AMETIST Review Meeting June 2003 Deliverable Axxom Case Study: Scheduling of Lacquer Production Contributors: Axxom Dortmund Twente Verimag

Overview Description of the case study MILP model and solution Timed automata model and analysis Twente work on general aspects of modelling Conclusions and future work

Motivation of the case study General focus of interest: supply chain management Supply chain: network of –production sites –warehouses –transportation systems Objective: minimisation of production and distribution costs, e.g. –delay costs –set up costs –resource costs –storage costs –transportation costs First problem: scheduling of lacquer production –Job shop problem with 29 jobs –Minimization of delay and storage costs

Lacquer Production Quality Check Pre-dispersion/ Dispersion Dose Spinner Mixing Vessel Filling Stations Pipeless multi-product plant Simplified production process

Pipeless Plant Mixing vessels move between stations Plant topology (paths, collisions) is not described and not modelled here. Multiple equal resource instances Mixing Vessel Pre-dispersionDispersionFilling stationQuality checkDose SpinnerFilling stationDose Spinner Mixing Vessel Mixing Vessel Mixing Vessel Mixing Vessel

Problem statement 3 Types of lacquers {uni-, metallic-, bronce-} lacquers 29 Jobs defined by (release date, due date, type, batch size) 6-8 Operations/Job, total number of operations: 202 Batch sizes vary from 11000L to 19000L 14 resources (incl. mixing vessels), different resource capacities Deterministic behaviour, no cycles in production chain Example of the uni- lacquer flow diagram

Additional problem characteristics Timing restrictions for pairs of operations Parallel allocation of mixing vessels Multiple equal resource instances -> machine allocation Lab is a non-bottleneck resource Example: uni lacquer

Timing constraints Additional restrictions for pairs of operations: –start-start restrictions –end-start restrictions –end-end restrictions Motivation:chemical reaction durations, spoiling of products, cleaning periods for mixing vessels allowed interval

Mathematical model formulation Continuous time representation Individual representation of time instances for machines as real numbers: no time slots Focused on tasks and machines Binary variables encode sequencing information and machine allocations. Products are not considered explicitly. Fixed batch sizes (no merging and splitting of batches) Model grows according to the number of tasks and not with the time horizon.

MILP formulation of the model: variables Real variables for starting and ending times of tasks Binary variables for the machine allocation –If task i is processed on machine k then: –Each task is processed on one machine: Binary variables for the sequencing of tasks –If task i is processed before task h on machine k then: Starting and ending dates for tasks i on machines k are modelled by real variables

Restrictions on binary variables If both tasks i and h are processed on machine k then either i is scheduled before h or vice versa Mutual exclusion of tasks: set iff task i is finished before task h and both are executed on machine k

Objective function Minimize too late and too early job completions Using weighting factors enables to give priorities to delays and too early completions of some tasks. Final tasks obtain highest weights Small weighting factors for other tasks

Additional heuristics Motivation: partial order reduction and stronger restriction of the feasible set 2-step solution procedure (next slide) Assume: operations i and h of 2 different jobs to be scheduled Non-overtaking of non-overlapping jobs Non-overtaking of equal-typed jobs Earliest Due Date (EDD)

2-step solution procedure Motivation: faster computation of first integer solutions better quality of integer solutions 1.Apply heuristics 1-3 (EDD) thus fixing some sequencing variables, solve the problem. 2.Fix only sequencing variables according to heuristics 1+2, relax the other fixed variables and solve the problem again, reusing the previous solution as an initial integer solution.

Results: optimal scheduling JobsObj EDDObjLBTardinessEarliness sec. CPU time limit for each of the two steps of the solution procedure.

Moving horizon solution procedure Motivation: Increasing number of jobs causes prohibitive computational effort poor integer solutions. Time horizon based decomposition strategy: Order all n jobs according to release dates, define a window size k, set i:=1 1.Solve the problem for jobs i...i+k-1 2.Fix all decision variables concerning job i 3.Set i:=i+1 4.If i+k-1>n then terminate 5.Else go to 1.

Results: moving horizon strategy Iter.Sched. Jobs Time EDD TimeObj EDDObjLB sec. CPU time limit for each step in every iteration

Example Gantt chart Moving horizon: 8 jobs/window, 29 jobs, 22 steps, 20 minutes time limit in each step, 176h accumulated delay, 1.5 GHz machine, 1GB Ram, Software: GAMS/CPLEX 20.5

Results Global scheduling strategy efficient for up to 10 jobs Zero-delay solutions can be computed in 20 min. for up to 20 jobs. First integer solutions are available by the EDD heuristics within a few minutes for all problem instances. Some problem instances are difficult and are not solved efficiently. Moving horizon strategy zero-delay solutions for up to 10 jobs For large critical instances better than global solution.

Timed Automata Approach Timed Automata models for scheduling problems exhibit a simple and intuitive structure and allow easy modelling. Jobs are modelled as acyclic automata, structured as chains of operations, represented by states and connected through restricted transitions. Resources can be modelled either as automata or using shared variables, in both cases multiple equal resource instances can be modelled by one automaton/variable. Tardiness is the only cost criterion. Finding a cost minimal schedule is achieved by searching for a cost-minimal path from the start state to the target state in which all jobs are completed.

Modelling efforts: IF no invariants on states 3 types of transitions: eager, lazy and delayed Synchronization with shared variables Simplified example: 3 operations, 3 resources modelled by variables, states represent allocations of resources

Heuristics: avoiding laziness Avoiding laziness of jobs: lazy runs immediately lead to special error states and abort the exploration of the path.

Additional heuristics Avoiding overtaking of jobs of the same type –jobs explicitly enable starting of successive jobs of the same type –pipeline-like execution scheme –The same heuristics was used in the mathematical model. Minimal separation times between operations are ensured to preserve feasibility of timing constraints between particular operations.

Computation of minimal cost paths using the software IF With heuristics described above, IF was able to compute a delay-free schedule for 29 jobs within 15 seconds. Not possible to construct the state space completely. Minimization of delays is the only cost criterion, no other cost terms were investigated so far. Detailed Uppaal models are also available, but no detailed computational studies have been carried out yet. TA Modelling and Analysis Results

Modelling practice: based on intuition, experience, creativity “model hacking preceeds model checking” Consequence: results are difficult to interprete different modelling & verification approaches are difficult to compare Goal: small-size models that can be analysed by tools documented knowledge about the aspects of the case study that are modelled Twente work on modelling

Create a DICTIONARY to make the vocabulary used as unambigous as possible! - Agree on the interpretation of the case with the case study provider, e.g. link types in diagrams: not common in computer science, informal descriptions do not always match the formal ones Make DESIGN DECISIONS explicit! - Which timing aspects are modelled? - What costs are modelled? - What stochastic parameters (availability) are modelled? - Why is the resulting model meaningful? Start with a DETAILED, GENERAL MODEL! - I nclude all relevant details for a basic model as reference point. Apply a number of ABSTRACTIONS to the general model to get small-size models, and make clear why the PROPERTIES of interest are PRESERVED. Systematic modelling

DICTIONARY: - Basic version complete, growing, ongoing discussion with AXXOM DESIGN DECISIONS: - Based on dictionary and identification of standard design patterns for modelling (growing) DETAILED, GENERAL MODEL: - UPPAAL models for lacquer productions including all different aspects of timing, no costs yet, no instantiation by orders yet ABSTRACTIONS & PROPERTIES of interest: - Will come after previous step is complete. MODEL CHECKING EXPERIMENTS & INTERPRETATION of the results - Will come after previous step has started Present status

Both modelling approaches: priced Timed Automata and Mathematical Modelling are suitable to model the problem. The TA approach allows graphical visualization, compositionality and is more intuitive. The Mathematical Programming approach is more powerful in terms of modelling restrictions and additional cost criteria. In both cases, the use of additional heuristics needed for an efficient computation of schedules. Example problems were not exactly the same, comparisons with several examples have to be made. Combination of both approaches (at the modelling and analysis level) could be promising. Conclusions and Future Work