Variability Basics
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
1
Variability Makes a Difference!
Little’s Law: TH = WIP/CT, so same throughput can be obtained
with large WIP, long CT or small WIP, short CT. The
difference?
Penny Fab One: achieves full TH (0.5 j/hr) at WIP=W0=4 jobs if
it behaves like Best Case, but requires WIP=27 jobs to achieve
95% of capacity if it behaves like the Practical Worst Case.
Why?
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
2
Tortise and Hare Example
Two machines:
• subject to same workload: 69 jobs/day (2.875 jobs/hr)
• subject to unpredictable outages (availability = 75%)
Hare X19:
• long, but infrequent outages
Tortoise 2000:
• short, but more frequent outages
Performance: Hare X19 is substantially worse on all measures than
Tortoise 2000. Why?
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
3
Variability Views
Variability:
• Any departure from uniformity
• Random versus controllable variation
Randomness:
• Essential reality?
• Artifact of incomplete knowledge?
• Management implications: robustness is key
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
4
Probabilistic Intuition
Uses of Intuition:
• driving a car
• throwing a ball
• mastering the stock market
First Moment Effects:
•
•
•
•
g
Throughput increases with machine speed
Throughput increases with availability
Inventory increases with lot size
Our intuition is good for first moments
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
5
Probabilistic Intuition (cont.)
Second Moment Effects:
• Which is more variable – processing times of parts or batches?
• Which are more disruptive – long, infrequent failures or short
frequent ones?
• Our intuition is less secure for second moments
• Misinterpretation – e.g., regression to the mean
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
6
Variability
Definition: Variability is anything that causes the system to depart
from regular, predictable behavior.
Sources of Variability:
•
•
•
•
•
•
setups
machine failures
materials shortages
yield loss
rework
operator unavailability
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
• workpace variation
• differential skill levels
• engineering change orders
• customer orders
• product differentiation
• material handling
http://www.factory-physics.com
7
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
http://www.factory-physics.com
8
Variability Classes in Factory Physics
Low variability
(LV)
0
Moderate variability
(MV)
0.75
High variability
(HV)
1.33
ce
Effective Process Times:
• actual process times are generally LV
• effective process times include setups, failure outages, etc.
• HV, LV, and MV are all possible in effective process times
Relation to Performance Cases: For balanced systems
• MV – Practical Worst Case
• LV – between Best Case and Practical Worst Case
• HV – between Practical Worst Case and Worst Case
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
9
Natural Variability
Definition: variability without explicitly analyzed cause
Sources:
•
•
•
•
operator pace
material fluctuations
product type (if not explicitly considered)
product quality
Observation: natural process variability is usually in the LV
category.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
10
Down Time – Mean Effects
Definitions:
t 0  base process time
c0  base process time coefficien t of variabili ty
r0 
1
 base capacity (rate, e.g., parts/hr)
t0
m f  mean time to failure
mr  mean time to repair
cr  coefficent of variabili ty of repair tim es ( r / mr )
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
11
Down Time – Mean Effects (cont.)
Availability: Fraction of time machine is up
A
mf
m f  mr
Effective Processing Time and Rate:
re  Ar0
te  t0 / A
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
12
Totoise and Hare - Availability
Hare X19:
Tortoise:
t0 = 15 min
0 = 3.35 min
c0 = 0 /t0 = 3.35/15 = 0.05
mf = 12.4 hrs (744 min)
mr = 4.133 hrs (248 min)
cr = 1.0
t0 = 15 min
0 = 3.35 min
c0 = 0 /t0 = 3.35/15 = 0.05
mf = 1.9 hrs (114 min)
mr = 0.633 hrs (38 min)
cr = 1.0
Availability:
A=
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
A=
http://www.factory-physics.com
13
Down Time – Variability Effects
Effective Variability:
te  t0 / A
2
2
  0  (mr   r )(1  A)t 0
2
σe    
A
 A
 e2
m
2
ce  2  c02  (1  cr2 ) A(1  A) r
t0
te
2
Conclusions:
•
•
•
•
•
Variability
depends on
repair times
in addition to
availability
Failures inflate mean, variance, and CV of effective process time
Mean (te) increases proportionally with 1/A
SCV (ce2) increases proportionally with mr
SCV (ce2) increases proportionally in cr2
For constant availability (A), long infrequent outages increase SCV
more than short frequent ones
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
14
Tortoise and Hare - Variability
Hare X19:
Tortoise 2000
te =
te =
ce2 =
ce2 =
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
15
Setups – Mean and Variability Effects
Analysis:
N s  average no. jobs between setups
t s  average setup duration
 s  std. dev. of setup time
s
cs 
ts
te  t0 
ts
Ns
σ  
2
e
c 
2
e
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
2
0
 s2
Ns

Ns 1 2
ts
2
Ns
 e2
te2
http://www.factory-physics.com
16
Setups – Mean and Variability Effects (cont.)
Observations:
• Setups increase mean and variance of processing times.
• Variability reduction is one benefit of flexible machines.
• However, the interaction is complex.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
17
Setup – Example
Data:
• Fast, inflexible machine – 2 hr setup every 10 jobs
t0  1 hr
N s  10 jobs/setup
t s  2 hrs
te  t0  t s / N s  1  2 / 10  1.2 hrs
re  1 / te  1 /(1  2 / 10)  0.8333 jobs/hr
• Slower, flexible machine – no setups
t0  1.2 hrs
re  1 / t0  1 / 1.2  0.833 jobs/hr
Traditional Analysis?
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
18
Setup – Example (cont.)
Factory Physics Approach: Compare mean and variance
• Fast, inflexible machine – 2 hr setup every 10 jobs
t0  1 hr
c02  0.0625
N s  10 jobs/setup
t s  2 hrs
cs2  0.0625
te  t0  t s / N s  1  2 / 10  1.2 hrs
re  1 / te  1 /(1  2 / 10)  0.8333 jobs/hr
 cs2 N s  1 
  0.4475
σ    t 

2 
Ns 
 Ns
ce2  0.31
2
e
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
2
0
2
s
http://www.factory-physics.com
19
Setup – Example (cont.)
• Slower, flexible machine – no setups
t0  1.2 hrs
c02  0.25
re  1 / t0  1 / 1.2  0.833 jobs/hr
ce2  c02  0.25
Conclusion:
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
20
Setup – Example (cont.)
New Machine: Consider a third machine same as previous machine
with setups, but with shorter, more frequent setups
N s  5 jobs/setup
t s  1 hr
Analysis:
re 
σ e2 
ce2 
Conclusion:
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
21
Other Process Variability Inflators
Sources:
•
•
•
•
•
operator unavailability
recycle
batching
material unavailability
et cetera, et cetera, et cetera
Effects:
• inflate te
• inflate ce
Consequences:
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
22
Illustrating Flow Variability
Low variability arrivals
t
High variability arrivals
t
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
23
Measuring Flow Variability
t a  mean time between arrivals
ra 
1
 arrival rate
ta
 a  standard deviation of time between arrivals
ca 
a
ta
 coefficien t of variation of interarriv al times
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
24
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
http://www.factory-physics.com
25
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
http://www.factory-physics.com
26
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
http://www.factory-physics.com
27
Variability Interactions
Importance of Queueing:
• manufacturing plants are queueing networks
• queueing and waiting time comprise majority of cycle time
System Characteristics:
•
•
•
•
•
•
•
•
Arrival process
Service process
Number of servers
Maximum queue size (blocking)
Service discipline (FCFS, LCFS, EDD, SPT, etc.)
Balking
Routing
Many more
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
28
Kendall's Classification
A/B/C
B
A: arrival process
B: service process
C: number of machines
A
Queue
M: exponential (Markovian) distribution
Server
G: completely general distribution
D: constant (deterministic) distribution.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
29
C
Queueing Parameters
ra = the rate of arrivals in customers (jobs) per unit
time (ta = 1/ra = the average time between
arrivals).
ca = the CV of inter-arrival times.
m = the number of machines.
re = the rate of the station in jobs per unit time = m/te.
Note: a station
can be
described
with 5
parameters.
ce = the CV of effective process times.
u = utilization of station = ra/re.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
30
Queueing Measures
Measures:
CTq = the expected waiting time spent in queue.
CT = the expected time spent at the process center, i.e., queue time
plus process time.
WIP = the average WIP level (in jobs) at the station.
WIPq = the expected WIP (in jobs) in queue.
Relationships:
CT = CTq + te
WIP = ra  CT
WIPq = ra  CTq
Result: If we know CTq, we can compute WIP, WIPq, CT.
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
31
The G/G/1 Queue
Formula:
CT q  V  U  t
 ca2  ce2
 
 2
 u 

 1  u t e

Observations:
•
•
•
•
Useful model of single machine workstations
Separate terms for variability, utilization, process time.
CTq (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
http://www.factory-physics.com
32
The G/G/m Queue
Formula:
CTq  V  U  t
 ca2  ce2  u 2( m 1) 1 

 
te


 2  m(1  u ) 
Observations:
•
•
•
•
Useful model of multi-machine workstations
Extremely general.
Fast and accurate.
Easily implemented in a spreadsheet (or packages like MPX).
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
33
setups
failures
basic data
VUT Spreadsheet
ra
Arrival CV
Natural Process Time (hr)
ca
t0
Natural Process SCV
Number of Machines
MTTF (hr)
MTTR (hr)
Availability
Effective Process Time (failures only)
c0
m
mf
mr
A
te'
Eff Process SCV (failures only)
Batch Size
Setup Time (hr)
ce '
k
ts
yield
2
2
2
2
Setup Time SCV
Arrival Rate of Batches
Eff Batch Process Time (failures+setups)
Eff Batch Process Time Var (failures+setups)
Eff Process SCV (failures+setups)
Utilization
measures
STATION:
MEASURE:
Arrival Rate (parts/hr)
cs
ra/k
te = kt0/A+ts
2
2
2
k*0 /A + 2mr(1-A)kt0/A+s
2
ce
u
2
Departure SCV
Yield
Final Departure Rate
cd
y
ra*y
Final Departure SCV
ycd +(1-y)
Utilization
Throughput
Queue Time (hr)
Cycle Time (hr)
Cumulative Cycle Time (hr)
WIP in Queue (jobs)
WIP (jobs)
Cumulative WIP (jobs)
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
2
u
TH
CTq
CTq+te
i(CTq(i)+te(i))
raCTq
raCT
i(ra(i)CT(i))
1
10.000
2
9.800
3
9.310
4
8.845
5
7.960
1.000
0.090
0.181
0.090
0.031
0.095
0.061
0.090
0.035
0.090
0.500
1
200
2
0.990
0.091
0.500
1
200
2
0.990
0.091
0.500
1
200
8
0.962
0.099
0.500
1
200
4
0.980
0.092
0.500
1
200
4
0.980
0.092
0.936
100
0.000
0.936
100
0.500
6.729
100
0.500
2.209
100
0.000
2.209
100
0.000
1.000
0.100
9.090
1.000
0.098
9.590
1.000
0.093
10.380
1.000
0.088
9.180
1.000
0.080
9.180
0.773
1.023
6.818
1.861
1.861
0.009
0.909
0.011
0.940
0.063
0.966
0.022
0.812
0.022
0.731
0.181
0.980
9.800
0.031
0.950
9.310
0.061
0.950
8.845
0.035
0.900
7.960
0.028
0.950
7.562
0.198
0.079
0.108
0.132
0.077
0.909
9.800
45.825
54.915
54.915
458.249
549.149
549.149
0.940
9.310
14.421
24.011
78.925
141.321
235.303
784.452
0.966
8.845
14.065
24.445
103.371
130.948
227.586
1012.038
0.812
7.960
1.649
10.829
114.200
14.587
95.780
1107.818
0.731
7.562
0.716
9.896
124.096
5.700
78.773
1186.591
http://www.factory-physics.com
34
Effects of Blocking
VUT Equation:
• characterizes stations with infinite space for queueing
• useful for seeing what will happen to WIP, CT without restrictions
But real world systems often constrain WIP:
• physical constraints (e.g., space or spoilage)
• logical constraints (e.g., kanbans)
Blocking Models:
• estimate WIP and TH for given set of rates, buffer sizes
• much more complex than non-blocking (open) models, often
require simulation to evaluate realistic systems
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
35
The M/M/1/b Queue
2
1
Note: there is room
for b=B+2 jobs in
system, B in the buffer
and one at each station.
Infinite
raw
materials
B buffer spaces
Model of Station 2
u
(b  1)u b 1
WIP ( M / M / 1 / b) 

1 u
1  u b 1
1 ub
TH ( M / M / 1 / b) 
r
b 1 a
1 u
CT ( M / M / 1 / b) 
WIP ( M / M / 1 / b)
TH ( M / M / 1 / b)
where u  t e (2) / t e (1)
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
Goes to u/(1-u) as b
Always less than WIP(M/M/1)
Goes to ra as b
Always less than TH(M/M/1)
Little’s law
Note: u>1 is possible; formulas valid for u1
http://www.factory-physics.com
36
Blocking Example
te(1)=21
te(2)=20
B=2
u  t e (2) / t e (1)  20 / 21  0.9524
WIP ( M / M / 1) 
M/M/1/b system has
less WIP and less TH
than M/M/1 system
u
 20 jobs
1 u
TH ( M / M / 1)  ra  1 / t e (1)  1 / 21  0.0476 job/min
1-u b
1  0.9524 4
TH(M/M/ 1/b) 
ra 
1-u b 1
1  0.9524 5
 1 
   0.039 job/min
 21 
18% less TH
u
(b  1)u b 1
5(0.9524 5 )
WIP ( M / M / 1 / b) 

 20 
 1.8954 jobs 90% less WIP
1 u
1  u b 1
1  0.9524 5
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
37
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
http://www.factory-physics.com
38
Variability Pooling
Basic Idea: the CV of a sum of independent random variables
decreases with the number of random variables.
Example (Time to process a batch of parts):
t 0  time to process single part
 0  standard deviation of time to process single part
0
c0 
t0
 CV of time to process single part
t 0 (batch)  nt0
 02 (batch)  n 02
 02 (batch)
2
2
2
n


c
c02 (batch)  2
 2 02  02  0
n
t 0 (batch) n t 0 nt0
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
 c0 (batch) 
http://www.factory-physics.com
c0
n
39
Basic Variability Takeaways
Variability Measures:
• CV of effective process times
• CV of interarrival times
Components of Process Variability
•
•
•
•
failures
setups
many others - deflate capacity and inflate variability
long infrequent disruptions worse than short frequent ones
Consequences of Variability:
•
•
•
•
variability causes congestion (i.e., WIP/CT inflation)
variability propagates
variability and utilization interact
pooled variability less destructive than individual variability
© Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http://www.factory-physics.com
40