Variability Basics

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• Variability Basics

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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 j/hr) at WIP
  W0 4 jobs if it behaves like Best Case, but
  requires WIP 57 jobs to achieve 95 of capacity
  if it behaves like the Practical Worst Case.
  Why?

Variability!
Variability!
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Tortoise 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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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

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Measuring Process Variability
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Variability Classes
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Natural Variability

• Definition variability without explicitly
  analyzed cause, variability inherent in natural
  process time.
• 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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Machine Downtimes

• Preemptive outages downtimes that occur whether
  we want them to or not (e.g., they can occur
  right in the middle of a job).
• Examples machine breakdowns, power outages,..
• Nonpreemptive outages downtimes that will
  inevitably occur but for which we have some
  control as to exactly when.
• Example process changeovers (setups)

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Down Time Mean Effects

• Definitions
• t0 mean of natural process time
• s0 standard deviation of natural process time
• c0 coefficient of variability of natural
  process time
• r0 natural capacity (rate)
• mf mean time to failure (MTTF)
• mr mean time to repair (MTTR)
• sr standard deviation of repair time
• cr coefficient of variability of repair time,

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Down Time Mean Effects (cont.)

• Availability Fraction of time machine is up
• Effective mean process time
• Effective capacity (rate)

m number of machines
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Tortoise 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

No difference between machines in terms of
availability.
re Ar0 0.75(4 jobs/hour) 3 jobs/hour
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Down Time Variability Effects

• te mean of effective process time
• se standard deviation of effective process
  time
• ce2 SCV of effective process time

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Tortoise and Hare - Variability

• Hare X19

Tortoise
Hare X19 is much more variable than Tortoise 2000!
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Setups Mean and Variability Effects

• Ns average number of jobs between setups
• ts mean setup time
• ss standard deviation of setup time
• cs CV of setup time

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Setup Example

• Machine 1 flexible machine, with no
  setups
• t0 1.2 hours
• c0 0.5
• Effective capacity

• Machine 2 2-hour setup every 10 jobs
• t0 1.0 hour
• c0 0.25
• Ns 10 jobs/setup
• ts 2 hours
• cs 0.25

No difference between machines in terms of
effective capacity.
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Setup Example (cont.)

• Machine 1 flexible machine, with no
  setups
• t0 1.2 hours
• c0 0.5
• Variability

• Machine 2 2-hour setup every 10 jobs
• t0 1.0 hour
• c0 0.25
• Ns 10 jobs/setup
• ts 2 hours
• cs 0.25

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Flow Variability
Low variability arrivals
t
smooth!
High variability arrivals
t
bursty!
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Measuring Flow Variability

• ta mean time between arrivals
• ra average arrival rate
• sa standard deviation of the time between
  arrivals
• ca CV of interarrival time
• td mean time between departures
• rd average departure rate
• cd departure CV

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Propagation of Variability
Station i
Station i1
Rates
re(i)
re(i1)
rd(i) ra(i1)
ra(i)
i
i1
ca(i)
cd(i) ca(i1)
ce(i)
ce(i1)
CVs
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Characterizing Variability in Flows

• Utilization fraction of time a workstation is
  busy over the long run..

u utilization of a workstation consisting of m
identical machines
Single Machine Station
Multi-Machine Station
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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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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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Batch Arrivals and Departures

• Batch arrivals Jobs are batched together for
  delivery to a station.
• Example
• 16 jobs arrive once per shift of 8 hours to a
  process center.
• One may think that since the arrivals always
  occur in this way with no randomness. sa2 ca2
  0.
• However, the interarrival time for the first job
  in the batch is 8 hours. For the next 15 jobs it
  is 0.
• ta 8 hours / 16 jobs 0.5
• sa2 Eti2 (Eti)2 (1/16)(82) - ta2
  3.75
• ca2 sa2 / ta2 3.75 / (0.5)2 15
• (In general, if k batch size, ca2 k - 1.)
• So, ca2 0 or ca2 15? In reality, it is
  somewhere in between.
• Batch arrivals increase variability.

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Variability Interactions-Queueing

• 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.)
  Routing
• Many more

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Kendall's Classification

• Characterization of a queueing station
• A / B / m / b
• A arrival process
• B service process
• m number of machines
• b maximum number of jobs
• that can be in the system
• M exponential (Markovian) distribution
• G completely general distribution
• D constant (deterministic) distribution.

B
A
m
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.
• b buffer size (i.e., maximum number of jobs
  allowed in system.)
• te mean effective process time.
• 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.

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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.

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The M/M/1 Queue
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The M/M/1 Queue-Example

• Tortoise
• ra 2.875 jobs/hour
• te 20 min 1/3 hour
• ce 1.0

Assume times between arrival are exponentially
distributed
Reasonable to use the M/M/1 model to represent
the Tortoise 2000.
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The G/G/1 Queue

• Formula
• Observations
• Refer to as Kingmans equation or VUT equation.
• Separate terms for variability, utilization,
  process time.
• CTq (and other measures) increase with ca2 and
  ce2 .
• Variability causes congestion!

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Example-Hare and Tortoise

• Hare X19
• ra 2.875 jobs/hour
• 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
• te 20 min 1/3 hour
• ce2 6.25

Tortoise ra 2.875 jobs/hour 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 te
20 min 1/3 hour ce2 1.0 CTq 7.6667 hours
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The G/G/1 Queue -Example

• Hare X19
• te 20 minutes
• ce2 6.25
• ca2 1.0
• u 0.9583

Assume times between arrival are exponentially
distributed
Comparison Tortoises CTq 7.67 hours
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The G/G/1 Queue -Example
Assume the Hare feeds the Tortoise, then ca2 for
the Tortoise is equal to cd2 for the Hare.
The expected queue time at the Tortoise
cd2 (1) ca2 (2) 5.82
1
2
Hare
Tortoise
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The M/M/m Queue

• Systems with multiple machines in parallel.
• All jobs wait in a single queue for the next
  available machine.

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The M/M/m Queue -Example

• Tortoise
• te 20 min 1/3 hour
• ce 1.0

• Job arrival
• ra 8.625 jobs/hour
• (207 jobs/day)
• ca 1.0

• Assume there are 3 machines with their own
  arrival stream.
• Then each machine sees 1/3 of the total demand,
  or 2.875 jobs/hour.
• The utilization of each machine is u 2.875/3
  0.958.
• CTq 7.67 hours

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The M/M/m Queue -Example

• Suppose there is a single queue that is serviced
  by the 3 machines in parallel.
• Utilization for each machine

Comparison case with separate queues CTq 7.67
hours
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The G/G/m Queue

• Formula
• Observations
• Useful model of multi-machine workstations
• Extremely general.
• Fast and accurate.
• Easily implemented in a spreadsheet (or packages).

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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., 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

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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
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)
Littles law
Note u gt 1 is possible formulas valid for u ? 1
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Blocking Example
te(1) 21
te(2) 20
b B 2 4
B 2
M/M/1/b system has less WIP and less TH than
M/M/1 system
18 less TH
90 less WIP
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Variability Pooling

• Variability pooling combine multiple sources of
  variability.
• Basic idea the CV of a sum of independent random
  variables decreases with the number of random
  variables.
• Example
• Batch processing
• Safety stock aggregation
• Queue sharing

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Conclusions

• Variability is a fact of life.
• There are many sources of variability in
  manufacturing systems.
• The coefficient of variation is a key measure of
  item variability.
• Variability propagates.
• Waiting time is frequently the largest component
  of cycle time.
• Limiting buffers reduces cycle time at the cost
  of decreasing throughput.
• Variability pooling reduces the effect of
  variability.