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

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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)

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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 .

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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

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r

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m

m

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σ

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Down Time---Mean Effects (cont.)

Availability: Fraction of time machine is up

Effective Processing Time and Rate:

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f

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Arr

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/0

0

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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

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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

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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

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15

2.5 (high variability!)

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15

1.0 (medium variability)

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Setups--Mean and Variability Effects

Analysis:

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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.

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Setup--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

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Setup--Example (cont.)

• Slower, flexible machine--no setups

Conclusion: flexibility reduces 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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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.

24

Illustrating Flow Variability

t

Low variability arrivals

t

High variability arrivals

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Measuring Flow Variability

timesalinterarriv of variationoft coefficien

arrivalsbetween timeofdeviation standard

rate arrival1

arrivalsbetween mean time

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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

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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

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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

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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.

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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.

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!

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34

Variability and Cycle Time Relationships

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35

The G/G/m Queue

Analysis:

Observations:• Extremely general.

• Fast and accurate.

• Easily implemented in a spreadsheet (or packages like MPX).

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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

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5

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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