Job selection in a heavily loaded shop

Scope and Purpose-In a recent paper. Slotnick and Morton introduce a problem of job selection in a heavily loaded shop. The problem is both practically relevant and theoretically interesting. As Slotnick and Morton point out, this kind of problems have just begun to receive attention in the research literature. For the problem at hand. Slotnick and Morton provide several algorithms (one exact and two approximate). They, however, do not establish the complexity status of the problem; nor do they provide any exact algorithm that is formally efficient or any approximate algorithm that guarantees performance within a specified bound. In this note, we attempt to address some of these unresolved issues. Abstract-Recently, Slotnick and Morton address a job selection problem in a heavily loaded shop, where a tradeoff is sought between the reward obtained when a job is accepted for processing and the lateness penalty incurred when such a job is actually delivered. They provide a branch and bound algorithm and a couple of heuristics for the problem's solution. They do not;however, resolve the issue of problem complexity. In this note. we first establish that the problem is NP-hard. We then go on to provide two pseudo-polynomial time algorithms which also show that the problem is solvable in polynomial time if either the job processing times or the job weights for the lateness penalty are equal. We further provide a fully polynomial time approximation scheme which always generates a solution within a specified percentage of the optimal.