Variability Basics
God does not play dice with the universe.
---Albert Einstein
Stop telling God what to do.
---Niels Bohr
1
Variability Makes a Difference!
Little’s Law:
TH 
WIP
CT
Consequence: Same throughput can be attained with small WIP and cycle
time or with large WIP and cycle time.
Difference: Variability!
2
Variability Views
Variability:
• Any departure from uniformity
• Random versus controllable variation
Randomness:
• Essential reality?
• Artifact of incomplete knowledge?
• Management implications: robustness is key
3
Probabilistic Intuition
Uses of Intuition:
• driving a car
• throwing a ball
• mastering the stock market
First Moment Effects:
•
•
•
•
Throughput increases with machine speed
Throughput increases with availability
Inventory increases with lot size
Our intuition is good for first moments
4
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
5
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
• workpace variation
• differential skill levels
• engineering change orders
• customer orders
• product differentiation
• material handling
6
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
7
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
8
Measuring Process Variability---Example
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
te
se
ce
ce2
Class
Machine 1
22
25
23
26
24
28
21
30
24
28
27
25
24
23
22
25.1
2.5
0.1
0.01
LV
Machine 2
5
6
5
35
7
45
6
6
5
4
7
50
6
6
5
13.2
15.9
1.2
1.4
MV
Machine 3
5
6
5
35
7
45
6
6
5
4
7
500
6
6
5
43.2
127.0
2.9
8.6
HV
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.
10
Φυσική Μεταβλητότητα - Παράδειγμα
• Μία εργασία απαιτεί μέσο όρο μία (1) ώρα (= 60 λεπτά) για να ολοκληρωθεί.
• t 0  60
c 02  1 / 2
 σ 02  1 / 2  60 2  1.800
• Εναλλακτικά, η εργασία μπορεί να διαιρεθεί σε δέκα (10) ξεχωριστές
υποεργασίες που η κάθε μια απαιτεί μέσο όρο 6 λεπτά επεξεργασίας.
c 02  1 / 2
 σ 02  1 / 2  6 2  18
• Υποεργασία: t 0  6
2
 c e2  180 / 60 2  0,05
• Εργασία: t e  10  6  60 σ e  10  18  180
• Ο συνολικός χρόνος επεξεργασίας των έχει μικρότερη μεταβλητότητα όταν
εκτελείται σαν 10 υποεργασίες από ότι όταν εκτελείται σαν μία εργασία.
(Εξήγηση: Στην πρώτη περίπτωση, για να τύχει ένας υπερβολικά μεγάλος
χρόνος επεξεργασίας πρέπει να είμαστε άτυχοι μια φορά. Στην δεύτερη
περίπτωση, για να τύχει ο ίδιος υπερβολικά μεγάλος χρόνος επεξεργασίας
πρέπει να είμαστε άτυχοι 10 φορές.
11
11
Down Time---Mean Effects
Definitions:
t0  base process time
σ 02  base process time variance
r0 
1
 base capacity (rate, e.g., parts/hr)
t0
m f  mean time between failures
mr  mean time to repair
12
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
13
Down Time---Variability Effects
Effective Variability:
te  t0 / A
2
2
  0  (mr   r )(1  A)t0
2
σe    
Amr
 A
2
c 
2
e
 e2
te2
 c02  (1  cr2 ) A(1  A)
mr
m
m
 c02  A(1  A) r  cr2 A(1  A) r
t0
t0
t0
Conclusions:
•
•
•
•
Failures inflate mean, variance, and CV of effective process time
Mean increases proportionally with 1/A
SCV increases proportionally with mr
For constant availability (A), long infrequent outages increase CV more
than short frequent ones
14
Down Time---Example
Data: Suppose a machine has
•
•
•
•
•
15 min stroke (t0 = 15min)
3.35 min standard deviation (0 = 3.35min)
12.4 hour mean time to failure(mf = 12.6  60 = 744min)
4.133 hour repair time (mr = 4.133  60 = 248min)
Unit CV of repair time (cr = 1.0)
Natural Variability:
c0 
3.35
 0.05 (very low variability)
15
15
Down Time---Example (cont.)
Effective Variability:
A
mf
m f  mr

744
 0.75
744  248
te  t0 / A  15 / 0.75  20 min
ce2  c02  (1  cr2 ) A(1  A)
mr
248
 (0.05) 2  (1  1)0.75(1  0.75)
 6.25
t0
15
ce  2.5 (high variability!)
Effect of Reducing MTTR: Suppose we can do frequent PM which
causes mf = 1.9 hrs (= 114min), mr = 0.633 hrs (= 38min).
A
mf
m f  mr

114
 0.75
114  38
ce2  c02  (1  cr2 ) A(1  A)
mr
38
 (0.05) 2  (1  1)0.75(1  0.75)  1.0
t0
15
ce  1.0 (medium variability)
16
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
2
0
 s2
Ns

Ns 1 2
ts
2
Ns
 e2
te2
17
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.
18
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: No difference!
19
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
2
0
2
s
20
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: flexibility reduces variability.
21
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  1 / te  1 /(1  1 / 5)  0.833 jobs/hr
 cs2 N s  1 
  0.2350
σ    t 

2 
Ns 
 Ns
ce2  0.16
2
e
2
0
2
s
Conclusion: Shorter, more frequent setups induce less variability.
22
Other Process Variability Inflators
Sources:
•
•
•
•
•
operator unavailability
recycle
batching
material unavailability
et cetera, et cetera, et cetera
Effects:
• inflate te
• inflate ce2
Consequences: effective process variability can be LV, MV,or HV.
23
Illustrating Flow Variability
Low variability arrivals
t
High variability arrivals
t
24
Measuring Flow Variability
t a  mean time between arrivals
ra 
1
 arrival rate
ta
σ a  standard deviation of time between arrivals
ca 
a
ta
 coefficien t of variation of interarriv al times
25
Propagation of Variability
cd2(i)
ca2(i+1)
i
i+1
Single Machine Station:
cd2  u 2ce2  (1  u 2 )ca2
where u is the station utilization given by u = rate
Multi-Machine Station:
u2 2
c  1  (1  u )(c  1) 
(ce  1)
m
2
d
2
2
a
where m is the number of (identical) machines and u 
ra te
m
26
Propagation of Variability
High Utilization Station
High Process Var
Low Flow Var
High Flow Var
Low Utilization Station
High Process Var
Low Flow Var
Low Flow Var
27
Propagation of Variability
High Utilization Station
Low Process Var
High Flow Var
Low Flow Var
Low Utilization Station
Low Process Var
High Flow Var
High Flow Var
28
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
29
Kendall's Classification
A/B/C
A: arrival process
B: service process
C: number of machines
M: exponential (Markovian) distribution
G: completely general distribution
D: constant (deterministic) distribution.
30
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.
31
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.
32
The G/G/1 Queue
Formula:
CTq  V  U  t
 ca2  ce2  u 

 
te
 2  1  u 
Observations:
• 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.
• Variance causes congestion!
33
Variability and Cycle Time Relationships
Flow Variabilit y
cd2  (1  u 2 )ca2  u 2 ce2
u  ra te
te , ce2
ra , ca2
ra , cd2
Queue Time
Process
 ca2  ce2  u 

CTq  
te
 2  1  u 
Time
te
34
The G/G/m Queue
Analysis:
CTq  V  U  t
 ca2  ce2  u 2( m1) 1 

 
te


 2  m(1  u ) 
Observations:
• Extremely general.
• Fast and accurate.
• Easily implemented in a spreadsheet (or packages like MPX).
35
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
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) 
u
 20 jobs
1 u
M/M/1/b system has
less WIP and less TH
than M/M/1 system
TH ( M / M / 1)  ra  1 / t e (1)  1 / 21  0.0476 job/min
1-u b
1  0.9524 4  1 
TH(M/M/ 1/b) 
r 
   0.039 job/min
b 1 a
5
1-u
1  0.9524  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
b 1
5
1 u
1 u
1  0.9524
37
Attacking Variability
General Strategies:
•
•
•
•
look for long queues (Little's law)
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
38
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 propogates
• variability and utilization interact
39