Variability
Definition: Variability is anything that causes the system to
depart from regular, predictable behavior.
Sources of Variability:
•
•
•
•
•
•
machine failures
setups
material shortages
yield loss
rework
operator unavailability
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
May be consequence
of manufacturing
practices
• workpace variation
• differential skill levels
• material handling
• demand fluctuations
• engineering change orders
• product variety
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May be consequence
of business strategy
1
Measuring Process Variability
te  mean process time of a job
σ e  standard deviation of process time
ce 
e
te
 coefficien t of variation , CV
Note: we often use the “squared
coefficient of variation” (SCV), ce2
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
2
Measuring Process Variability – Example
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
te
se
ce
Class
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
Machine 1
22
25
23
26
24
28
21
30
24
28
27
25
24
23
22
25.1
2.5
0.1
LV
Machine 2
5
6
5
35
7
45
6
6
5
4
7
50
6
6
5
13.2
15.9
1.2
MV
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Machine 3
5
6
5
35
7
45
6
6
5
4
7
500
6
6
5
43.2
127.0
2.9
HV
3
Flow Variability
Process Variability is bad enough…
• Inflates FT
• Inflates WIP
• Forces lower utilization of capacity
But, variability also propagates…
• Causes uneven arrivals downstream
• Inflates FT and WIP at other stations
• Forces lower utilization of capacity throughout the line
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
4
Illustrating Flow Variability
Low variability arrivals
t
High variability arrivals
t
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
5
Propagation of Variability –
Low Utilization Station
LV
HV
LV
HV
HV
HV
LV
LV
LV
HV
LV
HV
Conclusion: flow variability out of a low utilization station is
determined primarily by flow variability into that station.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
6
Propagation of Variability –
High Utilization Station
LV
HV
HV
HV
HV
HV
LV
LV
LV
HV
LV
LV
Conclusion: flow variability out of a high utilization station is
determined primarily by process variability at that station.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
7
Propagation of Variability
ce2(i)
ca2(i)
cd2(i) = ca2(i+1)
i
i+1
Single Machine Station:
c  u c  (1  u )c
2
d
2 2
e
2
2
a
where u is the station utilization given by u = rate
Multi-Machine Station:
departure var
depends on
arrival var
and process
var
u2 2
c  1  (1  u )(c  1) 
(ce  1)
m
rt
where m is the number of (identical) machines and u  a e
m
2
d
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
2
2
a
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8
Kendall's Classification
A/B/C
B
A: arrival process
B: service process
C: number of machines
A
Queue
M: exponential (Markovian) distribution
C
Server
G: completely general distribution
D: constant (deterministic) distribution.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
9
The G/G/1 Queue
Formula:
FTq  V  U  t
 ca2  ce2  u 

 
te
 2  1  u 
Observations:
•
•
•
•
Useful model of single machine workstations
Separate terms for variability, utilization, process time.
FTq (and other measures) increase with ca2 and ce2
Flow variability, process variability, or both can combine to inflate
queue time.
• Variability causes congestion!
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
10
Seeking Out Variability
General Strategies:
•
•
•
•
•
look for long queues (Little's law)
look for blocking
focus on high utilization resources
consider both flow and process variability
ask “why” five times
Specific Targets:
•
•
•
•
•
equipment failures
setups
rework
operator pacing
anything that prevents regular arrivals and process times
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
www.factoryphysics.com
11