In this paper, we give a polynomial algorithm for problem P | r j , p j = p | f j (C j), where f j is any non-decreasing function such that for any indices i and j, function f i − f j is monotonous, and a polynomial algorithm for problem P | r j , p j = p, D j | max ϕ j (C j), where ϕ j is any non-decreasing function for any j.

SUMMARYParallel machine problems with a single server are generalizations of classical parallel machine problems. Immediately before processing, each job must be loaded on a machine, which takes a certain set-up time. All these set-ups have to be done by a single server which can handle at most one job at a time. In this paper we continue studying the complexity aspects of server problems begun in Hall et al. (Discrete Appl. Math. 2000; 102:223) and Kravchenko and Werner (Math. Comput. Model. 1997; 26:1). New complexity results are derived for special cases.

We study the problem of preemptive scheduling n jobs with given release times on m identical parallel machines. The objective is to minimize the average flow time. We show that when all jobs have equal processing times then the problem can be solved in polynomial time using linear programming. Our algorithm can also be applied to the open-shop problem with release times and unit processing times. For the general case (when processing times are arbitrary), we show that the problem is unary NP-hard.

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