1. 1 Outline:  Outline of the algorithm  MILP formulation  Experimental Results  Conclusions and Remarks Advances in solving scheduling problems with timed automata and linear programming Sebastian Panek, Sebastian Engell and Olaf Stursberg Process Control Lab, University of Dortmund

2. 2 Motivation: what’s the problem? Optimization of Timed Systems: For given initial and target states and a model determine a sequence of steps that minimizes a given cost criterion. Cost criterion: Minimize accumulated costs along paths (loc. + trans.), or minimize the overall time required to reach the target state. Particular focus: job-shop scheduling.

3. 3 Combination of “Standard” Approaches  Timed Automata (TA) based:  Tree encodes possible evolutions  Search for the cost-optimal path (Reachability analysis + cost evaluation)  Mixed-Integer Programming (MIP):  Algebraic (in-)equalities for continuous and discrete variables  Solution by mathematical programming (or: constraint progr., evolutionary algorithms, …) Objective: combine both approaches

4. 4 Optimization tool for Timed Automata models, mainly scheduling problems. Input: scheduling problem specification. Automatic generation of (MI)LP models and PTA models. Optimization of PTA models with embedded linear programming: Computation of lower bounds of cost-to-go, Branch-and-bound techniques, Guiding of the search towards optimum. Our implemetation: TAopt

5. 5 The algorithm in detail I S T initial state target state with accumulated cost C‘ Use components of x S to compute priorities p(S‘) and p(S“) S‘S“ optimal objective value = C optimal solution = vector x S T C current cost C S p(S“)p(S‘) C = lower bound of cost-to-go creating a linear program by relaxation of the MILP PTA partLP part

6. 6 The “old” version (Munich 03) Problem specification PTA model LP model LP GeneratorPTA OptimizerCurrent State LP SolverSolution Lower bound to cut tree nodes, guiding of further search. Cost-optimal State

7. 7 Every detail of the PTA model is mapped into variables and equations in the MILP model → explicit time representation → thousands of variables and equations, → the MILP size becomes the limiting factor. Performance problem: solution of (many) relaxed MILP problems consumes much time and memory This disadvantage is not compensated by the reduction of explored nodes in TA optimization. Drawbacks of the “old” algorithm

8. 8 Replace generic by tailor-made MILP models i.e. for job- shop scheduling problems Advantage: only essential degrees of freedom are considered within the MILP model Disadvantage: loss of generality Our choice: The same MILP formulation as used to solve the Axxom CS (see also presentation Cassis 03) Automatic generation and relaxation of MILP models from a scheduling problem specification The idea: tailor-made MILPs

9. 9 Alternative Algorithm PTA Model LP Model LP GeneratorModel OptimizerCurrent State LP SolverSolution Cost-optimal State Abstract Problem Specification (i.e. job-shop) PTA Generator Small tailor-made LPs for job-shop problems Lower bound to cut nodes, Prediction of future evolutions

10. 10 Initial data: A time domain: A set of operations: A set of jobs: A set of resources: Assignment of resources: Assignment of jobs: Operation durations: Definition of operations on resources: The MILP model for job-shops

11. 11 Variables (shown here as mappings): Start dates of operations: Precedence variables: Meaning of those variables: for every pair of operations the following conditions must be satisfied: The MILP model for job-shops

12. 12 Equations: Operation dates: Job precedence constraints: No simultanous processing: The MILP model for job-shops

13. 13 Equations: Precedence constraints: The MILP model for job-shops

14. 14 The makespan objective function: s.t. equations described above. Any valid schedule can be described by values of s variables if there exist binary values for p variables which satisfy the constraints above. The MILP model for job-shops

15. 15 TAopt results: explored states SPS2 operations / 2 resources 3 operations / 3 resources Jobsnodestimenodestime 2210.01280.03 3790.091310.21 43420.566731.52 517533.78355910.76 61805553.828870117.6 795929358.752189701147.3 Experimental set-up: Left side:shortest-path search, non-laziness. Right side:branch-and-bound, best-lower-bound search, non- laziness, 30% relative optimality gap. BLS2 operations / 2 resources 3 operations / 3 resources Jobsnodestimenodestime 2120.03150.04 3340.09390.14 41200.471881.14 54102.46151114.79 6425437.018242109.8 713834159.28598951084.9

16. 16 Using tailor-made MILPs leads to competitive-size LP subproblems which can be solved with less effort. Cplex: 80-500 LPs/sec. Benefits: Branch-and-bound strategy to limit the search tree Best-lower-bound strategy to direct the search Alternative to best-lower-bound: relaxation-guided search - doesn’t work yet with the new MILP models. Conclusions

17. 17 Future research and development: Further improvements of the LP formulation, Search space reduction techniques on the TA side, Improvements of modeling capabilities. Objective: To solve the Axxom CS To investigate the trade-off between optimality and solution effort. We accept to achieve near-to-optimal solutions, how much can we reduce the solution effort? Outlook

18. 18 Thank You for your attention. Any questions? Discussion