Maximizing the Total Number of on TIME Jobs on Identical Machines

This paper studies the job-scheduling problem on m ≥ 2 parallel/identical machines.There are n jobs, denoted by Ji,1 ≤ i ≤ n. Each job Ji, has a due date di. A job has one or more tasks, each with a specific processing time. The tasks can’t be preempted, i.e., once scheduled, a task cannot be interrupted and resumed later. Different tasks of the same job can be scheduled concurrently on different machines. A job is on time if all of its tasks finish before its due date; otherwise, it is tardy. A schedule of the jobs specifies which task is scheduled on which machine at what time. The problem is to find a schedule of these jobs so that the number of on time jobs is maximized; or equivalently, the number of tardy jobs is minimized. We consider two cases: the case when each job has only a single task and the case where a job can have one or more tasks. For the first case, if all jobs have common due date we design a simple algorithm and show that the algorithm can generate a schedule whose number of on time jobs is at most (m-1) less than that of the optimal schedule. We also show that the modified algorithm works for the second case with common due date and has same performance. Finally, we design an algorithm when jobs have different due dates for the second case. We conduct computation experiment and show that the algorithm has very good performance.