1 Variability Basics God does not play dice with the universe. ---Albert Einstein Stop telling God...
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Transcript of 1 Variability Basics God does not play dice with the universe. ---Albert Einstein Stop telling God...
1
Variability Basics
God does not play dice with the universe.
---Albert Einstein
Stop telling God what to do.
---Niels Bohr
2
Variability Makes a Difference!
Little’s Law:
Consequence: Same throughput can be attained with small WIP and cycle time or with large WIP and cycle time.
Difference: Variability!
CT
WIPTH
3
Variability Views
Variability:• Any departure from uniformity
• Random versus controllable variation
Randomness:• Essential reality?
• Artifact of incomplete knowledge?
• Management implications: robustness is key
4
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
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
6
Variability
Definition: Variability is anything that causes the system to depart from regular, predictable behavior.
Sources of Variability:• setups • workpace variation
• machine failures • differential skill levels
• materials shortages • engineering change orders
• yield loss • customer orders
• rework • product differentiation
• operator unavailability • material handling
7
Measuring Process Variability
CV , variationoft coefficien
timeprocess ofdeviation standard
job a of timeprocessmean
e
ee
e
e
tc
σ
t
8
Variability Classes in Factory Physics
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
0.75
High variability(HV)
Moderate variability(MV)
Low variability(LV)
0 1.33ce
9
Measuring Process Variability---Example
Trial Machine 1 Machine 2 Machine 31 22 5 52 25 6 63 23 5 54 26 35 355 24 7 76 28 45 457 21 6 68 30 6 69 24 5 5
10 28 4 411 27 7 712 25 50 50013 24 6 614 23 6 615 22 5 5te 25.1 13.2 43.2se 2.5 15.9 127.0ce 0.1 1.2 2.9ce
2 0.01 1.4 8.6Class LV MV HV
10
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.
111 1
Φ υ σ ι κ ή Μ ε τ α β λ η τ ό τ η τ α - Π α ρ ά δ ε ι γ μ α
• Μ ί α ε ρ γ α σ ί α α π α ι τ ε ί μ έ σ ο ό ρ ο μ ί α ( 1 ) ώ ρ α ( = 6 0 λ ε π τ ά ) γ ι α ν α ο λ ο κ λ η ρ ω θ ε ί .
•
• Ε ν α λ λ α κ τ ι κ ά , η ε ρ γ α σ ί α μ π ο ρ ε ί ν α δ ι α ι ρ ε θ ε ί σ ε δ έ κ α ( 1 0 ) ξ ε χ ω ρ ι σ τ έ ς
υ π ο ε ρ γ α σ ί ε ς π ο υ η κ ά θ ε μ ι α α π α ι τ ε ί μ έ σ ο ό ρ ο 6 λ ε π τ ά ε π ε ξ ε ρ γ α σ ί α ς .
• Υ π ο ε ρ γ α σ ί α :
• Ε ρ γ α σ ί α :
• Ο σ υ ν ο λ ι κ ό ς χ ρ ό ν ο ς ε π ε ξ ε ρ γ α σ ί α ς τ ω ν έ χ ε ι μ ι κ ρ ό τ ε ρ η μ ε τ α β λ η τ ό τ η τ α ό τ α ν
ε κ τ ε λ ε ί τ α ι σ α ν 1 0 υ π ο ε ρ γ α σ ί ε ς α π ό ό τ ι ό τ α ν ε κ τ ε λ ε ί τ α ι σ α ν μ ί α ε ρ γ α σ ί α .
( Ε ξ ή γ η σ η : Σ τ η ν π ρ ώ τ η π ε ρ ί π τ ω σ η , γ ι α ν α τ ύ χ ε ι έ ν α ς υ π ε ρ β ο λ ι κ ά μ ε γ ά λ ο ς
χ ρ ό ν ο ς ε π ε ξ ε ρ γ α σ ί α ς π ρ έ π ε ι ν α ε ί μ α σ τ ε ά τ υ χ ο ι μ ι α φ ο ρ ά . Σ τ η ν δ ε ύ τ ε ρ η
π ε ρ ί π τ ω σ η , γ ι α ν α τ ύ χ ε ι ο ί δ ι ο ς υ π ε ρ β ο λ ι κ ά μ ε γ ά λ ο ς χ ρ ό ν ο ς ε π ε ξ ε ρ γ α σ ί α ς
π ρ έ π ε ι ν α ε ί μ α σ τ ε ά τ υ χ ο ι 1 0 φ ο ρ έ ς .
800.1602/1 220 σ600 t 2/12
0 c
60 t 2/120 c 1862/1 22
0 σ
60610 et 18018102 eσ 05,060/180 22 ec
12
Down Time---Mean Effects
Definitions:
repair tomean time
failuresbetween mean time
parts/hr)e.g., (rate,capacity base1
variance timeprocess base
timeprocess base
00
20
0
r
f
m
m
tr
σ
t
13
Down Time---Mean Effects (cont.)
Availability: Fraction of time machine is up
Effective Processing Time and Rate:
rf
f
mm
mA
Att
Arr
e
e
/0
0
14
Down Time---Variability Effects
Effective Variability:
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
0
2 2 22 0 0
22 2 2 2 2
0 020 0 0
/
( )(1 )
(1 ) (1 ) (1 ) (1 )
e
r re
r
e r r re r r
e
t t A
m A tσ
A Am
m m mc c c A A c A A c A A
t t tt
15
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:0
3.350.05
15 (very low variability)c
16
Down Time---Example (cont.)
Effective Variability:
Effect of Reducing MTTR: Suppose we can do frequent PM which causes mf = 1.9 hrs (= 114min), mr = 0.633 hrs (= 38min).
0
2 2 2 20
0
7440.75
744 248
/ 15 / 0.75 20min
248(1 ) (1 ) (0.05) (1 1)0.75(1 0.75) 6.25
15
2.5 (high variability!)
f
f r
e
re r
e
mA
m m
t t A
mc c c A A
t
c
2 2 2 20
0
1140.75
114 38
38(1 ) (1 ) (0.05) (1 1)0.75(1 0.75) 1.0
15
1.0 (medium variability)
f
f r
re r
e
mA
m m
mc c c A A
t
c
17
Setups--Mean and Variability Effects
Analysis:
2
22
22
220
2
0
1
timesetup of dev. std.
duration setup average
setupsbetween jobs no. average
e
ee
ss
s
s
se
s
se
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ss
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s
s
tc
tN
N
Nσ
N
ttt
tc
t
N
18
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.
19
Setup--Example
Data:• Fast, inflexible machine--2 hr setup every 10 jobs
• Slower,flexible machine--no setups
Traditional Analysis: No difference!
jobs/hr 8333.0)10/21/(1/1
hrs 2.110/21/
hrs 2
jobs/setup 10
hr 1
0
0
ee
sse
s
s
tr
Nttt
t
N
t
jobs/hr 833.02.1/1/1
hrs 1.2
0
0
tr
t
e
20
Setup--Example (cont.)
Factory Physics Approach: Compare mean and variance
• Fast,inflexible machine--2 hr setup every 10 jobs
31.0
4475.01
jobs/hr 8333.0)10/21/(1/1
hrs 2.110/21/
0625.0
hrs 2
jobs/setup 10
0625.0
hr 1
2
2
222
02
0
2
20
0
e
s
s
s
sse
ee
sse
s
s
s
c
N
N
N
ctσ
tr
Nttt
c
t
N
c
t
21
Setup--Example (cont.)
• Slower, flexible machine--no setups
Conclusion: flexibility reduces variability.
25.0
jobs/hr 833.02.1/1/1
25.0
hrs 2.1
20
2
0
20
0
cc
tr
c
t
e
e
22
Setup--Example (cont.)
New Machine: Consider a third machine same as previous machine with setups, but with shorter,more frequent setups
Analysis:
Conclusion: Shorter, more frequent setups induce less variability.
hr 1
jobs/setup 5
s
s
t
N
16.0
2350.01
jobs/hr 833.0)5/11/(1/1
2
2
222
02
e
s
s
s
sse
ee
c
N
N
N
ctσ
tr
23
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.
25
Measuring Flow Variability
timesalinterarriv of variationoft coefficien
arrivalsbetween timeofdeviation standard
rate arrival1
arrivalsbetween mean time
a
aa
a
aa
a
tc
σ
tr
t
26
Propagation of Variability
Single Machine Station:
where u is the station utilization given by u = rate
Multi-Machine Station:
where m is the number of (identical) machines and
22222 )1( aed cucuc
)1()1)(1(1 22
222 ead cm
ucuc
i i+1cd
2(i) ca2(i+1)
m
tru ea
27
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
28
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
29
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
30
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.
31
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.
ce = the CV of effective process times.
u = utilization of station = ra/re.
Note: a stationcan be describedwith 5 parameters.
32
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.
33
The G/G/1 Queue
Formula:
Observations:• Separate terms for variability, utilization, process time.
• CTq (and other measures) increase with ca2 and ce
2
• Flow variability, process variability, or both can combine to inflate queue time.
• Variance causes congestion!
eea
q
tu
ucc
tUV
12
CT
22
34
Variability and Cycle Time Relationships
ra , cd2ra , ca
2
te , ce2
eea
q tu
ucc
12CT
Time Queue22
ea
ead
tru
cucuc
22222 )1(
yVariabilit Flow
et
Time
Process
35
The G/G/m Queue
Analysis:
Observations:• Extremely general.
• Fast and accurate.
• Easily implemented in a spreadsheet (or packages like MPX).
e
mea
q
tum
ucc
tUV
)1(2
CT
1)1(222
36
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
37
Blocking Example
B=2
te(1)=21 te(2)=20
jobs 8954.19524.01
)9524.0(520
1
)1(
1)/1//(
job/min 039.021
1
9524.01
9524.01
1
11
job/min 0476.021/1)1(/1)1//(
jobs 201
)1//(
9524.021/20)1(/)2(
5
5
1
1
5
4
1
b
b
ab
b
ea
ee
u
ub
u
ubMMWIP
r-u
-u/b)TH(M/M/
trMMTH
u
uMMWIP
ttu M/M/1/b system has less WIP and less TH than M/M/1 system
18% less TH
90% less WIP
38
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
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 propogates• variability and utilization interact