The single-processor scheduling problem with time restrictions: complexity and related problems

We consider the single-processor scheduling problem with time restrictions (STR) in order to minimize the makespan, where a set of independent jobs has to be processed on a single processor, subject only to the following constraint: During any time period of length α, the number of jobs being executed is less than or equal to a given integer value B. First, we prove that the STR problem is binary NP-hard even for B= 2 by pointing out an interesting analogy of this problem to the parallel machine scheduling problem with a single server with equal processing times. Next, we give a formal description of the STR problem by providing two mixed integer programming formulations to solve it optimally. The first is based on time-indexed variables and the second is based on assignment and positional date variables (APV). Both models were tested on randomly generated instances, and the experimental results show that the APV model performs very well and can solve problem instances of up to n= 500 jobs.