Optimizing Assembly U-line Balance:
an Industrial Case Study*
Laurent CHAN1, Felix T. CHAN2, Van-Dat CUNG3 and Gerd Finke3
1. Dextus – Schneider Electric Consulting
2. Univ. of Hong Kong, Dept. of Industrial and Manufacturing Systems Engineering
3. Laboratory G-SCOP, [email protected]
Laboratoire G-SCOP
46, av Félix Viallet
38031 Grenoble Cedex
www.g-scop.fr
■
* This work is partially supported by
the Research Cluster GOSPI of the Rhône-Alpes region
Centre National de la Recherche Scientifique
■
Institut Polytechnique de Grenoble (Grenoble-INP)
■
Université Joseph Fourier
■
A U-line implant achieved by Dextus
• Line versus U-line
• Context of Lean
manufacturing and
Just-In-Time
production
• Workspace saving
• Reduce operators’
moving time
• U-line more flexible
(#operators,
#workstations,
diversity of the
production)
• U-line easier to supply
2
1
Three main objectives for Dextus
Buffer capacities
?
Takt & Cycle times
Repetitive Strain Injuries/
Cumulative Trauma Disorder
3
Dextus’ U-line design approach
I- Getting data & validation
II- Synoptic (production procedure)
III- Determine the line length
(1)
IV- Line balancing
(5)
(2)
1.
Production rate/#WS
2.
Additional constraints
(technical,
precedency, etc.)
3.
Improve the
production rate
4.
Storage constraints
5.
Ergonomic &
RSI/CTD Contraints
V- Production procedure validation
VI- Determine the buffer capacities
(3)
VII- Flow simulation
VIII- 3D implementation
(4)
4
2
Step IV: (U-)line balancing
5
Vocabulary and definitions
•
•
•
•
•
•
•
•
Processing task: an indivisible working unit i with a processing
time.
t , i ∈ 1,.., n
Precedence relations: precedence constraints between tasks. i
Workstation: line component where tasks are processed, one
operator or robot per WS.
S j , j ∈ {1,.., m}
Workstation load: Subset of tasks assigned to WS j.
T j = ti
Wokstation time: sum of the times ti of all tasks assigned to
WS j.
i∈S j
Cycle time: time interval between the processing of two
CT = Max(T j )
j∈WS
consecutive units.
CT − T j
Workstation idle time:
{
}
∑
Takt time: production rate (i.e. net available time to work/unit
produced), closely related to cycle time.
Classical objectives: minimize m for a desired CT
minimize CT given m.
6
3
Classical line balancing problem
[Salverson 1955]
6
6
1
4
2
Precedence constraint
7
X
2
8
4
3
5
4
4
10
9
1
10
Task Y with
processing time X
Y
5
5
6
S1
S2
S3
S4
S5
9
12
9
6
11
WS1
WS2
Cycle time=12
Idle time=13
Avg. time=9.4
12
10
8
6
4
2
0
WS3
WS4
WS5
7
Classical line balancing problem
(cont.)
6
1
6
4
2
Precedence constraint
7
2
8
4
3
5
4
4
10
9
1
10
X
Y
Task Y with
processing time X
5
5
6
Cycle time=12
Idle time=13
S1
S2
S3
S4
10
11
9
6
S5
11
Cycle time=11
Idle time=8
Avg. time=9.4
12
10
8
6
4
2
0
WS1
WS2
WS3
WS4
WS5
8
4
Taxonomy of ALBP
Assembly Line Balancing Problem (ALBP)
Assumtions:
-Straight line
SALBP
-Mono-product
-Precedence constraints only
-Deterministic Param.
-No buffer
-Becker
& Scholl(04)
GALBP
-Becker
& Scholl(06)
MALBP
UALBP
- Scholl(99)
- Miltenburg
& Wijngaard (DP)(94)
MUALBP
- Urban (MIP)(98)
- Sparling
& Miltenburg(98)
9
Line versus U-line
10
1
11
3
2
4
10
3
4
1
9
5
6
6
7
8
1
8
5
11
9
6
2
6
7
4
8
2
3
4
4
5
5
4
Cycle time=11
Idle time=8
10
6
2
10
1
9 10
5
6
10
7
10
9
10
8
9
Cycle time=10
Idle time=3
U-line increases possibilities
by allowing crossing-WS.
10
5
General Methodology in O.R.
•
Management of Transport, Production, Inventories, Design,
Ressource Planning , etc. in the areas such as Manufacturiing,
Energy, Information Technology, Nanatechology, Hospitals, etc.
•
Mathematical Programming (linear or not, multiobjective or not)
•
Graph theory, combinatorial optimization
•
Discret Event Systems (Petri Net, Queueing systems)
•
Complexity analysis of the problems and the algorithms
•
Exact (Branch&X) or approximative (GA, TS, SA, Ants, epsilonapprox.) methods
•
Tests, benchmarking with the data instances of the literature
•
Simulations, average analysis
•
On-site evaluation of the solutions
Applications
Modelling
Solving
Analysis &
evaluation
11
Adapted line ILP formulation [Urban 1998]
• Binary assignement x ∈ {0,1}, ∀i ∈ {1,.., n}, ∀j ∈ {1,.., m}
ij
variables
m
• Task assignment
xij = 1, ∀i ∈ {1,.., n}
∑
constraints
j =1
n
• Workstation time
T j = ∑ ti xij ≤ CT , ∀j ∈ {1,.., m}
constraints
i =1
m
• Precedence
(m − j + 1)(xrj − xsj ) ≥ 0, ∀(r , s )
∑
=
1
j
constraints (r,s)
• Adjacence
constraints (u,v)
xvj + xv ( j +1) ≥ xuj , ∀j ∈ {1,.., m − 1}, ∀(u , v )
12
6
Adapted U-line ILP formulation [Urban 1998]
• Adding a phantom reverse task graph
y variables
(lower-half of the U)
x variables
(upper-half of the U)
Idea: a task must belong only to one half of the U,
hence, either x or y is equal to 1.
13
Adapted U-line ILP formulation (cont.)
• Adding y binary
variables for the
reverse task graph
• Task assignment
constraints
• Workstation time
constraints
• Precedence
constraints (r,s)
• Adjacence
constraints (u,v)
yij ∈ {0,1}, ∀i ∈ {1,.., n}, ∀j ∈ {1,.., m}
∑ (x
m
j =1
ij
+ yij ) = 1, ∀i ∈ {1,.., n}
T j = ∑ ti (xij + yij ) ≤ CT , ∀j ∈ {1,.., m}
n
i =1
∑ (m − j + 1)(x
∑ (m − j + 1)(y
rj
− xsj ) ≥ 0, ∀(r , s )
sj
− yrj ) ≥ 0, ∀(r , s )
m
j =1
m
j =1
xvj + xv ( j +1) ≥ xuj , ∀j ∈ {1,.., m − 1}, ∀(u , v )
yvj + yv ( j −1) ≥ yuj , ∀j ∈ {2,.., m}, ∀(u, v )
14
7
Three balance objective functions (1/2)
• Minimize the
Cycle time (Max)
Min(CT )
• Minimize the CT
Min(CTmax + ε (CTmax − CTmin ))
and the gap
n
between
T j = ∑ ti (xij + yij ) ≥ CTmin , ∀j ∈ {1,.., m}
workstation times in=1
T j = ∑ ti (xij + yij ) ≤ CTmax , ∀j ∈ {1,.., m}
(Max-Min)
i =1
15
Three balance objective functions (2/2)


• Minimize the CT Min CT + ε ∑ (δ j+ + δ j− )
j =1

and the deltas to m 
the mean
Tj
∑
workstation time j =1 − T j = δ j+ − δ j− , ∀j ∈ {1,.., m}
m
(SumDeltas)
δ j+ ≥ 0, δ j− ≥ 0, ∀j ∈ {1,.., m}
m
16
8
Preliminary empirical results
•
•
•
•
OPL5.2+CPLEX10.2.
Pentium4 CPU, 3Ghz, 2Gbytes RAM, 10’ time limit.
Tests on given 2 to 11 workstations for S-line and U-line.
Task graph: 27 tasks, 37 precedencies, 2 adjacencies,
• The 3 objective functions give the same Cycle time in the
S-line configuration, respectively U-line.
• But…the U-line configuration can reduce Cycle time when
using 3 to 8 workstations compared to the S-line.
• The U-line allows to have better workstation times
balanced around the mean values w.r.t. SumDeltas and
Max-Min.
• SumDeltas > Max-Min > Max
17
Line balance comparisons
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
(Max-Min)%
Operator
Dextus
MIP S-line
MIP U-line
83%
46%
11%
3%
S-line balancing
U-line balancing
18
9
Impact on the flexibility of the line
45
40
35
Cadence (pc/h)
30
25
20
15
Opérateur
DEXTUS
PLNE
Equilibrage parfait
10
5
0
1
2
3
NbOpérateurs
4
5
6
7
The better the S/U-line is balanced, the closer is the performance
of the S/U-line to the optimal one.
19
On-going works
U-line optimization at Dextus
Mono-product
Present
situation
Coming-up
MIP
•Classical constraints
•Specific constraints
•Test in real line
conditions
•Integrating ergonomic
constraints
Buffer capacities
Simulation
•Mono-product
•Better integrating
ergonomic constraints
•Batch mixed-product
model
Multi-product
MIP
•Batch mixed-product
model
•Looking for real
cases
20
10
Ergonomic & RSI/CTD constraints
• Ergonomic (exclusive)
xuj + xvj
constraints
(u=sit down, v=stand up) not
in the same WS
• RSI/CTD constraints,
several types of efforts in
each WS according to the
assigned operations
≤ 1, ∀j ∈ {1,.., m − 1}, ∀(u, v )

 m
Min ∑ ∑ EffortCoeff t .etj 

 j =1 t∈EffortType
n
∑ TaskEffort ( x + y ) ≤ e , ∀j ∈ {1,.., m}, ∀t ∈ {EffortType}
etj ∈ {0,1}, ∀j ∈ {1,.., m}, ∀t ∈ {EffortType}
i =1
it
ij
ij
tj
21
Thank you for your attention
22
11