Using tropical optimization techniques to solve time-constrained bi-objective project scheduling problems

We consider a project that consists of a set of activities performed in parallel under constraints on their start and finish times, including start-finish precedence relationships, release start times, release end times and deadlines. The problems of interest are to schedule the activities to minimize both the maximum flow-time over all activities and the project makespan. We formulate and solve the problems in the framework of tropical mathematics, which investigates the theory and applications of algebraic systems with idempotent operations, as tropical bi-objective optimization problems. As a result, we derive complete Pareto-optimal solutions in a direct explicit form, ready for further analysis and straightforward computation. We examine the computational complexity of the solution, and give illustrative examples.