A Population Learning Algorithm for Discrete-Continuous Scheduling with Continuous Resource Discretisation

A problem of scheduling nonpreemptable tasks on parallel identical machines under constraint on discrete resource and requiring, additionally, renewable continuous resource to minimize the schedule length is considered in the paper. A continuous resource is divisible continuously and is allocated to tasks from given intervals in amounts unknown in advance. Task processing rate depends on the allocated amount of the continuous resource. The considered problem can be solved in two steps. The first step involves generating all possible task schedules and second - finding an optimal schedule among all schedules with optimal continuous resource allocation. To eliminate time consuming optimal continuous resource allocation, a problem Theta_Z with continuous resource discretisation is introduced. Because Theta _Z is NP-hard a population-learning algorithm (PLA) is proposed to tackle the problem. PLA belongs to the class of the population-based methods. Experiment results proved PLA to be competitive with known algorithms for solving the considered problem