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AMETIST Review Meeting June 2003

Deliverable 3.4.2

Axxom Case Study: Scheduling of Lacquer Production

Contributors:

AxxomDortmund

TwenteVerimag

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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

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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

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Lacquer Production

Quality Check

Pre-dispersion/Dispersion

Dose Spinner

Mixing Vessel

Filling Stations

• Pipeless multi-product plant• Simplified production process

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Pipeless Plant

• Mixing vessels move between stations• Plant topology (paths, collisions) is not described and not

modelled here.• Multiple equal resource instances

MixingVessel

Pre-dispersion DispersionFilling station

Quality checkDose Spinner

Filling stationDose Spinner

MixingVessel

MixingVesselMixing

Vessel

MixingVessel

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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

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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

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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

allowedinterval

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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.

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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

ii es ,

1ik

0,1 hikihk

1k

ik

ikiikikiik eess :,:

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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

hkikhikihk

))(1(

))(1()1(

hkikihkhkik

hkikihkhkik

Mmse

MMse

1ihk

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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

iiii

iiii

edeadline

deadlineeJ

0,max

0,maxmin

iiii ,,

01.0

1

_

_

STATIONFILLING

STATIONFILLING

0

0001.0

i

i

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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)

hkikhi sereleasedeadline thenif

hkikhihi setypetypedeadlinedeadline thenandif

hkikhi sedeadlinedeadline thenif

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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.

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Results: optimal scheduling

Jobs Obj EDD Obj LB Tardiness Earliness

10 0.6612 0.6612 0.6512 0.0012 0.66

14 7.2310 7.2310 1.2916 0.0010 7.2300

16 10.7110 10.7110 1.2920 0.0010 10.7100

18 13.9308 13.9308 2.0920 0.0008 13.9300

20 11.9314 11.9314 2.0924 0.0014 11.9300

22 214.0013 214.0013 2.3724 198.0313 15.9700

29 1383.8372 1383.8372 3.7830 1367.3172 16.5200

1200 sec. CPU time limit for each of the two steps of the solution procedure.

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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:=11. Solve the problem for jobs i...i+k-12. Fix all decision variables concerning job i3. Set i:=i+14. If i+k-1>n then terminate5. Else go to 1.

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Results: moving horizon strategy

Iter. Sched. Jobs

Time EDD

Time Obj EDD Obj LB

3 10 42.47 0.64 0.6512 0.6512 0.6512

7 14 250.97 107.23 123.7665 47.8566 47.8566

9 16 221.97 464.61 113.2653 51.2271 48.5842

11 18 75.03 600 58.0277 52.0271 49.2066

13 20 217.08 56.89 60.0283 52.0275 52.0275

15 22 600 600 154.2319 154.2319 61.1084

22 29 73.77 20.70 176.0516 176.0516 169.0578

600 sec. CPU time limit for each step in every iteration

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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

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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.

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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.

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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

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Heuristics: avoiding laziness

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

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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.

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• 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

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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 compareGoal:• 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

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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!- Include 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

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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

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• 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