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In this paper, the single machine scheduling problem with convex multi-resource dependent processing times, subject to meeting job deadlines is considered. The objective is to minimize the total cost, including the resource allocation costs and the fixed costs. We assume that the actual processing time of each job (task) is a function of the amount of resources allocated. Therefore, the decision variables of the model are: 1) resources allocated to the jobs, 2) total consumed resources, 3) processing times of the jobs, and 4) start/completion times of the jobs. We reformulate and solve the problem using a posynomial geometric programming model. In the proposed exact solution method based on the geometric programming, the original problem of any size is reduced to a two-variable unconstrainted optimization problem which can be easily solved by a simple grid search.

Single machine scheduling problem with convex multi-resource dependent processing times and job deadlines