The Effects of Process Variability 35E00100 Service Operations and Strategy #3 Fall 2015.

The Effects of Process Variability

35E00100 Service Operations and Strategy#3 Fall 2015

35E00100 Service Operations and Strategy #3 Aalto/BIZ Logistics2

Topics on Variability

Variability basics Measure of variability Process variability Flow variability Key points

The corrupting influence of variability Factory physics “laws” Batching Serial system Parallel system Transfer batching Ways to improve operations Key points

Useful material: Hopp, W. & Spearman, M. (2000), Factory Physics, Chapters 8, 9 and 15.3


Basics - The Concept of Variability

Variability Any departure from uniformity (regular, predictable behavior) Sources and causes Compared to randomness?

Use of intuition Measuring variability

Coefficient of variation (CV)

Classification based on the values of CV:

Natural process times have generally low variability (LV) Effective process times can be LV, MV, or HV

= mean process time of a job= standard deviation of process time

CV

e

e

ee

e

t

ct

0.75

High variability (HV)Moderate variability (MV)Low variability (LV)

0 1.33ce

Hopp and Spearman 2000, 248-254


Day 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 5 510 28 4 411 27 7 712 25 50 50013 24 6 614 23 6 615 22 5 5

Measuring VariabilityIllustrativeexample

What is the variability of each

machine?

mean 24,8 13,2 43,2st dev 2,6 15,9 127,0

CV 0,1 1,2 2,9


Natural Variability

Variability without explicitly analyzed cause(s) Sources in process

Operator pace Material fluctuations Product type (if not explicitly considered) Product quality

Observation Natural process variability is usually in the low variability

category

Hopp and Spearman 2000, 255



Mean Effects of Breakdowns

Definitions

Availability A is the fraction of time machine is up:

Effective process time te and rate re can be calculated as follows:

0

0

00

natural (base) process time

CV of natural process time

1 base capacity rate

mean time to failure (MTTF)

mean time to repair (MTTR)

f

r

t

c

r t

m

m

rf

f

mm

mA

00

ee

m mr A Ar

t t 0

et

tA


Which machine is better?

Two machines, Tortoise 2000 and Hare X19, are subject to the same average workload: 69 jobs per day operate 24 hours per day 2.875 jobs per hour have unpredictable breakdowns

Tortoise 2000 has long, infrequent breakdowns Hare X19 has short, more frequent breakdowns

How would you compare?

Example 1


Calculating Machine Availability

Tortoise 2000 t0 = 15 min

0 = 3.35 min

c02 = 0

2/t02= 3.352/152 =

0.05

mf = 12.4 hrs (744 min)

mr = 4.133 hrs (248 min)

cr = 1.0

Availability of the machine

Hare X19 t0 = 15 min

0 = 3.35 min

c02 = 0

2/t02= 3.352/152 = 0.05

mf = 1.9 hrs (114 min)

mr = 0.633 hrs (38 min)

cr = 1.0

Availability of the machine744

0.75744 248

f

f r

mA

m m

114

0.75114 38

f

f r

mA

m m


Example 1

No difference between the machines in terms of availability.


Assumptions Times between failures are exponentially distributed Time to repair follows some probability distribution

Effective variability

Conclusions Failures inflate mean, variance, and CV of effective process time Mean te increases proportionally with 1/A

For constant availability A, long infrequent breakdowns increase SCV more than short frequent ones

Variability Effects of Downtime

0

2202

22

0222

02

0

)1()1(

)1)((

/

t

mAAcc

tc

A

tAm

Aσ

Att

rr

e

ee

rre

e

Variability depends on repair times in addition

to availability



Estimating Variability

Tortoise 2000 Hare X19

0 1520 min

0.75et

tA

0 1520 min

0.75et

tA

2 2 20

0

(1 ) (1 )

2480.05 (1 1)0.75(1 0.75)

156.25

re r

mc c c A A

t

2 2 20

0

(1 ) (1 )

380.05 (1 1)0.75(1 0.75)

151.0

re r

mc c c A A

t

High variability Moderate variability

Example 1


Mean and Variability Effects of Setups

Analysis

Observations Setups increase the mean and variance of processing times Variability reduction is one benefit of flexible machines Interaction is complex

average number of jobs between setups (batch size)average setup durationstandard deviation of setup time

s

s

s

Nt

0e

tst tNs

22

2e

ee

ct

22 2 2

0 2

1s se s

s s

Nt

N N



Mean Effects of Setups

Two machines Fast, inflexible machine: 2 hour setup every 10 jobs

Slower, flexible machine: no setups

0

0

0

1 hr0.25 10 jobs/setup 2 hrs

21 1.2 hrs

101 2

1/(1 ) 0.8333 jobs/hr10

s

s

se

s

ee

tc

Nt

tt t

N

rt

0

0

0

1.2 hrs 0.51/ 1/1.2 0.833 jobs/hre

tcr t


Example 2

In traditional analysis there is no difference

between the machines.


Slower, flexible machine no setups

020

2

22 2 2

0 2

2

1 hr

0.0625

10 jobs/setup

2 hrs

0.0625

1

0.4475

0.31

s

s

s

s se s

s s

e

t

c

N

t

c

c Nσ t

N N

c

Variability Effects of Setups

020

0

2 20

1.2 hrs

0.25

1 10.833 jobs/hr

1.2

0.25

e

e

t

c

rt

c c

Flexibility can reduce variability.

Fast, inflexible machine 2 hour setup every 10 job

Example 2


Example 2

Variability Effects of SetupsThird Machine

New machine Otherwise same than the fast machine but more frequent setups

Analysis

Conclusion Shorter, more frequent setups induce less variability

22 2 2

0 2

2

1/ 1/(1 1/ 5) 0.833 jobs/hr

10.2350

0.16

e e

s se s

s s

e

r t

c Nσ t

N N

c

hr 1

jobs/setup 5

s

s

t

N


02 20

1 hr

0.25 0.0625

t

c

2 2

11 hrs5

0.25 0.0625

e

s

t

c


Inflators of Process Variability

Sources e.g. Operator unavailability Batching Material unavailability Recycle

Effects of process variability Inflate the mean processing time te

Inflate the CV of te

Effective process variability can be LV, MV, or HV


Flow Variability

t

Low variability arrivals

t

High variability arrivals


Propagation of Variability

i

Departure variance depends on arrival

variance and process variance

re(i)ra(i)


rd(i) = ra(i+1)

cd2(i) = ca

2(i+1)ce

2(i)ca

2(i)i+1

re(i+1)

ce2(i+1)

where station utilization u is given by u = rate

2 2 2 2 2(1 )d e ac u c u c

Departure SCV in single machine station

22 2 2 21 (1 )( 1) ( 1)d a e

uc u c c

m

where a er tu

m

Departure SCV in multi-machine station


Propagation of VariabilityLow Utilization Stations

High process VarLow flow Var Low flow Var

High process VarHigh flow Var High flow Var

Low process VarLow flow Var Low flow Var

Low process VarHigh flow Var High flow Var


Propagation of VariabilityHigh Utilization Stations

High process VarLow flow Var High flow Var

High process VarHigh flow Var High flow Var

Low process VarLow flow Var Low flow Var

Low process VarHigh flow Var Low flow Var


Variability Pooling

Basic idea CV of a sum of independent random variables decreases with

the number of random variables

Time to process a batch of parts

0 02 20 0

2 2 2 22 0 0 0 00 2 2 2 2

0 0 0

00

( )

( )

( )( )

( )

( )

t batch nt

batch n

batch n cc batch

nt batch n t nt

cc batch

n


0

0

time to process a single partstandard deviation of time to process a single part

t


Key Points

Variability Cannot be eliminated Causes congestion Propagates Interacts with utilization

Components of process variability Failures, setups and many others deflate capacity and inflate variability Long infrequent disruptions are worse than short frequent ones

Measure of variability: coefficient of variation (CV) Pooled variability is less destructive than individual variability


Notation ca

2 = SCV of the inter-arrival time ce

2 = SCV of the effective process time cr

2 = SCV of the repair times c0

2 = SCV of the base process time mf = mean time to failure mr = mean time to repair n = number of jobs or parts in a batch Ns = number of jobs or parts between setups ra = arrival rate re = service rate rd = departure rate r0 = base capacity rate ta = inter-arrival time te= process time ts= setup time t0 = base process time


Abbreviations Used

CV = coefficient of variation HV = high variability LV = low variability MTTF = mean time to failure MTTR = mean time to repair MV = moderate variability SCV = squared coefficient of variation

The Corrupting Influence of Variability


Factory Physics “Laws” Law 1: Variability Law

Increasing variability degrades the performance of a production system.

Law 2: Variability Buffering Law Systems w/ variability must be buffered by some combination of inventory, capacity and time.

Law 3: Product Flows Law In a stable system, over the long run, the rate out of a system will equal to the rate in, less any yield loss, plus any parts production within the

system.

Law 4: Capacity Law In steady state, all plants will release work at an average rate that is strictly less than the average capacity.

Law 5: Utilization Law If a station increases utilization without making any other changes, average WIP and cycle time will increase in a highly nonlinear fashion.

Law 6: Process Batching Law In stations with batch operations or significant changeover times minimum process batch size yielding a stable system may be over 1, cycle time at

the station will be minimized for some process batch size (may be greater than one), and as process batch size becomes large, average cycle time grows proportionally with batch size.

Law 7: Move Batching Law Cycle times over a segment of a routing are roughly proportional to transfer batch sizes used over that segment, provided there is no waiting for

the conveyance device.

Law 8: Assembly Operations Law The performance of an assembly station is degraded by increasing any of the following: the number of components being assembled, variability of

component arrivals, or lack of coordination between component arrivals.

Hopp and Spearman 2000


Variability Law

Increasing variability degrades the performance of a production system.

For example:

Higher demand variability requires more safety stock for same level of customer service. Higher cycle time variability requires longer lead time quotes to attain the same level of on-

time delivery.

”Law 1”



Variability Buffering Law

Systems with variability must be buffered by some combination of inventory,

capacity, and time.

Is variability always harmful?

”Law 2”



Variability Buffering Law

Systems with variability must be buffered by some combination of inventory, capacity, and time.

”Law 2”

Inventory

CapacityTime



Material Flow Laws

Product flowsIn a stable system, over the long run, the rate out of a system will equal to the rate in, less any yield loss, plus any parts production within the system.

CapacityIn steady state, all plants will release work at an average rate that is strictly less than the average capacity.

UtilizationIf a station increases utilization without making any other changes, average WIP and cycle time will increase in a highly nonlinear fashion.

”Laws 3-5”



Cycle Time versus Utilization

0

2

4

6

8

10

12

14

16

18

20

22

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Release Rate (entities/hr)

Cyc

le T

ime

(h

rs)

Capacity

High Variability

Low Variability


Process Batching Law

In stations with batch operations or significant changeover times

The minimum process batch size that yields a stable system may be greater than one.

Cycle time at the station will be minimized for some process batch size, which may be greater than one.

As process batch size becomes large, average cycle time grows proportionally with batch size.

”Law 6”



Recap: Forms of Batching

Serial batching Processes with sequence-dependent setups Batch size is the number of jobs between setups Reduces loss of capacity from setups

Parallel batching True batch operations Batch size is the number of jobs run together Increases the effective rate of process

Transfer batching Batch size is the number of parts that accumulate before being transferred to the

next station (not necessarily equal to the process batch lot splitting) Less material handling


Process Batch Versus Move BatchCase “Batch Size in a Dedicated Assembly Line”

Process batch Depends on the length of setup. The longer the setup, the larger the lot size required for the same capacity.

Move (transfer) batch: Why should it equal process batch? The smaller the move batch, the shorter the cycle time. The smaller the move batch, the more material handling.

Lot splitting: Move batch can be different from process batch.

1. Establish smallest economical move batch.

2. Group batches of like families together at bottleneck to avoid setups.

3. Implement using a “backlog”.


Batching and Process Performance

Impact of batching Flow variability Waiting inventory

Impact of lot splitting

2 2 2( 1) 1

CT2 (1 )

ma e

q ec c u

tm u


Serial Batching

Parameters

Effective process time

Arrival of batches

Utilization

For stability (u < 1)

2

serial batch size = 10

time to process a single part = 1

time to perform a setup = 5

SCV for batch (parts setup) = 0.5

arrival rate for parts = 0.4

CV of batch arrivals = 1.0

s

e

a

a

k

t

t

c

r

c

ttsra

ca

Queue of batches

Setupk

Forming batch

5 10 1 15e st t kt 0.4

0.0410

ar

k

5( ) ( ) 0.4 1 0.6

10a s

s ar t

u t kt r tk k

0.4 5.0 3.33

1 1 0.4 1.0a s

a

r tk

r t

Minimum batch size required for stability of system



split1

CT =2

10 116.875 5 (1.0) 27.375

2

q s split q sk

CT t WIBT t CT t t

Serial Batching

Average queue time at station

Average cycle time depends on move batch size Move batch = process batch

Move batch = 1

875.16156.01

6.0

2

5.01

12CT

22

e

eaq t

u

ucc

Splitting move batches reduces wait-in-batch time

Arrival CV of batches is assumed

ca regardless of batch size.


non splitCT ( )( 1)

16.875 5 10(1) 31.875

q e q s nonsplit

q s q s

CT t CT t WIBT tCT t k t t CT t kt

Effect of Batch Size on Average Total CT An analysis of a Series System

38


Cycle Time versus Batch Size in a Series System

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30 35 40 45 50

Batch size k

Cyc

le T

ime

Cycle Time versus Batch Size

Optimum batch size


Optimal Serial Process Batch SizesOne Product

Assumptions Identical product families in terms of process and setup times Poisson arrivals

Effective process time

Utilization

Good approximation of the serial batch size minimizing cycle time at a station is given by

( )a e ar t ru s kt

k k

0 0 0

a ar s r sk

u u u u

CT is minimized through finding the optimal station utilization.

Good approximation:

0u u


et s kt


Optimal Serial Process Batch SizesMultiple Products

Assumptions Multiple products Poisson arrivals

Eff. process time

Utilization

Good approximation of the serial batch size minimizing cycle time at a station is given by

1*

0

, where

n

ai i ii i

i

r s tL sk L s

t u u


1

( )n

aii i i

ii

ru s k t

k

11

( ), where n

ai ie i i i i n aji

i i

r kt s k t

rk


Parallel Batching

Parameters

Wait-to-batch time

Time to process a batch

Arrival rate of batches

Utilization

parallel batch size = 10time to process a batch = 90

effective CV for processing a batch = 1.0arrival rate for parts = 0.05

CV of batch arrivals = 1.0B = maximum batch size

e

a

a

kt

cr

c

t0k

Queue of batches

Forming batch

ra

ca

1 1 10 1 190

2 2 0.05a

kWTBT

r

90et t

0.05( ) 0.005

10a

ar

r batchk


0.005 90 0.45aru tk


Parallel Batching

Minimum batch size required for system stability (u<1)

Average queue + process time at station = CTq+ t

Total cycle time

0.05 90 4.5ak r t k

Batch size affects both WTBT and CTq.

2 2/1

2 2 1

90 130.5 220.5

q

a e

CT WTBT CT t

c k ck ut t t

ku u

2 2/

2 1

0.1 1 0.4590 90 130.5

2 1 0.45

a ec k c uCT t t

u

Effect of Batch Size on Average Total CT Analysis of a Parallel System

44


Cycle Time versus Batch Size Parallel System

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50 60 70 80 90 100 110

Nb

To

tal C

yc

le T

ime

Queue time due totoo high utilization Wait for batch time

B

Optimum Batch Size


Move Batching Law

Cycle times over a segment of a routing are roughly proportional to transfer batch sizes used over that segment, provided there is no waiting for the conveyance device.

Insights Queuing for conveyance device can offset cycle time reduction from

reduced move batch size. Move batching intimately related to material handling and layout

decisions.

”Law 7”



Effects of Transfer Batching

Two machines in series Machine 1

Receives individual parts at rate ra with CV of ca(1)

Mean process time of te(1) for one part with CV of ce(1)

Puts out batches of size k Machine 2

Receives batches of k

Mean process time of te(2) for one part with CV of ce(2)

Puts out individual parts How does cycle time depend on the batch size k?

single job

batch

Machine 1 Machine 2

kra

ca(1)te(1)ce(1)

te(2)ce(2)



Average time forming the batch:

Average time after batching:

Average total time spent at the 1st station:

Time between output of individual parts into the batch: ta

Time between output of batches of size k: kta

Variance of inter-output times of parts is cd2(1)ta

2, where

Variance of batches of size k:

2 2(1) (1) (1) 1 1CT(1) (1) (1) (1) CT (1)

2 1 (1) 2 (1) 2 (1)a e

e e e ec c u k k

t t t tu u u

1 1 1(1)

2 2 (1) ea

k kt

r u

2 2(1) (1) (1)(1) (1)

2 1 (1)a e

e ec c u

t tu

Transfer Batching – Machine 1

1st part waits (k-1)(1/ra), last part does not wait.

By definition CV cd

2(1)=d2/ta

2

Departures are independent variances add


2 2 2 2 2(1) (1 (1) ) (1) (1) (1)d a ec u c u c 2 2(1)d akc t


Transfer Batching - Machine 2

SCV of batch arrivals:

Time to process a batch of size k:

Variance of time to process a batch of size k:

SCV for a batch of size k:

Mean time spent in partial batch of size k:

Average time spent at the 2nd station:

k

c

tk

tkc e

e

ee )2(

)2(

)2()2( 2

22

22

)2(2

1et

k

)2(2

1)batching no CT(2,

)2()2(2

1)2(

)2(1

)2(

2

/)2(/)1()2(CT

22

e

eeeed

tk

ttk

ktu

ukckc

1st part doesn’t wait, last part waits (k-1)te(2)

2 2 2

2 2

(1) (1)d a d

a

kc t c

kk t


(2)ekt

2 2(2) (2)e ekc t


Transfer Batching – Total System

batch

(no batching)

(no batching)

CT CT(1) CT(2)

1 1CT (1) (2)

2 (1) 2

(1)1CT (2)

2 (1)

e e

ee

k kt t

u

tkt

u


Inflation factor dueto transfer batching


Assembly Operations Law

The performance of an assembly station is degraded by increasing any of the following

The number of components being assembled Variability of component arrivals

Lack of coordination between component arrivals

”Law 8”



Ways to Improve Operations

1. Increase throughput

2. Reduce queue time

3. Reduce batching delay

4. Reduce matching delay

5. Improve customer service



1. Increase Throughput

Throughput = P(bottleneck is busy) bottleneck rate

CTq = VUT

Reduce variability

Reduce utilization

Increase capacity •Add equipment• Increase operating time• Increase reliability•Reduce yield loss•Quality improvements

Reduce blocking/starving •Buffer with inventory (near bottleneck)

•Reduce system “desire to queue”



2. Reduce Queue Delay

u

u

1

Reduce variability •Process variability

- Repair times, setups•Arrival variability

- Decrease process variability in upstream

- Pull system- Eliminate batch releases

Reduce utilization • Increase bottleneck rate

- Decrease time to repair- Cross-training

•Reduce flow into bottleneck- Improve yield- Reduce rework, etc

=qCT VUT

2

22ea cc



3. Reduce Batching Delay

CTbatch = delay at stations + delay between stations

Reduce process batching •Optimize batch sizes•Reduce setups

- Stations where capacity is expensive

- Capacity versus WIP tradeoff

Reduce move batching •Move more frequently•Layout to support material handling

- E.g. cell manufacturing



4. Reduce Matching Delay

CTmatch = delay due to lack of synchronization

Reduce variability Improve coordination•Scheduling•Pull mechanisms•Modular designs

Reduce number of components

•E.g. product redesign



5. Improve Customer Service

LT = CT + zCT

Reduce CT variability (Generally same methods as for CT

reduction)

• Improve reliability• Improve maintainability•Reduce labor variability• Improve quality• Improve scheduling, etc.

Reduce avg CT•Queue time•Batch time•Match time

Reduce quoted LT•Assembly to order•Stock components•Delayed differentiation

Safety lead time



Variability Influences Cycle Times and Lead Times

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Cycle Time in Days

De

ns

itie

s

Lead Time = 14 days

Lead Time = 27 days

CT = 10CT = 3

CT = 10CT = 6


Key Points

Factory physics laws! Variability

Decreases performance Buffering through inventory, capacity, and time Interacts with utilization

Congestion effects multiply Nonlinear effects of utilization on cycle time

Batching In serial and parallel batching minimum feasible batch size may

be greater than one Cycle time increases proportionally with batch size

Without wait-for-batch time, cycle time decreases in batch size Lot splitting can reduce the effects of batching

Batching delay is essentially separate from a variability delay.


Notationce

2 = SCV of the effective process time (parts and setups)cd

2 = SCV of the departure timesCT = cycle timeD/d = demand k = serial batch sizeLT = lead time quoted to customern = number of products (i=index for products, i=1,…,n)Ns = number of jobs or parts between setupsra = arrival raterb = bottleneck ratere = service raterd = departure ratets= setup timet0 = time to process a partu0 = utilization without setupsWTBT = wait to batch timeWIBT = wait in batch time

The Effects of Process Variability 35E00100 Service Operations and Strategy #3 Fall 2015.

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Transcript of The Effects of Process Variability 35E00100 Service Operations and Strategy #3 Fall 2015.