interval Constraints Julia Chuzhoy Sudipto Guha Sanjeev Khanna Joseph (Seffi) Naor Abstract The problem of scheduling jobs with interval constraints is a well-studied
More informationThe Constrained Minimum Weighted Sum of Job Completion Times Problem 1
The Constrained Minimum Weighted Sum of Job Completion Times Problem 1 Asaf Levin 2 and Gerhard J. Woeginger 34 Abstract We consider the problem of minimizing the weighted sum of job completion times on
More informationOn-line Scheduling to Minimize Max Flow Time: An Optimal Preemptive Algorithm
On-line Scheduling to Minimize Max Flow Time: An Optimal Preemptive Algorithm Christoph Ambühl and Monaldo Mastrolilli IDSIA Galleria 2, CH-6928 Manno, Switzerland October 22, 2004 Abstract We investigate
More informationScheduling in an Assembly-Type Production Chain with Batch Transfer
This is the Pre-Published Version. Scheduling in an Assembly-Type Production Chain with Batch Transfer B.M.T. Lin 1,#, T.C.E. Cheng 2 and A.S.C. Chou 3 1 Department of Information and Finance Management
More informationPolynomially solvable and NP-hard special cases for scheduling with heads and tails
Polynomially solvable and NP-hard special cases for scheduling with heads and tails Elisa Chinos, Nodari Vakhania Centro de Investigación en Ciencias, UAEMor, Mexico Abstract We consider a basic single-machine
More informationA comparison of sequencing formulations in a constraint generation procedure for avionics scheduling
A comparison of sequencing formulations in a constraint generation procedure for avionics scheduling Department of Mathematics, Linköping University Jessika Boberg LiTH-MAT-EX 2017/18 SE Credits: Level:
More informationRecoverable Robustness in Scheduling Problems
Master Thesis Computing Science Recoverable Robustness in Scheduling Problems Author: J.M.J. Stoef (3470997) J.M.J.Stoef@uu.nl Supervisors: dr. J.A. Hoogeveen J.A.Hoogeveen@uu.nl dr. ir. J.M. van den Akker
More informationOn Two Class-Constrained Versions of the Multiple Knapsack Problem
On Two Class-Constrained Versions of the Multiple Knapsack Problem Hadas Shachnai Tami Tamir Department of Computer Science The Technion, Haifa 32000, Israel Abstract We study two variants of the classic
More informationNon-Preemptive and Limited Preemptive Scheduling. LS 12, TU Dortmund
Non-Preemptive and Limited Preemptive Scheduling LS 12, TU Dortmund 09 May 2017 (LS 12, TU Dortmund) 1 / 31 Outline Non-Preemptive Scheduling A General View Exact Schedulability Test Pessimistic Schedulability
More informationSingle processor scheduling with time restrictions
Single processor scheduling with time restrictions Oliver Braun Fan Chung Ron Graham Abstract We consider the following scheduling problem 1. We are given a set S of jobs which are to be scheduled sequentially
More informationSingle processor scheduling with time restrictions
J Sched manuscript No. (will be inserted by the editor) Single processor scheduling with time restrictions O. Braun F. Chung R. Graham Received: date / Accepted: date Abstract We consider the following
More informationOnline Scheduling with Bounded Migration
Online Scheduling with Bounded Migration Peter Sanders Universität Karlsruhe (TH), Fakultät für Informatik, Postfach 6980, 76128 Karlsruhe, Germany email: sanders@ira.uka.de http://algo2.iti.uni-karlsruhe.de/sanders.php
More informationHeuristics Algorithms For Job Sequencing Problems
Global Journal of Science Frontier Research Vol.10 Issue 4(Ver1.0),September 2010 P a g e 37 Heuristics Algorithms For Job Sequencing Problems K. Karthikeyan GJSFR Classification D (FOR) 091307,091305,010303
More informationAssigning operators in a flow shop environment
Assigning operators in a flow shop environment Imène Benkalai 1, Djamal Rebaine 1, and Pierre Baptiste 2 1 Université du Québec à Chicoutimi, Saguenay (Québec), Canada G7H-2B1. {imene.benkalai1,drebaine}@uqac.ca
More informationP C max. NP-complete from partition. Example j p j What is the makespan on 2 machines? 3 machines? 4 machines?
Multiple Machines Model Multiple Available resources people time slots queues networks of computers Now concerned with both allocation to a machine and ordering on that machine. P C max NP-complete from
More informationNo-Idle, No-Wait: When Shop Scheduling Meets Dominoes, Eulerian and Hamiltonian Paths
No-Idle, No-Wait: When Shop Scheduling Meets Dominoes, Eulerian and Hamiltonian Paths J.C. Billaut 1, F.Della Croce 2, Fabio Salassa 2, V. T kindt 1 1. Université Francois-Rabelais, CNRS, Tours, France
More informationarxiv: v1 [cs.ds] 15 Sep 2018
Approximation algorithms for the three-machine proportionate mixed shop scheduling Longcheng Liu Yong Chen Jianming Dong Randy Goebel Guohui Lin Yue Luo Guanqun Ni Bing Su An Zhang arxiv:1809.05745v1 [cs.ds]
More informationMachine scheduling with resource dependent processing times
Mathematical Programming manuscript No. (will be inserted by the editor) Alexander Grigoriev Maxim Sviridenko Marc Uetz Machine scheduling with resource dependent processing times Received: date / Revised
More informationEmbedded Systems 15. REVIEW: Aperiodic scheduling. C i J i 0 a i s i f i d i
Embedded Systems 15-1 - REVIEW: Aperiodic scheduling C i J i 0 a i s i f i d i Given: A set of non-periodic tasks {J 1,, J n } with arrival times a i, deadlines d i, computation times C i precedence constraints
More informationAn analysis of the LPT algorithm for the max min and the min ratio partition problems
Theoretical Computer Science 349 (2005) 407 419 www.elsevier.com/locate/tcs An analysis of the LPT algorithm for the max min and the min ratio partition problems Bang Ye Wu Department of Computer Science
More informationParallel machine scheduling with batch delivery costs
Int. J. Production Economics 68 (2000) 177}183 Parallel machine scheduling with batch delivery costs Guoqing Wang*, T.C. Edwin Cheng Department of Business Administration, Jinan University, Guangzhou,
More informationOn-line Bin-Stretching. Yossi Azar y Oded Regev z. Abstract. We are given a sequence of items that can be packed into m unit size bins.
On-line Bin-Stretching Yossi Azar y Oded Regev z Abstract We are given a sequence of items that can be packed into m unit size bins. In the classical bin packing problem we x the size of the bins and try
More informationFlow Shop and Job Shop Models
Outline DM87 SCHEDULING, TIMETABLING AND ROUTING Lecture 11 Flow Shop and Job Shop Models 1. Flow Shop 2. Job Shop Marco Chiarandini DM87 Scheduling, Timetabling and Routing 2 Outline Resume Permutation
More informationAn Approximate Pareto Set for Minimizing the Maximum Lateness and Makespan on Parallel Machines
1 An Approximate Pareto Set for Minimizing the Maximum Lateness Makespan on Parallel Machines Gais Alhadi 1, Imed Kacem 2, Pierre Laroche 3, Izzeldin M. Osman 4 arxiv:1802.10488v1 [cs.ds] 28 Feb 2018 Abstract
More informationNon-Work-Conserving Non-Preemptive Scheduling: Motivations, Challenges, and Potential Solutions
Non-Work-Conserving Non-Preemptive Scheduling: Motivations, Challenges, and Potential Solutions Mitra Nasri Chair of Real-time Systems, Technische Universität Kaiserslautern, Germany nasri@eit.uni-kl.de
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 P and NP P: The family of problems that can be solved quickly in polynomial time.
More informationThe Maximum Flow Problem with Disjunctive Constraints
The Maximum Flow Problem with Disjunctive Constraints Ulrich Pferschy Joachim Schauer Abstract We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative
More informationCS264: Beyond Worst-Case Analysis Lecture #15: Smoothed Complexity and Pseudopolynomial-Time Algorithms
CS264: Beyond Worst-Case Analysis Lecture #15: Smoothed Complexity and Pseudopolynomial-Time Algorithms Tim Roughgarden November 5, 2014 1 Preamble Previous lectures on smoothed analysis sought a better
More informationEmbedded Systems Development
Embedded Systems Development Lecture 3 Real-Time Scheduling Dr. Daniel Kästner AbsInt Angewandte Informatik GmbH kaestner@absint.com Model-based Software Development Generator Lustre programs Esterel programs
More informationOn-line Scheduling of Two Parallel Machines. with a Single Server
On-line Scheduling of Two Parallel Machines with a Single Server Lele Zhang, Andrew Wirth Department of Mechanical and Manufacturing Engineering, The University of Melbourne, VIC 3010, Australia Abstract
More informationThe Power of Preemption on Unrelated Machines and Applications to Scheduling Orders
MATHEMATICS OF OPERATIONS RESEARCH Vol. 37, No. 2, May 2012, pp. 379 398 ISSN 0364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/10.1287/moor.1110.0520 2012 INFORMS The Power of Preemption on
More informationDiscrete Applied Mathematics. Tighter bounds of the First Fit algorithm for the bin-packing problem
Discrete Applied Mathematics 158 (010) 1668 1675 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Tighter bounds of the First Fit algorithm
More informationA Robust APTAS for the Classical Bin Packing Problem
A Robust APTAS for the Classical Bin Packing Problem Leah Epstein 1 and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. Email: lea@math.haifa.ac.il 2 Department of Statistics,
More informationOperations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 27 Flow Shop Scheduling - Heuristics - Palmer, Campbell Dudek
More informationA PTAS for Static Priority Real-Time Scheduling with Resource Augmentation
A PTAS for Static Priority Real-Time Scheduling with Resource Augmentation Friedrich Eisenbrand and Thomas Rothvoß Institute of Mathematics École Polytechnique Féderale de Lausanne, 1015 Lausanne, Switzerland
More informationSPT is Optimally Competitive for Uniprocessor Flow
SPT is Optimally Competitive for Uniprocessor Flow David P. Bunde Abstract We show that the Shortest Processing Time (SPT) algorithm is ( + 1)/2-competitive for nonpreemptive uniprocessor total flow time
More informationHeuristics for the two-stage job shop scheduling problem with a bottleneck machine
European Journal of Operational Research 123 (2000) 229±240 www.elsevier.com/locate/orms Heuristics for the two-stage job shop scheduling problem with a bottleneck machine I.G. Drobouchevitch, V.A. Strusevich
More informationCost models for lot streaming in a multistage flow shop
Omega 33 2005) 435 450 www.elsevier.com/locate/omega Cost models for lot streaming in a multistage flow shop Huan Neng Chiu, Jen Huei Chang Department of Industrial Management, National Taiwan University
More informationOnline Scheduling of Parallel Jobs on Two Machines is 2-Competitive
Online Scheduling of Parallel Jobs on Two Machines is 2-Competitive J.L. Hurink and J.J. Paulus University of Twente, P.O. box 217, 7500AE Enschede, The Netherlands Abstract We consider online scheduling
More information