1. Transformation of Timed Automata into Mixed Integer Linear Programs Sebastian Panek

2. Overview  Motivation  First modeling approach  Syntax  Semantic  MILP-Formulation  Formulation of scheduling problems  Further work

3. Motivation: What? What do we want to do? 1.Model an (optimization) problem as Linearly Priced Timed Automata (LPTA) 2.Transform it into a Mixed-Integer Linear Program (MILP) 3.Solve it using MILP algorithms 4.Compare the results to other approaches in terms of computational effort and usability

4. Motivation: How?  We have some experience in building optimization models of hybrid systems and scheduling problems (Engell, Stursberg, Sand)  TA are simpler than hybrid automata, but there have been some open questions:  how to formulate parallel compositions?  how to formulate synchronization?  how to formulate continuous time?  how to exploit the simpler structure of TA  how to solve MILP models of TA faster?

5. Motivation: Why TA?  TA allow modeling complex behaviors quite easily (simple syntax)  Parallel compositions of TA allow the user specifying decomposed parts of a system separately  There exist graphical editors and languages for the modeling of TA  Powerful analysis tools are available

6. Motivation: Why MILP?  MILP is appropriate for problems in many application domains  A well-investigated MILP theory is known for many years  Many different MILP solution algorithms and heuristics are available  Powerful free and commercial MILP solvers have been developed  Modeling and debugging is very difficult!

7. First approach to the MILP formulation of TA  LPTA (costs on locations and transitions)  Continuous time in the MILP  Finite set of time points at which transitions may occur  Networks of TA are possible  Bidirectional synchronization using labels  More complex clock constraints for invariants and guards are supported (arbitrary polyhedra)

8. Syntax  Syntax of LPTA  Additionally:  Since MILP models are static, the automaton can‘t simply stop in the final state  Insert self-loops in all locations  Those loops allow the TA to do nothing without additional costs

9. Semantics: transitions  Bounded time horizon (an upper bound for all clock valuations must exist)  Finite number of time points n in the MILP implies finite number of transitions 2n in the TA  Each time point in the MILP corresponds with  1 delay transition  1 discrete transition

10. Semantics: synchronization  Semantic for synchronization: Synchronized transitions are taken if there are at least two automata waiting for the same label (bidirectional synchronization)

11. MILP formulation: variables  Example LPTA (Larsen et al.)  MILP model with n=4 time points  Real variables for clock vectors  Binary variables for locations and transitions 222

12. MILP formulation: constraints on binary variables  Real variables for time delays  Binary product variables for all combinations of locations and outgoing transitions  At every k the automaton is in one location:  At every k one transition is taken:

13. MILP formulation: computing products  Real product variables of clocks and locations  Real product variables of clocks and transitions

14. MILP formulation: clock constraints  Use polyhedral description to express clock constrains, i.e.  Enforce invariants for locations  Enforce guards for transitions

15. MILP formulation: evolution  Delay transitions:  Discrete transitions:  Clock resets on transitions:

16. MILP formulation: objective function  Define start state and final states fixing corresponding variables:  Define products for the objective function:  Objective function: minmize costs over all runs

17. TA formulation of scheduling problems  According to O. Maler and A. Fehnker  Use additional constraints to assert exclusive allocation of resources  Example: 2 jobs and 2 resources

18. Further work  Improve the MILP formulation  Test it on large scale models  Implement other types of TA (i.e. Uppaal-TA)  Build a parser which accepts common TA description languages and generates MILP code automatically  Compare the different approaches:  TA model, symbolic solution  MILP model, MILP solution,  TA model, MILP solution