Baptiste, P. (2000) Scheduling equal-length jobs on identical parallel machines, Discrete Applied Mathematics 103, 21-32.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....operation O mj is completed on Mm at time C mj = S 1j s mp. Thus, the problem reduces to the single machine problem 1jp j = p; r j jf for the last machine. Since for the objective functions f 2 f T j g the corresponding single machine problems are polynomially solvable (Baptiste [2] [3]) we conclude that problems F2; S1jp ij = p; s ij = 1; r j j polynomially solvable. Next we consider the case s p. For an arbitrary function f the property that an optimal permutation plan exists is no longer valid. A counterexample with the objective function U j can be found in ....

Baptiste, P. (2000) Scheduling equal-length jobs on identical parallel machines, Discrete Applied Mathematics 103, 21-32.

....problem; the cost of this optimal solution represents a lower bound on the initial problem. Unfortunately, most of the WCT problem relaxations along the natural dimensions are also NP hard. One exception is the WCT problem with release dates but without precedence constraints for which Baptiste [14] recently discovered a polynomial time algorithm. We outline the two key observations behind Baptiste s algorithm. First, the times at which jobs start and end in the optimal schedule belong to the set of release dates and their multiples (up to a factor of n) Next, a resource profile is ....

P. Baptiste. Scheduling equal-length jobs on identical parallel machines. Research Report UTC, 1998.

....a lower bound on the initial problem. Unfortunately, most of the WCT problem relaxations along the natural dimensions are also NP hard. One exception is the WCT problem with release dates but without precedence constraints for which Baptiste recently discovered a polynomial time algorithm [2]. We outline the two key observations behind Baptiste s algorithm. First, the times at which jobs start and end in the optimal schedule belong to the set of release dates and their multiples (up to a factor of n) Next, a resource profile is defined as a vector ( 1 ; 2 ; m ) such that ....

P. Baptiste. Scheduling equal-length jobs on identical parallel machines. Research Report UTC, 1998.

.... Thus, problem P; S1 j p j = 1; r j ; s j = s j P w j C j , problem P; S1 j p j = 1; r j ; s j = s j P w j U j and problem P; S1 j p j = 1; r j ; s j = s j P T j are polynomially solvable since the corresponding parallel and single machine problems are polynomially solvable (see Baptiste [1] [2]) 5 Finally, we consider problems P; S1 j p j = p; r j ; s j = s j C max and P; S1 j p j = p; r j ; s j = s j P C j : It is easy to prove by exchange arguments that these problems are solved by scheduling all jobs in the order of nondecreasing release dates. 4 Problem P2; S1 j p j = p j C ....

Baptiste, P. (1998) Scheduling equal-length jobs on identical parallel machines, Technical Report, Universite de Technologie de Compiegne, France.

....d j g meet these conditions, P jpmtn; p j = pj T j is in the class. Note that T j is a piecewise linear function. We assume that the jobs are in a linear order where for each pair of jobs J i and J j the functions f i f j are either strictly increasing or constant if i j. As it is shown in [2], such an order always exists. It can be trivially shown by the exchange argument that for any problem in the class there exists an optimal schedule where i j ) C i C j . Let us consider completion times C j for all j = 1; 2; n as deadlines. It is known [13] that a feasible schedule ....

....O(n [2(2m 1) This implies that the problem can be solved by dynamic programming in polynomial time. 9 Pmjr j ; p j We describe a decomposition scheme that follows the technique used earlier for 1jr j ; p j = pj w j U j [1] Pmjr j ; p j = pj w j C j , Pmjr j ; p j = pj T j [2] and a suggestion in [10] Let us de ne a resource pro le to be a vector a = a 1 ; a 2 ; am ) with integral components in T such that a max a min p, and the components associated with the machines M 1 ; M 2 ; Mm . We assume that a min is the completion time of a job, a i = a ....

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Ph. Baptiste, Scheduling equal-length jobs on identical parallel machines, Discrete Applied Mathematics 103 (2000) 21-32.

....0; 0) Note that this dynamic programming scheme does not run in polynomial time since the variables t i can take any integer value. However, it s easy to see that on an active schedule, tasks start and are completed in T = ft j 9r i ; 9l 2 f0; ng; t = r i lpg: See for instance [1] for a proof of a similar claim. Since there is an optimal active schedule and since there are O(n 2 ) time points in T , we have no more than O(n 2m ) possible tuples of processor variables to consider. On top of that, there are O(n m ) tuples of task variables. So we have O(n 3m ) ....

Ph. Baptiste. Scheduling Equal-Length Jobs on Identical Parallel Machines. Discrete Applied Mathematics, 103:21-32, 2000.

....or can be reduced to such functions. The dynamic programming algorithms for serial and parallel problems are Batching Identical Jobs 5 described in x5 and x6. Finally in x7 we show that our approach can be extended to handle T max and we draw some conclusions. 2 Starting Times As shown in [3, 4, 9, 10, 12], one of the reasons why it is easy to schedule equal length jobs, either on a single or on parallel machines, is that there are few possible starting times. Indeed, left shifted schedules are dominant and thus starting times are equal to a release date modulo p. As we will see later on the ....

....end in P, where P = fr p; 2 f1; ng; 2 f0; ngg (2) Proof. Similar to the proof of Theorem 1. Notice that jPj = O(n 2 ) 3 Ordered Objective Functions Ordered objective functions form a particular class of functions. A strict subclass of this class has been studied in [4], where it has been shown that scheduling identical jobs on a xed number of machines can be done in polynomial time for some objective functions including P w i C i ; P T i but not P w i U i . The class of ordered objective functions is more general (in particular, it includes P w i U i ) ....

Ph. Baptiste, Scheduling Equal-Length Jobs on Identical Parallel Machines, Technical Report 98-159(1998), Universite de Technologie de Compiegne (submitted).

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P. Baptiste, Scheduling equal-length jobs on identical parallel machines, Technical Report, Universite de Technologie de Compiegne, France, 1998.

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