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Optimization of Schedules of a
Multitask Production Cell
Karin Thörnblad
September 2011
• Part of Volvo Group
• Develops and produces aircraft and rocket engine components
• About 3000 employees
9510KT, Karin Thörnblad
Volvo Aero Corporation Proprietary Information.
This information is subject to restrictions on first page.
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Parts processed in the Multi Task Cell
~ 8 compressor rear frames for different aero engines and gas turbines.
About 30 different jobs are processed in the Multi Task Cell.
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The Multitask Production Cell
Central tool storage
Multipurpose machines
3
Stocker crane
1
Setup stations
2
Manual deburring
Deburring cell
4
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Input/Output
conveyor
Stocker crane
Transports between storage and route operations
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3 setup stations
Mount/demount in and out of fixtures
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5 multitask machines
Drilling, milling and turning
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Automatic deburring cell
Robot deburring
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The routing of a part
Every production order follows a routing in the planning system
One job in the multitask cell ↔ 3-5 route operations
Multitask job
Job processed elsewhere
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The queue of parts
Planned
order
Processed
elsewhere
MT-cell
vjq , planned lead time from
completion of job j to arrival
at MT-cell for job q
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v0j, planned lead time
from current position
to arrival at MT-cell
Stock
checked-in
Problem decomposition
Time (h)
Multitaskmachines
Deburring
and setup
stations
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Notation
Multipurpose machines
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Notation
Multitask job
Job to be processed
elsewhere
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Notation
Multitask job
Job to be processed
elsewhere
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Notation
Multitask job
Job to be processed
elsewhere
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Notation
Multitask job
Job to be processed
elsewhere
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Time-indexed formulation
l, length of the discrete time interval
time interval u starts at time ul hours
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Decision variables
Binaries:
Time variables
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Test case No 3 (20 jobs): 9300
Time–indexed formulation with nail variables
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A planned lead time
on the same part
0
Job j
Job q
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has to elapse between jobs
T
tid
Time–indexed formulation with nail variables
Objective: Minimize the weighted sum of
completion times and tardiness.
One job is scheduled only once
Each job can only be assigned to an
allowed resource k
Only one job at a time can be processed
on resource k
Planned lead time between jobs on same
part
Definition of completion time
Definition of tardiness
Job j cannot start in resource k before the
job’s release date or before k is available
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The engineer’s model
(with continuous time variables)
Binaries:
Time variables
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Test case No 3 (20 jobs): 2100
weights (A=B=1)
Big number
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Objective: Minimize the weighted sum
of completion times and tardiness.
weights (A=B=1)
One job is scheduled only once
Each job can only be assigned to an
allowed resource k
Only one job at a time can be
processed on resource k
Big number
Job j cannot start in resource k before
the job’s release date or before k is
available
Planned lead time between jobs on
same part
Definition of completion time
Definition of tardiness
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Comparison between the precedence
constraints for the set
Constraints:
Test case No 3:
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372
4
Comparison between the constraints for
only one job at a time
Constraints:
Test case No 3:
465
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6000
Symmetric constraints versus
ordering constraints
(the engineer’s model)
These two constraints regulates the
ordering of the jobs in resource k
The ordering constraints
can be replaced by
constraints symmetric to
”the big M-constraints”:
• The symmetric constraints are common in text books since the ordering
constraints are special for problems with multiple machines.
• Computational tests indicate that the model with ordering constraints
has shorter computation times than a model using the symmetric big Mconstraints.
2010-07-14
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What is the size of a realistic scenario?
Number of storage locations without fixture:
Number of parts arriving during the coming shift:
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45 jobs
30
15
Real production scenarios
• 6 real scenarios – based on real
production data extracted from the
Volvo Aero ERP-system during the
autumn of 2010
• The jobs were ordered according
to increasing release dates
• From each scenario test instances
were created with 5,10,15,…, 70 of
the first jobs in the sorted list of
jobs (i.e. the queue of jobs)
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Realistic release dates
The release date from the ERP system may be negative (i.e. in the past).
Therefore a realistic estimate on the part arrival time at the multitask cell is
calculated using the knowledge of the part’s actual position at time t0.
The release date is then calculated by the formula
Standard queue time
rj = max {realistic estimate; ERP release date - 0.8υj}
If order checked-in: rj = 0
rq = rj +
rl = rj +
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+ pq
Postprocessing with real ”undiscrete” data
Real release date = 17.5 h
Time-indexed model with l = 1 h: 18 h
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Mean differences between optimal objective
values after postprocessing of data
Mean diff in optimal value
0.20%
Compared with the time-indexed
model with l = 0.25 h
l=2h
0.15%
disc int=0.25h
disc int=0.5h
0.10%
disc int=1h
disc int=2h
0.05%
l 0.04%
=1h
l (l=≤0.5h
1 h)
0.00%
10
15
20
25
30
Compared with the engineer’s model # jobs
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35
40
45
50
Mean computation times
Test case No 3 (20 jobs):
l = 0.5h
>7200 s (mipgap 0.66%)
l = 1h
l = 2h
l = 0.25h
Realistic scenario: <1 min
Test case No 3 (20 jobs):
0.3 s (same schedule found)
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Evolution of computation times
Comparison between different models’ CPU times (seconds)
~3 months
(s)
1000000
100000
~8 hours
Full engineer's model
10000
Divided engineer's model
~15 min
1000
< 2 min
100
Current time-indexed +
engineer's feas model
10
Feasibility model: ~100 s
1
5
10
15
Number of jobs
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Old time-indexed +
engineer's feas model
20
Current time-indexed
model only < 0.5 s
The feasibility model
Fixed to solution from
machining problem
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different weights for different
setup stations
Current detail planning of the multitask cell
Manual planning based on
•
•
•
•
Earliest Due Date (EDD) priority list
Other priorities based on the current logistical situation
The FIFO priority rule (First In First Out) is used in other parts of the factory
SPT (shortest processing time) is a priority rule known to produce good
schedules
Time (h)
00
10
10
MC1
MC1
Multitask
machines
96
7
MC2
MC2
8 1010 11
9
1 16
1
13
MDM1
9
17
16
19
20
14
2
5
5
6
9 6
13
113
1
14
2
5
3 6
3 2
17
5
14
17
1310
13 16 1
67 10 8 11
9 12
101 11
1613
1 1514
3 15 8 5
7 1711
2 12
9 12 9 36 11
1613
1 12 2 7
5
4 15 4 15 5 17 61614
182 4 18 4 14
5
3 6
3
17
21918
19 2018
20
14 19
19
17
20
The deburring and set-up stations are scheduled by the use of
the feasibility model.
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60
60
20
19
17
3
148
1112 2
18
18
50
50
8
8
MDM2 10 9 7
MDM3
4
40
40
1010
DBR
DBR
Deburring
and setup
stations
7
3
MC4
MC4
ManGr
ManGr
4
13
MC5
MC5
30
30
15
15
12
MC3
MC3
20
20
20
EDD, FIFO and SPT versus
mathematical optimization
21 scenarios with real production data
Collected during April – August 2010
Real scenarios April - August 2010
Time distribution for data collections
17
16
15
Time
14
13
12
11
10
9
8
1
Mo
2
Tu
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3
We
Weekday
4
Th
5
Fr
Work load variation
The variation in number of jobs checked-in indicate how the work load
has varied during the period.
# jobs checked-in
20
High work load J>10
15
10
Low work load J≤10
5
0
Scenarios
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Mean differences between completion
times and tardiness results
The schedules resulting from the use of the priority rules are compared
to the optimal values found by the time-indexed model with l=1h in
sequence with the feasibility model.
22% higher tardiness
using FIFO
12% higher tardiness
using EDD
6% higher completion
times using SPT
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Shortsighted scheduling
No knowledge about which jobs are on the way to the multitask cell
(or further down in the priority list)
t0
t1
t2
time
Job 1 - MC1 & MC2
Job 2 - MC1 & MC 2
Job 3 - MC 2
t0
MC1
t1
time
Job 1 - MC 1 & MC2
MC2
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t2
Job 2 – MC1 & MC 2
Job 3 – MC 2
Looking into the future…
The optimization model takes all jobs in the queue into account
t0
t1
t2
time
Job 1 - MC1 & MC2
Job 2 - MC1 & MC 2
Job 3 - MC 2
t0
t1
Job 1 - MC 1 & MC2
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time
Job 2 – MC 1 & MC 2
MC1
MC2
t2
Job 3 – MC 2
Looking into the future…
t0
t1
t2
t3
time
Job 1 - MC1 & MC2
Job 2 - MC1 & MC 2
Job 3 - MC 2
Job 4 - MC1 & MC2
t0
MC1
t1
t2
t3
Job 1 - MC 1 & MC2
Job 4 - MC1 & MC2
Job 2 – MC1 & MC 2
MC2
time
Job 3 – MC 2
Lost capacity
t0
t1
Job 1 - MC 1 & MC2
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t3
time
Job 2 – MC 1 & MC 2
MC1
MC2
t2
Job 3 – MC 2
Job 4 - MC1 & MC2
An optimal schedule versus a schedule
created using the EDD rule
Timeindexed
model
EDD
Lost capacity?
9% higher tardiness using EDD and
25% higher completion times!
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Coping with reality
As soon as the production schedule is optimized – something changes!
Expected events
• New parts in the queue
• Variances in the planned lead time
Unexpected events
•
•
•
•
Machine breakdown
Operator sick
Part with non-conformance leaves queue
etc.
Optimized
schedule
Expected events
Unexpected
events
Frequency:
CPU time
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Reschedule
shiftnecessary
when
< 5 min
Future research
• Compare results to more sophisticated scheduling algorithms
• Make a pilot test of using the schedules in the multitask cell
•
•
•
•
•
Constraint programming
Lagrangian relaxation to get better lower bounds
Column generation
Find stronger formulations of the problem
Robust optimization
• More realistic model: fixtures,
personnel etc.
• Find better objective functions
• Develop a time-indexed
formulation for the feasibility
problem
• …
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More tests
More theory
Better model
Questions and comments?
Thank you
Karin Thörnblad, industrial PhD student,
[email protected]
Financial support from Volvo Aero, The Swedish Research Council,
NFFP (Swedish National Aeronautics Research Programme)
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