Variability Basics

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Variability Basics
God does not play dice with the universe.
Albert Einstein
Stop telling God what to do.
Niels Bohr
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Variability Makes a Difference!

• Littles 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 jobs/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?

Variability!
Variability!
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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?

Variability!
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Variability Views

• Variability
• Any departure from uniformity
• Random versus controllable variation
• Randomness
• Essential reality?
• Artifact of incomplete knowledge?
• Management implications robustness is key

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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.

g
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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

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Variability

• Definition Variability is anything that causes
  the system to depart from regular, predictable
  behavior.
• Sources of Variability
• setups work pace variation
• machine failures differential skill levels
• materials shortages engineering change orders
• yield loss customer orders
• rework product differentiation
• operator unavailability material handling

May be consequence of business strategy
May be consequence of manufacturing practices
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Measuring Process Variability
Note we often use the squared coefficient of
variation (SCV), ce2
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Variability Classes in Factory Physics
High variability (HV)
Moderate variability (MV)
Low variability (LV)

• 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

ce
0.75
0
1.33
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Measuring Process Variability Example
Question can we measure ce this way?
Answer No! Wont consider rare
events properly.
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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.

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Down Time Mean Effects

• Definitions

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Down Time Mean Effects (cont.)

• Availability Fraction of time machine is up
  (working)
• Effective Processing Time and Rate

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Totoise and Hare - Availability

• Hare X19
• 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
• Availability

• Tortoise
• 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

A
A
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Machine Failures in Effective Process Time
Notation
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Machine Failures in Effective Process Time
Preliminaries
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Machine Failures in Effective Process Time ET
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Machine Failures in Effective Process Time ET2
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Machine Failures in Effective Process Time ET2
(Cont.)
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Machine Failures in Effective Process Time VarT

• If repair times are exponential, then cr 1, so

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Down Time Variability Effects

• Effective Variability
• Conclusions
• 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

Variability depends on repair times in addition
to availability
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Tortoise and Hare - Variability

• Hare X19
• te
• ce2

• Tortoise 2000
• te
• ce2

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Impact of Variability
2.875 jobs/hr
2.875 jobs/hr
Tor- toise
Hare

• Hare X19
• CT 28 hours
• WIP 81 jobs

• Tortoise 2000
• CT 8 hours
• WIP 23 jobs

Conclusion Capacity and arrival variability are
the same, but CT and WIP are greatly inflated by
process variability due to failures.
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Setups (and Other Non-Preemptive Outages)
Notation

• Examples of other non-preemptive outages include
  PMs, breaks, meetings, shift changes, rework, and
  failures that are based on number of runs (not
  run time).

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Setups Analysis
X and S are independent random variables
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Setups Mean and Variability Effects

• Effective Variability
• Conclusions
• Setups increase mean and variance of processing
  times.
• Variability reduction is one benefit of flexible
  machines.
• However, the interaction is complex.

Capacity Effect setups inflate average process
time Variability Effect setups also inflate
process time variability
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Setups (Non-Preemptive Outages) Example

• Data
• Fast, inflexible machine (2 hr setup every 10
  jobs)
• Slower, flexible machine (no setups)
• Traditional Analysis?

No difference!
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Setup Example (cont.)

• Factory Physics Approach Compare mean and
  variance
• Fast, inflexible machine 2 hr setup every 10
  jobs
• Slower, flexible machine no setups
• Conclusion

Flexibility can reduce variability.
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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.
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Formulas for Effective Process Time
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Other Process Variability Inflators

• Sources
• operator unavailability
• rework
• batching
• material unavailability
• others
• Effects
• inflate te
• inflate ce
• Consequences

Effective process variability can be LV, MV,or HV.
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Flow Variability

• Process Variability is bad enough
• Inflates CT
• Inflates WIP
• Forces lower utilization of capacity
• But, variability also propagates
• Causes uneven arrivals downstream
• Inflates CT and WIP at other stations
• Forces lower utilization of capacity throughout
  the line

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Illustrating Flow Variability
Low variability arrivals
t
smooth!
High variability arrivals
t
bursty!
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Measuring Flow Variability
Do not confuse these with the number of arrivals
per unit time!
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Propagation of Variability
departure variability depends on both arrival
variability and process variability
ce2(i)
cd2(i) ca2(i1)
ca2(i)
i
i1

• Single-Machine Station
• where u is the station utilization given by u
  ra/re rate
• Multi-Machine Station
• where m is the number of (identical) machines and

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Propagation of Variability High Utilization
Station
Conclusion flow variability out of a high
utilization station is determined primarily by
process variability at that station.
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Propagation of Variability Low Utilization
Station
Conclusion flow variability out of a low
utilization station is determined primarily by
flow variability into that station.
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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

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Kendalls Classification

• A/B/C
• A arrival process
• B service process
• C number of machines
• M exponential (Markovian or memoryless)
  distribution
• G unknown (completely general) distribution
• D constant (deterministic) distribution.

B
A
C
Queue
Server
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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 lt 1.

Note a station can be described with 5
parameters.
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Queueing Measures

• Measures
• CTq the expected time spent waiting in the
  queue.
• CT the expected time spent at the workstation
• (waiting time plus process time).
• WIP the expected WIP level (in jobs) at the
  workstation.
• WIPq the expected WIP level (in jobs) waiting
  in the queue.
• Relationships
• CT CTq te
• WIP ra ? CT
• WIPq ra ? CTq
• Result If we know CTq, we can compute WIP, WIPq,
  CT.

Littles law
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The M/M/1 Queue
ra
ra
ra
ra
ra
ra

0
1
2
3
4
5
re
re
re
re
re
re
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The M/M/1 Queue (Cont.)
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The M/M/1 Queue (Cont.)
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The G/G/1 Queue

• Formula
• Observations
• Known as Kingmans equation (after its inventor).
• Useful model of single-machine workstations.
• Exact for M/M/1 queues (in fact, for all M/G/1
  queues).
• Separate terms for variability, utilization, and
  process time.
• CTq (and other measures) increase with ca2 and
  ce2.
• Flow variability and process variability can
  combine to inflate queue time.
• Variability causes congestion!

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The G/G/1 Queue (Cont.)

• Other useful approximations include

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The M/M/m Queue

• With m identical machines in the workstation

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The M/M/m Queue (Cont.)
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The M/M/m Queue (Cont.)
Parts waiting in queue, not in service.
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The M/M/m Queue (Cont.)
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The G/G/m Queue

• Formula
• Observations
• Known as Sakasegawas equation (after its
  inventor).
• Useful model of multi-machine workstations.
• Equivalent to G/G/1 formula when m 1.
• Extremely general.
• Fast and accurate.
• Easily implemented in a spreadsheet.

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VUT Spreadsheet
basic data
failures
setups
yield
measures
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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., kanban cards)
• 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

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The M/M/1/b Queue
Model of Station 2
Note there is room for b B 2 jobs, B in
the buffer and one at each station.
Infinite raw materials
2
1
B buffer spaces
ra
ra
ra
ra
ra
ra

0
1
2
B
b1
b
re
re
re
re
re
re
Note u gt 1 is possible.
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The M/M/1/b Queue u 1 Case
Goes to ra as b ? ?. Always lt ra TH(M/M/1).
Goes to ? as b ? ?. Always lt ? WIP(M/M/1).
Goes to ? as b ? ?. Always lt ? CT(M/M/1).
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The M/M/1/b Queue u ? 1 Case
If u lt 1, goes to ra as b ? ?, but always lt ra
TH(M/M/1). If u gt 1, goes to re as b ? ?. Goes to
zero as u ? ?.
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The M/M/1/b Queue u ? 1 Case (Cont.)
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The M/M/1/b Queue u ? 1 Case (Cont.)
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The M/M/1/b Queue u ? 1 Case (Cont.)
If u lt 1, goes to u/(1 u) as b ? ?, but always
lt u/(1 u) WIP(M/M/1). If u gt 1, goes to ? as
b ? ?. Goes to b as u ? ?.
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The M/M/1/b Queue u ? 1 Case (Cont.)
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The M/M/1/b Queue u ? 1 Case (Cont.)
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The M/M/1/b Queue u ? 1 Case (Cont.)
If u lt 1, goes to te/(1 u) as b ? ?, but
always lt te/(1 u) CT(M/M/1). If u gt 1, goes
to ? as b ? ?. Goes to teb as u ? ?.
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The M/M/1/b Queue u ? 1 Case (Cont.)
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Blocking Example M/M/1/? vs. M/M/1/b
te(1)21
te(2)20
M/M/1/b has 18 less TH than M/M/1
B 2 so b 4
91 less WIP
88 less CT
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The G/G/1/b Queue
WIP with no blocking
Corrected utilization
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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
• Specific Targets
• equipment failures
• setups
• rework
• operator pacing
• anything that prevents regular arrivals and
  process times

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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 n parts)

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Pooling Example

• PCs consist of 6 components (CPU, HD, CD ROM,
  RAM, removable storage device, keyboard)
• 3 choices of each component 36 729 different
  PCs
• Each component costs 150 (900 material cost per
  PC)
• Demand for all models is Poisson distributed with
  mean 100 per year
• Replenishment lead time is 3 months (0.25 years)
• Use base stock policy with fill rate of 99

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Pooling Example - Stock PCs

• Base Stock Level for Each PC ? 100 ? 0.25
  25, so using Poisson formulas,
• G(R 1) ? 0.99 R 38 units
• Average On-Hand Inventory for Each PC
• I(R) R ? B(R) 38 25 0.0138
  13.0138 units
• Value of Total On-Hand Inventory
• 13.0138 ? 729 ? 900 8,538,358

70
Pooling Example - Stock Components
729 models of PC. 3 types of each component.

• Necessary Service for Each Component
• S (0.99)1/6 0.9983
• Base Stock Level for Components ? (100 ?
  729/3) ? 0.25 6075, so
• G(R 1) ? 0.9983 R 6306
• Average On-Hand Inventory Level for Each
  Component
• I(R) R ? B(R) 6306 6075 0.0363
  231.0363 units
• Value of Total On-Hand Inventory
• 231.0363 ? 18 ? 150 623,798

93 reduction!
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Formulas for Effective Process Time
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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 (bullwhip effect)
• variability and utilization interact
• pooled variability less destructive than
  individual variability