Scheduling n Independent Jobs on m Uniform Machines with both Flowtime and Makespan Objectives: A Parametric Analysis

We consider the problem of scheduling n jobs without precedence constraints on m uniform machines (i.e., the machines are identical except for speed), with preemptions allowed at no cost. We are interested in generating the entire tradeoff curve of schedules which are Pareto-optimal (undominated) for the flowtime and makespan objectives. To achieve this, we first develop an O(mn) algorithm that produces a schedule with minimum flowtime, subject to a fixed makespan deadline. This algorithm alternates between the Shortest Processing Time on Fastest Machine (SPT-FM) rule and the Longest Remaining Processing Time on Fastest Machine (LRPT-FM) rule. We then investigate how the behavior of the algorithm changes as the deadline is varied parametrically. Our knowledge of the structure of optimal schedules allows us to characterize breakpoints on the (piecewise linear) tradeoff curve, and then to compute all of the O(mn) breakpoints in O(mn) time. Our analysis yields various useful sensitivity results as a by-product.