This study considers the identical parallel machines operational fixed job scheduling problem with machine-dependent job weights. A job is either processed in a fixed interval or is not processed at all. Our aim is to maximise the total weight of the processed jobs. We show that the problem with machine eligibility constraints resides as a special case of this problem. We identify some special polynomially solvable cases and propose a branch-and-bound (BB) algorithm that employs efficient bounding schemes and dominance conditions. Computational experience on large-sized problem examples reveals the satisfactory performance of the BB algorithm.