Abstract: 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. Recently, Yuan, Cheng and Ng   showed that this problem is NP-hard in the binary sense, but the   exact complexity of the problem is still open. We show in this paper  that this problem is strongly NP-hard.