Bicriterion Single Machine Scheduling with Resource Dependent Processing Times

A bicriterion problem of scheduling jobs on a single machine is studied. The processing time of each job is a linear decreasing function of the amount of a common discrete resource allocated to the job. A solution is specified by a sequence of the jobs and a resource allocation. The quality of a solution is measured by two criteria, F1 and F2. The first criterion is the maximal or total (weighted) resource consumption, and the second criterion is a regular scheduling criterion depending on the job completion times. Both criteria have to be minimized. General schemes for the construction of the Pareto set and the Pareto set ε-approximation are presented. Computational complexities of problems to minimize F1 subject to F_2≤ K$ and to minimize F2 subject to $F_1≤ K$, where K is any number, are studied for various functions F1 and F2. Algorithms for solving these problems and for the construction of the Pareto set and the Pareto set $ε$-approximation for the corresponding bicriterion problems are presented.