NP-hardness of the single-variable-resource scheduling problem to minimize the total weighted completion time

Baker and Nuttle studied the following single-variable-resource scheduling problem: sequencing n jobs for processing by a single-resource to minimize a function of job completion times, when the availability of the resource varies over time. When the objective function to be minimized is the total weighted completion time, Baker and Nuttle conjectured that the problem is NP-hard. We show in this note that the conjecture is true.