Integrating Mixed-Integer Optimisation and Satisfiability Modulo Theories: Application to Scheduling

One way to address multi-scale optimisation problems is by integrating logic and optimisation. For example, a scheduling problem may have two levels: (i) assigning orders to machines and (ii) sequencing orders on each machine. In a minumum cost model, assigning orders to machines is a mixed-integer optimisation problem, sequencing orders is a constraint satisfaction problem. The entire problem may be reformulated as either an optimisation or logic problem, but this misses the chance to use optimisation and logic synergistically. Hybrid optimisation/logic approaches have been developed combining mixed-integer linear programming (MILP) and constraint programming (CP), but CP requires specialised, bespoke constraints. We consider modifying the hybrid method by replacing CP with satisfiability modulo theories (SMT); SMT is a constraint satisfaction technique combining propositional satisfiability with a background theory. We find that a logic-based Benders decomposition approach combining MILP and SMT works very well on a minimum cost model for scheduling and performs significantly better than either MILP or SMT alone. But the hybrid MILP/SMT method is weaker than either MILP or SMT on a minimum makespan model.