Planning and Scheduling to Minimize Tardiness

We combine mixed integer linear programming (MILP) and constraint programming (CP) to minimize tardiness in planning and scheduling. Tasks are allocated to facilities using MILP and scheduled using CP, and the two are linked via logic-based Benders decomposition. We consider two objectives: minimizing the number of late tasks, and minimizing total tardiness. Our main theoretical contribution is a relaxation of the cumulative scheduling subproblem, which is critical to performance. We obtain substantial computational speedups relative to the state of the art in both MILP and CP. We also obtain much better solutions for problems that cannot be solved to optimality. We address a planning and scheduling problem that occurs frequently in manufacturing and supply chain contexts. Tasks must be assigned to facilities and scheduled on each facility subject to release dates and due dates. Tasks assigned to a given facility may run in parallel if desired, subject to a resource constraint (cumulative scheduling). We consider two objectives: minimizing the number of late tasks, and minimizing total tardiness. The problem can be formulated entirely as a constraint programming (CP) problem or a mixed integer/linear programming (MILP) problem. However, these models are hard to solve. By linking CP and MILP in a hybrid method, we obtain significant speedups relative to the state of the art in both MILP and CP. The linkage is achieved by logic-based Benders decomposition. The facility assignment problem becomes the master problem and is solved by MILP, while the scheduling problem becomes the subproblem (slave problem) and is solved by CP. The primary theoretical contribution of this paper is a linear relaxation of the cumulative scheduling subproblem. We find that including such a relaxation in the master problem is essential to the success of the Benders method. We solve problem instances in which tasks have the same release date and different due dates, although the the method is valid for different release dates as well. We obtain substantial speedups on nearly all instances relative to MILP (as represented by CPLEX), which in turn is generally faster than CP (as represented by the ILOG Scheduler). On larger instances, the hybrid method generally achieves speedups of two or three orders of magnitude when minimizing the number of late tasks, and it solves significantly more problems to optimality. There is a lesser but still significant speedup when minimizing total tardiness, and even when the hybrid method cannot obtain provably optimal solutions, it obtains much better solutions than provided by MILP in the same time period.