Variability Basics God does not play dice with the universe.

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Transcript Variability Basics God does not play dice with the universe.

Variability Basics

God does not play dice with the universe.

– Albert Einstein

Stop telling God what to do.

– Niels Bohr

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Variability Makes a Difference!

Little’s Law:

TH = WIP/CT, so same throughput can be obtained with large WIP, long CT or small WIP, short CT. The difference?

Variability!

Penny Fab One

:

achieves full TH (0.5 j/hr) at WIP=W 0 =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!

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

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!

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Variability Views

Variability:

• Any departure from uniformity • Random versus controllable variation

Randomness:

• Essential reality?

• Artifact of incomplete knowledge?

• Management implications: robustness is key

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

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

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

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

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

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 7

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Measuring Process Variability

t e

 mean process time of a job

σ e

 standard deviation of process time

c e

 

e t e

 coefficien t of variation , CV

Note: we often use the “squared coefficient of variation” (SCV), c

e 2 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Variability Classes in Factory Physics

0 Low variability (LV) Moderate variability (MV) 0.75

1.33

High variability (HV)

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

c e

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Measuring Process Variability – Example

Trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

t e s e c e

Class

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000

Machine 1 22 25 23 26 24 28 21 30 24 28 27 25 24 23 22 25.1

2.5

0.1

LV Machine 2 5 6 5

6 6 5 4 7

6 6 5 13.2

15.9

1.2

http://www.factory-physics.com

Machine 3 5 6 5

6 6 5 4 7

500

6 6 5 43.2

127.0

2.9

Question: can we measure c e this way?

Answer: No! Won’t consider “rare” events properly.

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.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Down Time – Mean Effects

Definitions:

c t

0  base process time  base process time coefficien t of variabili ty 0

m r

0  1  base capacity (rate, e.g., parts/hr)

0  mean time to failure

f m r c r

  mean time to repair coefficent of variabili ty of repair tim es ( 

m r

)

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Down Time – Mean Effects (cont.)

Availability:

Fraction of time machine is up



m f m f



m r

Effective Processing Time and Rate:

r e



t e



0 /

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Totoise and Hare - Availability

Hare X19:

0  0 = 15 min = 3.35 min

0 =  0 /

0 = 3.35/15 = 0.05

f = 12.4 hrs (744 min)

r = 4.133 hrs (248 min)

r = 1.0

Tortoise:

0  0 = 15 min = 3.35 min

0 =  0 /

0 = 3.35/15 = 0.05

f = 1.9 hrs (114 min)

r = 0.633 hrs (38 min)

r = 1.0

Availability:

m f m f



m r



744 744



248



0 .

m f m f



m r



114 114





0 .

No difference between machines in terms of availability.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Down Time

–

Variability Effects

Effective Variability:

t e



0 /

A σ e

2  

0 2  (

m r

2  

2 )( 1 

Am r A

)

Variability depends on repair times

Conclusions:

c e

2  

t e

2 

0 2  ( 1 

c r

2 )

( 1 

)

m r t

in addition to availability

• Failures inflate mean, variance, and CV of effective process time • Mean (

e ) increases proportionally with 1/

• SCV (

c e

2 ) increases proportionally with

m r

• SCV (

c e

2 ) increases proportionally in

r 2 • For constant availability (

), long infrequent outages increase SCV more than short frequent ones 15

http://www.factory-physics.com

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000

Tortoise and Hare - Variability

Hare X19: Tortoise 2000

e =



15 0 .



20 min

e =



15 0 .



20 min

e 2 =

0 2



( 1



c r

2 )

( 1



)

m r t



0 .



( 1



1 ) 0 .

75 ( 1



0 .

75 ) 248 15



6 .

25 high variabilit y

e 2 =

0 2



( 1



c r

2 )

( 1



)

m r t



0 .



( 1



1 ) 0 .

75 ( 1



38 0 .

75 ) 15



1 .

0 moderate variabilit y

Hare X19 is much more variable than Tortoise 2000!

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Setups

–

Mean and Variability Effects

Analysis:

N s



t s s c s

 average no.

jobs between setups  average setup duration  std.

dev.

of setup time  

s t s t e σ e

c e

2 

0   0 2 

t s N s

 

N s s

2  

t e

2 

N s N s

 1 2

t s

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

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.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Setup

–

Example

Data:

• Fast, inflexible machine – 2 hr setup every 10 jobs

0  1 hr

N s t r e e

 10 jobs/setup

t s

 2 hrs  

0 1 /



t s



N s

 1 /( 1  1  2 / 2 / 10 ) 10   1 .

2 hrs 0 .

8333 jobs/hr • Slower, flexible machine – no setups

0  1.2

hrs

r e

 1 /

0  1 / 1 .

2  0 .

833 jobs/hr

Traditional Analysis?

No difference!

http://www.factory-physics.com

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000

Setup

–

Example (cont.)

Factory Physics ® Approach:

Compare mean

and

variance • Fast, inflexible machine

–

2 hr setup every 10 jobs

0  1 hr

0 2

N s

 0 .

0625  10 jobs/setup

t s

 2 hrs

c s

2  0 .

0625

t e r e



0  1 /



t s e

N s

  1 /( 1  1  2 / 2 / 10 ) 10   1 .

2 hrs 0 .

8333 jobs/hr

σ e

2   0 2 

t s

2  

s N s



N s N s

2  1    0 .

4475

c e

2  0 .

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Setup

–

Example (cont.)

• Slower, flexible machine

–

0  1 .

2 hrs no setups

0 2  0 .

r c e

 1 /

0 

0 2  1 /  0 .

25 1 .

2  0 .

833 jobs/hr

Conclusion:

Flexibility can reduce variability.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Setup

–

Example (cont.)

New Machine:

Consider a third machine same as previous machine with setups, but with shorter, more frequent setups

N s t s

 5 jobs/setup  1 hr

Analysis:

r e σ e

2  1 /

t e

  0 2  1 /( 1 

t s

2  

c s

N s

 1 /  5 )

N s N

 0 .

833 jobs/hr

 1 2    0 .

2350

c e

2  0 .

Conclusion:

Shorter, more frequent setups induce less variability.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Other Process Variability Inflators

Sources:

• operator unavailability • recycle • batching • material unavailability • et cetera, et cetera, et cetera

Effects:

• inflate

t e

• inflate

c e

Consequences:

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

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Illustrating Flow Variability

Low variability arrivals

smooth!

High variability arrivals

bursty!

t t © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Measuring Flow Variability

t a

 mean time between arrivals

r a

 1

t a

 arrival rate 

 standard deviation of time between arrivals

c a

 

a t a

 coefficien t of variation of interarriv al times

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

c a

2 (

)

Propagation of Variability

c e

2 (

)

c d

2 (

) =

c a

2 (

Single Machine Station:

c d

2 

c e

2  ( 1 

2 )

c a

2 where

is the station utilization given by

r a t e

Multi-Machine Station:

 1  ( 1 

2 )(

c a

2  1 ) 

(

c e

2  1 ) where

is the number of (identical) machines and



r a t e m i+1

departure var depends on arrival var and process var

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Propagation of Variability – High Utilization Station

HV HV HV HV HV LV LV LV HV LV

Conclusion: flow variability out of a high utilization station is determined primarily by process variability at that station.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Propagation of Variability – Low Utilization Station

HV LV HV HV HV LV LV LV HV LV

Conclusion: flow variability out of a low utilization station is determined primarily by flow variability into that station.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

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

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

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.

Queue B Server C

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Queueing Parameters

r a

= the rate of arrivals in customers (jobs) per unit time (

t a

= 1/

r a

arrivals). = the average time between

c a

= the CV of inter-arrival times.

= the number of machines.

r e

= the rate of the station in jobs per unit time =

m/t e

c e

= the CV of

effective

process times.

Note: a station can be described with 5 parameters.

= utilization of station =

r a /r e

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

Queueing Measures

Measures:

= 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. WIP

= the expected WIP (in jobs) in queue.

Relationships:

CT = CT

WIP =

r a

 +

t e

CT WIP

r a

 CT

Result:

If we know CT

, we can compute WIP, WIP

, CT.

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

The G/G/1 Queue

Formula:





  

c a

2 

c e

2 2    

1 

 

t e

Observations:

• Useful model of single machine workstations • • Separate terms for variability, utilization, process time.

• CT

(and other measures) increase with

c a

2 and

c e

2 • Flow variability, process variability, or both can combine to inflate queue time.

Variability causes congestion!

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

The G/G/m Queue

Formula:





  

c a

2 

c e

2 2   

2 (

 1 )  1

( 1 

) 

t e

Observations:

• Useful model of multi-machine workstations •

Extremely

general.

• Fast and accurate.

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

© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

MEASURE:

Arrival Rate (parts/hr) Arrival CV Natural Process Time (hr) Natural Process SCV Number of Machines MTTF (hr) MTTR (hr) Availability Effective Process Time (failures only) Eff Process SCV (failures only) Batch Size Setup Time (hr) Setup Time SCV Arrival Rate of Batches Eff Batch Process Time (failures+setups) Eff Batch Process Time Var (failures+setups) Eff Process SCV (failures+setups) Utilization Departure SCV Yield Final Departure Rate Final Departure SCV Utilization Throughput Queue Time (hr) Cycle Time (hr) Cumulative Cycle Time (hr) WIP in Queue (jobs) WIP (jobs) Cumulative WIP (jobs)



0 2 /A 2 + 2m r (1-A)kt 0 /A+



s 2 c e 2 u c d 2 y r a *y yc d 2 +(1-y) u TH CT q CT q +t e



i (CT q (i)+t e (i)) r a CT q r a CT



i (r a (i)CT(i)) 0.936

100 0.000

1.000

0.100

9.090

0.773

0.009

0.909

0.181

0.980

9.800

0.198

0.909

9.800

45.825

54.915

458.249

549.149

1 10.000

1.000

0.090

0.500

1 200 2 0.990

0.091

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0.936

100 0.500

1.000

0.098

9.590

1.023

0.011

0.940

0.031

0.950

9.310

0.079

0.940

9.310

14.421

24.011

78.925

141.321

235.303

784.452

2 9.800

0.181

0.090

0.500

1 200 2 0.990

0.091

6.729

100 0.500

1.000

0.093

10.380

6.818

0.063

0.966

0.061

0.950

8.845

0.108

0.966

8.845

14.065

24.445

103.371

130.948

227.586

1012.038

3 9.310

0.031

0.095

0.500

1 200 8 0.962

0.099

2.209

100 0.000

1.000

0.088

9.180

1.861

0.022

0.812

0.035

0.900

7.960

0.132

0.812

7.960

1.649

10.829

114.200

14.587

95.780

1107.818

4 8.845

0.061

0.090

0.500

1 200 4 0.980

0.092

2.209

100 0.000

1.000

0.080

9.180

1.861

0.022

0.731

0.028

0.950

7.562

0.077

0.731

7.562

0.716

9.896

124.096

5.700

78.773

1186.591

5 7.960

0.035

0.090

0.500

1 200 4 0.980

0.092

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

The M/M/1/b Queue

Infinite raw materials 1

B buffer spaces

2 Note: there is room

for b=B+2 jobs in system, B in the buffer and one at each station .

Model of Station 2

WIP

(

/ 1 /

) 

1 

 (

 1 )

u b

 1 1 

u b

 1

Goes to u/(1-u) as b



Always less than WIP(M/M/1)

(

/ 1 /

)  1 

u b

1 

u b

 1

r a CT

(

/ 1 /

) 

WIP

(

M TH

(

/ 1 /

) ) where



t e

( 2 ) /

t e

( 1 )

Goes to r a as b



Always less than TH(M/M/1) Little’s law

Note: u>1 is possible; formulas valid for u



Blocking Example

t e (1)=21 t e (2)=20 B=2



t e

( 2 ) /

t e

( 1 )  20 / 21  0 .

9524

WIP

(

M TH

(

/ /

/ 1 ) 

1 

u M

/ 1 ) 

r a

 20 jobs  1 /

t e

( 1 )  1 / 21  0 .

0476 job/min

M/M/1/b system has less WIP and less TH than M/M/1 system

TH(M/M/

/b)

 1

-u b

 1

r a

 1  0 .

9524 4 1  0 .

9524 5   1 21    0 .

039 job/min

WIP

(

/ 1 /

) 

1 

 (

 1 )

u b

 1 1 

u b

 1  20  5 ( 1  0 .

9524 0 .

9524 5 ) 5  1 .

8954 jobs

18% less TH 90% less WIP

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 • ask “why” five times

Specific Targets:

• equipment failures • setups • rework • operator pacing • anything that prevents regular arrivals and process times

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 parts):



0 0  time to process single part  standard deviation of time to process single part   0  CV of time to process single part

0 

0 

0 (

batch

) 0 2 (

batch

) 

 0 2

0 2 (

batch

)   0 2 (

batch

)

0 2 (

batch

) 

 2 0

0 2   0 2

0 2 

0 2



0 (

batch

) 

Safety Stock Pooling Example

• • • • • •

PC’s consist of 6 components (CPU, HD, CD ROM, RAM, removable storage device, keyboard) 3 choices of each component: 3 6 = 729 different PC’s Each component costs $150 ($900 material cost per PC) Demand for all models is normally distributed with mean 100 per year, standard deviation 10 per year Replenishment lead time is 3 months, so average demand during LT is



= 25 for computers and



= 25(729/3) = 6075 for components Use base stock policy with fill rate of 99%

Pooling Example - Stock PC’s

Base Stock Level for Each PC:

R =



+ z s



= 25 + 2.33(



25) = 37

cycle stock safety stock

On-Hand Inventory for Each PC:

I(R) = R -



+ B(R)



R -



= z s



= 37 - 25 = 12 units

Total (Approximate) On-Hand Inventory :



729



$900 = $7,873,200

Pooling Example - Stock Components

Necessary Service for Each Component:

S = (0.99) 1/6 = 0.9983

z s = 2.93

Base Stock Level for Each Component:

R =



+ z s



= 6075 + 2.93(



cycle stock safety stock

6075) = 6303

On-Hand Inventory Level for Each Component:

I(R) = R -



+ B(R)



R -



= z s



= 6303-6075 = 228 units

Total Safety Stock:

228



$150 = $615,600

92% reduction!

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 • variability and utilization interact • pooled variability less destructive than individual variability