Bicriterion scheduling with truncated learning effects and convex controllable processing times

This paper investigates single-machine scheduling in which the processing time of a job is a function of its position in a sequence, a truncation parameter, and its resource allocation. For a convex resource consumption function, we provide a bicriteria analysis where the first is to minimize total weighted flow (completion) time, and the second is to minimize total resource consumption cost. If the weights are positional-dependent weights, we prove that three versions of considering the two criteria can be solved in polynomial time, respectively. If the weights are job-dependent weights, the computational complexity of the three versions of the two criteria remains an open question. To solve the problems with job-dependent weights, we present a heuristic (an upper bound) and a branch-and-bound algorithm (an exact solution).