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Optimization of Schedules of a
Multitask Production Cell
Karin Thörnblad, Ann-Brith Strömberg,
Michael Patriksson, Torgny Almgren
September 2010
• 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
Multitask job
Job processed elsewhere
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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
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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
Current detail planning of the multitask cell
Manual planning based on
• Earliest Due Date 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
Time (h)
0
10
MC1
Multitask
machines
6
MC2
16
MC3
10
13
2
Real production case
5
10
DBR
MDM1
1
5
3
2
17
14
8
MDM2 10
MDM3
9 6
20
14
8
1112
60
19
3
13
50
18
17
9
MC5
40
4
7
1
ManGr
30
15
MC4
Deburring
and setup
stations
20
6
13 16 1
9 10 1613 1 3 15 8 5 12 9 6 11 7 171112 2 7
5
4 15
18 4 14
17
3
2 19
2018
14 19
20
The route operations of the remaining resources are set in a feasible schedule.
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Sets and indices
Multi Task job
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Sets and indices
Multi Task job
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Parameters
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Parameters cont’d
If order checked-in: rj = rq = rl = 0
Else: rj = rq = rl = max (date available (v0j); planned release date)
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Variables
• Binary variables
• Time variables
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The optimization model of the MT-cell
To be cont’d…
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The optimization model of the MT-cell
weight (A=1)
The sum of completion times and tardiness,
i.e. every job is done as early as possible
and tardiness is punished.
One route operation is scheduled
only once
Operation assigned to an allowed
resource k
These two constraints regulates
the ordering of the operations for a
resource k
Starting time (p,q) after compl. time
(i,j) if same k
Big number
To be cont’d…
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The optimization model cont’d
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The optimization model cont’d
Operation processed and transported
before next
Job may be started after release date
Resource k available first time
Job q may be started after completion of
job j + planned lead time
Definition of completion time
Definition of tardiness
Positive starting times
Binary variables
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Division into two models
Too high CPU times for the whole model (AMPL-CPLEX11)
• The processing times of the machining resources >> other route operations
• Machining resources most heavy investments
The model divided into two models:
• The machining model optimizes the schedule of the machining resources MC1-5
• The feasibility model finds a feasible schedule for the rest of the route operations
MC1
MC2
MC3
MC4
MC5
Man Gr
DBR
MDM1 1
MDM2 8
MDM3 3
3
1
11
4
5
10
8
13
1
11
5
8
13
10
1 14
9
5 12
6
3
24 13
28
21
26
22
27
29
30
6
12 14
20 19 15 30
9
12
16
4
18
20
11
25
15
10
4
2
23
17
9
10 11
1
4
19
16
24
3
4
Mount operation
Demount operation
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6
14
2
12
20
14 2 15
9
20
19 10
17
12
30
6
18
26
24
23
16
17 23 18 25 21
19
23
25
15 21
20 29
19 30
17
26 28
28
28
25
22
25
18
21
23
22
22
27
29
27
22
28
27
The machining problem
v mjq
v jq t1q
nj
p pm
j
pij
i 3
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The feasibility model
different weights for different
setup stations
Fixed to solution from
machining problem
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Discrete machining model
0
MC1
10
8 11
20
16
15
MC2
19
MC3
12 7
10
30
MC4
4
20
6
17
14
MC5
9
1
50
3
18
13
40
5
2
The time horizon of the schedule is divided into T+1 discrete time steps.
Variables:
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60
The discrete machining problem
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The discrete machining problem
Objective: Minimize the 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
Job q may be started after completion of job j + planned lead
time between the jobs on the same part
Release date
Resource availability
Definition of completion time
Definition of tardiness
Binary variables
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Comparison
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Test scenarios
All 20 jobs are assumed to be checked-in into the Multitask cell.
•
Real production case – one day in March 2010
(20 jobs, all jobs late at t=0)
1. As is
2. short jobs (25 %)
3. long jobs (25 %)
•
High volume case –
created from prognosis by the market department
(20 jobs, approximately half of the jobs late at t=0)
1. As is
2. short jobs (25 %)
3. long jobs (25 %)
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Computational results – CPU times
All computations have been carried out on a
4 Gb quad-core Intel Xeon 3.2 GHz system using AMPL-CPLEX12
Comparison of CPU times (seconds)
(s)
1000000
~3 months
100000
~8 hours
10000
Full engineer's model
1000
Divided engineer's model
~15 min
100
Discrete machining +
engineer's feas model
10
1
5
10
15
Number of jobs
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20
Computational results
Results given as a mean per job, and the differences are relative
to the completion time of the optimal solution.
Scenario
Scheduling
algorithm
Real
OPT
22.9
0
0%
0
0%
prod
FIFO
26.9
4.0
18.0%
4.0
18.0%
case
EDD
26.3
2.4
10.4%
2.4
10.4%
High
OPT
25.4
0
0%
0
0%
volume
FIFO
33.9
8.5
33.9%
5.7
22.4%
case
EDD
32.5
7.1
28.9%
2.9
11.7%
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Completion
time (h)
Diff from
optimal
solution (h)
Completion
time diff
(%)
Tardiness
diff (h)
Tardiness
diff (%)
Job tardiness results
Tardiness results from high volume long jobs scenario
90
80
70
Tardiness (h)
60
50
Opt
EDD
40
FIFO
30
20
10
0
1
2
3
4
5
6
7
8
9
10
11
Job number
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12
13
14
15
16
17
18
19
20
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
Scenario: high volume long jobs
0
MC1
10
1615
20
1
14
MC2
9
9
DBR
MC1
19
1
MDM3
MC4
2
17
20
18
MDM1
MC2
MDM2
MC3
10
3
10
1
7
30
11
6
40
3
13
2
9
1 7
MC5
9
1
313 41218
3
8
5
17
207 14
5
11
6
17 6 20 3
11
9
1
8
14
10
70
2
20
10
15 19 16
ManGr
60
12
7
5
DBR
4
50
13
4
16151615 9 1219 181114
19
Opt results
5
18
70
8
11
19
60
6
4
MC4
0
50
13
7
MC5
40
20
12
MC3
ManGr
30
810 13
42
18
5
17
6
8
10
2
Earliest due date
17
19
4
3
5
2
6
18 7 11
13
8
17
10
MDM1
MDM2
1
MDM3
4
9
2 5
6 3
1
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9
19
7 15194 11163 151813516 8 2
6 1710 18 7
11
12
1314
8
17
2012 1014
20
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
Unexpected events
Frequency:
CPU time
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Reschedule
shiftnecessary
when
< 15 min
Continued research
• Compute results on a broader spectra of scenarios –
based both on realistic data and high volume cases
• Compare results to more sophisticated scheduling algorithms
• Evaluate the results with an existing simulation model
•
•
•
•
Constraint programming
Lagrangian relaxation to get better lower bounds
Column generation
Development of heuristics
• More realistic model: fixtures,
manpower etc.
• Find better objective functions
• …
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More tests
More theory
Better model
Questions and comments?
Thank you
Karin Thörnblad, industrial PhD student,
karin.thornblad@volvo.com
Financial support from Volvo Aero, The Swedish Research Council,
NFFP (Swedish National Aeronautics Research Programme)
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