Abstract
We considered the online scheduling problem of simple linear deteriorating job families on m parallel batch machines to minimize the makespan, where the batch capacity is unbounded. In this paper, simple linear deteriorating jobs mean that the actual processing time p j of job J j is assumed to be a linear function of its starting time s j , i.e., p j = α j s j , where α j > 0 is the deterioration rate. Job families mean that one job must belong to some job family, and jobs of different families cannot be processed in the same batch. When m = 1 , we provide the best possible online algorithm with the competitive ratio of ( 1 + α max ) f , where f is the number of job families and α max is the maximum deterioration rate of all jobs. When m ≥ 1 and m = f , we provide the best possible online algorithm with the competitive ratio of 1 + α max .
Highlights
BackgroundAll jobs arrive over time, i.e., each job has an arrival time. Before the jobs arrive, we do not know any information, including arrival time, processing time, deterioration rate, etc
Job families mean that one job must belong to some job family, and jobs of different families cannot be processed in the same batch
To minimize the makespan of the online scheduling problem on m parallel machines with linear deteriorating jobs, Cheng et al [19] constructed an algorithm and proved that the bound of the competitive ratio of the algorithm is tight, where m = 2 and the largest deterioration rate of jobs is known in advance
Summary
All jobs arrive over time, i.e., each job has an arrival time. Before the jobs arrive, we do not know any information, including arrival time, processing time, deterioration rate, etc. For a minimization scheduling problem, we define the competitive ratio of the online algorithm A as:. Fu et al [3] studied the online algorithm on a single machine to minimize the makespan. Research is lacking on the parallel-batch online scheduling with incompatible deteriorating job families. The scheduling problem of deteriorating jobs was first introduced by Browne and Yechiali [8] and Gupta and Gupta [9], independently Both considered minimizing makespan on a single machine. To minimize the makespan of the online scheduling problem on m parallel machines with linear deteriorating jobs, Cheng et al [19] constructed an algorithm and proved that the bound of the competitive ratio of the algorithm is tight, where m = 2 and the largest deterioration rate of jobs is known in advance.
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