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Scheduling problems requires to explicitly account for control considerationsin their optimisation. The literature proposes two traditional ways to solvethis integrated problem: hierarchical and monolithic. The monolithic approachignores the control level's objective and incorporates it as a constraint intothe upper level at the cost of suboptimality. The hierarchical approachrequires solving a mathematically complex bilevel problem with the schedulingacting as the leader and control as the follower. The linking variables betweenboth levels belong to a small subset of scheduling and control decisionvariables. For this subset of variables, data-driven surrogate models have beenused to learn follower responses to different leader decisions. In this work,we propose to use ReLU neural networks for the control level. Consequently, thebilevel problem is collapsed into a single-level MILP that is still able toaccount for the control level's objective. This single-level MILP reformulationis compared with the monolithic approach and benchmarked against embedding anonlinear expression of the neural networks into the optimisation. Moreover, aneural network is used to predict control level feasibility. The case studiesinvolve batch reactor and sequential batch process scheduling problems. Theproposed methodology finds optimal solutions while largely outperforming bothapproaches in terms of computational time. Additionally, due to well-developedMILP solvers, adding ReLU neural networks in a MILP form marginally impacts thecomputational time. The solution's error due to prediction accuracy iscorrelated with the neural network training error. Overall, we expose how - byusing an existing big-M reformulation and being careful about integratingmachine learning and optimisation pipelines - we can more efficiently solve thebilevel scheduling-control problem with high accuracy.

Bilevel optimisation with embedded neural networks: Application to scheduling and control integration