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

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

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 TAopt results: explored states SPS2 operations / 2 resources 3 operations / 3 resources Jobsnodestimenodestime 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

16 Using tailor-made MILPs leads to competitive-size LP subproblems which can be solved with less effort. Cplex: 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 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 Thank You for your attention. Any questions? Discussion