An Island-Based Evolution Algorithm for Discrete-Continuous Scheduling with Continuous Resource Discretisation

A problem of scheduling nonpreemtable tasks on parallel identical machines under an additional continuous resource to minimize the schedule length is considered in the paper. A continuous resource is divisible continuously and is allocated to tasks fromgiven 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. The second step can be formulated as a convex programming problem with linear constraints that can be solved by specialized solvers. To eliminate time consuming optimal continuous resource allocation a problem Theta(Z) with continuous resource discretisation is introduced. Because Theta(Z) is NP-hard an island-based evolution algorithm (IBEA) is proposed to tackle the problem. Experimental results proved IBEA to be competitive with known algorithms for solving the considered problem.