A heuristic approach to allocating the continuous resource in discrete–continuous scheduling problems to minimize the makespan

A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemptable and independent and all are available at the start of the process. The total amount of the continuous resource available at a time is limited and the resource is a renewable one. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. The processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation which minimize the makespan. A heuristic approach to allocating the continuous resource is proposed. A tabu search algorithm to solve the considered problem is presented and the results for the algorithms with exact and heuristic procedures for allocating the continuous resource are compared on the basis of some computational experiments.